Applied Probability and Stochastic Processes [1st ed.] 9789811559501, 9789811559518

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Infosys Science Foundation Series in Mathematical Sciences

V. C. Joshua S. R. S. Varadhan Vladimir M. Vishnevsky  Editors

Applied Probability and Stochastic Processes

Infosys Science Foundation Series Infosys Science Foundation Series in Mathematical Sciences

Series Editors Irene Fonseca, Carnegie Mellon University, Pittsburgh, PA, USA Gopal Prasad, University of Michigan, Ann Arbor, USA Editorial Board Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur, India Weinan E, Princeton University, Princeton, USA Chandrashekhar Khare, University of California, Los Angeles, USA Mahan Mj, Tata Institute of Fundamental Research, Mumbai, India Ritabrata Munshi, Tata Institute of Fundamental Research, Mumbai, India S. R. S. Varadhan, New York University, New York, USA

The Infosys Science Foundation Series in Mathematical Sciences is a sub-series of The Infosys Science Foundation Series. This sub-series focuses on high-quality content in the domain of mathematical sciences and various disciplines of mathematics, statistics, bio-mathematics, financial mathematics, applied mathematics, operations research, applied statistics and computer science. All content published in the sub-series are written, edited, or vetted by the laureates or jury members of the Infosys Prize. With this series, Springer and the Infosys Science Foundation hope to provide readers with monographs, handbooks, professional books and textbooks of the highest academic quality on current topics in relevant disciplines. Literature in this sub-series will appeal to a wide audience of researchers, students, educators, and professionals across mathematics, applied mathematics, statistics and computer science disciplines.

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V. C. Joshua • S. R. S. Varadhan • Vladimir M. Vishnevsky Editors

Applied Probability and Stochastic Processes

Editors V. C. Joshua Department of Mathematics CMS College Kottayam, India

S. R. S. Varadhan Courant Institute of Mathematical Sciences New York University New York, NY, USA

Vladimir M. Vishnevsky Institute of Control Sciences Russian Academy of Sciences Moscow, Russia

ISSN 2363-6149 ISSN 2363-6157 (electronic) Infosys Science Foundation Series ISSN 2364-4036 ISSN 2364-4044 (electronic) Infosys Science Foundation Series in Mathematical Sciences ISBN 978-981-15-5950-1 ISBN 978-981-15-5951-8 (eBook) https://doi.org/10.1007/978-981-15-5951-8 Mathematics Subject Classification: 60B10, 60J65, 60K20, 60K25, 60K30, 62H99, 68M18, 90B05, 90B15, 91B05 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is a collection of selected papers presented at the International Conference on Advances in Applied Probability and Stochastic Processes (ICAAP&SP@CMS 2019) held during 7–10 January 2019 at CMS College Kottayam, Kerala, India, in honour of Prof. Dr. A. Krishnamoorthy, Emeritus Professor and Former Head of Department of Mathematics, Cochin University of Science and Technology, Kerala, India. It is the first conference in the series of conferences decided to be conducted biannually, aiming to promote high-quality research in the field of applied probability that would keep pace with the advancements in science and technology. It focuses on applied probability techniques in modelling and analysis of systems evolving in time. Stochastic modelling plays a key role in analysing real-life situations such as queueing theory, reliability, inventory problems, biology, medicine, and finance. The conference was a great success, attracting 145 delegates from 14 countries. Professor S. R. S. Varadhan, FRS (Abel Laureate) delivered the keynote address. It had 9 plenary speakers and 26 invited speakers. It provided a platform to researchers, academicians, practitioners, industrialists, and so on from various countries, interested in the theory and applications of applied probability and stochastic processes to share their views, discuss prospective developments, and pursue collaborations in these areas. The conference had 104 papers for presentation. Out of 104 papers, 30 papers are selected for publication in a book form as Springer proceedings in accordance with the conducted peer reviews. These papers are revelations of strong theoretical as well as practical foundations of probabilistic modelling tools. Its target audience includes researchers and practitioners of probability theory, stochastic process, and mathematical modelling.

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Preface

We thank all the authors for the contributions, reviewers for peer- reviewing, members of the program committee for timely help, and sponsors for financial support. Kottayam, India New York, NY, USA Moscow, Russia January 2020

V. C. Joshua S. R. S. Varadhan Vladimir M. Vishnevsky

Contents

Shift-Coupling and Maximality . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Hermann Thorisson

1

Diffusion Approximation Analysis of Multihop Wireless Networks: Quality-of-Service and Convergence of Stationary Distribution.. . . . . . . . . . . K. S. Ashok Krishnan and Vinod Sharma

15

Analysis of Retrial Queue with Heterogeneous Servers and Markovian Arrival Process . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Liu Mei and Alexander Dudin

29

What is Standard Brownian Motion?.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Krishna B. Athreya

51

Busy Period Analysis of Multi-Server Retrial Queueing Systems . . . . . . . . . . Srinivas R. Chakravarthy

61

Steady-State and Transient Analysis of a Single Channel Cognitive Radio Model with Impatience and Balking . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Alexander Rumyantsev and Garimella Rama Murthy Applications of Fluid Queues in Rechargeable Batteries . . . . . . . . . . . . . . . . . . . . Shruti Kapoor and S. Dharmaraja

77 91

Analysis of BMAP /R/1 Queues Under Gated-Limited Service with the Server’s Single Vacation Policy .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 Souvik Ghosh, A. D. Banik, and M. L. Chaudhry A Production Inventory System with Renewal and Retrial Demands. . . . . . 129 G. Arivarignan, M. Keerthana, and B. Sivakumar A Queueing System with Batch Renewal Input and Negative Arrivals . . . . 143 U. C. Gupta, Nitin Kumar, and F. P. Barbhuiya

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Contents

Asymptotic Analysis Methods for Multi-Server Retrial Queueing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 159 Ekaterina Fedorova, Anatoly Nazarov, and Alexander Moiseev On the Application of Dynamic Screening Method to Resource Queueing System with Infinite Servers . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179 Michele Pagano and Ekaterina Lisovskaya “Controlled” Versions of the Collatz–Wielandt and Donsker–Varadhan Formulae . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 199 Aristotle Arapostathis and Vivek S. Borkar An (s, S) Production Inventory System with State Dependant Production Rate and Lost Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 215 S. Malini and Dhanya Shajin Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent Phase-Type Claims and a Constant Barrier . . . . . . . . . . . . . . 235 A. S. Dibu and M. J. Jacob A PH Distributed Production Inventory Model with Different Modes of Service and MAP Arrivals . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 263 Salini S. Nair and K. P. Jose On a Generalized Lifetime Model Using DUS Transformation .. . . . . . . . . . . . 281 P. Kavya and M. Manoharan Analysis of Inventory Control Model for Items Having General Deterioration Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 V. P. Praveen and M. Manoharan A Two-Server Queueing System with Processing of Service Items by a Server . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 307 A. Krishnamoorthy and Divya V. A Two-Stage Tandem Queue with Specialist Servers . . . .. . . . . . . . . . . . . . . . . . . . 335 T. S. Sinu Lal, A. Krishnamoorthy, V. C. Joshua, and Vladimir Vishnevsky The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer Induced Interruption and Retrial of Customers . . . . . . . . . . . . . . . . . 355 Jomy Punalal and S. Babu Valuation of Reverse Mortgage . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 371 D. Kannan and Lina Ma Stationary Distribution of Discrete-Time Finite-Capacity Queue with Re-sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 399 Rostislav Razumchik and Lusine Meykhanadzhyan The Polaron Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 415 Chiranjib Mukherjee and S. R. S. Varadhan

Contents

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Batch Arrival Multiserver Queue with State-Dependent Setup for Energy-Saving Data Center .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 421 Tuan Phung-Duc Weak Convergence of Probability Measures of Trotter–Kato Approximate Solutions of Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . 441 T. E. Govindan Stochastic Multiphase Models and Their Application for Analysis of End-to-End Delays in Wireless Multihop Networks . .. . . . . . . . . . . . . . . . . . . . 457 Vladimir Vishnevsky and Andrey Larionov Variance Laplacian: Quadratic Forms in Statistics . . . . . .. . . . . . . . . . . . . . . . . . . . 473 Garimella Rama Murthy On the Feynman–Kac Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 491 B. Rajeev Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 507 Ekaterina Lisovskaya, Svetlana Moiseeva, Michele Pagano, and Ekaterina Pankratova

About the Editors

V. C. Joshua is Associate Professor at the Department of Mathematics, CMS College, Kerala, India. He received his Ph.D. in Mathematics from the Cochin University of Science and Technology, Kerala, India, in 2003. His research interests include stochastic modelling analysis and applications, queuing theory, inventory, and reliability. In addition to having authored one book and 30 research papers, he is also a reviewer for a number of international journals. He has been a participating scientist of two international bilateral scientific research projects and has organized three international conferences on applied probability and stochastic processes. S. R. S. Varadhan is the Frank Jay Gould Professor of Science at Courant Institute of Mathematical Sciences, New York University, USA, and is a renowned mathematician. He completed his Ph.D. in Mathematics from Indian Statistical Institute, Kolkata, India. He is known for his fundamental contributions to probability theory and for creating a unified theory of large deviations. He is a recipient of the National Medal of Science (in 2010) from President Barack Obama, the highest honour bestowed by the Government of the United States of America on scientists, engineers, and inventors. The Government of India awarded him the Padma Bhushan (in 2008). He received the Abel Prize (in 2007) for his work on large deviations with Monroe D. Donsker. He was also awarded the Leroy P. Steele Prize for Seminal Contribution to Research (in 1996) by the American Mathematical Society for his work with Daniel W. Stroock on diffusion processes, Margaret and Herman Sokol Award of the Faculty of Arts and Sciences, New York University (in 1995), and Birkhoff Prize (in 1994). He also has two honorary degrees from Université Pierre et Marie Curie, Paris, France, and from Indian Statistical Institute, Kolkata, India. He is a member of the National Academy of Sciences, Washington, and the Norwegian Academy of Science and Letters, Oslo, Norway. He has been an elected Fellow of the American Academy of Arts and Sciences, Cambridge, USA; the World Academy of Sciences, Trieste, Italy; the Institute of Mathematical Statistics; Royal Society, London, UK; the Indian Academy of Sciences, Bangalore, India; the

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About the Editors

Society for Industrial and Applied Mathematics, Philadelphia, USA; and American Mathematical Society, Providence, USA. His areas of research include probability theory and its relation to analysis, various aspects of stochastic processes and their connections to certain classes of linear and nonlinear partial differential equations. Vladimir M. Vishnevsky is Head of the Telecommunication Networks Laboratory at the V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences (ICS RAS), Moscow, Russia. Earlier, he was Assistant Head of the Institute of Information Transmission Problems of RAS, from 1990 to 2010, and Assistant Head of Laboratory with ICS RAS, from 1971 to 1990. He also served as Full Professor at ICS RAS from 1989 and the Moscow Institute of Physics and Technology from 1990. He earned his Ph.D. in queuing theory and telecommunication networks and D.Sc. in telecommunication networks from ICS RAS in 1974 and 1988, respectively. He has authored over 300 research papers in queuing theory and telecommunications, 10 monographs, and 20 patents for inventions. His areas of research include computer systems and networks, queuing systems, telecommunications, discrete mathematics (extremal graph theory and mathematical programming), and wireless information transmission networks. He is a co-chair of a number of IEEE conferences and project leader of several international research projects related to the research and development of the next-generation 5G/IMT-2020 networks. In 2019, by a decree of the President of the Russian Federation, he was awarded the title “Honored Scientist of the Russian Federation”.

Shift-Coupling and Maximality Hermann Thorisson

Abstract We consider shift-coupling on groups. The theory is based on a key maximality result that does not rely on the group condition. Keywords Coupling · Shift-coupling · Invariant sets · Cesaro asymptotics · Total variation AMS MSC 2010 60G60, 60G57, 60G55, 60B10

1 Introduction This paper presents basic theory of shift-coupling on locally compact second countable Hausdorff topological groups, involving invariant sets and Cesaro total variation asymptotics. The key part, concerning the existence of shift-couplings, is based on a maximality result that is proved in the latter half of the paper. That result does not rely on the group condition. Shift-coupling dates back to the 1979 monograph [2] by Berbee, where the following result (Theorem 4.3.3 in [2]) for one-sided random sequences (Xk )k≥0 and (Yk )k≥0 on a Borel space, is proved: there exist copies (Xˆ k )k≥0 of (Xk )k≥0 and (Yˆk )k≥0 of (Yk )k≥0 (coupling) and finite random times R and S (shifts) such that (Xˆ R+k )k≥0 = (YˆS+k )k≥0

(1.1)

H. Thorisson () University of Iceland, Reykjavík, Iceland e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_1

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2

H. Thorisson

if and only if the distributions of (Xk )k≥0 and (Yk )k≥0 converge in Cesaro total variation    n−1  1 n−1    1   P (Xi+k )k≥0 ∈ · − P (Yi+k )k≥0 ∈ ·  n  → 0, n i=0

n → ∞.

i=0

Here  ·  is the total variation norm defined for bounded signed measures ν by ν = sup ν − inf ν. Note that if ν = P − Q, where P and Q are probability measures, then sup ν = − inf ν so P − Q = 2 sup(P − Q) = 2 sup |P − Q|.

(1.2)

Berbee’s result is a counterpart of earlier results linking exact coupling (coupling such that (1.1) holds with R = S, invented by Doeblin [4] to prove the basic limit theorem of Markov chains) to total variation convergence through a coupling inequality and a maximal coupling; see [5, 6, 15]; see also Theorem 4.3.2 in [2]. In [5] exact coupling is further linked to the tail σ -algebra. The term ‘shift-coupling’ for ((Xˆ k )k≥0 , (Yˆk )k≥0 , R, S) is from the 1993 paper [1] where a link to the invariant σ -algebra is established. In [17] the view was extended to continuous time and a shift-coupling inequality presented providing the Cesaro result. In that paper epsilon-couplings (for each  > 0 there is a shift-coupling such that |R − S| < , see [13, 14]) were also linked to a smooth total variation convergence through -coupling inequalities, and to a smooth tail σ -algebra. In [18] the view was further extended to a group setting, where the Borel-space condition turns out to be not needed. Applications in Palm theory to simple point processes on Rd were presented in the 1999 paper [19]. Those applications cover both coin tosses and the Poisson process in Rd as special cases, but no explicit constructions were given, only the abstract existence of shift-coupling. The first explicit construction of a shift-coupling was the surprising Extra Head Scheme for doubly infinite coin tosses and for the Poisson process on the line, presented by Liggett in the 2002 paper [12]. Then in [7, 8] and [3] came equally surprising constructions for point processes on Rd . In [11] the Palm theoretic view was extended from point processes on Rd to random measures on groups, and applications to local time of two-sided Brownian motion (and Lévy processes) followed in [9, 16] and [10]. Those applications involved unbiased Skorokhod embedding, embedding the Brownian bridge into the path of Brownian motion, and unbiased embedding of excursions. See the notes and references in these papers (and in [20]) for a more complete background.

Shift-Coupling and Maximality

3

2 Shift-Coupling and the Invariant σ -Algebra I Let G be a locally compact second countable Hausdorff topological group with Borel sets G. Let (E, E) be a measurable space equipped with a measurable flow θt : E → E,

t ∈ G.

This means that the map (t, x) → θt x is measurable with respect to G ⊗ E and E, that with e the identity of G the map θe is the identity on E, and that θs θt = θst for all s, t ∈ G. Let (, F , P) be the probability space on which all the random elements in this paper are defined. We allow (, F , P) to be extended by introducing new independent random elements or new random elements with specified regular conditional distributions given a random element that already is defined on (, F , P); see Sect. 5. Throughout the paper let X and Y be random elements in (E, E). These random elements could be random measures on (G, G) or random fields indexed by G. For instance, if G = Rd , then for a random measure X = X(·) we have θt X = X(t + ·) while for a random field X = (Xs )s∈Rd we have θt X = (Xt +s )s∈Rd . Definition Say that X and Y admit shift-coupling if there exists (possibly after extension) a random element T in (G, G) such that D

θT X = Y. D

Here = denotes identity in distribution. Let I be the invariant σ -algebra, I = {A ∈ E : θt−1 A = A, t ∈ G}. Lemma 2.1 If A ∈ I and T is a random element in (G, G), then {θT X ∈ A} = {X ∈ A}. Proof From A ∈ I we obtain the second step in {θT X ∈ A} =



{T = t, θt X ∈ A} =

t ∈G



{T = t, X ∈ A} = {X ∈ A}.

t ∈G

The only-if direction in the following theorem is easy. The if-direction relies on the maximality result proved in Sect. 4. Theorem 2.2 The random elements X and Y admit shift-coupling if and only if P(X ∈ A) = P(Y ∈ A),

A ∈ I.

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H. Thorisson D

Proof If θT X = Y and A ∈ I, then by Lemma 2.1 P(X ∈ A) = P(θT X ∈ A) = P(Y ∈ A),

A ∈ I.

For the converse claim, see Corollary 5.1 at the end of the paper.



3 Shift-Coupling Inequality and Cesaro Asymptotics Let λ be right-invariant Haar measure on (G, G). For B ∈ G with 0 < λ(B) < ∞ let UB be a random element in (G, G) that is independent of X and Y and has distribution λ(·|B), P(UB ∈ · | X, Y ) = λ(· ∩ B)/λ(B). Note that the distribution of θUB X can be written on Cesaro-average form as follows:  P(θUB X ∈ ·) =

 P(θs X ∈ ·)λ(ds) λ(B). B

Let denote the symmetric difference of two sets, B C = (B \ C) ∪ (C \ B). D

Theorem 3.1 (The Shift-Coupling Inequality) If θT X = Y , then

   P(θU X ∈ ·) − P(θU Y ∈ ·)  ≤ E λ(B BT ) . B B λ(B)

(3.1)

Proof Take A ∈ E and let the uniform UB be independent of T , X, Y . Due to D

θT X = Y , P(θUB X ∈ A) − P(θUB Y ∈ A) = P(θUB X ∈ A) − P(θUB θT X ∈ A). Thus P(θUB X ∈ A) − P(θUB Y ∈ A) = E[1{θUB X ∈ A} ] − E[1{θUB θT X ∈ A} ] 

  =E 1{θs X∈A} λ(ds) − 1{θs θT X∈A} λ(ds) λ(B). B

B

Shift-Coupling and Maximality

Right-invariance of λ yields

5

 B

1{θs θT X ∈ A} λ(ds) =

 BT

1{θs X ∈ A} λ(ds) and thus

 P(θUB X ∈ A) − P(θUB Y ∈ A) = E 1{θs X ∈ A} λ(ds) 

B

− BT

 1{θs X ∈ A} λ(ds)

λ(B).

The B ∩ BT parts of the integrals cancel and dropping what then remains of the negative integral yields  P(θUB X ∈ A) − P(θUB Y ∈ A) ≤ E

 B\BT

1{θs X ∈ A} λ(ds)

λ(B).

Now 1{θs X ∈ A} ≤ 1 and thus

λ(B \ BT ) P(θUB X ∈ A) − P(θUB Y ∈ A) ≤ E . λ(B)

(3.2)

Use the right-invariance of λ for the second identity in λ(B BT ) = λ(B \ BT ) + λ(BT \ B) = 2 λ(B \ BT ). Thus taking supremum over A ∈ E in (3.2) and consulting (1.2) yields (3.1).



In order to obtain Cesaro total variation convergence we need to assume that G is amenable. This means that there exist Følner sets, namely a family of bounded sets of positive λ-measure, Br ∈ G, r > 0, expanding to G in such a way that ∀t ∈ G :

λ(Br Br t) → 0, λ(Br )

r → ∞.

For instance, if G = Rd (under addition, with λ the Lebesgue measure) and B is a convex set of positive finite volume containing 0 in its interior, then Br = r · B, r > 0, are Følner sets. Applying the shift-coupling inequality under the Følner condition adds a limit result to the equivalence in Theorem 2.2. Theorem 3.2 Suppose there exist Følner sets Br ∈ G, r > 0. Then X and Y admit shift-coupling if and only if    P(θU X ∈ ·) − P(θU Y ∈ ·) → 0, Br Br

r → ∞.

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H. Thorisson D

Proof If θT X = Y , apply Theorem 3.1 and bounded convergence to obtain

   P(θU X ∈ ·) − P(θU Y ∈ ·)  ≤ E λ(Br Br T ) → 0, Br Br λ(Br )

r → ∞.

  Conversely, assume that  P(θUBr X ∈ ·) − P(θUBr Y ∈ ·) → 0, r → ∞. Take A ∈ I and apply Lemma 2.1 to obtain the first step in      P(X ∈ A) − P(Y ∈ A)  =  P(θU X ∈ A) − P(θU Y ∈ A)  → 0, Br Br

r → ∞.

Thus |P(X ∈ A) − P(Y ∈ A)| = 0, that is, P(X ∈ A) = P(Y ∈ A) for A ∈ I. Now apply Theorem 2.2 to obtain that X and Y admit shift-coupling.

tv

Remark 3.3 (Application in Palm Theory) Let → denote total variation converD gence. If Y is stationary (that is, θt Y = Y, t ∈ G), then the limit claim in Theorem 3.2 becomes tv

θUBr X → Y,

r → ∞.

Let η be a stationary random measure (e.g. point process) with finite intensity and ξ be its Palm version. Take X = ξ and Y = η. Assume ergodicity, P(η ∈ ·) = 0 or 1 on I. Then it is readily checked that P(ξ ∈ ·) = P(η ∈ ·) on I. Thus according to Theorem 2.2, ξ and η admit shift-coupling. Moreover according to Theorem 3.2, tv

θUBr ξ → η,

r → ∞,

if Følner sets Br , r > 0, exist.

4 The Key Theorem and a Corollary In the following basic maximality theorem we drop the condition that G is a group and θs , s ∈ G, a flow. We only assume that (G, G, λ) is some σ -finite measure space and that θt : E → E,

t ∈ G,

is some collection of mappings such that (t, x) → θt x is measurable with respect to G ⊗ E and E.

Shift-Coupling and Maximality

7

ˆ Yˆ in (E, E) and R, ˆ Sˆ in (G, G), events Theorem 4.1 There are random elements X, ˆ D, ˆ and a set A ∈ E, such that C, D Xˆ = X

 E G

and

D Yˆ = Y

(4.1)

ˆ = P({θ ˆ Yˆ ∈ ·} ∩ D) ˆ P({θRˆ Xˆ ∈ ·} ∩ C) S

(4.2)



1{θs X∈A}∩ λ(ds) = 0 = E 1 λ(ds) . ˆ Cˆ c {θs Yˆ ∈Ac }∩Dˆ c

(4.3)

G

Proof Let T1 be a random element in (G, G) with  distribution P having the same null sets as the σ -finite λ. For instance, P = n λ(·|Bn )2−n , where the Bn are a disjoint covering of G with 0 < λ(Bn ) < ∞. Let T1 be independent of a quadruple (X1 , Y1 , C1 , D1 ), where X1 , Y1 are random elements in (E, E), C1 , D1 are events, and (a) P(X1 ∈ ·) = P(X ∈ ·) and P(Y1 ∈ ·) = P(Y ∈ ·), (b) P({θT1 X1 ∈ ·} ∩ C1 ) = P({θT1 Y1 ∈ ·} ∩ D1 ), (c) ∃ A1 ∈ E : P({θT1 X1 ∈ A1 } ∩ C1c ) = 0 = P({θT1 Y1 ∈ Ac1 } ∩ D1c ). In order to obtain this (X1 , Y1 , C1 , D1 ), let X1 , Y1 satisfy (a), let μ be the common component of the measures ν = P(θT1 X1 ∈ ·) and η = P(θT1 Y1 ∈ ·), that is, dν dη dμ = ∧ , d(ν + η) d(ν + η) d(ν + η) and then introduce (by extension, see Remark 5.2) splitting events C1 and D1 cutting out the common component μ from P(θT1 X1 ∈ ·) and P(θT1 Y1 ∈ ·), P({θT1 X1 ∈ ·} ∩ C1 ) = μ = P({θT1 Y1 ∈ ·} ∩ D1 ). This yields (b) and also (c) because P(θT1 X1 ∈ ·) − μ and P(θT1 Y1 ∈ ·) − μ have no mass in common (are mutually singular). Repeat this recursively to obtain i.i.d T1 , T2 , . . . that are independent of a sequence of independent quadruples (Xk , Yk , Ck , Dk )1≤k 1     c c (d) P(Xk ∈ ·) = P Xk−1 ∈ ·|Ck−1 and P(Yk ∈ ·) = P Yk−1 ∈ ·|Dk−1 ,       (e) P θTk Xk ∈ ·  ∩ Ck = P θTk Yk ∈ · ∩ Dk  ,   (f) ∃ Ak ∈ E : P {θTk Xk ∈ Ak } ∩ Ckc = 0 = P θTk Yk ∈ Ack ∩ Dkc . Now put K = inf{k ≥ 1 : 1Ck = 1} and N = inf{k ≥ 1 : 1Dk = 1}.

8

H. Thorisson

By induction P(X ∈ ·) = P(XK ∈ ·, K ≤ k) + P(Xk+1 ∈ ·)P(K > k)

(4.4)

because for k = 1 we obtain (4.4) by summing the results of the following calculations: P(XK ∈ ·, K ≤ k) = P(X1 ∈ ·, C1 ) = μ P(Xk+1 ∈ ·)P(K > k) = P(X2 ∈ ·)P(C1c ) = P(X1 ∈ ·, C1c ) = P(X ∈ ·) − μ while if (4.4) holds for some k ≥ 1, then it holds with k replaced by k + 1 since P(X ∈ ·) − P(XK ∈ ·, K ≤ k) = P(Xk+1 ∈ ·)P(K > k)

(by the induction assumption)

c )P(K > k) = P(Xk+1 ∈ ·, Ck+1 )P(K > k) + P(Xk+2 ∈ ·)P(Ck+1

= P(XK ∈ ·, K = k + 1) + P(Xk+2 ∈ ·)P(K > k + 1). Send k → ∞ in (4.4) to obtain P(X ∈ ·) ≥ P(XK ∈ · , K < ∞). Let X∞ be independent of (Tk , Xk , Yk , Ck , Dk )1≤k k).

Shift-Coupling and Maximality

9

This and P(XK ∈ ·, K = ∞) ≤ P(XK ∈ ·, K ≥ k) yield P(XK ∈ ·, K = ∞) ≤ P(Xk ∈ ·)P(K ≥ k). D

Use this and T∞ = Tk (and the independence assumptions) to obtain P(θT∞ XK ∈ ·, K = ∞) ≤ P(θTk Xk ∈ ·)P(K ≥ k).

(4.6)

Due to (c) and (f) we have P({θTk Xk ∈ Ak } ∩ Ckc ) = 0. This and (4.6) yield P(θT∞ XK ∈ Ak , K = ∞) ≤ P({θTk Xk ∈ Ak } ∩ Ck )P(K ≥ k). Thus P(θT∞ XK ∈ Ak , K = ∞) ≤ P(Ck )P(K ≥ k) = P(K = k) so    P θT∞ XK ∈ Ak , K = ∞ ≤ P(n ≤ K < ∞) → 0,

n → ∞.

n≤k 0 G

and note that 



B ⊆ x∈E: c

G

 1{θs x∈Ac } λ(ds) > 0 .

Since λ is right-invariant we have B ∈ I. From (4.3) and B ∈ I we obtain P({Xˆ ∈ B} ∩ Cˆ c ) = 0

and P({Yˆ ∈ B c } ∩ Dˆ c ) = 0.

From (4.7), B c ∈ I and P({Yˆ ∈ B c } ∩ Dˆ c ) = 0 we further obtain P({Xˆ ∈ B c } ∩ Cˆ c ) = 0. D Thus P(Cˆ c ) = 0 and similarly P(Dˆ c ) = 0. This and (4.2) yield θRˆ Xˆ = θSˆ Yˆ .



5 Transfer from Xˆ and Yˆ to X and Y The transfer method works as follows. Let Z and Zˆ be identically distributed random elements in some measurable space (E, E) and Vˆ be a random element in a Borel space (H, H). Let Q be the joint distribution of Zˆ and Vˆ , that is, Q is a probability measure on (E, E) ⊗ (H, H) and   ˆ Vˆ ∈ ·) = Q. P (Z, Since (H, H) is Borel there exists a regular version Q(· | ·) of P(Vˆ ∈ · | Zˆ = ·). This can be used to extend the underlying probability space (, F , P) to support a new random element V as follows: define a measure Q on (, F ) ⊗ (H, H) through  Q(A × B) =

Q(B | Z) dP,

A ∈ F, B ∈ H ;

A

ˆ ˆ for (ω, v) ∈ ×H set V (ω, v) = v, Z(ω, v) = Z(ω), Z(ω, v) = Z(ω), Vˆ (ω, v) = D Vˆ (ω); rename the extended space (, F , P). Then Z = Zˆ still holds and moreover P(V ∈ · | Z = ·) = Q(· | ·) = P(Vˆ ∈ · | Zˆ = ·).

Shift-Coupling and Maximality

11

Thus D ˆ ˆ (Z, V ) = (Z, V ).

We have transferred Vˆ from Zˆ to Z to obtain V . In the following corollary we complete the proof of Theorem 2.2. Corollary 5.1 Let G be a locally compact second countable Hausdorff topological group with Borel sets G. If P(X ∈ B) = P(Y ∈ B),

B ∈ I,

then there exists (possibly after extension) a random element T in (G, G) such that D

θT X = Y. D D D ˆ Proof Due to Corollary 4.2 we have θRˆ Xˆ = θSˆ Yˆ , where Xˆ = X, Yˆ = Y and R, ˆ ˆ S are random elements in (G, G). Since (G, G) is Borel we can first transfer R from Xˆ to X to obtain R such that D ˆ R) ˆ (X, R) = (X,

and then transfer Sˆ from Yˆ to Y to obtain S such that D ˆ (Y, S) = (Yˆ , S). D Since θRˆ Xˆ = θSˆ Yˆ this implies that D

θR X = θS Y. Now transfer S from θS Y to θR X to obtain a V such that D

(θR X, V ) = (θS Y, S). Since θS −1 θS Y = Y (where S −1 denotes the group inverse of S) this implies that D

θV −1 θR X = Y. D

Put T = V −1 R to obtain the desired result, θT X = Y .



Remark 5.2 (Splitting, Used in the Proof of Theorem 4.1) Let Z be a random element in (E, E) and let μ be a measure that is a component of the distribution

12

H. Thorisson

of Z, that is, P(Z ∈ ·) ≥ μ. Let Zˆ be a random element in (E, E) and Vˆ take the values 0 and 1. Let Zˆ and Vˆ have the following joint distribution: P(Zˆ ∈ ·, Vˆ = 1) = μ,

P(Zˆ ∈ ·, Vˆ = 0) = P(Z ∈ ·) − μ.

Then Zˆ has the same distribution as Z so we can transfer Vˆ from Zˆ to Z to obtain ˆ Vˆ ). Thus V such that (Z, V ) has the same distribution as (Z, P(Z ∈ ·, V = 1) = μ, that is, we have introduced a splitting event C = {V = 1} cutting out the component μ from the distribution of Z.

References 1. Aldous, D., Thorisson, H.: Shift-coupling. Stoch. Proc. Appl. 44, 1–14 (1993) 2. Berbee, H.C.P.: In: Random walk with Stationary Increments and Renewal Theory. Mathematical Centre Tracts, vol. 112. Mathematisch Centrum, Amsterdam (1979) 3. Chatterjee, S., Peled, R., Peres, Y., Romik, R.: Gravitational allocation to Poisson points. Ann. Math. 172, 617–671 (2010) 4. Doeblin, W.: Exposé de la théorie des chaînes simple constantes de Markov à un nobre fini d’états. Rev. Math. Union Interbalkan. 2, 77–105 (1938) 5. Goldstein, S.: Coupling methods for Markov processes. Z. Wahrscheinlichkeitsth. 46, 193–204 (1979) 6. Griffeath, D.: Coupling methods for Markov processes. In: Studies in Probability an Ergodic Theory. Advances in Mathematics. Supplementary Studies, vol. 2 (1978) 7. Holroyd, A.E., Peres, Y.: Extra heads and invariant allocations. Ann. Probab. 33, 31–52 (2005) 8. Hoffman, C., Holroyd, A.E., Peres, Y.: A stable marriage of Poisson and Lebesgue. Ann. Probab. 34, 1241–1272 (2006) 9. Last, G., Mörters, P., Thorisson, H.: Unbiased shifts of Brownian motion. Ann. Probab. 42, 431–463 (2014) 10. Last, G., Tang, W., Thorisson, H.: Transporting random measures on the line and embedding excursions into Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. 54, 2286–2303 (2018) 11. Last, G., Thorisson, H.: Invariant transports of stationary random measures and massstationarity. Ann. Probab. 37, 790–813 (2009) 12. Liggett, T.M.: Tagged particle distributions or how to choose a head at random. In: Sidoravicious, V. (ed.) In and Out of Equilibrium. Progress in Probability, vol. 51, pp. 133–162. Birkhäuser, Boston (2002) 13. Lindvall, T.: On coupling of continuous-time renewal processes. J. Appl. Probab. 19, 82–89 (1982) 14. Ney, P.: A refinement of the coupling method in renewal theory. Stoch. Proc. Appl. 11, 11–26 (1981) 15. Pitman, J.: Uniform rates of convergence for Markov chains transition probabilities. Z. Wahrscheinlichkeitsth. 29, 193–227 (1974)

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16. Pitman, J., Tang, W.: The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift. Electron. J. Probab. 20, 1–28 (2015) 17. Thorisson, H.: Shift-coupling in continuous time. Prob. Theo. Rel. Fields 99, 477–483 (1995) 18. Thorisson, H.: Transforming random elements and shifting random fields. Ann. Probab. 24, 2057–2064 (1996) 19. Thorisson, H.: Point-stationarity in d dimensions and Palm theory. Bernoulli 5, 797–831 (1999) 20. Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)

Diffusion Approximation Analysis of Multihop Wireless Networks: Quality-of-Service and Convergence of Stationary Distribution K. S. Ashok Krishnan and Vinod Sharma

Abstract Consider a multihop wireless network, with multiple source–destination pairs. We obtain a channel scheduling policy which can guarantee end-to-end mean delay for different traffic streams. We show the stability of the network for this policy by convergence to a fluid limit. It is intractable to obtain the stationary distribution of this network. Thus, we also provide a diffusion approximation for this scheme under heavy traffic. We further show that the stationary distribution of the scaled process of the network converges to that of the Brownian limit. This theoretically justifies the performance of the system. We verify the theoretical properties by means of simulations. Keywords Multihop wireless network · Quality-of-service · Diffusion approximation

1 Introduction and Literature Review A multihop wireless network is constituted by nodes communicating over a wireless channel. Some of the nodes, called source nodes, have data to be sent to other nodes, called receivers. In general, the data will have to be transmitted across multiple hops. The data, originating from different applications, may have different qualityof-service (QoS) requirements, such as delay or bandwidth constraints. Therefore, we need to design routing and link scheduling algorithms that can meet all these requirements. Network performance has been studied using various mathematical techniques. Stability of flows in a network is a minimum QoS requirement. Algorithms based

K. S. Ashok Krishnan () · V. Sharma Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_2

15

16

K. S. Ashok Krishnan and V. Sharma

on backpressure [7] are throughput optimal, which means that they stabilize the network if it is possible by any other policy. Another approach is to use the framework of Markov decision processes [17]. The problem of minimizing power while simultaneously providing mean and hard delay guarantees is studied in [12]. However, knowledge of system statistics is required, and the scheme is not throughput optimal. In [13], an algorithm using per hop queue length information is presented, along with a low complexity approximation that stabilizes a fraction of the capacity region. In [18], the problem of routing and scheduling transient flows in a multihop network is studied. They also provide schemes for optimal routing. The analysis of fluid scaling of networks was pioneered in works such as [16] and [4], where it was demonstrated that stability of the fluid limit of the network implies the stability of the network. Further, one may obtain bounds on moments of asymptotic values of the queues using these techniques [5]. A comprehensive treatment of work in this direction is provided in [14]. A delay-based scheduling scheme is proposed in [9], where the analysis of stability uses fluid limits. Diffusion approximation of networks [20] has been used to study the behaviour of the system under a scaling corresponding to the functional central limit theorem [1]. The weak limit of the diffusion scaled systems under heavy traffic is generally a reflected Brownian motion [8], which under certain assumptions on the scaling rate has a limiting stationary distribution. This distribution may be used as a proxy for the actual distribution of the system state. The diffusion approximation of the MaxWeight algorithm is studied in [19], using properties of certain fluid scaled paths to obtain properties of the diffusion scaled paths, as in [2]. Of these, [19] deals with a discrete time switch under the MaxWeight policy. To obtain the behaviour of the network under stationarity, one also needs to show the convergence of the stationary distribution of the network to that of the limiting network. Sufficient conditions for these have only recently been studied, in [6] and [3], in the case of Jackson networks. An important requirement for the exchange of limits in [3] to hold is the Lipschitz continuity of an underlying Skorokhod map, which may not always hold in general. A recent concise survey of diffusion approximations and convergence of stationary distributions is given in [15]. Our main contributions in this work are summarized below. – We propose a new link scheduling algorithm to guarantee end-to-end mean delay for different traffic flows. This algorithm is close to the one proposed in [10] and has the same fluid limit. Hence, it is also throughput optimal. – We obtain a reflected Brownian motion (with drift) as the weak limit of the system under diffusion scaling. This Brownian motion exhibits state space collapse. – We also show that the stationary distribution of our network converges to the stationary distribution of the limiting Brownian network. This allows us to approximate the stationary distribution of our network by that of the limiting network which is explicitly available. While diffusion approximations have been used to traditionally study networks, the proof of convergence of stationary distributions is still not known in many systems. Our work proves this in a

Diffusion Approximation

17

controlled multihop wireless system with a general scheduling policy with QoS provisions. However, our proof does not require Lipschitz continuity of the Skorokhod map, unlike [3]. The rest of the paper is organized as follows. In Sect. 2, we describe the system model and formulate the control policy used in the network. In Sect. 3, we describe the two scaling regimes in which we study the network and prove the existence of the Brownian limit. In Sect. 4, we show that the stationary distribution of the limit of the scaled process is the stationary distribution of the limiting Brownian process. In Sect. 5, we provide simulation results, followed by the conclusions in Sect. 6.

2 System Model and Control Policy We consider a multihop wireless network (Fig. 1). The network is a connected graph G = (V, E) with V = {1, 2, . . . , N} being the set of nodes and E being the set of links on V. The system evolves in discrete time denoted by t ∈ {0, 1, 2, . . .}. The links are directed, with link (i, j ) from node i to node j having a time varying channel gain Hij (t) at time t. Denote the channel gain vector at time t by H (t), evolving as independent and identically distributed (i.i.d.) process across slots with distribution γ over a finite set H. Let Eh (t) denote the cumulative number of slots till time t when the channel state was h ∈ H. The vector of all Eh (t) is denoted by E(t). f At a node i, Ai (t) denotes the cumulative (in time) process of exogenous arrival of packets destined to node f . The packets arrive as an i.i.d sequence across slots, f f f with mean arrival rate λi and variance σi . Let λ denote the vector of all λi . All traffic in the network with the same destination f is called flow f ; the set of all flows is denoted by F . Each flow has a fixed route to follow to its destination. At each node f there are queues, with Qi (t) denoting the queue length at node i corresponding to f flow f ∈ F at time t. For a queue Qi with i = f , we have the queue evolution given by, f

f

f

f

f

Qi (t) = Qi (0) + Ai (t) + Ri (t) − Di (t), Fig. 1 A simplified depiction of a wireless multihop network

m

(1)

Him(t)

i

Qji (t) Aji (t)

j p l

n

k

18

K. S. Ashok Krishnan and V. Sharma f

where Ri (t) is the cumulative arrival of packets by routing (i.e., arrivals from f f other nodes), and Di (t) is the cumulative departure of packets. Let Sij (t) be the cumulative number of packets of flow f transmitted over link (i, j ). We write f

Ri (t) =



f

f

Ski (t), and Di (t) =

k=i



f

(2)

Sij (t).

j =i

We assume that the links are sorted into M interference sets I1 , I2 , . . . , IM . At any time, only one link from an interference set can be active. A link may belong to multiple interference sets. We also assume that each node transmits at unit power. Then, the rate of transmission between node i and node j is given by an achievable rate function, which depends on H (t) and the schedule at time t. The vector of queues at time t is denoted by Q(t). Similarly, we have the vectors f A(t), R(t), D(t) and S(t). Consider a vector = [ ij ](i,j )∈E ,f ∈F . Define S to be the set of all that satisfies, f

1. ij ∈ {0, 1} ∀i, j, f,  f 2. f ij ≤ 1, ∀(i, j ) ∈ E,   f 3. (i,j )∈Im f ij ≤ 1, m = 1, . . . , M. f

Such a vector is called a schedule. Clearly, any ∈ S has elements ij , which, if one, indicates that flow f is to be sent over link (i, j ). The constraints listed above represent the fact that no two flows can be transmitted simultaneously on a link at any time. Furthermore, no two links in an interference set can transmit at the same time. For any schedule and channel state h, we assume there exists a channel rate f function μ = [μij ](i,j )∈E ,f ∈F , where, f

μij = F(h, ),

(3)

where F is some achievable rate function. We want to develop scheduling policies such that the different flows obtain f f their end-to-end mean delay deadline guarantees. Define Qij = max(Qi −  f f Qj , 0), Qf (t) = i Qi (t). Let M(t) = {F(H (t), ) : ∈ S} be the set of feasible rates at time t. Our network control policy is as follows. At each t, given the region of feasible rates M(t), we obtain the optimal allocation μ∗ , μ∗ = argμ∈M(t ) max



f

f

f

α(Qf (t), Q )Qij (t)μij ,

(4)

i,j,f f

f

assuming Qij > 0 for at least one link flow pair (i, j ), f . If all Qij are zero, we define the solution to be μ∗ = 0. We optimize a weighted sum of rates, with more weight given to flows with larger backlogs, with α capturing the delay requirement

Diffusion Approximation

19 f

of the flow. The weights α are functions of Qf (t), and Q denotes a desired value for the queue length of flow f . We use, α(x, x) = 1 +

a1 . 1 + exp(−a2 (x − x))

(5)

Thus, flows requiring a lower mean delay would have a higher weight compared to flows needing a higher mean delay. Flows whose mean delay requirements are not f met should get priority over the other flows. The Q are chosen, using Little’s law, f as Q = λf D, where D is the target end-to-end mean delay and λf is the arrival rate of flow f . Note that we will often use α(x) instead of α(x, x) ¯ for simplicity of notation. Let GhI ijf (t) be the number of slots till time t, in which channel state was h, the schedule was I and flow f was scheduled over (i, j ). Denote the vector of all GhI ijf (t) by G(t). Define the process, Z = (A, E, G, D, R, S, Q),

(6)

where we have A = (A(t), t ≥ 0) (and likewise for the other processes). This process describes the evolution of the system. The state of the system at time t is Q(t), which takes values in a state space Q. Define the capacity region as follows. Definition 1 The capacity region Λ of the network is the set of all λ for which a stabilizing policy exists. We denote the set of real numbers by R, and the set of integers by Z. We use C [0, ∞) to denote the set of all continuous functions from [0, ∞) to R, and D[0, ∞) the set of all right continuous functions with left limits (RCLL) from [0, ∞) to R. We use ⇒ to denote weak convergence. For a vector x, |x| j denotes its norm (modulus). The vector of variables of the form xi over all i and j j will be denoted by (xi )i,j . For any two vectors x and y, we denote their inner product by x, y. For a vector x = (x1 , . . . , xn ) and scalar t, xt will be the product (x1 t, . . . , xn t). We will also need the following definition. Definition 2 A sequence of functions ζn is said to converge uniformly on compact sets (u.o.c) to ζ if ζn → ζ uniformly on every compact subset of the domain.

3 Fluid and Diffusion Limits Now we describe the behaviour of Z under two scaling regimes, fluid and diffusion.

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K. S. Ashok Krishnan and V. Sharma

3.1 Fluid Scaling For the process Z, define the scaled continuous time process, zn (t) =

Z(nt) , n

(7)

where · represents the floor function. This is called the fluid scaled process. Note that the time argument t on the left side is continuous, while that on the right is discrete. Whether a time argument is discrete or continuous will be generally clear from the context. Let zn denote the process (zn (t), t ≥ 0). We have zn = (a n , en , g n , d n , r n , s n , q n ),

(8)

with the scaling in (7) being applied to each component of Z. Note that a n = f,n (ai )i,f , and a similar notational convention holds for all the constituent functions of z. The limit of zn , as n → ∞, offers insight into the behaviour of the system under the scheduling policy in (4). The following result can be shown for our policy. Lemma 1 The algorithm described by the slot-wise optimization in (4) stabilizes the system for all arrival rate vectors λ in the interior of Λ. Here, stability implies that the Markov chain Q(t) is positive recurrent. The proof of this lemma proceeds on the lines of the proof of Theorems 1 and 3 in [10]. It can be shown that there exists a subsequential limit z = (a, e, g, d, r, s, q) for the family {zn , n ≥ 0}. This z is called the fluid limit, and the convergence of the processes is u.o.c. The limiting functions are also Lipschitz continuous, and hence almost everywhere differentiable. The points t at which it is differentiable are called regular points. In addition, the limiting functions satisfy the following properties (see [11]): a(t) = λt, f

ri (t) =



f

sj i (t),

e(t) = γ t, f

di (t) =



j

 I

(9) f

sij (t),

(10)

j

q(t) = q(0) + a(t) + r(t) − d(t),

(11)

˙ q(t) ˙ = λ + r˙ (t) − d(t),

(12)

f

hI gijf (t) = eh (t), sij (t) =



t 0

f

s˙ij (τ )dτ,

(13)

Diffusion Approximation

21

and s˙ (t) satisfies 

f

f

α(q f (t))qij (t)˙sij (t) = max μ¯

i,j,f



f

f

α(q f (t))qij (t)μ¯ ij ,

(14)

i,j,f

 where the dot indicates derivative, at regular t and μ¯ = h γh μ(h, S), where μ(h, S) is an achievable rate when channel is in state h and schedule is S. Using the Lyapunov function,  L1 (q(t)) = −

∞

exp(t − τ )

t



f

f

α(q f (τ ))qi (τ )q˙i (τ )dτ,

i,f

we can establish that the fluid system is stable, and consequently, so is the stochastic system. We can also show that the draining time of the system, which is the time for all the fluid queues to go to zero, is of the form T |x| , where T is a finite quantity,|x| is the initial norm of the fluid queues and  denotes the distance of λ to the boundary of Λ. Studying the fluid limit gives us insights into the stability properties of the system. However, it only proves the existence of a stationary distribution. In order to predict the behaviour of the system, one needs the stationary distribution, or some approximation to the same. However, explicitly computing the stationary distribution for our system is not feasible. Thus, we define the heavy traffic regime, and the associated diffusion scaling, below. We will also show that the stationary distribution of our system process converges to that of the limiting Brownian network. This will provide us an approximation of the stationary distribution under heavy traffic, the scenario of most practical interest.

3.2 Diffusion Scaling Consider a sequence of systems, Z n . Each system differs from the other in its arrival rate, λn . The λn are chosen such that, as n → ∞, λn → λ∗ , and, lim nψ, λn − λ∗  = b∗ ∈ R,

n→∞

(15)

where λ∗ is a point on the boundary of Λ, and ψ denotes the outer normal vector to Λ at the point λ∗ . This is known as heavy traffic scaling. We will also assume that λ∗ falls in the relative interior of one of the faces of the boundary of Λ. For this sequence of systems, we define the diffusion scaling, given by, zˆ n (t) =

Z n (n2 t) . n

(16)

22

K. S. Ashok Krishnan and V. Sharma

Let zˆ n denote the process (ˆzn (t), t ≥ 0). As before, we have zˆ n = (aˆ n , eˆn , gˆ n , dˆn , rˆ n , sˆn , qˆ n ). Define the system workload W n (t) in the direction ψ as, W n (t) = ψ, Qn (t),

(17)

and, wˆ n (t) =

W (n2 t) . n

Denote wˆ n = (wˆ n (t), t ≥ 0). Define an invariant point to be a vector φ that satisfies, for some k > 0, α(φ)φ = kψ,

(18)

where α(φ) is the vector of all α(φj ), with α defined in (5), and the product of the vectors is element-wise. Then, we have the following result, which characterizes the weak convergence of the diffusion scaled processes. Theorem 1 Consider {ˆzn , n ∈ N }, under heavy traffic scaling satisfying (15),and N a sequence of positive integers n increasing to infinity. Assume that the fluid f scaled z = (a, e, g, d, r, s, q) has components a = (ai )i,f and e = (eh )h∈H that satisfy, with probability one, as m → ∞, for any T > 0, for all i, j , f , c ∈ H, f

f

f

max

sup |ai (m,  + ) − ai (m, ) − λi | → 0,

(19)

max

sup |ec (m,  + ) − ec (m, ) − γc | → 0.

(20)

0≤≤mT 0≤≤1 0≤≤mT 0≤≤1

Further, assume that, qˆ n (0) ⇒ cφ,

(21)

where c is a non-negative real number. Then, the sequence {wˆ n , n ∈ N } converges weakly to a reflected Brownian motion wˆ as n → ∞, in D[0, ∞). Further, {qˆ n , n ∈ N } converges weakly to φ w. ˆ The existence of the Brownian limit is demonstrated as follows. We write the scaled workload wˆ n as the sum of two terms, one of which converges to a Brownian motion, and the second as its corresponding regulating process. Together, they act as a reflected Brownian motion. The detailed proof is available in [11].

Diffusion Approximation

23

Having established the existence of a limiting Brownian motion, we proceed to demonstrate that the stationary distributions of the scaled systems converge to the stationary distribution of the Brownian motion, in the next section.

4 Convergence of Stationary Distributions In order to establish the convergence of stationary distributions, we use the following result, which is a consequence of Theorems 3.2, 3.3 and 3.4 of [3]. Lemma 2 Assume that, for all nodes i, j , flows f , for any n ≥ 1, t ≥ 0, we have, for some B < ∞,  E

  2  f,n  f,n sup Ai (k) − a¯ i (k) ≤ Bt,

(22)

  2  f,n  f,n sup Ri (k) − r¯i (k) ≤ Bt,

(23)

  2  f,n  f,n ¯ sup Di (k) − di (k) ≤ Bt.

(24)

0≤k≤t

 E  E

0≤k≤t

0≤k≤t

Further, assume that there exists T such that for all t ≥ T , we have, lim sup

|x|→∞ n

1 E|qˆx (n, t|x|)|2 = 0. |x|2

(25)

Then the sequence of distributions {πn } is tight. It can be shown that the above conditions are satisfied in our case, as stated below. Lemma 3 In our system model, conditions (22)–(24) hold. Further, there exists T such that (25) holds. Consequently, the sequence {πn } is tight.



Proof See [11]. As a consequence of the above two lemmas, we have the following result. Theorem 2 As n → ∞, qˆ n (∞) ⇒ φ w(∞), ˆ as n → ∞,

(26)

where the time argument being infinity denotes the respective stationary distributions. Proof See [11].



24

K. S. Ashok Krishnan and V. Sharma

The Brownian motion wˆ obtained as the limit of wˆ n is a unidimensional reflected Brownian motion, having drift b∗ < 0. The distribution of w(∞) ˆ is given by [8], P[w(∞) ˆ < y] = 1 − exp(2b ∗ y/σ 2 ).

(27)

This therefore becomes an approximation for the queue length distribution of the system under heavy traffic.

5 Numerical Simulations For simulations, we consider two topologies. In both cases, the slot-wise allocation is done by performing the optimization (4). We compute the solution numerically by means of an exhaustive search. Since the search space of the optimization increases exponentially in the number of channel states, we limit the channel states to take values over a finite set of small size. Example 1 We consider a star network topology (Fig. 2). There are two Poisson distributed arrival processes, one arriving at node 1, with node 4 as its destination. The other arrives at node 2, with node 5 as destination. Two links which share a common node interfere with each other. Thus, there is one interference set, which contains all the links. Consequently, only one link can be active at a time. We assume that the channels are independent and identically distributed, with the distribution being uniform over the values {0, 1, 2, 3}. The arrival vector (λ1 , λ2 ) = (λ, λ), i.e., increasing along the line of unit slope. Under heavy traffic, it is easy to see that, given the interference constraints, it is optimal to schedule the link with the highest channel gain. From simulations, the maximum arrival rate that can be supported by scheduling the link with the highest channel gain yields λ∗ = (0.65, 0.65). From the diffusion approximation and (27), we can see that the mean of the Brownian σ2 motion corresponding to the queue can be approximated by the vector φ 2b ∗ . The Brownian motion is a limit of the scaled process of the form n, we may approximately write,

Fig. 2 Example 1: the network

Q(n2 t)



σ2 nφ 2b ∗.

Q(n2 t ) n .

For a large

If we run the simulations for a

1

4

3

2

5

Diffusion Approximation Table 1 Approximation of queues

25

Arrival rate λ 0.64 0.641 0.642 0.643 0.644 0.645 0.646 0.647

Mean queue length 233 263 319 367 381 479 517 568

Approximation 232 258 290 332 387 465 581 775

The mean queue length of the flow 1 → 3 → 5 corresponding to various arrival rates is displayed, along with the numerical approximation Table 2 Mean queue length target and obtained, for both flows

λ 0.63 0.64 0.641

Mean queue length asked (250, 100) (250, 100) (250, 100)

Queue length obtained (213, 98) (264, 110) (292, 120)

time n, we may further also approximately write b∗ = n|λ − λ∗ |. Hence, we have the approximation, Q(∞)  φ

σ2 . 2|λ − λ∗ |

(28)

We will be looking at the total queue length of the flow 1 → 3 → 4. The value of ¯ for both σ 2 is 2λ + σˆ 2 . The vector φ is approximately ( √1 , √1 ). (The value of Q 2

2

queues is set at 100.) We take σˆ 2  8. The values of the total queue length of the flow 1 → 3 → 5 are listed in Table 1 (owing to symmetry both queue lengths are same), for simulation runs of length 105, averaged over 20 simulations. It can be seen that the approximations follow the queue length closely. In order to demonstrate that the algorithm can satisfy different QoS requirements, we simulate the network at three points in the interior of the capacity region. The mean queue length asked from the flows is 250 and 100, respectively. We also pick a2 in the expression of α for the second flow to be 4, since it requires a tighter constraint to be met. In Table 2, the first column gives the arrival rate, the second shows the target queue length for the two flows and the final column shows the queue length obtained. We see that the end-to-end mean queue length requirement is met for both the flows till rate 0.64. The capacity boundary is at 0.65. Thus, our algorithm can provide QoS under heavy traffic as well.

26 Fig. 3 Example 2: the network

K. S. Ashok Krishnan and V. Sharma 1

5

3

4

2

8

6

7

9

Example 2 Consider the network in Fig. 3. The arrival process, channel state distribution and interference constraints are the same as in Example 1. There are three flows, 1 → 3 → 4 → 6 → 8, 2 → 3 → 4 → 5 and 7 → 4 → 6 → 9. They will be called flow 8, flow 5 and flow 9. From simulations, the boundary of the capacity region, λ∗ ≈ (0.59, 0.59, 0.01). We take arrival rates close to this point and show the values of total queue length of flow 8 obtained by simulations and the numerical approximations (using (28)), in Table 3. For calculating the approximation, we use σˆ 2 ≈ 9. In this case also, the approximations track the queue lengths well. Just as in the previous case, we provide an example to show how the queue length values meet targets, in Table 4. These are simulated at the arrival rate (0.55.0.55, 0.01), which is in the interior of the capacity region. In the weight function α, we use a1 = 5, a2 = 1 to give weights to flows. Since flows 8 and 5 are competing for network resources, delays of both cannot be reduced simultaneously. This is also clear from the simulations. Table 3 Entries of the form (a, b) indicate delay target a, delay achieved b

Table 4 Entries of the form (a, b) indicate delay target a, delay achieved b

Arrival rate λ 0.5 0.54 0.56 0.57 0.58 0.582 0.584 0.585

Mean queue length 21 52 99 119 253 331 403 457

Approximation 26 47 79 144 239 299 399 479

Mean delay (slots) for each flow Flow 8 Flow 5 Flow 9 (50, 52) (100, 112) 9 (40, 46) (100, 114) 9 (100, 139) (50, 53) 21 Arrival rate is (0.55.0.55, 0.01)

Diffusion Approximation

27

6 Conclusion We have presented an algorithm for scheduling in multihop wireless networks that guarantees end-to-end mean delays of the packets transmitted in the network. The algorithm is throughput optimal. Using diffusion scaling, we obtain the Brownian approximation of the algorithm. We also prove theoretically that the stationary distribution of the limiting Brownian motion is the distribution of a sequence of scaled systems, and is consequently a good approximation for the stationary distribution of the original system. Using these relations, we obtain an approximation for queue lengths, and demonstrate via simulations that these are accurate.

References 1. Billingsley, P.: Convergence of Probability Measures. Wiley, London (1968) 2. Bramson, M.: State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30(1–2), 89–140 (1998) 3. Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34(1), 45–56 (2009) 4. Dai, J.G.: On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5, 49–77 (1995) 5. Dai, J.G., Meyn, S.P.: Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Autom. Control 40(11), 1889–1904 (1995) 6. Gamarnik, D., Zeevi, A., et al.: Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab. 16(1), 56–90 (2006) 7. Georgiadis, L., Neely, M.J., Tassiulas, L., et al.: Resource allocation and cross-layer control in wireless networks. Found. Trends® Netw. 1(1), 1–144 (2006) 8. Harrison, J.: Brownian Motion and Stochastic Flow Systems. Wiley, London (1985) 9. Ji, B., Joo, C., Shroff, N.B.: Delay-based back-pressure scheduling in multihop wireless networks. IEEE/ACM Trans. Netw. 21(5), 1539–1552 (2013) 10. Krishnan, A., Sharma, V.: Distributed control and quality-of-service in multihop wireless networks. In: 2018 IEEE International Conference on Communications (ICC), pp. 1–7. IEEE, Piscataway (2018) 11. Krishnan, A., Sharma, V.: Quality-of-service in multihop wireless networks: diffusion approximation (2018). arXiv:1810.12209 12. Kumar, S.V., Sharma, V.: Joint routing, scheduling and power control providing hard deadline in wireless multihop networks. In: 2017 Information Theory and Applications Workshop (ITA). San Diego (2017) 13. Li, B., Srikant, R.: Queue-proportional rate allocation with per-link information in multihop wireless networks. Queueing Syst. 83(3–4), 329–359 (2016) 14. Meyn, S.: Control Techniques for Complex Networks. Cambridge University Press, Cambridge (2008) 15. Miyazawa, M.: Diffusion approximation for stationary analysis of queues and their networks: a review. J. Oper. Res. Soc. Jpn. 58(1), 104–148 (2015) 16. Rybko, A.N., Stolyar, A.L.: Ergodicity of stochastic processes describing the operation of open queueing networks. Problemy Peredachi Informatsii 28(3), 3–26 (1992) 17. Singh, R., Kumar, P.: Throughput optimal decentralized scheduling of multi-hop networks with end-to-end deadline constraints: unreliable links (2016). arXiv:1606.01608

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18. Siram, V., Varma, K., et al.: Routing and scheduling transient flows for QoS in multi-hop wireless networks. In: 2018 International Conference on Signal Processing and Communications (SPCOM). Bangalore (2018) 19. Stolyar, A.L., et al.: Maxweight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic. Ann. Appl. Probab. 14(1), 1–53 (2004) 20. Williams, R.J.: Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Syst. 30(1–2), 27–88 (1998)

Analysis of Retrial Queue with Heterogeneous Servers and Markovian Arrival Process Liu Mei and Alexander Dudin

Abstract Multi-server retrial queueing system with heterogeneous servers is analyzed. Customers arrive to the system according to the Markovian arrival process. Arriving primary customers and customers retrying from orbit occupy available server with the highest service rate, if any. Otherwise, the customers move to the orbit having an infinite capacity. Service times have exponential distribution. The total retrial rate infinitely increases when the number of customers in orbit increases. Behavior of the system is described by multi-dimensional continuous-time Markov chain which belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows to derive simple and transparent ergodicity condition and compute the stationary distribution of the chain. Presented numerical results illustrate the dynamics of some performance indicators of the system when the average arrival rate increases and the importance of account of correlation in the arrival process. Keywords Retrial queue · Heterogeneous servers · Markovian arrival process

1 Introduction Theory of retrial queues is an important part of queueing theory that takes into account the effect of retrials. Capacity of the system is finite and some customers cannot be accepted for service immediately upon arrival due to the temporal Research is supported by “RUDN University Program 5–100” and the grant F19KOR-001 of Belarusian Republican Foundation for Fundamental Research. L. Mei Belarusian State University, Minsk, Belarus A. Dudin () Belarusian State University, Minsk, Belarus Peoples Friendship University of Russia (RUDN University), Moscow, Russia e-mail: [email protected]

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_3

29

30

L. Mei and A. Dudin

unavailability of the capacity. In contrast to the queues with buffers where such customers are placed to a buffer and, then, are picked up for service according to some disciplines and the queues with losses where such customers are lost, in retrial queues such customers move to some virtual place called orbit and try to get access to service after random intervals of time. Due to their high practical interests, retrial queues attract a lot of attention of researchers. Field of applications of the theory of retrial queues includes various telecommunication systems with disciplines of multiple access, databases, call centers, etc. For references to the state of the art in research in retrial queues the books [1] and [9] are recommended. Due to the state-inhomogeneous behavior of the Markov chains that describe behavior of retrial queues, their analysis is essentially more involved than analysis of queues with buffers or losses. The most essential difficulties arise in analysis of multi-server retrial queues even in the simplest assumptions about the arrival, service, and retrial processes, see, e.g., the study of the M/M/N retrial queue with the classical retrial policy presented in [9]. The difficulties essentially increase if more realistic assumptions about the arrival and service process are imposed. In [2], the BMAP /P H /N type retrial queue is studied. Here, the BMAP stands for the batch Markovian arrival process introduced in [13] as a potentially useful descriptor of the correlated bursty flows in modern telecommunication networks. For more information about the BMAP and related research see [3, 20]. In our paper, we assume that the arrival process is the MAP which is the particular case of the BMAP when no batch arrival is allowed. Abbreviation P H denotes phasetype distribution, see [15]. This class of distributions is quite wide and includes, in particular, exponential, Erlangian, Coxian distributions, and their mixtures. In consideration of multi-server queues, usually it is assumed that the servers are identical and an arbitrary idle server is engaged with equal probability for the service when the new customer arrives. Much less investigated are the queues with heterogeneous servers which are more interesting subject for research. Often, quite non-trivial optimization problems relating to assigning the servers to arriving customers, depending of relations of the means service rates and costs of their use, arise. The problem of an optimal allocation of jobs between heterogeneous servers aiming to minimize the mean number of jobs in the ordinary queueing system was considered in [5, 11, 12, 16–19]. It was shown that the optimal policy belongs to a class of monotone policies, i.e., threshold policies, which use a slow server only when the queue length exceeds a certain threshold. In paper [6], it is shown for the retrial queue with heterogeneous customers and the classical retrial policy that a threshold policy is also optimal for retrial queues and an algorithm, which allows to construct optimal policies for a versatile class of queueing systems, is proposed. Analogous analysis is given in [6] for the case of the constant retrial rate. Multi-server retrial queues, in which the servers are homogeneous, however the new arriving customer does not select an arbitrary idle server with equal probability but is addressed for the service to the certain concrete server, are considered, e.g., in [8] and [14]. In this paper, we address the multi-server retrial queue of MAP /Mˆ N /N type. The symbols Mˆ N mean that distributions of service time at the servers are

Retrial Queue with Heterogeneous Servers

31

exponential with different service rate. We assume that the servers are enumerated in the order of decreasing the rates, i.e., the server-1 is the fastest, . . . , the server-N is the slowest. Known results about the structure of the optimal control, see, e.g., [7], assume that the decision-maker has an opportunity to observe the number of customers in the orbit and activates a new, slower, server if this number exceeds the definite threshold. In our paper, we make an assumptions that: (1) the number of customers in the orbit is not observable that takes place in the majority of real-world systems because the orbit is a virtual place and indeed the waiting customers are distributed in some, probably very wide, area and are invisible; (2) service discipline is conservative. This means that if the customer from the orbit makes an attempt and not all servers are busy, this customer will be accepted for service. The problem of choosing a concrete server from the set of available servers is quite difficult. Its solution should be prefaced by formulation of some economic criterion including, e.g., costs of waiting of customers in orbit (or sojourn time in the system) and costs of using available servers per unit of time. In the borders of this paper, we do not account the economic aspects (this is planned to in further research) and examine the discipline of servers assignment as: the fastest server should be used first. Change of the server is not allowed during the service of any customer. The structure of the paper is as follows. In Sect. 2, mathematical model is formulated. In Sect. 3, dynamics of the considered system is described by the multi-dimensional continuous-time Markov chain. The generator of this chain is presented. It is shown that this Markov chain belongs to the class of asymptotically quasi-Toeplitz Markov chains. In Sect. 4, sufficient conditions for ergodicity and non-ergodicity of this Markov chain are presented. Section 5 contains a short comment about computation of the stationary distribution of the Markov chain and some performance measures of the system. Section 6 is devoted to brief description of the numerical results. Section 7 concludes the paper.

2 The Mathematical Model We consider an N-server queueing system. The primary customers arrive to the system according to a Markovian arrival process (MAP ). We denote the directing process of the MAP by νt , t ≥ 0. The state space of the irreducible continuoustime Markov chain νt is {0, 1, . . . , W }. The intensities of transitions of the process νt are defined as the entries of matrices (D0 ,D1 ) of size W¯ = W + 1. The matrix D(1) = D0 + D1 is an infinitesimal generator of the process νt . The vector θ that is the unique solution to the system of equations θ D(1) = 0, θe = 1 defines the stationary distribution of the process νt . Here and thereafter e is a column vector of an appropriate size consisting of 1’s and 0 is a row-vector of an appropriate size consisting of 0’s. The average (fundamental) arrival rate λ of the MAP is defined as λ = θ D1 e. The coefficient cvar of variation of intervals between customer arrivals is defined by

32

L. Mei and A. Dudin

cvar = 2λθ (−D0 )−1 e−1. The coefficient of correlation cvar of successive intervals 2 . between arrivals is computed as ccor = (λθ (−D0 )−1 D1 (−D0 )−1 e − 1)/cvar Service time distribution is assumed to be exponential. Different servers have different service rates, respectively, μ1 , μ2 , · · · , μN . Here, we assume that the servers are enumerated in such a way as the inequalities μ1 > μ2 > · · · > μN are fulfilled. However, in a future the obtained results can be used for solving the problem of the optimal numeration of the servers taking into account not only the service rates, but also the costs of the use of the servers. If the arriving customer meets all servers being idle, the customer enters the first server to receive the service. If the first server is busy, then the customer enters the idle server with the minimum number. If all servers are busy, then the customer goes to the orbit. Capacity of the orbit is unlimited. These customers are said to be repeated customers. These customers try their luck later until they will be served. We assume that the total flow of retrials is such that the probability of generating the retrial attempt in the interval (t, t + Δt) is equal to αi Δt + o (Δt) when the orbit size (the number of customers on the orbit) is equal to i, i > 0, αi = 0 when i = 0. We do not fix the explicit dependence of the intensities αi on i. We assume the infinitely increasing retrial rate: lim αi = ∞. This holds true, in particular, for i→∞

classic retrial strategy where αi = iα and the linear strategy αi = iα + γ . Our goal is to derive the stationary state distribution of the system.

3 The Process of the System States Let, at the moment t, t > 0, – it be the number of customers on the orbit, it ≥ 0; (n) – ξt be the state of the service on the nth server, n = 1, N :  ξt(n)

=

0, if the nth server is idle; 1, if the nth server is busy;

– νt be the state of the directing process of the MAP , νt = 0, W . Consider the continuous-time multi-dimensional process (1)

(N)

ζt = {it , ξt , . . . , ξt

, νt }, t ≥ 0.

It is easy to see that this process is an irreducible Markov chain. Let us assume that the stationary probabilities of this Markov chain   (1) (N) π(i, r (1) , . . . , r (N) , ν) = lim P it = i, ξt = r (1) , . . . , ξt = r (N) , νt = ν t →∞

exist for any i ≥ 0, r (n) = 0, 1, n = 0, N , ν = 0, W .

Retrial Queue with Heterogeneous Servers

33

Enumerate the states of the chain ζt , t ≥ 0, in lexicographic order and form the row-vector π(i, r (1) , . . . , r (N) ) = (π(i, r (1), . . . , r (N) , 0), . . . , π(i, r (1) , . . . , r (N) , W )) of the stationary probabilities π(i, r (1) , . . . , r (N) , ν) , and the row-vectors π i , consisting of the vectors π(i, r (1) , . . . , r (N) ), i ≥ 0. Note that the size of the vectors π i is equal to K = (W + 1) 2N . Define also the infinite-dimensional probability vector π = (π 0 , π 1 , π 2 , . . .). For the use in the sequel, introduce the following notation: • I is an identity matrix of appropriate dimension (when needed the dimension is identified with a suffix); • On denotes zero matrices of size n; • ⊗ and ⊕ are the symbols of the Kronecker product and sum of matrices, S ⊗l = S ⊗ · · · ⊗ S , l ≥ 1;    l

• J is the square matrix of size 2N given by J = diag{0, . . . , 0, 1}; • diag{. . . } means the diagonal matrix with the diagonal entries given in the brackets; ⎛ ⎞ 0 0 0 0 ⎜ μN −μN ⎟ 0 0 ⎟; • G=⎜ ⎝μN−1 0 −μN−1 ⎠ 0 0 μN−1 μN −μN−1 − μN $ % O2k+1 O2k+1 • Iˇk(n) = I2n−k−2 ⊗ , k = 1, n − 2 , n = 2, N − 1; I2k+1 −I2k+1 $ % $ % 0 1 • a1 = , a2 = , b1 = (0, 1) , b2 = (1, 0) . 1 0 Lemma 1 If the vector π of stationary probabilities exists, then it satisfies the equilibrium equations π Q = 0, πe = 1 where 0 is the infinite row-vector consisting of zeroes and the matrix Q, which is the infinitesimal generator of the chain ζt , t ≥ 0, has the following structure: ⎛

Q00 ⎜ Q10 ⎜ ⎜ Q=⎜ 0 ⎜ 0 ⎝ .. .

Q01 Q11 Q21 0 .. .

0 Q12 Q22 Q32 .. .

0 0 Q23 Q33 .. .

⎞ ··· ···⎟ ⎟ ···⎟ ⎟ ···⎟ ⎠ .. .

(1)

34

L. Mei and A. Dudin

where the blocks Qij , i, j ≥ 0, j = {max {0, i − 1} , i, i + 1} of the matrix Q have size K and are defined as follows: Qi,i+1 = J ⊗ D1 , Qi,i−1 = αi I˜β where ⎛ O2N −1 ×2N −1 ⎜ ⎜O2N −2 ×2N −1 ⎜ ⎜O N −3 N −1 ⎜ 2 ×2 ⎜ .. ˜ Iβ = ⎜ ⎜ . ⎜ ⎜ O 1 N −1 ⎜ 2 ×2 ⎜ ⎝ O20 ×2N −1 O1×2N −1

a2 ⊗ I2N −2 a1 ⊗ a2 ⊗ I2N −3 · · · a1 ⊗(N−2) ⊗ a2 a1 ⊗(N−1) a2 ⊗ I2N −3 O2N −3 ×2N −3 .. . O21 ×2N −3 O20 ×2N −3 O1×2N −3

O2N −2 ×2N −2 O2N −3 ×2N −2 .. . O21 ×2N −2 O20 ×2N −2 O1×2N −2

⎧ ⎪ μ " (b ⊗(m−1) ⊗ b2 ⊗ I2N −r−1 ) ⊗ IW¯ , ⎪ ⎪ r +1 1 ⎪ ⎪ ⎪ ⎪ ⎪ μr " +1 b1 ⊗(m−1) ⊗ IW¯ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D0 ⊕ ΔN−r−1 − αi IW¯ ·2N −r−1 , ⎪ ⎪ ⎨ N (Qi,i )r,r "  ⎪ ⎪ D − μk IW¯ , 0 ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⊗(l−1) ⎪ ⊗ a2 ⊗ I2N −r " −1 ⊗ D1 , ⎪ ⎪ a1 ⎪ ⎪ ⎪ ⊗(l−1) ⎩ ⊗ D1 , a1

⎞

⎟ · · · a1 ⊗(N−3) ⊗ a2 a1 ⊗(N−2) ⎟ ⎟ · · · a1 ⊗(N−4) ⊗ a2 a1 ⊗(N−3) ⎟ ⎟ ⎟ .. .. .. ⎟⊗I , . W ⎟ . . ⎟ ⎟ ··· a2 a1 ⎟ ⎟ ⎠ ··· O20 ×20 1 ··· O1×20 0 r " = r − m, m = 1, r, r = 0, N − 1, r " = r − m, m = 1, r, r = N, r " = r, r = 0, N − 1, r " = r, r = N, r " = r + l, l = 1, N − r − 1, r = 0, N , r " = r + l, l = N − r, r = 0, N .

Here ⎞ N−2  − μ 0 k ⎟ ⎜ k=1 ⎟, μk , Δ 1 = ⎜ N−2 ⎠ ⎝  μN − μk − μN ⎛

Δ0 = −

N−1  k=1

k=1

Δn = I2n−2 ⊗ G +

n−2  k=1

(n) Iˇk μN−k−1 −

N−n−1 

μk I2n , n = 2, N − 1.

k=1

Proof of the lemma consists of careful analysis of transitions of the Markov chain ζt during the interval of an infinitesimal length. Corollary 1 Markov chain ζt belongs to the class of asymptotically quasi-Toeplitz Markov chains, see [10].

Retrial Queue with Heterogeneous Servers

35

Proof According to the definition of the asymptotically quasi-Toeplitz Markov chains given in [10], we have to prove the existence of the limits Y0 = lim Ri −1 Qi,i−1 , Y2 = lim Ri −1 Qi,i+1 , Y1 = lim Ri −1 Qii + I i→∞

i→∞

i→∞

where Ri is a diagonal matrix with diagonal entries defined as the moduli of the corresponding diagonal entries of the matrix Qii , i ≥ 0. It can be easily verified (n) that Ri is matrix with the diagonal blocks Ti , n = 0, N, defined as follows: (n) Ti

⎧ ⎨Λ ⊕ Zn + αi IW¯ ·2N−n−1 , n = 0, N − 1, N  = ⎩ Λ+ μk IW¯ , n = N, k=1

where Λ, Zn are diagonal matrices with diagonal entries defined by the diagonal entries of the matrices −D0 , ΔN−n−1 , n = 0, N, respectively. Then, by direct calculations, it can be verified that ⎛

⎛ ⎞ O OO ⎜O O O⎟ ⎜ ⎟ .. ⎟ , Y = ⎜ .. .. ⎜. . ⎟ 2 .⎟ ⎜ ⎝O O ⎠ O ··· O O Γ1 Γ2 · · · ΓN Ψ OO

O ⎜O ⎜ ⎜ Y0 = I˜β , Y1 = ⎜ ... ⎜ ⎝O

O O .. .

··· ··· .. .

O O .. .

⎞ O O⎟ ⎟ .. ⎟ .⎟ ⎟ · · · O O⎠ ··· O Φ ··· O ··· O . . .. . .

where * Γn = μn b1 ⊗(N−n) ⊗ C, n = 1, N , C = Λ +

N 

+−1 μk IW¯

,

k=1

* Ψ = C D0 −

N 

+ μk IW¯

+ I,

Φ = CD1 .

k=1

Corollary 1 is proven.

4 Ergodicity Condition Theorem 1 The Markov chain ζt is ergodic if the inequality λ


N 

μk .

(3)

k=1

Here λ is the fundamental rate of the MAP . Proof It follows from [10] that the sufficient condition for ergodicity of the Markov chain ζt is the fulfillment of the inequality yY0 e > yY2 e

(4)

where the vector y is the unique solution of the system of linear algebraic equations y(Y0 + Y1 + Y2 ) = y, ye = 1.

(5)

Let us represent the vector y in the form y = (y0 , . . . , yN ) and solve the system (5) by means of sequential multiplication of the vector y by the corresponding block columns of the matrix Y = Y0 + Y1 + Y2 . Multiplying this vector by the first block column, we have the relation y0 = yN Γ1 = yN μ1 HN−1 where Hr = (O, . . . , O, C), r = 0, N − 1. Let us note that all the block entries of    2r

the vector y0 , except the last one, are equal to zero. Multiplying vector y by the second block column of the matrix Y , we have the relation y1 = y0 ((a2 ⊗ I2N−2 ) ⊗ IW¯ ) + yN Γ2 . % I2N−2 . Because all the block entries of the vector y0 , except Here a2 ⊗ I2N−2 = O2N−2 the last one, are equal to zero while the last block of the vector a2 ⊗ I2N−2 equals to zero, we conclude that y0 ((a2 ⊗ I2N−2 ) ⊗ IW¯ ) = 0 and, hence, $

y1 = yN Γ2 = yN μ2 HN−2 . Analogously, we sequentially derive relations: yk = yN μk+1 HN−k−1 , k = 0, N − 1.

(6)

Retrial Queue with Heterogeneous Servers

37

Finally, by multiplying vector y by the last block column of the matrix Y , we have the relation N−1 

yN =

yk ((a1 ⊗(N−k−1) ⊗ IW¯ ) + yN (Ψ + Φ)

k=0

that can be rewritten as *N ++ * N     ⊗(N−k) = 0. yN μk HN−k a1 ⊗ IW¯ + C D0 + D1 − μk IW¯ k=1

k=1

Because a1 ⊗(N−k) ⊗ IW¯ = (O, . . . , O, IW¯ )T , we have that HN−k ((a1 ⊗(N−k) ⊗    2N−k

IW¯ )) = C. Therefore, the equation for the vector yN is rewritten in the form yN C(D0 + D1 ) = 0. This implies that yN C = gθ or yN = gθ C −1

(7)

where the vector θ defines the stationary distribution of the underlying process of the MAP νt and g is positive constant. Formulas (6) and (7) completely define the components of the vector y and we can substitute them into inequality (4). The left-hand side of (4) is computed as: yY0 e =

N−1 

yk e = yN

k=0

= gθ

N−1 

μk+1 HN−k−1 e = yN

k=0

N  k=1

μk e = g

N 

N−1  k=0

μk .

k=1

The right-hand side of (4) is computed as: yY2 e = yN Φe = gθ C −1 CD1 e = gλ.

μk+1 Ce

38

L. Mei and A. Dudin

Therefore, inequality (4) is rewritten in the form gλ < g

N 

μk that is equivalent

k=1

to condition for ergodicity (2). Condition (3) for non-ergodicity analogously follows from condition yY0 e < yY2 e proven in [10]. The theorem is proven. Remark 1 Condition of ergodicity (2) is intuitively clear. Usually condition of ergodicity consists of requirement that, in overloaded system, arrival rate is less than the service rate. In the considered model, when it is overloaded, i.e., huge number of N  μk . customers stay in the orbit, all servers are busy. Thus, the total service rate is k=1

In what follows we assume that condition (2) is fulfilled. Hence the vectors π i , i ≥ 0, defined above exist. They satisfy the system of equilibrium equations π Q = 0, πe = 1. This system is infinite and the problem of its solution is quite difficult. The system can be solved using the algorithm developed in [10] and more recent and efficient algorithm from [4].

5 Performance Measures As soon as the vectors π i , i ≥ 0, have been calculated, we are able to find various performance measures of the system. The average number Lorbit of customers in the orbit is computed by Lorbit =

∞ 

iπ i e.

i=1

The probability Pimm that an arbitrary customer will start service immediately upon arrival is computed by Pimm = λ−1

∞ 

π i ((I2N − J ) ⊗ D1 )e.

i=0

Let R = {(r (1) , . . . , r (N) ) : r (n) = 0, 1, n = 1, N }. The average number Nbusy of busy servers is computed by Nbusy =

∞ 



i=0 (r (1) ,...,r (N) )∈R

    r (1) + . . . + r (N) π i, r (1) , . . . , r (N) e.

Retrial Queue with Heterogeneous Servers

39

(n)

The probability Pbusy that the nth server is busy at an arbitrary moment is computed by (n) Pbusy =

∞ 



  π i, r (1) , . . . , r (N) e, n = 1, N .

i=0 (r (1) ,...,r (N) )∈R, r (n) =1

Remark 2 For control of accuracy of computation of the vectors π i , i ≥ 0, it is useful to check the fulfillment of the following equalities: ∞ 

π i (e2N ⊗ IW¯ ) = θ and

i=0

N 

(n) μn Pbusy = λ.

n=1

6 Numerical Results To illustrate the feasibility and outcome of the presented algorithms as well to show the effect of correlation in arrival process, we briefly consider the following example. Let initially the MAP -input be characterized by the matrices $ D0 =

% −1.35164 0 , 0 −0.04387

$ D1 =

% 1.34265 0.00899 . 0.02443 0.01944

This arrival process has the coefficient of correlation of two successive intervals between arrivals ccor = 0.2, and the squared coefficient of variation of the intervals between customer arrivals cvar = 13.4. In the presented experiment, we will vary the average rate of the MAP λ that is done by multiplying the matrices D0 and D1 by the appropriate scalar. In parallel, we present the results of computation for the model where the arrival flow is defined as the stationary Poisson process with the same intensity. Let us assume that the total number N of servers be equal to 4 and service rates at the corresponding servers be μ1 = 4, μ2 = 3, μ3 = 1, and μ4 = 0.5, correspondingly. The retrial rates are defined by α0 = 0, αi = iα, α = 1, i > 0. Figures 1, 2, and 3 show the behavior of the value Lorbit depending on the input rate λ when N = 2 (the slowest fourth and third servers are not used), N = 3 (the slowest, fourth, server is not used), and N = 4 for two considered arrival processes. Table 1 shows the value of Lorbit for several values of λ when the arrival process is the MAP . Figure 4 shows the behavior of the value Lorbit depending on the input rate λ under different numbers of servers for the arrival process MAP .

40

L. Mei and A. Dudin

N=2

500

M MAP

450 400 350

250

L

orbit

300

200 150 100 50 0 1

2

3

4

5

6

7

Fig. 1 Dependence of Lorbit on the input rate λ when N = 2 for flows with different correlation

Figures 5 and 6 show the behavior of the value Pimm depending on the input rate λ when N = 3 and N = 4 for two considered arrival processes. Table 2 shows the value of Pimm for several values of λ when the arrival process is the MAP . Figure 7 shows the behavior of the value Pimm depending on the input rate λ under different numbers of servers for the arrival process MAP .

Retrial Queue with Heterogeneous Servers

41

N=3

600

M MAP

500

300

L

orbit

400

200

100

0 1

2

3

4

5

6

7

8

Fig. 2 Dependence of Lorbit on the input rate λ when N = 3 for flows with different correlation

42

L. Mei and A. Dudin

N=4

1200

M MAP

1000

L orbit

800

600

400

200

0 1

2

3

4

5

6

7

8

9

Fig. 3 Dependence of Lorbit on the input rate λ when N = 4 for flows with different correlation

Table 1 The values of Lorbit for several values of λ and N = 2, 3, 4 when the arrival process is the MAP

N 2 3 4

λ Lorbit Lorbit Lorbit

1 0.0656 0.0115 0.0020

2 0.5323 0.1754 0.0672

3 2.0526 0.8670 0.4692

4 6.4419 2.9030 1.8211

5 20.0501 8.5491 5.5925

6 7 8 70.0549 24.8427 81.1391 16.0165 46.4190 205.5134

Retrial Queue with Heterogeneous Servers

43

1200 N=2 N=3 N=4

1000

L orbit

800

600

400

200

0 1

2

3

4

5

6

Fig. 4 Dependence of Lorbit on the input rate λ under N = 2, 3, 4

7

8

9

44

L. Mei and A. Dudin

N=3

1

M MAP

0.9 0.8 0.7

Pimm

0.6 0.5 0.4 0.3 0.2 0.1 0 1

2

3

4

5

6

7

8

Fig. 5 Dependence of Pimm on the input rate λ when N = 3 for flows with different correlation

Retrial Queue with Heterogeneous Servers

45

N=4

1

M MAP

0.9 0.8 0.7

Pimm

0.6 0.5 0.4 0.3 0.2 0.1 0 1

2

3

4

5

6

7

8

9

Fig. 6 Dependence of Pimm on the input rate λ when N = 4 for flows with different correlation

Table 2 The values of Pimm for several values of λ and N = 2, 3, 4 when the arrival process is the MAP

N 2 3 4

λ Pimm Pimm Pimm

1 0.9467 0.9895 0.9975

2 0.8184 0.9319 0.9724

3 0.6406 0.8109 0.8990

4 0.4386 0.6355 0.7388

5 0.2601 0.4360 0.5426

6 7 8 0.1318 0.2664 0.1388 0.3517 0.2061 0.0809

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L. Mei and A. Dudin

1 N=2 N=3 N=4

0.9 0.8 0.7

Pimm

0.6 0.5 0.4 0.3 0.2 0.1 0 1

2

3

4

5

6

7

8

9

Fig. 7 Dependence of Pimm on the input rate λ under N = 2, 3, 4

Figure 8 shows the behavior of the value Nbusy when N = 4 for two considered arrival processes. Table 3 show the value of Nbusy for several values of λ when the arrival process is the MAP . Figure 9 shows the behavior of the value Nbusy for different numbers of service for the arrival process MAP . It is evidently seen from the presented figures that correlation in arrival process essentially impacts on the values of the performance measures. Positive correlation worsens performance measures of the system.

Retrial Queue with Heterogeneous Servers

47

N=4

4

M MAP

3.5

Nbusy

3

2.5

2

1.5

1

0.5 1

2

3

4

5

6

7

8

9

Fig. 8 Dependence of Nbusy on the input rate λ when N = 4 for flows with different correlation

Table 3 The values of Nbusy for several values of λ and N = 2, 3, 4 when the arrival process is the MAP

N 2 3 4

λ Nbusy Nbusy Nbusy

1 0.6673 0.7449 0.7777

2 0.8347 1.1552 1.4252

3 0.9358 1.4465 2.0214

4 1.0272 1.6001 2.2901

5 1.1128 1.7623 2.4619

6 7 8 1.1845 1.9351 2.1014 2.6897 2.9261 3.1585

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L. Mei and A. Dudin

3.5 N=2 N=3 N=4

3

Nbusy

2.5

2

1.5

1

0.5 1

2

3

4

5

6

7

8

9

Fig. 9 Dependence of Nbusy on the input rate λ under N = 2, 3, 4

7 Conclusion We analyzed retrial queueing model with heterogeneous servers and MAP arrival process. The results can be used for solving various optimization problems related, in particular, to enumeration of the servers. The results are planned to be extended to the case of phase type distribution of service time.

References 1. Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems: A Computational Approach. Springer, Berlin (2008) 2. Breuer, L., Dudin, A.N., Klimenok, V.I.: A retrial BMAP /P N/N system. Queueing Syst. 40, 433–457 (2002) 3. Chakravarthy, S.R.: The batch Markovian arrival process: a review and future work. In: Krishnamoorthy, A., Raju, N., Ramaswami, V. (eds.) Advances in Probability Theory and Stochastic Processes. Notable Publications, New Jersey, pp. 21–29 (2001)

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4. Dudin, S., Dudina, O.: Retrial multi-server queueing system with PHF service time distribution as a model of a channel with unreliable transmission of information. Appl. Math. Model. 65, 676–695 (2019) 5. Efrosinin, D.V.: Controlled Queueing Systems with Heterogeneous Servers. Ph.D. Dissertation, Trier University, Germany (2004) 6. Efrosinin, D., Breuer, L.: Threshold policies for controlled retrial queues with heterogeneous servers. Ann. Oper. Res. 41(1), 139–162 (2006) 7. Efrosinin, D., Sztrik, J.: Performance analysis of a two-server heterogeneous retrial queue with threshold policy. Quality Technol. Quant. Manag. 8(3), 211–236 (2011) 8. Falin, G.: Stability of the multiserver queue with addressed retrials. Ann. Oper. Res. 196(1) 241–246 (2012) 9. Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman & Hall, London (1997) 10. Klimenok V., Dudin, A.: Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Syst. 54(4), 245–259 (2006) 11. Lin, W., Kumar, P.R.: Optimal control of a queueing system with two heterogeneous servers. IEEE Trans. Autom. Control 29, 696–703 (1984) 12. Luh, H.P., Viniotis, I.: Optimality of Threshold Policies for Heterogeneous Server Systems. North Carolina State University, Raleigh (1990) 13. Lucantoni, D.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat. Stoch. Models 7, 1–46 (1991) 14. Mushko, V.V., Jacob, M.J., Ramakrishnan, K.O., Krishnamoorthy, A., Dudin, A.N.: Multiserver queue with addressed retrials. Ann. Oper. Res. 141, 283–301 (2006) 15. Neuts M. Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore (1981) 16. Nobel, R., Tijms, H.C.: Optimal control of a queueing system with heterogeneous servers. IEEE Trans. Autom. Control 45(4), 780–784 (2000) 17. Rosberg, Z., Makowski, A.M.: Optimal routing to parallel heterogeneous servers-small arrival rates. Trans. Autom. Control 35(7), 789–796 (1990) 18. Rykov, V.V.: Monotone control of queueing systems with heterogeneous servers. Queueing Syst. 37, 391–403 (2001) 19. Rykov, V.V., Efrosinin, D.V.: Numerical analysis of optimal control polices for queueing systems with heterogeneous servers. Inf. Process. 2(2), 252–256 (2002) 20. Vishnevskii, V.M., Dudin, A.N.: Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks. Autom. Remote Control 78, 1361–1403 (2017)

What is Standard Brownian Motion? Krishna B. Athreya

Abstract In this expository note, we explain several different historical approaches to the construction of standard Brownian motion. Keywords Standard Brownian motion · Gaussian process · Probability measure · Stopping time · Martingale

1 Introduction An easy and quick answer to the question posed in the title is that it is a Gaussian process {X(t) : t ∈ [0, ∞)} with index set T ≡ [0, ∞), mean function m(t) ≡ 0 for all t ∈ T , and covariance function c(s, t) ≡ min(s, t) for all s, t ∈ T . That is, for each positive integer k and any increasing k-tuple 0 ≤ t1 < t2 < . . . < tk < ∞ the random vector (X(t1 ), . . . , X(tk )) has a k-variate normal distribution in Rk with mean vector (0, . . . , 0) and variance–covariance matrix Cov(X(ti ), X(tj )) = min(ti , tj ). That is, the probability distribution of the vector(X(t1), . . . , X(tk )) on Rk (with the standard Borel σ -algebra) is absolutely continuous with respect to Lebesgue and has probability density function φk,(t1,t2 ,...,tk )(x1 ,...,xk ) ≡ √

1 2π

% $ 1 1 −1 T xΣ x exp k ||Σk ||1/2 2 k

where x = (x1 , . . . , xk ) ∈ Rk

K. B. Athreya () Department of Mathematics and Statistics, College of Liberal Arts and Sciences, Iowa State University, Ames, IA, USA e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_4

51

52

K. B. Athreya

and Σk = min(ti , tj )1≤i,j ≤k , and xT is the transpose of x. A natural question is whether such a process exists. Indeed it does. This follows from Kolmogorov’s consistency theorem (see, for example, [[1], Theorem 6.3.1]). Indeed, if ΩT ≡ RT ≡ the set of real-valued functions on T = [0, ∞) and FT is the σ -algebra generated by the class of sets Wt1 ,...,tk ;A = {w ∈ ΩT : (w(t1 ), w(t2 ), . . . , w(tk )) ∈ A}, where ti is as above and A is a Borel subset of Rk , there is a probability measure PT on (ΩT , FT ) such that the stochastic process {w(t) : t ≥ 0} is the process X(t) described above. One of the problems with the above construction is that the sample space ΩT = RT is too large but the σ -algebra FT is too small. It can be shown that FT coincides with the σ -algebra defined as follows: given a countable subset I ⊂ T , let πI denote the projection from ΩT to RI , and consider the collection of sets πI−1 (B), where B is a Borel subset of RI , and I ranges over all countable subsets of T. In particular, the subset C[0, 1] of real-valued continuous functions is not a member of FT . It can be shown that under the measure PT the trajectories are in C[0, 1] with probability 1 but that this event is not in FT and hence is not measurable. Similarly, the function M(w) ≡ sup{w(t) : 0 ≤ t ≤ 1} is not FT -measurable. An approach pioneered by J. L. Doob is the notion of separable stochastic processes. Another approach pioneered by Kolmogorov and Skorokhod is to restrict the sample space to functions that are continuous or which are right continuous and have left limits (see, for example, Billingsley [2]). If {X(t) : t ≥ 0} is a Gaussian process as defined above, it can be shown that for any t, h > 0, the increment X(t + h) − X(t) is stochastically independent of {X(u) : u ≤ t} and has variance |h|. This suggests that X(t + h) − X(t) goes to zero as h → 0 if t is fixed, i.e., that the trajectory should be continuous at t. However, before one gets hopes too high, one can show that the trajectories, while continuous, are not differentiable.

2 Another Construction N. Weiner solved this problem of measurability in a different way (see, e.g., Karatzas and Shreve [5]). Let {ηi (ω)}i≥1 be a sequence of independent, identically distributed (i.i.d.) N(0, 1) random variables on a probability space (Ω, B, P ). It can be shown that (Ω, B, P ) can be chosen to be the standard Lebesgue space

What is Standard Brownian Motion?

53

[0, 1] with the Borel σ -algebra B, and P standard Lebesgue measure. Let {ψj (.)}j ≥1 be a complete orthonormal basis in L2 ([0, 1], P ), and for each positive integer N, let BN (t, ω) =

N 

 ηj (ω)

t

ψj (u)du, for 0 ≤ t ≤ 1, ω ∈ Ω

0

j =1

Then for each N, and ω ∈ Ω, {BN (t, ω) : 0 ≤ t ≤ 1} is a well-defined function in C0 [0, 1], the Banach space of continuous real-valued functions f on [0, 1] with f (0) = 0. It can be shown that the sequence {BN (·, ω)}N≥0 is a Cauchy sequence in C0 [0, 1] for almost all ω, and since C0 [0, 1] is a complete metric space (when equipped with the sup norm), there exists a process {B(t, ω) : 0 ≤ t ≤ 1} such that BN (·, ω) → B(·, ω) in C0 [0, 1] for almost all ω. It can be further shown that {B(t, ω) : 0 ≤ t ≤ 1} is a Gaussian process with mean function m(t) ≡ 0 for all 0 ≤ t ≤ 1 and covariance function C(s, t) = min(s, t); 0 ≤ s, t ≤ 1. Thus, the standard Brownian motion (SBM) on [0, 1] is Gaussian process with continuous trajectories on [0, 1]. This is the definition we will use, instead of that from 1.

3 Definition of SBM on [0, ∞) In 2, we defined SBM on [0,1] using Weiner’s approach. Now we extend it to the whole positive real line [0, ∞) as follows. Let {B (j ) (t, ω) : 0 ≤ t ≤ 1} be i.i.d copies of the SBM described in 2. Given t ∈ [0, ∞), let n = t. Define B(t, ω) ≡

n 

B (j ) (1, ω) + B (n+1) (t − n, ω).

j =1

Then it can be verified that {B(t, ω) : t ≥ 0} satisfies: 1. B(0, ω) = 0 for all ω. 2. The function t → B(t, ω) is continuous in t for all ω. 3. It is a Gaussian process with mean function m(t) ≡ E(B(t, ω)) = 0 for all t and covariance function C(s, t) = min(s, t), 0 ≤ s, t < ∞. We will refer to this process as standard Brownian motion (SBM) on [0, ∞).

54

K. B. Athreya

4 Donsker’s Invariance Principle A construction due to Donsker [3] gives another way of constructing SBM on [0, 1]. Theorem 1 (Donsker’s Invariance Principle) Let {Xi }∞ i=1 be i.i.d random variables with P (Xi = 1) = P (Xi = −1) = 1/2. Let 1  Yn (j/n) = √ , j = 0, 1, 2, . . . , n. n j

i=1

and define Yn (t) for 0 ≤ t ≤ 1 by linear interpolation. Let μn (·) denote the measure on C0 [0, 1] supported on the set of realizations of Yn (t), i.e., for measurable A ⊂ C0 [0, 1], μn (A) = P (Yn (·) ∈ A). Then there is a probability measure μ on C0 [0, 1] so that μn → μ in the weak-* topology, that is, for any functional f : C0 [0, 1] → R, 

 C0 [0,1]

f dμn →

C0 [0,1]

f dμ

Remark 1 The limiting probability measure μ on C0 [0, 1] is the same as the Wiener measure constructed in 2. Remark 2 The above theorem of Donsker extends to the case when {Xi } are i.i.d. mean 0 random variables with variance E(X12 ) = 1. This√is useful in studying the limit behavior in the Kolmogorov–Smirnov statistic Tn = nsup0≤x≤1|Fn (x) − x|, where Fn (x) = n1 ni=1 I{Ui ≤x} is the empirical distribution function of the sample U1 , . . . , Un , where Ui are i.i.d. uniform [0,1] random variables. It can be shown that for each x ∈ R, P (Tn ≤ x) → P (T ≤ x), where T ≡ sup0≤t ≤1|Y (t) − tY (1)| and Y (·) is SBM on [0, 1], i.e., Tn → T in distribution as n → ∞.

5 Some Basic Properties of SBM on [0, ∞) Let {B(t, ω) : t ≥ 0} satisfying the conditions from 3, that is, 1. B(0, ω) = 0 for all ω.

What is Standard Brownian Motion?

55

2. The function t → B(t, ω) is continuous in t for all ω. 3. It is a Gaussian process with mean function m(t) ≡ E(B(t, ω)) = 0 for all t and covariance function C(s, t) = min(s, t), 0 ≤ s, t < ∞. Then we claim it has the following properties: Scaling For each (deterministic) c > 0, let 1 Bc (t, ω) ≡ √ B(ct, ω). c Then {Bc (t, ω) : t ≥ 0} is also an SBM. To prove this, note that conditions (1) and (2) are immediate, and condition (3) is an easy computation. Reflection If {B(t, ω) : t ≥ 0}is an SBM, then so is ˜ ω) = −B(t, ω). B(t, This is also a straightforward verification of the above conditions. Time Inversion For t > 0, set ˜ ω) = tB B(t,

$

% 1 ,ω , t

˜˜ ω) : t ≥ 0} is also an SBM. To prove this, it ˜ and set B(0, ω) = 0. Then {B(t, is straightforward to verify that it is a Gaussian process with the specified mean and covariance functions, and that it is continuous on the open interval (0, ∞). It remains to verify that ˜˜ ω) = 0. lim B(t,

t →0

˜˜ ω) : t ≤ t ≤ t } with probability 1. To prove this, fix 0 < t1 < t2 < ∞. Then {B(t, 1 2 is a Gaussian process with continuous trajectories and has the same distribution as {B(t, ω) : t1 ≤ t ≤ t2 }. Hence X1 ≡ sup{B(t, ω) : t1 ≤ t ≤ t2 } and   ˜˜ ω) : t ≤ t ≤ t . X2 ≡ sup B(t, 1 2 have the same distribution. As t1 → 0, X1 and X2 converge with probability 1 to X1∗ (t2 ) ≡ sup{B(t, ω) : 0 ≤ t ≤ t2 }

56

K. B. Athreya

and   ˜˜ ω) : t ≤ t ≤ t X2∗ (t2 ) ≡ sup B(t, 1 2 respectively, and thus, these have the same distribution. Now as t2 → 0, X1∗ (t2 ) converges to limsupt →0 B(t, ω) which is 0 with probability 1. This implies that X2∗ (t2 ) → 0 with probability 1 as t2 → 0. Thus ˜˜ ω) = 0, lim B(t,

t →0

with probability 1. As a corollary, we obtain that lim

t →∞

B(t) = 0, t

with probability 1.

6 Translation Invariance 6.1 Translation Invariance after a Deterministic Time As above, let {B(t, ω) : t ≥ 0} be an SBM. Fix t0 > 0 and for t ≥ 0, let Bt0 (t, ω) ≡ B(t0 + t, ω) − B(t0 , ω). Then it is easy to check that {Bt0 (t, ω) : t ≥ 0} is also an SBM.

6.2 Translation Invariance after Stopping Times T A random variable T ∼ = T (ω) with values in [0, ∞) is a stopping time with respect to an SBM if for any deterministic t0 ≥ 0, the event {T (ω) ≤ t0 } is determined by the history {B(u, ω) : u ≤ t0 }, i.e., the set {ω : T (ω) ≤ t0 } belongs to the σ algebra generated by {B(u, ω) : u ≤ t0 }. We have the following result on translation invariance: Theorem 2 Let T be a stopping time with respect to an SBM {B(u, ω) : u ≥ 0}. Then the stochastic process {B(u + t, ω) − B(T , ω) : u ≥ 0} is also an SBM and is independent of the σ -algebra σT ≡ {A : A ∩ {T ≤ t} ∈ σ (B(u, ω) : u ≤ t)}. For a proof of this theorem, see, for example, Karatzas and Shreve [5].

What is Standard Brownian Motion?

57

Examples of Stopping Times Two important examples of stopping times are 1. Fix a ∈ R. Let Ta ≡ inf {t ≥ 0 : B(t, ω) = a} be the first hitting time of a. 2. Fix −∞ < a < 0 < b < ∞. Let Ta,b ≡ inf {t ≥ 0 : B(t, ω) ∈ / (a, b)}.

6.3 The Reflection Principle Consider the stopping time Ta defined above. Then P (Ta ≤ t) = P (Ta ≤ t, B(t, ω) > a) + P (Ta ≤ t, B(t, ω) < a), since P (B(t, ω) = a) = 0. By the continuity of trajectories of SBM, B(Ta ) = a on the event {Ta ≤ t}. We then have P (Ta ≤ t, B(t) < a) = P (Ta ≤ t, B(t) − B(Ta ) < 0) = P (Ta ≤ t, B(t) − B(Ta ) > 0). So $ %  a , P (Ta ≤ t) = 2P (Ta ≤ t, B(t) > a) = 2 1 − Φ √ t where Φ(.) is the standard N(0, 1) cdf. So for every a > 0, Ta has an abso2

lutely continuous distribution with probability density function p E(Ta )

< ∞ for all p
0. Then P (M(t, ω) > a) = P (Ta ≤ t) = 2P (B(t) > a) = P (|B(t, ω)| > a) since B(t, ω) has an N(0, t) distribution, which is symmetric about 0. Thus, M(t, ω) has the same distribution as |B(t, ω)|. A similar argument shows that m(t, ω) ≡ min{B(u, ω) : 0 ≤ u ≤ t} has the same distribution as −|B(t, ω)|.

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8 Sample Path Properties of SBM Theorem 3 ([4]) Let f : R → R be integrable with respect to Lebesgue measure, and {B(u) : u ≥ 0} SBM on [0, ∞). Then 1 N(t)



t

t →∞

f (B(u))du −−−→

0



∞ −∞

f (u)du,

where N(t) is the number of times the SBM {B(u) : u ≥ 0} hits level 1 and then hits level 0 at a later time before getting back to level 1 in the time interval [0, t].

8.1 Nondifferentiability Theorem 4 With probability 1, the SBM {B(u) : u ≥ 0} is nowhere differentiable on [0, ∞).

8.2 Increments Theorem 5 Let {B(u) : u ≥ 0} be SBM and set Δ≡

n 

 2 sup B(tj ) − B(tj −1 )

0

where the supremum is taken over all partitions {t0 < t1 . . . < tn } of [0, 1]. Then Δ = 1 with probability 1.

8.3 Martingale Properties Theorem 6 Let{B(u) : u ≥ 0} be SBM. Then • (a) {B(u) : u ≥ 0} is a martingale • (b) {B 2 (u) − u : u ≥ 0} is a martingale θ2

• (c) For all θ ∈ R, {eθB(u)− 2

u

: u ≥ 0} is a martingale.

Acknowledgments I want to thank Prof. A. Krishnamoorthy and Dr. Varghese C. Joshua for inviting me to this conference and to let me present this paper.

What is Standard Brownian Motion?

59

References 1. Athreya, K.B., Lahiri, S.N.: Measure Theory and Probability Theory, Springer Texts in Statistics. Springer, New York (2006). MR2247694 2. Billingsley, P.: Convergence of probability measures. In: Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999). A Wiley-Interscience Publication, MR1700749 3. Donsker, M.D.: An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 6, 12 (1951). MR0040613 4. Kallianpur, G., Robbins, H.: Ergodic property of the Brownian motion process. Proc. Nat. Acad. Sci. USA 39, 525–533 (1953). https://doi.org/10.1073/pnas.39.6.525. MR0056233 5. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd ed. In: Graduate Texts in Mathematics, vol. 113. Springer, New York (1991). MR1121940

Busy Period Analysis of Multi-Server Retrial Queueing Systems Srinivas R. Chakravarthy

Abstract The literature on the busy period analysis in queueing theory is very limited due to the inherent complexity in its study. Recently, using the simulation approach the busy period for the classical multi-server queueing systems was studied by this author and some interesting observations were reported. In this paper we carry out a similar analysis but on a smaller scale in the case of multiserver retrial queueing systems. It should be pointed out that while the literature on retrial queueing system is vast, the same cannot be said about the busy period analysis in retrial queueing systems. Only a few papers with restricted assumptions are available in the literature. This paper is an attempt to fill the void. Keywords Retrial · Queueing · Busy period · Simulation

1 Introduction In general, the busy period analysis in queueing systems is very involved and complicated. This is not due to the choice but rather due to the difficulty inherent in its study [10]. Realizing this difficulty, Chakravarthy [10] used simulation to record some interesting observations on the busy period of classical queueing systems. We refer the reader to [10] for details including a literature survey on the busy period analysis of the classical queues. In this paper, which can be considered as a sequel to [10], we look at multiserver retrial queueing systems with the arrivals governed by a point process, general services, and general retrial times. This point process also includes a Markovian arrival process (MAP ). Recall that a retrial queueing system (see, e.g., [2]) is such that an arriving customer finding all servers busy will enter into a retrial orbit and

S. R. Chakravarthy () Departments of Industrial and Manufacturing Engineering and Mathematics, Kettering University, Flint, MI, USA e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_5

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attempt to capture a free server at random times. Such a system is studied as a leveldependent queue unless the retrial rate is independent of the number in the retrial orbit. While a number of retrial queueing models have been studied in the literature (see, e.g., [1, 2, 14, 22]) placing restrictions on the retrial distribution, only a few papers deal with more complex distributions for the retrial attempts (see, e.g., [8] and the references therein). While in the classical queueing systems, the busy period is defined in terms of the servers’ (which will coincide with that of the busy system in the case of a single server) busy time, the busy period for the retrial queues should be defined in terms of the system being busy. This is due to the fact that in (continuous-time) retrial queueing systems, the server will always alternate between an idle period and a service period. This can be seen immediately by noting that there is no queue in front of the server and that the next service starts either with a new arrival or with a successful attempt from the retrial orbit by a customer. Thus, the non-trivial busy period analysis involves that of the system which will be the case in this paper too. There are very few papers that deal with busy period analysis in retrial queueing systems. In the context of MAP /M/c with a finite buffer for retrial customers, Artalejo et al., [4] derive the Laplace–Stieltjes transform of the busy period as well as the probability generating function of the number of customers served during a busy period. Recently, Kim [13] derived expressions (but without any computational procedures) for the first and the second moments of the duration of the busy period for M X /G/1 retrial queue. However, to our knowledge, there is no paper dealing with the busy period analysis for a multi-server retrial queueing system in a general context. Thus, the objective of this paper is to present some insight into the study of the busy period analysis of a general multi-server retrial queueing system. The paper is organized as follows. The model under study is described in Sect. 2. After validating the simulated model for the M/G/1 retrial queueing model [3] in Sect. 3, we report some interesting observations based on our simulated results in Sect. 4. Some concluding remarks are mentioned in Sect. 5.

2 Model Description In this paper, we assume that the customers arrive according to a point process including a MAP with (irreducible) representation matrices (D0 , D1 ) of dimension m. Let D = D0 + D1 be the underlying generator with steady-state probability vector π . That is, π D = 0 and π e = 1 so that the arrival rate is given by λ = π D1 e. The MAP is a versatile class of point processes introduced by Neuts [17] and studied extensively by Neuts and his colleagues in the context of a variety of queueing, inventory, and reliability models among others. Modeling the inter-arrival times with MAP has many advantages including capturing any correlation that may be present between two successive inter-arrival times. We refer the reader to [5– 7, 9, 15–17, 19–21] for details on MAP and other key references.

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63

There are c servers in the system and the service times are assumed to be λ generally distributed with mean μ1 . Let ρ = cμ . An arriving customer finding a free server will enter into service immediately; however, an arriving customer finding all the servers busy will enter into a retrial buffer of infinite capacity. Each customer entering into the retrial orbit will attempt to capture a free server (independently of the others waiting in the orbit) after a random period of time. This random variable is assumed to be generally distributed with a finite mean, say, ξ1 . We assume that the service time and the retrial time have a finite variance. We also assume that the interarrival times, the service times, and the retrial times are all mutually independent. Since we are considering retrial queueing models of the GI /G/c-type as well as MAP /G/c-type with general retrial times and since our main focus in this paper is the busy period analysis, we will resort to simulation. It should be pointed out that our main purpose is to motivate the need for studying the busy period of retrial queueing models as in this era of technology the customers’ queries such as billing enquiries, fixing appointments, and the status details in service sectors force customers to make repeated attempts when all servers are busy. The study of general retrial queueing models becomes very complex even for obtaining the standard measures such as the mean waiting time, leave alone a study on the busy period of the system. We will use ARENA, a powerful simulation software [12] in our study here. Simulation should be considered as an important tool and also a way to get insight into obtaining theoretical results. Hence, in this paper a few selected retrial queueing models will be simulated and their key results will be summarized. In this paper, we will be focusing on the following three measures: (a) the mean busy period of the system; (b) the coefficient of variation of the busy period; and (c) the number of busy periods during the simulation period. While the first two measures are very standard and easy to explain the need for them, the measure given in (c) needs some justification. It is known in queueing theory literature the effect of the variability (as well as the correlation) in the inter-arrival times on the system performance measures. For example, in [10] it is shown that hyperexponential interarrival times appear to have a larger mean busy period compared to that of Erlang inter-arrival times in GI /G/c-type queues, and positively correlated inter-arrival times appear to yield a higher mean busy period as compared to the corresponding negatively correlated ones in the context of MAP /G/c-type queues. However, in retrial queues, since the server alternates between being idle and offering only one service, not only the mean busy period is of interest but also the number of busy periods.

3 Validation It is important to validate our simulated model by comparing the simulated results with those of the published analytical results for some well-known retrial queueing models in the literature. However, to the best of our knowledge, we found only one paper [3] dealing with the busy period analysis in the context of M/G/1 queue

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with exponential retrial times that has illustrative numerical examples reporting the second moment of the busy period. In [3], the authors investigate a single server queue with Poisson arrivals, general services, and exponential retrial times, and point out the limitations in using the expressions (given in the form of Laplace– Stieltjes transforms) for the busy period, and offer a direct way to compute the second moment of the busy period. Thus, we validate our model by comparing our simulated results with the numerical results based on the analytical expressions given in [3]. First, it is worth mentioning that in [3] the authors report the numerical values for the second moment of the busy period for the following four sets of data: (1) hyperexponential services by considering three values, namely 1.25, 1.50, 1.75, for the coefficient of variation of the service times and by varying the arrival rate. For all these scenarios, the mean service time is fixed at 1.0 and the retrial rate is fixed at 0.5; (2) hyperexponential services by considering three values, namely 0.25, 0.50, 0.75, for the arrival rate, and by varying the retrial rate. For all these scenarios, the mean service time is fixed at 1 and the coefficient of variation to be 1.25. Note that taking the retrial rate to be ∞ reduces the retrial model to the classical queueing model; (3) Erlang services by looking at three values, namely 2, 4, 6, for the order of the Erlang distribution, and the arrival rate is varied. The other parameters are set as in case (1); (4) the service time is Erlang of order 3 by varying the retrial rate, and considering three values for the arrival rate. The parameter values are as given in (2). Also, it should be pointed out that a number of measures (other than the busy period ones) for the retrial queueing models in a more general setup have been validated and reported in [8]. Unless otherwise specified, all our simulation involves five replicates with each replicate for 500,000 units. Only in some cases like when λ = 0.1, we had to simulate for longer periods of times so as to get the simulated values close enough to the analytical results when validating. In Table 1 below, we display the representative error percentages of our simulated results with the numerical results (based on analytical formulas) for the mean and the coefficient of variation of the busy period for the M/G/1 retrial queue with exponential retrial times. For services, the authors in [3] consider (for numerical purposes) Erlang and hyperexponential distributions under different sets   of parameters. The error  o et al.−Simulat ed  percentage is calculated as 100 Art alejArt %. alej o et al. As can be seen from the table below (and the other error percentages not displayed here due to lack of space) the simulated values agree with the analytical ones very well (the largest error percentage is less than 10%).

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Table 1 Error percentages using Artalejo et al. [3]

λ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Hyperexponential services CV = 1.25 CV = 1.50 1.77% 1.79% 0.89% 0.90% 2.01% 1.02% 0.51% 0.59% 0.87% 0.44% 2.43% 0.23% 0.21% 0.95% 0.57% 0.24% 0.97% 1.10% 1.27% 0.20% 0.22% 1.64% 0.98% 1.64% 1.85% 0.32% 2.29% 0.78% 3.13% 3.00% 2.54% 3.90% 4.71% 4.28%

CV = 1.75 2.30% 2.23% 0.11% 0.95% 1.94% 2.14% 1.76% 0.04% 0.07% 1.42% 0.48% 2.76% 3.01% 1.88% 1.07% 2.49% 0.11%

Erlang services k=2 k=4 0.56% 5.31% 0.37% 0.56% 2.47% 0.26% 0.81% 1.56% 0.27% 0.31% 0.21% 0.58% 1.51% 2.22% 0.19% 0.58% 0.93% 0.76% 1.11% 0.02% 0.76% 0.33% 0.32% 2.47% 2.18% 1.96% 0.19% 2.61% 3.31% 2.76% 9.08% 0.34% 3.65% 0.43%

k=6 2.08% 0.16% 0.48% 0.44% 1.14% 0.35% 1.07% 0.17% 0.73% 1.84% 1.31% 1.58% 1.34% 2.27% 3.30% 4.11% 0.45%

4 Simulated Results In this section, we will discuss a few illustrative examples based on simulation. Towards this end, we consider ten different arrival processes consisting of eight MAP s, a Weibull, and a constant; five different types of service time; and three retrial time distributions. While it is clear (see, e.g., [6]) that Erlang, exponential, and hyperexponential are very special cases of a MAP , a few other MAP s considered here are based on the construction (for numerical purposes) originally described in [7] and elaborated with more details in [11]. Before we display the arrival processes, we set some notation. Define ⎞

⎛

−λ1 λ1 ⎜ −λ1 λ1 ⎜ α(m) = (1, 0, · · · , 0), T (λ1 , m) = ⎜ .. ⎜ . ⎝

⎟ ⎟ ⎟, ⎟ ⎠ −λ1

+

* D0 (λ1 , λ2 , m) =

⎞ 0 ⎜ . ⎟ ⎜ . ⎟ . ⎟ T 0 (λ1 , m) = ⎜ ⎜ ⎟, ⎝ 0 ⎠ λ1

T (m) 0 0 −λ2

⎛

* , D1 (λ1 , λ2 , p1 , p2 , m) =

p1 T 0 α q1 T 0 q2 λ2 α p2 λ2

+ ,

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where qi = 1 − pi and 0 < pi < 1, i = 1, 2. Note that the representation (α(m), T (m)) of dimension m is for Erlang distribution of order m (see e.g., [18]). The representation (D0 (λ1 , λ2 , m), D1 (λ1 , λ2 , p1 , p2 , m)) of dimension m + 1 is for a MAP . This form of MAP will be used for arrival processes in our illustrative examples below. Suppose that r denotes the 1-lag correlation coefficient of this MAP . It is shown in [11] that r is obtained explicitly in terms of the parameters of the MAP as r=

q1 q2 (1 − q1 − q2 )(λ1 − mλ2 )2

. q1 λ21 (q1 + 2q2 ) − 2mq1 q2 λ1 λ2 + mq2 λ22 [q2 + (m + 1)q1 ]

A. Arrival Process: We consider the following distributions for the arrival processes. Note that these will be normalized so as to have a specific arrival rate, λ, for comparison purposes. Also, these are qualitatively different in that they have different correlation and variance structure. 1. Erlang (ERA): This is Erlang of order 5 with parameter 5λ. 2. Hyperexponential (H EA): This is hyperexponential with mixing probability vector taken as (0.7, 0.25, 0.05) with the corresponding rate vector given by λ(8.2, 0.82, 0.082). 3. MAP with negative correlation (NC1): Here we consider the MAP with representation (D0 (1.25, 2.5, 2), D1(1.25, 2.5, 0.01, 0.01, 2)) of dimension 3. Note that r = −0.32667. 4. MAP with negative correlation (NC2): Here we consider the MAP with representation (D0 (2.25, 4.5, 4), D1(2.25, 4.5, 0.01, 0.01, 4)) of dimension 5. Note that r = −0.57855. 5. MAP with negative correlation (NC3): Here we consider the MAP with representation (D0 (4.75, 9.5, 9), D1(4.75, 9.5, 0.01, 0.01, 9)) of dimension 10. Note that r = −0.78022. 6. MAP with positive correlation (P C1): Here we consider the MAP with representation (D0 (1.25, 2.5, 2), D1(1.25, 2.5, 0.99, 0.99, 2)) of dimension 3. Note that r = 0.32667. 7. MAP with positive correlation (P C2): Here we consider the MAP with representation (D0 (2.25, 4.5, 4), D1(2.25, 4.5, 0.99, 0.99, 4)) of dimension 5. Note that r = 0.57855. 8. MAP with positive correlation (P C3): Here we consider the MAP with representation (D0 (4.75, 9.5, 9), D1(4.75, 9.5, 0.99, 0.99, 9)) of dimension 10. Note that r = 0.78022. 9. Constant (CT A): Here we consider constant inter-arrival times with a value of 1 . λ 10. Weibull (W BA): We consider a 2-parameter (where one is fixed to be 0.5) x 0.5 Weibull whose CDF is given by FW BA (x, θ ) = 1 − e−( θ ) , x ≥ 0, θ > 0. B. Service Times We consider the following for distributions for the services. Note that these will be normalized so as to have a specific service rate, μ, for comparison

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purposes. Also, these are qualitatively different in that they have a different variance structure. 1. Erlang (ERS): This is Erlang of order 5 with parameter 5μ. 2. Hyperexponential (H ES): This is hyperexponential with mixing probability vector taken as (0.9, 0.1) with the corresponding rate vector given by μ(1.9, 0.19). 1 3. Constant (CT S): Here we consider constant services with a value of . μ 4. Weibull (W BS): We consider a 2-parameter (where one is fixed to be 0.5) Weibull whose CDF is given by FW BS (x, η) = 1 − e

−( xη )0.5

, x ≥ 0, η > 0.

C. Retrial Times We consider the following for distributions for the trials. Note that these will be normalized so as to have a specific retrial rate, ξ , for comparison purposes. Also, these are qualitatively different in that they have different variance structure. 1. Erlang (ERR): This is Erlang of order 5 with parameter 5ξ . 2. Exponential (EXR): Here we consider exponential retrials with parameter ξ . 3. Hyperexponential (H ER): This is hyperexponential with mixing probability vector taken as (0.9, 0.1) with the corresponding rate vector given by ξ(1.9, 0.19). 1 Example 1 In this example, we fix λ = 1, μ = 0.95c , ξ = 0.5, and look at different scenarios by varying the arrival process, the service times, the retrial times, and the number of servers, c, in the system. In Figs. 1 and 2, respectively, we display the Ln(Mean busy period) vs Ln(Mean number of busy periods) for the renewal and the correlated arrivals, and in Figs. 3 and 4, respectively, we display the coefficient of variation of the busy period for the renewal and the correlated arrivals. These figures are under various scenarios.

Some key observations, keeping in mind the sampling errors due to simulation, from these figures are summarized below. 1. When c = 1, the scenario corresponding to ERA arrivals and ERS services appears to yield a small mean busy period along with a small mean number of busy periods. Note that the traffic load of the queue is high here. Hence, this is counterintuitive but can be explained. In another example, we will discuss this by looking at smaller to medium values for the traffic load. The small values for the mean number of periods along with the small mean busy period triggered us to look into running the simulation even for a longer period of time from the current one of 500,000 units. However, when simulating this scenario for 100,000,000 units, we still noticed a similar phenomenon. So, this indicates that when the inter-arrival and service times have a smaller variability the mean busy period lasts for a longer period of time. Due to the sampling error, the mean busy period is obtained based on a very few busy periods and after which the simulation ends with no more busy period completed. This results in small numbers for these two

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Fig. 1 Ln(Mean busy period) vs Ln(Mean number of busy periods) under various scenarios for the renewal arrivals

measures and hence one should not interpret that ERA appears to yield smaller mean busy periods. On the contrary, the mean busy period is large for ERA as compared to, say, H EA, when the traffic load is high. 2. As the number, c, of servers is increased, we see that the mean busy period becomes larger and the mean number of busy periods pretty much staying small only for scenarios wherein the variability in the arrivals and also the variability in the services are small (i.e., ERA and ERS combination). 3. In the case of H EA, noting that this has a higher variability in the inter-arrival times, we see the system getting free more often and also getting busy more often. This is the case for all scenarios involving hyperexponential arrivals. This

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Fig. 2 Ln(Mean busy period) vs Ln(Mean number of busy periods) under various scenarios for the correlated arrivals

can be, somewhat, explained intuitively as follows. Due to a large variability in the inter-arrival times (for H EA case), we see the customers arrive with short inter-arrival times and then once in a way it takes a longer time for an arrival to occur. During this longer interval, customers from orbit occupy the server at a faster rate. 4. When comparing the correlated arrivals, we notice that almost for all scenarios the negatively correlated arrivals have a higher mean busy period compared to those of the positively correlated ones. However, when looking at the mean number of busy periods, we notice that it is the positively correlated arrivals that have a higher value compared to those of the negatively correlated ones.

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Fig. 3 Coefficient of variation of the busy period under various scenarios for the renewal arrivals

5. Having a high variability in either the inter-arrival times or in the services appears to have a larger coefficient of variation of the busy period. 6. Looking at the coefficient of variation of the busy period, we notice that the positively correlated arrivals appear to have a higher value when compared with the negatively correlated ones. 7. In the case of the constant arrivals (CT A) as well as for the Weibull arrivals (W T A) we see the behavior of the three measures that are similar to the hyperexponential arrivals (H EA). Note that here the types of services are different from the ones seen for the H EA case. However, the main thing here

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Fig. 4 Coefficient of variation of the busy period under various scenarios for the correlated arrivals

is to show that even for the heavy-tailed distribution (either in the arrivals and or in services) like Weibull considered here behaves differently as compared to the Erlang ones. Finally, a few additional comments for which the figures are not included here due to lack of space. We looked at other scenarios involving Erlang arrivals (ERA). These include CT S, W BS, with the combinations of ERR, H ER and c = 1, 2, 5. We noticed that constant service (CT S) indicated that (under all combinations for retrials and the number of servers), during the entire simulation period, there was at least one customer waiting in the retrial orbit with probability ranging from 0.98 to

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1.0, while the average number of customers in the orbit ranged from 7.2679 through 38.6894 (based on the type of retrial times and the number of servers). In the case of W BS, the range for the probability is from 0.9186 through 0.9964 and the average number in the orbit from 45.7554 to 86.1952. Due to these additional observations it is no wonder that for some scenarios involving ERA arrivals, the busy period looked “almost” endless even after simulating for a long period of time. As we saw earlier, the scenario corresponding to ERA − ERS − ERR (i.e., Erlang arrivals, Erlang services, and Erlang retrials) produces a larger mean busy period and a smaller number of busy periods especially when the traffic intensity is higher. In the next example, we investigate the effect of the traffic intensity and the order of Erlang arrivals on the two of the three measures under consideration. Example 2 In this example, we investigate the effect of the traffic intensity and the order of Erlang arrivals on the three measures under consideration. Towards this end, we fix μ = 1, ξ = 0.5, c = 1, vary λ = 0.1, . . . , 0.8, and consider Erlang for arrivals, services, and for retrials. While we fix the order of the Erlang for the services and the retrials to be 5, we vary the order of the Erlang for the arrivals from m = 2, . . . , 9. In Fig. 5, we display Ln(Mean busy period) and Ln(mean number of busy periods), under various scenarios. Looking at this figure with two plots, we notice the following interesting observations. 1. For λ taking values up to 0.6, we see that the mean busy period decreases as m increases; however, when λ > 0.6, we see this trend is reversed indicating that the mean busy period increases as m is increased. 2. With respect to the number of busy periods, we notice that this measure increases as m is increased for λ up to 0.6, and then the trend is reversed. Thus, we notice that when arrivals and services occur with less variability, then as the traffic load increases, the mean busy period becomes large and the number of busy periods decreases. Example 3 In this example, we investigate the effect of the retrial rate for Erlang arrivals on the three measures under consideration. Towards this end, we fix λ = 1 1, μ = 0.95 , c = 1, vary ξ = 1, 2, 5, 10, and consider various services and retrial times. In Fig. 6, we display Ln(Mean busy period), Ln(mean number of busy periods), and the coefficient of variation of the busy period under various scenarios. Looking at this figure, we notice the following interesting observations. 1. As ξ is increased, we notice that the mean busy period decreases and the mean number of busy periods increases. This is to be expected as the retrial queueing model approaches the corresponding classical queueing models. 2. As ξ is increased, we observe that the coefficient of variation increases. Furthermore, for a fixed ξ , H ES has a higher coefficient of variation compared to the corresponding case for ERS. This is true for both retrial distributions.

Busy Period Analysis of Multi-Server Retrial Queueing Systems

Fig. 5 Graphs of the two measures as functions of λ and m for Erlang arrivals

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Fig. 6 Graphs of various measures for Erlang arrivals

5 Concluding Remarks In this paper, we looked at the busy period in the context of multi-server retrial queueing systems. As is known, the study of the busy period even in the classical queueing systems is complex due to the inherent difficulty in its study. This is further compounded in the case of retrial queueing systems due to the server alternating between idle and service periods even when the system has customers waiting in the orbit. Hence, we used simulation in this paper to carry out the busy period analysis. We showed how the variability in the arrival/service/retrial times as well as the correlation in the inter-arrival times affects the busy period. Furthermore, we

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offered some insight into the behavior of the busy period when arrivals, services, and retrials are all modeled using Erlang distribution. Even though the few illustrative examples provided some insights and interesting results, more examples need to be simulated and this will be a topic for future research.

References 1. Artalejo, J.R.: Accessible bibliography on retrial queues. Math. Comput. Model. 30, 1–6 (1999) 2. Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems: A Computational Approach. Springer, Berlin (2008) 3. Artalejo, J.R., Lopez-Herrero, M.J.: On the busy period of the M/G/1 retrial queue. Nav. Res. Logist. 47, 115–127 (2000) 4. Artalejo, J.R., Chakravarthy, S.R., Lopez-Herrero, M.J.: The busy period and the waiting time analysis of a MAP /M/c queue with finite retrial group. Stoch. Anal. Appl. 25, 445–469 (2007) 5. Artalejo, J.R., Gomez-Correl, A., He, Q.M.: Markovian arrivals in stochastic modelling: a survey and some new results. SORT 34(2), 101–144 (2010) 6. Chakravarthy, S.R.: The batch Markovian arrival process: a review and future work. In: Krishnamoorthy, A. et al. (ed.) Advances in Probability Theory and Stochastic Processes. Notable Publications, New Jersey, pp. 21–39 (2001) 7. Chakravarthy, S.R.: Markovian arrival processes. In: Wiley Encyclopedia of operations research and management science. Wiley, New York (2010) 8. Chakravarthy, S.R.: Analysis of MAP /P H /c retrial queue with phase type retrials— simulation approach. In: Dudin, A. et al. (eds.) BWWQT 2013, CCIS 356, pp. 37–49 (2013) 9. Chakravarthy, S.R.: Matrix-analytic queueing models, Chapter 8. In: Narayan Bhat, U. (ed.) An Introduction to Queueing Theory, 2nd edn, Birkhauser/Springer, New York (2015) 10. Chakravarthy, S.R.: Busy period analysis of GI /G/c and MAP /G/c queues. In: Deep, K. et al. (eds.) Performance Prediction and Analysis of Fuzzy, Reliability and Queueing Models, Asset Analytics, pp. 1–31 (2019). https://doi.org/10.1007/978-981-13-0857-4_1 11. Chakravarthy, S.R.: Queueing models in services—analytical and simulation approach. In: Anisimov, V. Prof., Limnios, N. Prof. (eds.) To Appear in Advanced Trends in Queueing Theory. Mathematics and Statistics. Sciences, ISTE/Wiley, London (2020) 12. Kelton, W.D., Sadowski, R.P., Swets, N.B.: Simulation with ARENA, 5th edn., McGraw-Hill, New York (2010) 13. Kim, J.: Busy period distribution of a batch arrival retrial queue. Commun. Korean Math. Soc. 32(2), 425–433 (2017). https://doi.org/10.4134/CKMS.c160106 14. Kim, J., Kim. B.: A survey of retrial queueing systems. Ann. Oper. Res. 247(1), 3–36 (2016). https://doi.org/10.1007/s10479-015-2038-7 15. Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Stoch. Model. 7, 1–46 (1991) 16. Lucantoni, D.M., Meier-Hellstern, K.S., Neuts, M.F.: A single-server queue with server vacations and a class of nonrenewal arrival processes. Adv. Appl. Probl. 22, 676–705 (1990) 17. Neuts, M.F.: A versatile Markovian point process. J. Appl. Prob. 16, 764–779 (1979) 18. Neuts, M.F.: Matrix-geometric solutions in stochastic models: an algorithmic approach. The Johns Hopkins University, Baltimore (1981). [1994 version is Dover Edition] 19. Neuts, M.F.: Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, New York (1989)

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20. Neuts, M.F.: Models based on the Markovian arrival process. IEICE Trans. Commun. E75B, 1255–1265 (1992) 21. Neuts, M.F.: Algorithmic Probability: A Collection of Problems. Chapman and Hall, New York (1995) 22. Phung-Duc, T.: Retrial queueing models: a survey on theory and applications. In: Dohi, T. et al. (eds.) Stochastic Operations Research in Business and Industry. World Scientific, Singapore (2017). http://infoshako.sk.tsukuba.ac.jp/~tuan/papers/Tuan_chapter_ver3.pdf

Steady-State and Transient Analysis of a Single Channel Cognitive Radio Model with Impatience and Balking Alexander Rumyantsev and Garimella Rama Murthy

Abstract In this paper, motivated by an increasing interest in Cognitive Radio wireless transmission systems, we study a stochastic model of a single node of such a system with underlay transmission and balking. The considered model is essentially a single-server system with an ON–OFF type environment governing the service time intensity and triggering the balking events. We utilize the matrix analytic method for steady-state analysis, and perform transient analysis by Complete Level Crossing Information approach. The results of analysis are validated and illustrated by simulation. Keywords Radio · Structured Markov chain · Transient analysis · Matrix analytic method · Complete level crossing information

1 Introduction In recent years, due to proliferation of wireless networks, demand for electromagnetic spectrum is increasing. However, measurements in an urban environment show that the spectrum utilization can be relatively low both spatially and temporally. Thus, spectrum sharing solutions are used to increase wireless networks efficiency. One of widely used solutions is the so-called Cognitive Radio (CR) wireless network technology. CR allows to share the wireless transmission channel among the licensed users (known as Primary Users, PU) and unlicensed users (or Secondary

A. Rumyantsev () Institute of Applied Mathematical Research of the Karelian Research Centre of R.A.S., Petrozavodsk, Russia Petrozavodsk State University, Petrozavodsk, Russia e-mail: [email protected] G. Rama Murthy Mahindra Ecole Centrale, Bahadurpally, Hyderabad, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_6

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Users, SU). The latter are obliged to vacate the channel or decrease the transmission activity in favor of the former. In general, one of the following methods of spectrum sharing is used [15]: – Interweaving: either PU or SU can use the channel at a time. – Underlay: PU/SU share the channel given low interference at PU receiver. – Overlay: PU and SU cooperatively improve signal/noise ratio at PU receiver. The framework of stochastic modeling and queueing theory is a common tool for researchers in the field of CR [15], and specifically matrix analytic method (MAM) is widely adopted to study the steady-state performance of CR systems under various assumptions [5, 14, 20]. MAM involves analysis of the structured discrete space continuous time Markov chain (CTMC) describing the system dynamics. In many cases, the CTMC belongs to a class of two-dimensional Markov chains, known as quasi-birth-and-death (QBD) processes, which exhibit a matrix-geometric solution for the steady-state probability distribution [6, 11, 13] and allows (under some restrictions) to obtain the steady-state performance explicitly. While steady-state performance allows to capture the long-run system behavior, transient analysis delivers insights on the time-dependent system evolution and sensitivity of the system to various management parameters, which is of high practical value. Such an analysis in general requires to obtain the solution of an (infinite) system of differential equations and is more complicated than steadystate analysis. At the same time, special structure of the CTMC under study (e.g., QBD processes) allows to obtain the transient solution explicitly in terms of Laplace transform, using the so-called Complete Level Crossing Information (LCI) approach [1, 7, 18]. In this research paper, we consider CR networks with a single channel. This channel is licensed for utilization by PU. Opportunistically, the channel is accessed by a pool of SU. There are many practically interesting wireless networks in which such assumption holds true, e.g. Mobile Ad-Hoc Networks with identical wireless handsets employed in civilian applications. The interleaving paradigm is more easy for hardware implementation and more attractive for analysis. Compared to the latter, the underlay and overlay methods allow to achieve higher spectrum utilization. However, both methods are underrepresented in the literature [15] due to higher complexity of the models. This motivates us to study the single channel underlay CR system. At the same time, to keep analytical tractability, we focus on the exponential distributions of governing sequences. The contribution of this paper is threefold. First, we study the model with underlay paradigm and randomized balking, which generalizes the models studied in [5, 14, 19, 20] (where the SU are always required to leave the system once the PU arrives at the system). Second, we obtain the steady-state distribution explicitly by the method developed in [8]. Finally, we perform transient analysis of the system, illustrating the results with the help of a simulation model. The results of numerical study verify the applicability of the approach. We stress that transient analysis is performed using Laplace transform of the desired performance measures. To obtain

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the transient performance in time domain, inverse transform is required, which may be performed numerically. This research paper is organized as follows. In Sect. 2, QBD model of a single channel wireless CR is discussed in detail. In Sect. 3, steady-state analysis is performed, followed by transient analysis in Sect. 4. Simulation results are presented in Sect. 5. The research paper concludes in Sect. 6.

2 Model Description In what follows we formulate and study a queueing-theoretic model of the CR system. We adopt the following notions common in the queueing theory: server (corresponding to a wireless transmission channel), PU/SU customer (PU/SU transmission packet), queue (backlog of SU packets waiting for transmission), service (transmission time of PU/SU packet). The CR wireless transmission system is a single-server model capable of serving two types of customers, PU and SU, with PU having absolute priority over SU. The SU arrive at epochs of Poisson input of rate λs and are waiting to be served in a single first-come-first-served queue. Service times of SU are independent identically distributed (iid.) random variables (r.v.) having exponential distribution of rate μs . At any time an arriving PU can reclaim the server for an exponentially distributed service time of rate μp . We assume that a new PU arrives after an exponentially distributed time, of rate λp , passes since the departure epoch of the previous PU. Thus, the server state is indeed the so-called ON–OFF process with exponential ON and OFF period duration (of different rates), which is a common assumption [15]. Upon arrival of a PU, the interrupted SU either returns back to the first position in the queue (with probability α ∈ [0, 1]), or balks immediately (with probability 1 − α). Such a randomized balking policy is a continuous-time generalization of the so-called α-retry policy [19]. When PU is present in the system, the SU are served at a different service rate β. We can think of such a service as an underlay transmission which requires the SU to use reduced signal strength once PU is present in the system. (The periods of PU transmission may also be considered as working breakdowns, or reduced energy/performance states.) The service rate μs is restored at the PU departure epoch, and the SU being served at rate β (if any) immediately starts a new service time at the rate μs at the departure epoch of PU. (Note that β may be also thought as the rate of impatience of SU waiting in the first position of the queue during the PU service.) Thus, service/interarrival times of a PU are in fact periods of a binary environment state modifying the parameter of service time distribution of SU. We may also think of the PU arrivals as an input following a rate-λp Poisson process with no waiting space for PU. Thus, given the model assumptions, the model is a state-dependent single-server M/M/1-type system with randomized balking. The model assumptions allow us to study the system dynamics as a two-dimensional discrete space CTMC {X(t), J (t)}t 0 ,

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Fig. 1 State transition diagram of the CTMC model {X(t), J (t)}t0 of cognitive radio with impatience and randomized balking

where X(t)  0 is the number of SU in the system, and J (t) ∈ {0, 1} is the number of PU being served at time t  0. Note that such a structured CTMC has two states at each level (i, ·), i  0. To simplify comprehension, we illustrate the transition rate diagram on Fig. 1. It is easy to see that the process {X(t), J (t)}t 0 a QBD process with infinitesimal generator of the following block-tridiagonal form ⎛

⎞ A0 0 0 . . . A1 A0 0 . . . ⎟ ⎟ . ⎟ A2 A1 A0 . . ⎟ ⎟, .. ⎟ 0 A2 A1 . ⎟ ⎠ .. .. .. . . . 0 0

A0,0 ⎜ A2 ⎜ ⎜ ⎜ Q=⎜ 0 ⎜ ⎜ 0 ⎝

(1)

where the matrix A0,0 corresponds to transitions between boundary states (0, ·), matrix A2 corresponds to level decreasing transitions (departures of SU and balking at arrivals of PU), A0 corresponds to arrivals of SU, and A1 is related to PU arrival/service completion. Below we define these matrices explicitly. $ A0 = λs I, $ A0,0 =

A1 =

−c1 αλp μp −c2

−λs − λp λp μp −λs − μp

$

% ,

A2 =

μs (1 − α)λp 0 β

% ,

(2)

% ,

(3)

Steady-State and Transient Analysis of a Single Channel Cognitive Radio. . .

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where c1 = λp + λs + μs and c2 = λs + μp + β. The diagonal elements of A1 follow from balance condition A1 = 0, where $ A = A0 + A1 + A2 =

−λp λp μp −μp

% .

(4)

It is interesting to note that A2 is full rank matrix if β > 0, which is in focus of our analysis. If β = 0 and α = 1, A2 has only one nonzero element, and the model corresponds to a classical M/M/1 queue with breakdowns. The stability criterion follows from the celebrated Neuts ergodicity condition [11] αA2 1 > αA0 1,

(5)

where α is derived from the following system: 

αA = 0, α1 = 1.

(6)

From (4) and (6), α readily follows: $ α=

μp λp , λp + μp λp + μp

% .

(7)

Thus, the stability criterion (5) reduces to λs
0 and A2 (·, j ) > 0, i.e. j -th columns of matrices A0 and A2 are nonzero; – a single state (i, k, j ), if Ak (·, j ) > 0 and A2−k (·, j ) = 0, k = 0, 2; – a single state (i, 1, j ), if A0 (·, j ) = 0 and A2 (i, j ) = 0. Then transition rates of the new infinitesimal generator are defined as ˆ Q([i, k, j ], [i " , k " , j " ]) := Q([i, j ], [i " , j " ]), if one of the following conditions holds: – i " = i − 1 and k " = 2; – i " = i + 1 and k " = 0; – i " = i and k " is minimal s.t. (i " , k " , j " ) is a state. Note that the boundary states (0, j ) cannot receive transitions from below, thus, they are replaced with a single state (i, k, j ), where k = 2 if A2 (·, j ) > 0 and k = 1 otherwise. We use the three-dimensional lexicographically ordered index of the matrix Qˆ and two-dimensional index of Q, which simplifies comprehension compared to sequentially renumbered states in the state space. We select the minimal k " w.o.l.o.g., since the destination for inter-level transitions can be chosen arbitrarily. More details on LCI-completeness and the transformations can be found in [18]. We specifically use the second component as opposed to third component used in [18], since this allows to preserve lexicographical order without the need of reordering.

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Fig. 2 State transition diagram of the CTMC model {X(t), D(t), J (t)}t0 of cognitive radio with impatience and randomized balking satisfying LCI-completeness property

In Fig. 2 we illustrate the mutation of the state space performed to satisfy the LCIcompleteness (for two consecutive states). After such a partition, the new state space allows each state (i, k, j ) to receive transitions either from levels i+1 or from i−1 only. Thus, the infinitesimal generator Qˆ of the new process has the following bidiagonal form known as LCI-complete canonical form: ⎞ ⎛ B0 B1 0 0 . . . ⎜ 0 C B 0 ...⎟ ⎟ ⎜ ⎜ .. ⎟ ⎟ ⎜ Qˆ = ⎜ 0 0 C B . ⎟ , (14) ⎟ ⎜ ⎜ 0 0 0 C ... ⎟ ⎠ ⎝ .. .. .. 0 0 . . .

Steady-State and Transient Analysis of a Single Channel Cognitive Radio. . .

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where all matrices, except B0 and B1 , are square matrices of order m (recall m is the number of states at non-boundary levels). B0 is an (m0 + m) × (m0 + m − m1 ) matrix, where m0 is the number of states at boundary level and m1 is the number of states at non-boundary level that do not allow downward transitions to the boundary level, while B1 is (m0 + m) × m. Note that after the transformation of CR model CTMC, m = 4, m0 = 2, m1 = 2. Hereafter we define the blocks of Qˆ for CR model explicitly: ⎛

μs (1 − α)λp ⎜ 0 β C=⎜ ⎝ μs (1 − α)λp 0 β

−c1 μp 0 μp

⎞ αλp −c2 ⎟ ⎟, αλp ⎠

⎛

0 ⎜ 0 B=⎜ ⎝ −c1 0

0

0 0 0 −c2

λs 0 λs 0

⎞ 0 λs ⎟ ⎟, 0⎠ λs (15)

$ B0,0 =

−(λp + λs ) λp λs 0 μp −(μp + λs ) 0 λs

%

$ ,

B1 =

0 B

%

$ , B0 =

B0,0 C

% . (16)

The new matrix Qˆ allows to obtain the steady-state distribution of the LCIcomplete QBD as a solution of a system of linear equations, since the matrix Qˆ is bidiagonal [1]. Note that the method of obtaining a solution is somewhat similar to the linear solution (12), however, the interrelation of these methods is beyond the scope of this paper and is left for future research. On the contrast, to obtain transient solution, the following system is solved: dπ(t) ˆ = π(t)Q, dt

(17)

with boundary condition π(0) = π0 . Performing a componentwise Laplace transform of (17) leads to the following system: Π(u)(Qˆ − uI ) = −π0 ,

(18)

where I is the identity matrix, and 

∞

Π(u) =

π(t)e−ut dt,

Reu  0.

0

ˆ − uI are obtained from It is easy to obtain from (14) that blocks of matrix Q blocks of Qˆ by replacing ci with ci (u) := ci + u, i = 1, 2. We use the notation C(u), B(u), Bi (u), i = 1, 2, and B0,0 (u) to refer to the corresponding submatrices of (18).

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It follows from (15) that |C(u)| = c1 (u)c2 (u)βμs > 0. After some algebra, C −1 (u) can be obtained explicitly ⎛ C

−1

1 − c1k(u)

⎜ μp ⎜ (u) = ⎜ βc1 (u) ⎝ − c 1(u) 1 0

k2 1 c2 (u) μs

k1 c1 (u) μp − βc1 (u) 1 c1 (u)

0 0 −1 c2 (u)

+

− μk1p − 1 β

k2 c2 (u)

0

⎞ ⎟ ⎟ ⎟, ⎠

1 c2 (u)

0

where k1 = (1 − α)λp μp /(βμs ) and k2 = αλp /μs The transient solution Π(u) is then obtained in matrix-geometric form by the following recursion: Πn+1 (u) = Πn (u)W (u),

(19)

where W (u) = −B(u)C −1 (u) can be found explicitly: ⎛

λs c1 (u)

⎜ ⎜ 0 ⎜ W = ⎜ λs ⎜ ⎝ c1 (u) − k1 μp c2 (u) βc1 (u)

s − c1λ(u)

0 λs c2 (u) k2 c1 (u) c1 (u) c2 (u) μs λs c2 (u)

0

−

⎟ ⎟ ⎟ . (u) ⎟ ⎟ − k1 cμ1p(u) − k2c2c1(u) ⎠ c2 (u) λs β − c2 (u) s − c2λ(u)

0 + k1 −

⎞

λs c1 (u)

μp c2 (u) βc1 (u)

4.1 Boundary Value Problem To obtain the Laplace transform of transient state vectors for boundary states, Π0 (u), Π1 (u), the following linear system has to be solved: $

B0 (u) B1 (u) [Π0 (u) Π2 (u)] 0 C(u)

% = −π0 ,

(20)

where Π0 (u) is m0 + m-component vector (corresponding to the boundary state 0 as well as the level 1), and Π2 (u) has m components. However, the system has multiple solutions and requires m1 more equations to find the solution explicitly. This requirement is satisfied by obtaining the eigenvalues of matrix W (u) which are on or outside of the unit circle, and eliminating the corresponding unstable modes from the solution [18]. Some straightforward algebra allows to deduce the following characteristic equation for the eigenvalues of W (u):

φ(ξ ) = (ξ − 1) ξ 3 + a2 ξ 2 + a1 ξ + a0 = 0,

Steady-State and Transient Analysis of a Single Channel Cognitive Radio. . .

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where % c2 (u) c1 (u) a2 = − 1 − − − k1 , β μs $

a1 =

λs (c1 (u) + c2 (u) − λs ) λ2 , a0 = − s . βμs βμs

It can be seen that ξ = 1 is the root of φ(ξ ), which is as expected. To obtain the largest real root of φ(ξ ), we use a trigonometric formula for cubic equation: $ %% $ √ q a2 1 arccos − , ξ0 = 2 −p cos √ 3 2p −p 3 where p=

3a1 − a22 , 9

q=

2a23 − 9a1a2 + 27a0 . 27

The corresponding right eigenvector r0 is then any solution of the following linear system: (W − ξ0 I )r0 = 0, and normalization condition r0 1 = 1 can be used to obtain unique r0 . Similarly, r1 is the right eigenvector corresponding to eigenvalue ξ1 = 1. This allows to extend the system (20) with the following equations: $ [Π0 (u) Π2 (u)]

0 0 r0 r1

% = 0.

(21)

For simplicity, we assume an initially empty system, that is, π0 = (1, 0, . . . , 0). Now equations (20) and (21), together with recursion (19), allow to obtain Π(u) explicitly. To obtain the transient solution π(t), an inversion of the Laplace transform Π(u) is required, that in general is done numerically. Finally, the performance measures of the system may be also obtained in terms of Laplace transform inversion. In particular, the Laplace transform of the transient mean number of SU in the system is X(u) =

∞ 

iΠi (u)1 = Π1 (u)1 + Π2 (u)P (u)(2I − Λ(u))(I − Λ(u))−2 P −1 (u)1,

i=1

where Π1 (u) is the vector of m rightmost components of Π0 corresponding to level 1, P (u) is the matrix of right eigenvectors of W (u), P −1 (u) is the matrix of left eigenvectors of W (u), and Λ(u) is the diagonal matrix of eigenvalues of W (u) with zero values replacing the eigenvalues ξ0 > 1 and ξ1 = 1. The details of obtaining the Laplace transform of such a performance measure may be found in [18].

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5 Simulation Results

2.0 1.0

1.5

E X(t)

2.5

3.0

3.5

For simulation purpose, we created a discrete event simulation model of the system in R language [16]. We have validated the steady-state distribution and performance measures with analytical solution for trajectories up to 108 arrivals long, and the difference is of order 10−4 . We also performed validation of the transient model, and numerical errors for performance measures are reasonable. However, the sensitivity of the transient solution to parameters and the numerical stability has to be studied separately. To illustrate the approach, we performed a simulation of the CR system model with the following arbitrary chosen parameters: λp = 3, μs = 2, μp = 5, α = 0.2, β = 4. We selected λs = 3.285 such that the stability criterion is satisfied, but the system load is relatively large. Steady-state analysis allows to obtain the mean stationary number of SU in the system as EX ≈ 3.34. At the same time, we simulated the system in transient state for t  100. To obtain the estimates  of EX(t) in simulation model, we calculated simple ensemble averages as N1 N i=1 Xi (tj ), where Xi (tj ) is the number of SU in the system at i-th trajectory (simulation run) at j -th time point, tj = j, j = 1, . . . , 100. We performed N = 105 simulations to obtain these point estimators, and performed Laplace inversion to obtain the analytical result. The results of simulation depicted on Fig. 3 illustrate good adequacy of analytical and simulation results.

0.5

Transient Steady−state Simulation 0

5

10

15

20

time, t

Fig. 3 Convergence of transient performance measure EX(t) to steady-state value EX for analytical and simulation models

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6 Conclusion We performed steady-state and transient analysis of the QBD model of CR wireless transmission system node with randomized balking and underlay transmission. However, the proposed approach can be generalized to the so-called G/M/1-type processes, which would require additional effort (in particular, the steady-state solution would require deriving the eigenvalues numerically). Such a generalized model can incorporate the features of the so-called Spread Spectrum (CDMA) Cognitive Radio Networks [4], where the SU utilize orthogonal signature sequences as in direct sequence spread spectrum wireless communication systems. In such a system there is a positive probability that multiple secondary users can successfully transmit their packets without collision (sharing the channels efficiently). Furthermore, it would be interesting to use the game-theoretic approach where SU are considered as players, and the strategy could be seen as the pair (α, β) for each player. However, these generalizations are beyond the scope of this paper, and we leave this for future research. Acknowledgments The authors thank Sergey Astafiev and anonymous referees for their suggestions that helped to improve the paper. This work is supported by Russian Foundation for Basic research, projects No 18-07-00147, 18-07-00156, 18-37-00094, 19-07-00303, 19-57-45022.

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8. Garimella, R.M., Rumyantsev, A.: On an exact solution of the rate matrix of Quasi-Birth-Death process with small number of phases. In: Proceedings: 31st European Conference on Modelling and Simulation (ECMS 2017), 23rd–26th May 2017, pp. 713–719. Budapest, Hungary (2017). https://doi.org/10.7148/2017-0713 9. Garimella, R.M., Rumyantsev, A.: On an exact solution of the rate matrix of G/M/1—type Markov process with small number of phases. J. Parallel Distrib. Comput. 119, 172–178 (2018). https://doi.org/10.1016/j.jpdc.2018.04.013, http://linkinghub.elsevier.com/retrieve/pii/ S074373151830282X 10. Gillent, F., Latouche, G.: Semi-explicit solutions for M/PH/1-like queuing systems. Eur. J. Oper. Res. 13(2), 151–160 (1983). https://doi.org/10.1016/0377-2217(83)90077-2 11. He, Q.M.: Fundamentals of Matrix-Analytic Methods. Springer, New York (2014) 12. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia (1999) 13. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University, Baltimore (1981) 14. Oklander, B., Sidi, M.: Modeling and analysis of system dynamics and state estimation in cognitive radio networks. In: 2010 IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications Workshops, pp. 49–53. IEEE, Istanbul (2010). https://doi. org/10/ctk99w. http://ieeexplore.ieee.org/document/5670521/ 15. Paluncic, F., Alfa, A.S., Maharaj, B.T., Tsimba, H.M.: Queueing models for cognitive radio networks: a survey. IEEE Access 6, 50801–50823 (2018). https://doi.org/10/gfvxsg. https:// ieeexplore.ieee.org/document/8445574/ 16. R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2018). https://www.R-project.org/ 17. van Leeuwaarden, J., Winands, E.: Quasi-birth-and-death processes with an explicit rate matrix. Stoch. Model. 22(1), 77–98 (2006). https://doi.org/10.1080/15326340500481747 18. Zhang, J., Coyle, E.J.: Transient analysis of quasi-birth-death processes. Commun. Stat. Stoch. Model. 5(3), 459–496 (1989). https://doi.org/10.1080/15326348908807119. http:// www.tandfonline.com/doi/abs/10.1080/15326348908807119 19. Zhao, Y., Jin, S., Yue, W.: A novel spectrum access strategy with α-retry policy in cognitive radio networks: a queueing-based analysis. J. Commun. Networks 16(2), 193–201 (2014). https://doi.org/10.1109/JCN.2014.000030 20. Zhu, D.B., Wang, H.M., Xu, Y.N.: Performance analysis of CSMA in an unslotted cognitive radio network under non-saturation condition. In: 2012 Second International Conference on Instrumentation, Measurement, Computer, Communication and Control, pp. 1122–1126. IEEE, Harbin (2012). https://doi.org/10/gfxg66. http://ieeexplore.ieee.org/document/6429100/

Applications of Fluid Queues in Rechargeable Batteries Shruti Kapoor and S. Dharmaraja

Abstract In this paper, the transient solution of the amount of charge in a rechargeable battery of finite capacity is obtained. The level of charge in the battery is governed by different input and output processes and are dependent on the level of charge in the battery. This model has been already discussed in Jones et al. (Fluid queue models of battery life. In: IEEE 19th international symposium on modeling, analysis and simulation of computer and telecommunication systems (MASCOTS), pp. 278–285, 2011), and the distribution of the hitting time was found numerically. In this paper, the method chosen is based on probabilistic approach which allows us to achieve a closed form solution for the distribution of the level of charge in the battery at any time t. Numerical illustrations are presented to verify the analytical results. Keywords Fluid queue · Transient distribution · Recurrence relations · Battery life time · Markovian queues

1 Introduction An important issue in the energy-constrained ad-hoc wireless networks is to obtain ways that increase their lifetime. The communication protocols for these wireless networks have to be developed such that they are aware of the state of the battery charge. The stored energy in these batteries is limited and should be effectively utilized, thereby increasing the battery lifetime. In this paper, we propose a fluid queue model for the charge in a battery.

S. Kapoor Department of Mathematics, Jesus and Mary College, University of Delhi, New Delhi, India S. Dharmaraja () Department of Mathematics, IIT Delhi, New Delhi, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_7

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The first stochastic models for batteries were developed by Chiasserini and Rao [2]. They describe two models for a battery of a mobile communication device which transmitting packets. Further extensions were made to improve these model in Chiasserini and Rao [3, 4]. The remaining paper is organized as follows. Section 2 presents the literature survey and related work. Section 3 describes the fluid queue model in detail; Sect. 4 presents the transient analysis of the distribution of the battery charge. In Sect. 5 we numerically illustrate the buffer content distribution and throughput in steady state and finally, Sect. 5.1 shows the sensitivity analysis, which can help in reducing the probability of zero charge. Pointers to further research and conclusion are given in Sect. 6.

2 Literature Survey and Related Work Fluid queue models have been applied to various real life situations and provide extensive information. The existing literature shows the diverse fields where fluid queue models have been applied. In Arunachalam et al. [1] the performance of the IEEE 802.11 protocol is modeled using the fluid queues. The outage probability performance of a new relay selection scheme is investigated in Liu [8] for the energy harvesting relays based on the wireless power transfer. A recent paper by Tunc and Akar [10] describes a fluid queue model to prolong the lifetime of Internet of Things (IoT) device with energy harvesting rechargeable batteries. This paper analyzes the use of fluid queues in rechargeable batteries which are used in digital cameras, mobile phones, remote sensors, and communication satellites. By applying fluid models to a rechargeable battery, we study the amount of charge in the battery at any time t. Chiasserini and Rao [3] model the life of a battery using fluid queues. Anupam and Dharmaraja [5] developed the fluid queue model for the battery life of a DRX mechanism in LTE-A networks and the cumulative distribution function of the battery life is derived. In this paper, our aim is to develop a fluid model to describe the charge in a battery. By analyzing the fluid model, we hope to improve the life of the batteries. Transient solutions for the buffer content distribution help in analyzing the system at any time. Various methodologies have been studied in literature to obtain the transient buffer content distribution of fluid queues. Transient analysis for fluid queue driven by chain sequence BDP with catastrophes was discussed in Vijayalakshmi and Thangaraj [11]. The closed form expression for the transient solution of fluid queue model driven by a birth death process is found in Kapoor and Dharmaraja [7]. For a fluid queue driven by an M/M/1 Queue with disaster and subsequent repair, the exact stationary solution is obtained in Vijayashree and Anjuka [12]. In this paper, we analyze the application of fluid queue models in a battery. We consider the model discussed in Jones et al. [6], and use simple probability concepts to obtain an exact solution for the amount of charge in a battery at any time.

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3 Model Description In today’s world the wireless age is expanding to include not just the smart phones, tablets, and laptops, but also the cars, homes, offices, and even whole communities. The ubiquitous battery is why we can carry immense computing power in our pocket. Thus, in this paper the charge level of a battery, which is subject to random charging and discharging periods, has been presented. The battery can be charged to a finite capacity B. The background process is determined by a switch, that is, an on-off model. Assume that the transition rates of on to off and off to on are λ and μ, respectively. When the switch is on, the battery gets charged at a rate α and when the switch is off, the battery is being discharged. The rate at which the battery gets discharged depends on the level of charge in the battery. The rate of discharge is either βh if level of charge is above the level V or βl if it is below V . Let {X(t), t ≥ 0} represent the state of the background process with state space S = {0, 1}, generator matrix Q = [qij ] and C(t) represent the level of charge in the battery at any time t. The fluid model is the pair {(X(t), C(t)), t ≥ 0} and the corresponding distribution function is defined as Fi (t, x) = P {X(t) = i, C(t) ≤ x};

t ≥ 0, x ≥ 0, i ∈ S.

(1)

The net inflow rate is a vector depending on level of battery charge and the state of the background process, defined as:

rk,i

⎧ −βl , ⎪ ⎪ ⎨ α − βl = ⎪ −βh , ⎪ ⎩ α − βh ,

0 < x ≤ V, i = 0 0 < x ≤ V, i = 1 . V < x ≤ B, i = 0 V < x ≤ B, i = 1

(2)

where k can take value l or h depending on 0 < x ≤ V or V < x ≤ B, respectively. The system of partial differential equations governing the above fluid queue model is given by ∂Fi (t, x) ∂Fi (t, x)  + rk,i = qij Fj (t, x), i ∈ S. ∂t ∂x j ∈S

We assume the generator matrix takes the form Q=



−λ λ . μ −μ

Without loss of generality, we assume that λ > μ.

(3)

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We consider rates given by Eq. (3) in two intervals (0, V ) and (V , B) and define the corresponding distribution function be Fil (t, x) and Fih (t, x), respectively. The system of partial differential equations reduces to ∂F0l (t, x) ∂F l (t, x) − βl 0 = −λF0l (t, x) + μF1l (t, x) ∂t ∂x ∂F1l (t, x) ∂F l (t, x) + (α − βl ) 1 = λF0l (t, x) − μF1l (t, x) ∂t ∂x

(4)

∂F0h (t, x) ∂F h (t, x) − βh 0 = −λF0h (t, x) + μF1h (t, x) ∂t ∂x ∂F1h (t, x) ∂F h (t, x) + (α − βh ) 1 = λF0h (t, x) − μF1h (t, x). ∂t ∂x

(5)

Charge Level

In Fig. 1, a sample path of the fluid queue model is shown. In charging periods, the level of charge in the battery increases. Further, it is observed from the figure that, when the level of charge in the battery reduces to less than the threshold level V , the rate of discharge increases.

Time

Fig. 1 Sample path of the charge level vs time

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4 Transient Analysis To simplify calculations, we follow the uniformization method and letting one-step transition probability matrix P = I + Q μ . Let Z = {Zn : n = 0, 1, . . .} be a time-homogeneous discrete time Markov chain (DTMC) with finite state space S. Let {N(t), t ≥ 0} be a time-homogeneous Poisson process with parameter μ. Assume that, it is independent of Z. Hence, X(t) = ZN(t ) for t ≥ 0. Given the number of transitions n in the interval (0, t), we define the probability vector p(t) = [p0 (t), p1 (t)] as p(t) =

∞ 

e−μt

n=0

(μt)n πP n n!

where pi (t) is the probability that the stochastic process {X(t), t ≥ 0} is in ith state at time t, π is the probability vector at t = 0, satisfying πQ = 0 and P the uniformized matrix. We assume t1 , t2 , . . . , tn be the n transition times. Thus splitting the interval (0, t) into n+1 subintervals with lengths t1 , t2 −t1 , . . ., t −tn . A net input rate is associated with each of these intervals, based on the state of the stochastic process in that interval. For the proposed model, the net input rate vector is r = [r1 , r2 , r3 , r4 ] = [−βl , α − βl , −βh , α − βh ]. Given n transitions have occurred, let, ki = the number of intervals associated with net input rate ri . Here, k = (k1 , k2 , k3 , k4 ) is called a partition of n+1, i.e., ||k|| = k1 +k2 +k3 +k4 = n + 1. Thus, by condition on the number of n transitions and k partition, we obtain P [C(t) > x] =

∞ 

e−μt

n=0

=

∞  n=0

e−μt

(μt)n n!



G[n, k]M(t, x, n, k)

||k||=n+1

n (μt)n  G[n, k]M(t, x, n, k) n!

(6)

k1 =0

M(t,  x, n, k) = P [C(t) > x|n transitions and partition k] and G[n, k] = i∈S Gi [n, k]. Here, given n transitions and k partition, Gi [n, k] is the probability that the state visited after the last transition is i. Suppose that, if i and j are the states visited after the last (n − 1)th and nth transitions, then k is equal to the previous partition +1 at the entry corresponds to the net rate associated with the state j .

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Therefore, the recurrence relations for the model described can be given by   Gl0 [n, k] = Gl1 [n − 1, k2 − 1] + Gh0 [n − 1, k3 − 1] + Gh1 [n − 1, k4 − 1]   Gl1 [n, k] = Gl0 [n − 1, k] + Gh0 [n − 1, k3 − 1] +Gl0 [n − 1, k1 − 1]

μ λ+μ

λ λ+μ

μ λ+μ

  Gh0 [n, k] = Gl1 [n − 1, k2 − 1] + Gh1 [n − 1, k4 − 1]

μ λ+μ

λ μ μ + Gl1 [n − 1, k] + Gl0 [n − 1, k1 − 1] λ+μ λ+μ λ+μ μ . +Gh0 [n − 1, k3 − 1] λ+μ

Gh1 [n, k] = Gl0 [n − 1, k]

From the above, the function Gi [n, k] recursively satisfies the initial conditions (0)

G1 [1, (1, 0)] = π0

Gi [0, (0, 1)] = πi(0) for i ∈ S \ {1}. Now, in order to find the function M(t, x, n, k), first we assume that n transitions yield k partition. Let U1 , U2 , . . . , Un be iid random variables each having uniform distribution in the interval (0, 1). Therefore, U(1), U(2) ,. . . , U(n) be their order statistics such that U(0) = 0 and U(n+1) = 1. Then, τi , i.e., the distribution of time of the ith transition has the identical distribution as tU(i) (refer [9]). Thus, we have Y1 ≡ tU(1) , Y2 ≡ t (U(2) − U(1) ), . . . , Yn+1 ≡ t (1 − U(n) ). We note that Yi ’s are exchangeable random variables, hence by rearranging the intervals, we let first k1 intervals be associated with the rate r1 . Then, next k2 intervals be associated with the rate r2 . Then, k3 intervals be associated with the rate r3 . Finally, the last k4 intervals be associated with the rate r3 . Now, C(t) is the cumulative buffer during the interval (0, t). Hence, the required event {C(t) > x given n transitions, and k partitions } can be given as {C(t) > x | n transitions, k partitions} = {r1 (Y1 + . . . + Yk1 ) +r2 (Yk1 +1 + . . . + Yk2 ) +r3 (Yk2 +1 + . . . + Yk3 ) +r4 (Yk3 +1 + . . . + Yn+1 ) > x}.

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Substituting values of ri and solving, we obtain P (C(t) > x | n transitions, k partitions) = P ((−αt)(Uk1 + Uk3 ) +t (βh − βl + α)Uk2 > x + t (βh − α)). Now, we find obtain M(t, x, n, k). For that, first we need to obtain the distribution of a linear combination of uniform distributed order statistics on the interval (0, 1). A solution for this was presented in [13]. Using the result given in [13], we get M(t, x, n, k) =

 f (ki −1) (ri , k) i (ki − 1)! r t >x i

where fi(ki −1) is the (ki − 1)st derivative of the following function: fi (y, k) = ,1

(y − x)n

j =0 (y j =i

− rj )kj

.

5 Numerical Illustration The following graphs illustrate the model numerically. Values for some parameters have been fixed to obtain the graphs, thereby analyzing the model in transient as well as steady state. Variation of the measure, average buffer content in steady state versus threshold is presented graphically. Table 1 describes the parameters and their values assumed for numerical purpose. Table 1 List of parameters and their values

Rates λ μ α βl βh B

Meaning Value Arrival rate of X(t) 1 Departure rate of X(t) 2 Charge rate 12 Discharge rate in regime l 5 Discharge rate in regime h 8 Maximum buffer level 500

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x=100 x=200 x=300

0.8

P(C(t)>x)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1000

2000

3000

4000

5000 6000 Time (t)

7000

8000

9000

10000

Fig. 2 Complement cumulative buffer content distribution

In Fig. 2, the complement buffer content is plotted against time. It can be seen from the graph that as time increases the complement buffer content decreases and will eventually approach 0. In Fig. 3, the average buffer content in steady state has been plotted against the threshold (V ). It shows that as the threshold increases the average buffer content also increases.

5.1 Sensitivity Analysis In Figs. 4 and 5 the 3 dimensional graph of the probability of system being empty against the discharge rates βl and βh has been plotted for V = 200 and V = 100. It can be seen that the graph resembles a paraboloid and thus the probability of the system being empty can be minimized. Also, with decrease in the value of the threshold, the minimum probability of the system being empty is further reduced which is in accordance with the numerical illustration.

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50 45

Average Buffer Content

40 35 30 25 20 15 V = 100 V = 80 V = 70

10 5 0

0

10

20

30

40

50 60 Threshold (V)

70

80

90

100

Fig. 3 Average buffer content vs threshold

1.4 1.2

P(empty system)

1 0.8 0.6 0.4 0.2 0 12 10

12

8

10 6

8 6

4

4

2 E

h

2 0

0

E

l

Fig. 4 Probability of the system is empty versus discharge rate for V = 200

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0.7 0.6

P(empty system)

0.5 0.4 0.3 0.2 0.1 0 12 10

12

8

10 6

8 6

4

4

2 E

2 0

0

h

E

l

Fig. 5 Probability of the system is empty versus discharge rate for V = 100

6 Conclusion and Future Work In this paper, we study a fluid queue model driven by a two state birth death process in which the net flow rate of fluid into buffer is dependent on the state of birth death process. The aim of the paper is to determine the amount of charge in a battery at any time t. Using the fluid queue model, a new methodology for finding the distribution of the level of charge in the battery at any time t is presented. This can be used in increasing battery life of a device and reducing energy consumption. In future, we plan to study the battery life distribution for a device modulated by a general Markov process. Acknowledgments Authors are thankful to the editor and two anonymous reviewers for their valuable suggestions and comments which helped improve the manuscript to great extent and are grateful for the financial support received from the Department of Telecommunications (DoT), India.

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References 1. Arunachalam, V., Gupta, V., Dharmaraja, S.: A fluid queue modulated by two independent birth-death processes. Comput. Math. Appl. 60(8), 2433–2444 (2010) 2. Chiasserini, C.F., Rao, R.R.: Pulsed battery discharge in communication devices. In: Proceedings of the 5th Annual ACM/IEEE International Conference on Mobile Computing and Networking, pp. 88–95 (1999) 3. Chiasserini, C.F., Rao, R.R.: A model for battery pulsed discharge with recovery effect. In: IEEE Wireless Communications and Networking Conference, vol. 2, pp. 636–639 (1999) 4. Chiasserini, C.F., Rao, R.R.: Improving battery performance by using traffic shaping techniques. IEEE J. Sel. Areas Commun. 19(7), 1385–1394 (2001) 5. Gautam, A., Dharmaraja, S.: An analytical model driven by fluid queue for battery life time of an user equipment in LTE-A networks. Phys. Commun. 30, 213–219 (2018) 6. Jones, G.L., Harrison, P.G., Harder, U., Field, T.: Fluid queue models of battery life. In: IEEE 19th International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS), pp. 278–285 (2011, July) 7. Kapoor, S., Dharmaraja, S.: On the exact transient solution of fluid queue driven by a birth death process with specific rational rates and absorption. Opsearch 52, pp. 746–755 (2015) 8. Liu, K.H.: Performance analysis of relay selection for cooperative relays based on wireless power transfer with finite energy storage. IEEE Trans. Veh. Technol. 65(7), 5110–5121 (2016) 9. Rubino, G., Sericola, B.: Markov Chains and Dependability Theory. Cambridge University Press, Cambridge (2014) 10. Tunc, C., Akar, N.: Markov fluid queue model of an energy harvesting IoT device with adaptive sensing. Perform. Eval. 111, 1–16 (2017) 11. Vijayalakshmi, T., Thangaraj, V.: Transient analysis of a fluid queue driven by a chain sequenced birth and death process with catastrophes. Int. J. Math. Oper. Res. 8(2), 164–184 (2016) 12. Vijayashree, K.V., Anjuka, A.: Exact stationary solution for a fluid queue driven by an M/M/1 queue with disaster and subsequent repair. Int. J. Math. Oper. Res. 15, 92–109 (2019) 13. Weisberg, H.: The distribution of linear combinations of order statistics from the uniform distribution. Ann. Math. Stat. 42(2), 704–709 (1971)

Analysis of BMAP /R/1 Queues Under Gated-Limited Service with the Server’s Single Vacation Policy Souvik Ghosh, A. D. Banik, and M. L. Chaudhry

Abstract This paper deals with the finite-buffer single server vacation queues with batch Markovian arrival process (BMAP ). The server follows gated-limited service discipline, i.e., the server can serve a maximum of L customers out of those that are waiting at the start of the busy period or all the waiting customers, whichever is minimum. It has been assumed that the server can take only one vacation, i.e., if no customers are found at the end of a vacation, the server remains idle until a batch of customers arrives. The service time and vacation time distributions are considered to possess rational Laplace–Stieltjes transform. The queue-length distribution at postdeparture, arbitrary, and pre-arrival epochs has been obtained. Various performance measures like mean queue-length, mean waiting time of an arbitrary customer, and mean length of busy and idle periods have been derived for this model. Numerical results have been presented based on the analysis done. Keywords Queueing · Batch Markovian arrival process · Gated-limited service · Single vacation · Roots method

S. Ghosh Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv-Yafo, Israel A. D. Banik () School of Basic Science, Indian Institute of Technology Bhubaneswar, Permanent campus, Argul, Khurda, Odisha, India e-mail: [email protected] M. L. Chaudhry Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ON, Canada e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_8

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1 Introduction In today’s connected society we often encounter system with server’s vacation. A server may go for vacation due to routine maintenance or utilization of its idle time. Queueing models with server’s vacation have found many applications in today’s life. Vacation queueing models can be classified based on the number of vacations taken by the server, i.e., single vacation (SV ) and multiple vacations (MV ) queueing models. Further, vacation queueing models can be classified depending on the starting and termination rules of the service time, namely, exhaustive, limited, gated, exhaustive-limited (E-limited), gated-limited (G-limited), and many more. For more details on different vacation models, readers are referred to Doshi [7], Takagi [21], and Tian and Zhang [22]. In the analysis of queueing systems, generally it is assumed that arrival follows Poisson process. However, modern day’s correlated arrivals can be described by Markovian arrival process (MAP ) and batch Markovian arrival process (BMAP ), see Neuts [16] and Lucantoni [14]. Lucantoni et al. [15] considered MAP /G/1 vacation queueing system and obtain stationary queue-length distributions using matrix-analytic approach. The MAP /G/1 queueing system under N-policy with and without vacations has been studied by Kasahara et al. [13]. Alfa [1] considered a discrete-time MAP /P H /1 vacation queueing system under gated time and limited service. In addition to these infinite-buffer capacity queueing models, Gupta and Sikdar [10] have studied the finite-buffer MAP /G/1/N queueing system with the server’s single as well as multiple vacations policy. Vacation queueing models with BMAP arrivals have also drawn a great attention of the researchers, see, e.g., Banik et al. [3], Saffer and Telek [18], Vishnevsky et al. [23], and the references therein. Recently, Banik and Ghosh [2], and Ghosh [9] have studied both the finite- and infinite-buffer BMAP /R/1/N(∞) multiple vacation queueing system under G-limited service discipline. In the G-limited service discipline, the server can serve a maximum of L customers out of those that were waiting at the start of the busy period or all the waiting customers, whichever is minimum. The queue-length distribution of the BMAP /R/1/N − SV queueing system under G-limited service discipline at postservice-completion and post-vacation-termination epochs has been briefly discussed by Banik and Ghosh [2]. In this paper, we have discussed the detailed computation procedure of the queuelength distributions at various epochs of the BMAP /R/1 − SV queueing system under G-limited service discipline. The computation of the queue-length distribution at post-service-completion or post-vacation-termination epochs, is based on the determination of the roots of the characteristic equation, see Chaudhry et al. [5, 19]. Further details on root method may be found in Gupta et al. [11] and Singh et al. [20]. One may note here that the queue-length distribution at postservice-completion or post-vacation-termination epochs may also be obtained using matrix-analytic methods, see Neuts [17]. Further, considering remaining service time of a customer and remaining vacation time of the server as supplementary variables, we have described the determination of the queue-length distribution of

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the queueing system at an arbitrary epoch. A detailed discussion on supplementary variable technique can be found in Hokstad [12] and Choi et al. [6]. Organization of the paper is as follows. Section 2 describes the model. In Sect. 3, we have analyzed the queue-length distribution at post-vacation-termination, postservice-completion, and post-departure epochs. Queue-length distributions at arbitrary and pre-arrival epochs have been studied in Sect. 4. Section 5 deals with various performance measures of the model. The computation procedure of the queue-length distribution at different epochs has been discussed in Sect. 6. Finally, in Sect. 7, numerical results have been presented based on the analytical expression obtained in the previous sections.

2 Model Description Let us consider a single server queueing system with infinite-buffer capacity where the customers are arriving in batches following a BMAP . BMAP is a generalization of the batch Poisson process where arrivals are governed by an underlying m-state Markov chain. Let Na (t) denote the number of arrivals in (0, t] and J (t) be the state of the underlying Markov chain at time t with state space {j : 1 ≤ j ≤ m}. Then {Na (t), J (t)} is a two-dimensional Markov process of BMAP with state space {(k, j ) : k ≥ 0, 1 ≤ j ≤ m}. The arrival process is characterized by the matrices D k (k ≥ 0) of order m × m. The (i, j )-th (1 ≤ i, j ≤ m) entry of D 0 denotes the state transition rate from state i to state j in the underlying Markov chain without an arrival and the (i, j )-th entry of D k (k ≥ 1) represents the state transition rate from state i to state j in the underlying Markov chain with an arrival of batch size k. Theinfinitesimal generator of the underlying Markov chain {J (t)} is given by D = ∞ π is the stationary k=0 D k . If probability vector of the BMAP , then π D = 0 and π em = 1, where em is a column vector of order m with all its entries equal to 1. Throughout the paper the subscript m is not used and the vector is presented as e. However, when e’s dimension is other than m it has been mentioned as its subscript. The average arrival rate λ and average ∞ batch arrival rate λ of the stationary BMAP are given by λ = π kD k e and b k=1  λb = π ∞ k=1 D k e, respectively. Let P (n, t) be a square matrix of order m whose (i, j )-th element is the conditional probability defined as Pi,j (n, t) = P {Na (t) = n, J (t) = j | Na (0) = 0, J (0) = i}, 1 ≤ i, j ≤ m, n ≥ 0, t ≥ 0. ∞ k For k=0 D k z and P (z, t) = ∞ |z| ≤ 1 nand t ≥ 0, we define D(z) = D(z)t . For more details on n=0 P (n, t)z . It may be shown that P (z, t) = e BMAP , see Lucantoni [14].

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In this paper, we have assumed that the server is entitled to take a single vacation only and the service discipline is G-limited with a limit, say L. In a single vacation queueing system, the server goes for a vacation whenever the system becomes empty and after completing the vacation the server remains dormant until a batch arrives in the system. In a G-limited service system, customers who have arrived during the vacation period are considered for service and those customers who are arriving during the busy period are excluded from service in the current busy period. These newly arrived customers have to wait till the next vacation period terminates. Further, in a queueing system under G-limited service with service limit L, the server serves a maximum of L customers in a busy period even if he finds more than L customers in the queue after termination of the vacation period. However, the server serves all the waiting customers in the busy period, if he finds not more than L number of waiting customers in the queue after termination of the vacation period. One may note that for large L, i.e., L → ∞, gated-limited systems can be expressed as exhaustive gated systems, whereas L = 1 presents pure limited systems, see [22]. Let B, B(x), b(x), and b∗ (s) (R(s) ≥ 0) be the random variable (RV), distribution function (DF), probability density function (pdf), and Laplace–Stieltjes transformation (LST) of the service-time distribution of a customer, respectively. Similarly, let V , V (x), v(x), and v ∗ (s) (R(s) ≥ 0) be the respective RV, DF, pdf, and LST of the vacation time distribution of the server. So the expected service time is given by E(B) = −b∗(1)(0) and the expected vacation time is given by E(V ) = −v ∗(1) (0), where f ∗(i) (η) is the i-th (i ≥ 1) derivative of f ∗ (x) at x = η. Hence the traffic intensity of the queueing system is given by ρ = λE(B). Any arbitrary cycle time for this queueing system may be considered as a busy period - denote the RV with L services followed by the vacation period of the server. Let B for the whole cycle time duration. Hence, the expected length of the cycle time is - = LE(B) + E(V ). Let us denote - then the stability given by E(B) ρ = λE(B), criterion for this gated-limited queueing system will be ρ - < L, which is equivalent λE(V ) to L(1−ρ) < 1.

3 Analysis of Queue-Length Distribution at Post-Vacation-Termination and Post-Service-Completion Epochs In this section, BMAP /G/1 − SV queueing system under G-limited service discipline has been analyzed at post-vacation-termination and post-service-completion epochs. The states of the system at an arbitrary time epoch t has been presented by the following random variables:

BMAP /R/1 − SV Queueing System Under G-Limited Service Discipline

ξ(t)

107

⎧ ⎪ 0, the server is dormant, ⎪ ⎪ ⎪ ⎨1, the server is on vacation, ' ⎪ (r, l), the server is serving the r-th customer during a busy period ⎪ ⎪ ⎪ ⎩ started with l customers in the queue, 1 ≤ r ≤ L, l ≥ r.

N(t)

'

number of customers present in the queue.

J (t)

'

state of the underlying Markov chain.

. B(t)

'

remaining service time of the customer in service.

.(t) V

'

remaining vacation time of the server.

Let us observe the system at post-service-completion or post-vacation-termination epochs which are taken as embedded points. Let ti (i = 0, 1, 2, · · · ) be the time epoch at which either a service is completed or a vacation terminates. The state of the system at the time epoch ti (i = 0, 1, 2, · · · ) is defined as {N + (ti ), ξ + (ti ), J + (ti )}. Further, just after a service-completion epoch or a vacation-termination epoch, we define the following steady-state probabilities: (r,l)+ πn,j = lim P {N + (ti ) = n, ξ + (ti ) = (r, l), J + (ti ) = j }, i→∞

1 ≤ r ≤ L, l ≥ r, n ≥ l − r, 1 ≤ j ≤ m, + ωn,j = lim P {N + (ti ) = n, ξ + (ti ) = 1, J + (ti ) = j }, i→∞

1 ≤ j ≤ m, n ≥ 0.

Let us define the row vectors π (r,l)+ and ω+ n n , whose j -th (1 ≤ j ≤ m) component (r,l)+ + is πn,j and ωn,j , respectively. We also define an m × m order An (V n ) matrix whose (i, j )-th entry represents the joint probability that n (≥ 0) customers arrive during a service (vacation) period and at the end of the service (vacation) period, the phase of the underlying Markov chain is j which was i at the beginning of the service (vacation) period. Now, from the definition of P (n, t), matrices An and V n can be obtained as ∞ An =

∞ P (n, t) dB(t)

0

and

Vn =

P (n, t) dV (t). 0

Let us denote h+ n,j as the joint probability that there are n (≥ 0) customers in the system just after the end of a busy period ( which also includes service-completion instants) with state of the arrival process being j (1 ≤ j ≤ m). Further, we may + denote h+ n as a row vector whose j -th (1 ≤ j ≤ m) component is given by hn,j .

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Hence, from the definition of a gated-limited service discipline, we have h+ 0,j =

L 

(l,l)+ π0,j ⇒ h+ 0 =

l=1

h+ n,j =

L 

L 

π (l,l)+ , 0

(1a)

l=1 (l,l)+

πn,j

+

l=1

L+n 

πn(L,l)+ ⇒ h+ n =

l=L+1

L 

π (l,l)+ + n

l=1

L+n 

π (L,l)+ , n ≥ 1. n

l=L+1

(1b) Considering the two consecutive embedded Markov points, it may be shown that (r,l)+ the probability vectors ω+ (1 ≤ r ≤ L, l ≥ r, n ≥ l − r, 1 ≤ n (n ≥ 0) and π n j ≤ m) satisfy the following set of equations: ω+ n =

n 

h+ k V n−k ,

n ≥ 0,

(2a)

k=0 + π (1,l)+ = (ω+ n 0 D l + ωl )An−l+1 , n+1 

π (r,l)+ = n

(r−1,l)+

πk

l ≥ 1, n ≥ l − 1,

An−k+1 ,

(2b)

2 ≤ r ≤ L, l ≥ r, n ≥ l − r,

(2c)

k=l−r+1

where D l is the phase transition matrix of the underlying Markov process during the idle time which terminates with the arrival of a batch of size l and can be computed (r)+ as D l = (−D 0 )−1 D l . Let us denote π n as a row vector whose j -th (1 ≤ j ≤ m) entry gives the joint probability that there are n customers in the system immediately after the r-th (1 ≤ r ≤ L) a service completion during a busy period and the state of the underlying Markov chain is j . Then the probability vector at r-th post-servicecompletion epoch can be given as π (r)+ n

=

n+r 

π (r,l)+ , n

1 ≤ r ≤ L.

(3)

l=r

For |z| ≤ 1, we define the vector-generating functions (V GF s) H + (z) = ∞ + n ∞ (r)+ (z) = ∞ π (r)+ zn , Π + (z) = + n hn z , W + (z) = n n=0 ωn z , Π n=0  n=0  (r)+ n ∞ L ∞ ∞ n , and V (z) = n π z , A(z) = A z V n=0 r=1 n n=0 n n=0 n z . Hence, using the properties of P (n, t), it can be shown that ∞ A(z) =

∞ P (z, t) dB(t)

0

and

V (z) =

P (z, t) dV (t). 0

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109

Lemma 1 The V GF of the probability vectors at r-th (1 ≤ r ≤ L) post-servicecompletion epoch can be given as Π

(r)+



1 (z) = r z

r−1    −1 ω+ (−D ) D l zl D(z) − 0 0 l=0

 r−1   r   + l + ωl z A(z) . + W (z) −

(4)

l=0

Proof Using the expression of the probability vectors at r-th (1 ≤ r ≤ L) postservice-completion epoch and the definition of the V GF , one may compute Π (r)+ (z) =

∞ 

π (r)+ zn = n

n=0

∞  n+r 

π (r,l)+ zn . n

(5)

n=0 l=r

Hence, for r = 1, Eq. (5) can be rewritten as Π (1)+ (z) =

∞  n+1 

π (1,l)+ zn = n

n=0 l=1

=

1 z

∞ 

∞  n+1  + n (ω+ 0 D l + ωl )An−l+1 z n=0 l=1

l ω+ 0 Dl z +

l=1

∞ 

l ω+ l z

∞  

l=1

An−l+1 zn−l+1

n=l−1

    1 + ω0 (−D 0 )−1 D(z) − D 0 + W + (z) − ω+ = A(z). 0 z Similarly, for r = 2, Eq. (5) can be rewritten as Π

(2)+

(z) =

n+2 ∞  

π (2,l)+ zn n

n=0 l=2

=

n+2  n+1 ∞  

=

n+2  n+1 ∞  

π (1,l)+ An−k+1 zn k

n=0 l=2 k=l−1 + n (ω+ 0 D l + ωl )Ak−l+1 An−k+1 z

n=0 l=2 k=l−1

=

∞ ∞ ∞   1  + + l k−l+1 (ω D + ω )z A z An−k+1 zn−k+1 l k−l+1 l 0 z2 l=2

k=l−1



=

   1 + −1 l ω (−D ) D z D(z) − 0 l 0 z2 1



l=0

1   2   l ω+ z . A(z) + W + (z) − l l=0

n=k−1

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Again, for r = 3, Eq. (5) can be rewritten as n+3 ∞  

Π (3)+ (z) =

π (3,l)+ zn = n

n=0 l=3

π (2,l)+ An−k+1 zn k

n=0 l=3 k=l−2

∞  n+3  n+1  k+1 

=

n+3  n+1 ∞  

(1,l)+

πj

Ak−j +1 An−k+1 zn

n=0 l=3 k=l−2 j =l−1 ∞  n+3  n+1  k+1 

=

+ n (ω+ 0 D l + ωl )Aj −l+1 Ak−j +1 An−k+1 z

n=0 l=3 k=l−2 j =l−1 ∞ ∞  1  + + l (ω D + ω )z Aj −l+1 zj −l+1 0 l l z3

=

j =l−1

l=3

∞ 

×

Ak−j +1 zk−j +1

k=j −1

∞ 

An−k+1 zn−k+1

n=k−1



   1 + −1 l D(z) − (−D ) D z ω 0 l 0 z3

=

2

l=0

 2   3   + l + A(z) . + W (z) − ωl z l=0

For 4 ≤ r ≤ L, proceeding similarly as above, i.e., using Eq. (2) successively in Eq. (5), one may obtain the following result: Π

(r)+

1 (z) = r z

 −1 ω+ 0 (−D 0 )

 r−1 r−1       + l l + D(z) − D l z + W (z) − ωl z

 r × A(z) , 1 ≤ r ≤ L.

l=0

l=0

(6)



Lemma 2 The V GF of the probability vectors at post-vacation-termination epoch can be given as W + (z) = H + (z)V (z). Proof The result follows directly from Eq. (2a) and the definition of the V GF .

(7)



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Lemma 3 The V GF of the busy period completion epoch probability vectors may be expressed as H + (z) =

L−1  

+ ω+ 0 D l + ωl

 l

+ Π (L)+ (z). A(z)

(8)

l=1

Proof Using Eq. (1b) and the definition of the V GF , we have H + (z) =

∞ 

+ n h+ n z = h0 +

n=0

=

L 

π (l,l)+ 0

+

∞  L 

∞ 

* L 

n=1

l=1

π (l,l)+ zn + n

n=0 l=1

=

=

π (l,l)+ n

L+n 

+

+ π (L,l)+ n

zn

l=L+1

∞ L+n  

π (L,l)+ zn n

n=1 l=L+1

∞ L−1  

π (l,l)+ zn + n

∞ 

π (L,L)+ zn + L

n=0 l=1

n=0

∞ L−1  

∞ L+n  

π (l,l)+ zn + n

n=0 l=1

=

n h+ nz

n=1

l=1

=

∞ 

∞ L+n  

π (L,l)+ zn n

n=1 l=L+1

π (L,l)+ zn n

n=0 l=L

L−1 ∞ 

π (l,l)+ zn + n

l=1 n=0

∞ 

π (L)+ zn . n

n=0

Now, we may compute ∞ 

π (1,1)+ zn n

∞  + + + n = (ω+ 0 D 1 + ω1 )An z = (ω0 D 1 + ω1 )A(z).

n=0

n=0

Similarly, one may compute ∞ 

π (2,2)+ zn = n

n=0

∞  n+1 

π (1,2)+ An−k+1 zn k

n=0 k=1

=

∞  n+1  + n (ω+ 0 D 2 + ω2 )Ak−1 An−k+1 z n=0 k=1

(9)

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S. Ghosh et al.

+ = (ω+ 0 D 2 + ω2 )

∞ 

Ak−1 zk−1

k=1

 2 + = (ω+ D + ω ) A(z) . 2 0 2

∞ 

An−k+1 zn−k+1

n=0

Hence, proceeding similarly, for l ≥ 1, one may express ∞ 

 l + π (l,l)+ zn = (ω+ n 0 D l + ωl ) A(z) .

(10)

n=0



Thereafter, Eq. (8) directly follows from Eqs. (9) and (10). Theorem 1 The V GF of

ω+ n ’s

(n ≥ 0) satisfy the following equation:

L−1

    L L −1 l (−D ) D(z) − D z W (z) z I m − (A(z)) V (z) = ω+ 0 l 0 +

l=0

−

L−1 

 l (A(z))L ω+ z l

l=0

+

L−1 

ω+ 0 Dl

+

ω+ l



A(z) z V (z), l

L

l=1

(11) where I m is the identity matrix of order m. Proof The Theorem directly follows from Lemmas 1–3.



ω+ n s’

It may be noted from Theorem 1, that the V GF of (n ≥ 0) is completely dependent on L unknown vectors ω+ s’ (0 ≤ n ≤ L − 1) which consists of mL n + unknown joint probabilities, namely, ωn,j as 1 ≤ j ≤ m. Now, we may define p + n as a row vector of order m whose j -th (1 ≤ j ≤ m) element is the joint probability that there are n (≥ 0) customers in the queue and + phase of the arrival process ∞ is +j at departure epoch of a customer. Since p n is + proportional to π n and n=0 pn e = 1, we may compute p+ n =

π+ n . L ∞   (r)+ πn e n=0 r=1

(12)

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113

4 Analysis of Queue-Length Distribution at Arbitrary and Pre-arrival Epochs (r,l) Let πn,j (x, t) denote the joint pdf of having n (≥ 0) number of customers in the queue at time t with state of the underlying Markov process is j (1 ≤ j ≤ m) when the server is serving the r-th customer in a busy period which was started with l waiting customers in the queue and the remaining service time is x. Similarly, we denote ωn,j (x, t) as the joint pdf of having n (≥ 0) number of customers in the queue at time t with state of the underlying Markov process is j (1 ≤ j ≤ m) when the server is on vacation and the remaining vacation time is x. Hence, using the random variables, we may write (r,l) . ≤ x + Δx, ξ(t) = (r, l)}, πn,j (x, t)Δx = P {N(t) = n, J (t) = j, x ≤ B(t)

1 ≤ r ≤ L, l ≥ r, n ≥ l − r, 1 ≤ j ≤ m, x ≥ 0, .(t) ≤ x + Δx, ξ (t) = 1}, ωn,j (x, t)Δx = P {N(t) = n, J (t) = j, x ≤ V 1 ≤ j ≤ m, x ≥ 0. (r,l)

Hence, in steady-state, the above pdfs’ can be rewritten as πn,j (x) and ωn,j (x). Moreover, we denote ν0,j (t) as the probability that the server is dormant at the time epoch t and the phase of the underlying Markov process is j (1 ≤ j ≤ m), i.e., ν0,j (t) = P {N(t) = 0, J (t) = j, ξ (t) = 0},

1 ≤ j ≤ m.

Hence, in steady-state, the above probability can be rewritten as ν0,j . Further, we define the row vectors π (r,l) n (x), ωn (x), and ν 0 whose j -th (1 ≤ j ≤ m) components (r,l) are πn,j (x), ωn,j (x), and ν0,j , respectively. Further, in steady-state, we have ν 0 = ω0 (0)[−D 0 ]−1 .

(13)

Now, relating the state of the system at two consecutive time epochs t and (t + Δt) and using probabilistic arguments, we may obtain a set of partial differential equations for each phase j (1 ≤ j ≤ m). Thus, in steady-state, we have the following set of equations: −

d (1,l) (1,l) (x) = π l−1 (x)D 0 + ωl (0)b(x) + ν 0 D l b(x), π dx l−1

−

n  d (1,l) (1,l) (x) = π k (x)D n−k , π dx n k=l−1

l ≥ 1, n ≥ l,

l ≥ 1,

(14a) (14b)

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−

n  d (r,l) (r−1,l) π (r,l) π n (x) = k (x)D n−k + π n+1 (0)b(x), dx k=l−r

2 ≤ r ≤ L, l ≥ r, n ≥ l − r,

(14c)

 (l,l) d π 0 (0)v(x), ω0 (x) = ω0 (x)D 0 + dx L

−

(14d)

l=1

−

n L L+n    d ωk (x)D n−k + π (l,l) π (L,l) (0)v(x), n ≥ 1, ωn (x) = n (0)v(x) + n dx k=0

l=1

l=L+1

(14e) where π (r,l) n (0) and ωn (0) are the respective service completion rate of customers and vacation termination rate of the server. Now, for R(s) ≥ 0, we define the LST of π (r,l) n (x) and ωn (x) as  π (r,l)∗ (s) = n

s 0

e−sx π (r,l) n (x) dx

and ω∗n (s) =



∞

e−sx ωn (x) dx,

0

so that  = π (r,l)∗ (0) = π (r,l) n n

∞ 0

π (r,l) n (x) dx

and ωn = ω∗n (0) =



∞

ωn (x) dx. 0

(r,l)

We may define π n as a row vector whose j -th (1 ≤ j ≤ m) component represents the joint probability of having n customers in the queue when the state of the arrival process is j and the server is serving the r-th customer in a busy period which was started with l waiting customers in the queue. Whereas ωn may be defined as a row vector whose j -th (1 ≤ j ≤ m) component represents the joint probability that there are n customers in the queue when the arrival process is in state j and the server is on (r,l)∗ (r,l) vacation. Let the V GF of π r,l (s), ω∗n (s), π n , and ωn have been n (0), ωn (0), π n ∗ ∗ denoted by Π(z, 0), W (z, 0), Π (z, s), W (z, s), Π(z), and W (z), respectively. From the definition of ωn and π n , one may observe that W (z) = W ∗ (z, 0) and Π(z) = Π ∗ (z, 0). Now, multiplying equations (14a)–(14e) by e−sx (R(s) ≥ 0) and then integrating w.r.t.x over 0 to ∞, we get π l−1 (0) − sπ l−1 (s) = π l−1 (s)D 0 + ωl (0)b∗ (s) + ν 0 D l b∗ (s), (1,l)

(1,l)∗

(1,l)∗

l ≥ 1, (15a)

(1,l)∗ (s) = π (1,l) n (0) − sπ n

n  k=l−1

π (1,l)∗ (s)D n−k , n

l ≥ 1, n ≥ l,

(15b)

BMAP /R/1 − SV Queueing System Under G-Limited Service Discipline

(r,l)∗ π (r,l) (s) = n (0) − sπ n

n 

(r−1,l)

π (r,l)∗ (s)D n−k + π n+1 n

115

(0)b∗(s),

k=l−r

2 ≤ r ≤ L, l ≥ r, n ≥ l − r, ω0 (0) − sω∗0 (s) = ω∗0 (s)D 0 +

L 

(15c)

∗ π (l,l) 0 (0)v (s),

(15d)

l=1

ωn (0) − sω ∗n (s) =

n 

ω∗k (s)D n−k +

L 

k=0

+

∗ π (l,l) n (0)v (s)

l=1

L+n 

π (L,l) (0)v ∗ (s), n

n ≥ 1.

(15e)

l=L+1

Hence, setting s = 0 in Eq. (15a)–(15e) and noting that b ∗ (0) = 1 and v ∗ (0) = 1, we may obtain (1,l)

(1,l)

π l−1 (0) = π l−1 D 0 + ωl (0) + ν 0 D l , π n(1,l) (0) =

n 

l ≥ 1,

(16a)

l ≥ 1, n ≥ l,

π n(1,l)D n−k ,

(16b)

k=l−1

π (r,l) n (0) =

n 

(r−1,l) π (r,l) n D n−k + π n+1 (0),

2 ≤ r ≤ L, l ≥ r, n ≥ l − r,

k=l−r

(16c) ω0 (0) = ω0 D 0 +

L 

π (l,l) 0 ,

(16d)

l=1

ωn (0) =

n 

ωk D n−k +

k=0

L 

(l,l)

π0

l=1

+

L+n 

π (L,l) (0), n

n ≥ 1.

(16e)

l=L+1

Now, we may define pn as a row vector whose j -th (1 ≤ j ≤ m) entry denotes the joint probability of having n (≥ 0) customer in the queue at an arbitrary time and the state of the arrival process is j . Moreover, we may denote the V GF of pn by P (z). It is clear from the context that P (z) = W (z) + Π(z) + ν 0 . Lemma 4 The V GF W (z) satisfy the following equation: W (z, 0) − W (z)D(z) =

L  ∞  l=1 n=0

n π (l,l) n (0)z

+

∞ 

∞ 

l=L+1 n=l−L

π (L,l) (0)zn . n

(17)

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S. Ghosh et al.

Proof Multiplying equation (16e) by zn and summing over n (≥1) along with Eq. (16d) we have ∞ 

ωn (0)zn = ω 0 D 0 +

∞  n 

n=0

ωk D n−k zn +

n=1 k=0

+

∞ L+n  

∞ L+n  

(l,l)

π 0 (0) +

l=1

π (L,l) (0)zn = n

n=1 l=L+1

+

L 

∞  n 

∞  L 

n π (l,l) n (0)z

n=1 l=1 ∞  L 

ωk D n−k zn +

n=0 k=0

n π (l,l) n (0)z

n=0 l=1

π (L,l) (0)zn . n

(18)

n=1 l=L+1

Using the definition of W (z, 0) and changing the order of summation in the first term of the right-hand side of Eq. (18), we can get W (z, 0) =

∞ 

ω k zk

k=0

∞ 

L  ∞ 

D n−k zn−k +

n=k

n π (l,l) n (0)z +

l=1 n=0

∞ 

∞ 

π (L,l) (0)zn . n

l=L+1 n=l−L

(19) Hence using the definition of W (z) and D(z), Eq. (17) directly follows from Eq. (19).

Lemma 5 The (V GF ) Π(z) may be computed as Π(z) =

   



−1 1 Π(z, 0) z − 1 − W (z) − ν 0 D(z) D(z) . z

(20)

Proof Multiplying equation (15a) by zl−1 and Eqs. (15b)–(15c) by zn and then summing them over n (≥0), r (≤L) and l (≥r), one can get ∞  ∞ L  

* n π (r,l) n (0)z

=s

r=1 l=r n=l−r

∞  ∞ L  

+ π (r,l)∗ (s)zn n

r=1 l=r n=l−r

+

L  ∞  ∞ n  

(r,l)∗

πk

(s)D n−k zn

r=1 l=r n=l−r k=l−r

+

∞ ∞ L   

n−1 ∗ π (r,l) b (s) n (0)z

r=1 l=r+1 n=l−r

+

∞  (ωl (0) + ν 0 D l )zl−1 b∗ (s). l=1

(21)

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117

Now, changing the order of summation and using the definitions of Π(z, 0) and Π ∗ (z, s), we can get ∗

Π(z, 0) = sΠ (z, s) +

* L ∞ ∞  

+ π (r,l)∗ (s)zk k

r=1 l=r k=l−r

+

∞ 

D n−k zn−k

n=k

(ωl+1 (0) + ν 0 D l+1 )zl b∗ (s) +

* L ∞ ∞  

π (r,l) n (0)

r=1 l=r n=l−r

l=0

−

∞ 

L  ∞ 

∞ 

π (r,r) n (0) +

r=1 n=0

∞ 

π (L,l) (0) n

+ 

zn−1 b∗ (s).

(22)

l=L+1 n=l−L

Hence using Lemma 4 and the definition of Π ∗ (z, s), D(z), and W (z, 0), we have Π(z, 0) = sΠ ∗ (z, s) + Π ∗ (z, s)D(z) +

1 W (z, 0) − ω0 (0) z

∞ L ∞   1    (r,l) + ν 0 D(z) − D 0 b∗ (s) + π n (0)zn b∗ (s) z r=1 l=r n=l−r

−

1 z

W (z, 0) − W (z)D(z) b ∗ (s).

(23)

Thereafter, setting s = 0 in Eq. (23), some simple algebraic calculation along with Eq. (13) leads to Eq. (20).

Theorem 2 The relation among the V GF s at post-vacation (W + (z)), post-service (Π + (z)), and arbitrary epoch probabilities (P (z)) is given by P (z) =

  1 1 (z − 1)σ Π + (z) + W + (z) − Υ + (z) [D(z)]−1 + (z + 1)ν 0 , z z

(24)

where σ =

∞  ∞ L  

π (r,l) n (0)e +

r=1 l=r n=l−r

Υ + (z) =

L  ∞  l=1 n=0

π (l,l)+ zn + n

∞ 

ωn (0)e,

n=0 ∞ 

∞ 

π (L,l)+ zn . n

l=L+1 n=l−L

Proof Using Lemma 5 and the relation among P (z), Π(z), and W (z), we have P (z) =

1 1 (z − 1)[Π(z, 0) + W (z)D(z)][D(z)]−1 + (z + 1)ν 0 . z z

(25)

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One may note that π n (0) = σ π n and ωn (0) = σ ω + n , where σ is L ∞  (r,l)+ ∞ a  proportional constant. Now, using the fact that e r=1 l=r n=l−r π n ∞ + + n=o ωn e = 1, we have (r,l)

(r,l)+

(r,l)

π n (0) , 1 ≤ r ≤ L, l ≥ r, n ≥ l − r, σ ωn (0) ω+ , n ≥ 0. n = σ

π (r,l)+ = n

(26a) (26b)

Hence, using the definitions of Π + (z), W + (z), Π(z, 0), and W (z, 0), from Lemma 4, we have * W (z)D(z) = σ W + (z) −

∞ L  

π (l,l)+ zn − n

l=1 n=0

Now, denoting Υ + (z) = may express

L ∞ l=1

n=0

∞ 

∞ 

+ π (L,l)+ zn . n

(27)

l=L+1 n=l−L (l,l)+ n z

πn

+

∞

∞

(L,l)+ n z , n=l−L π n

l=L+1

  W (z)D(z) = σ W + (z) − Υ + (z) .

we

(28)

Hence Eqs. (25) and (28) yield the relation among P (z), Π + (z), and W + (z), i.e., Eq. (24).

Now, we define p − n as a row vector whose j -th (1 ≤ j ≤ m) component is given − by pn,j which represents the joint probability that a batch arrival finds n (≥ 0) customers in the queue and the arrival process is in state j . Then p− n is given by p− n =

pn

∞

k=1 D k

λb

,

n ≥ 0.

(29)

Corollary 1 The mean number of entrances to the vacation state per unit of time is equal to the mean number of departure from the vacation state per unit of time. L  ∞  l=1 n=0

π (l,l) n (0)e +

∞ 

∞ 

l=L+1 n=l−L

π (L,l) (0)e = n

∞ 

ωn (0)e.

(30)

n=1

Proof Let us set z = 1 in Eq. (17) and then multiply these equations by e. Hence, the result directly follows from the fact that W (1)D(1)e = 0.



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Corollary 2 The traffic intensity (ρ) of this queueing system satisfies the following equations: * E(B)

L  ∞  ∞ 

+

r=1 l=r n=l−r

*

E(V )

=

π (r,l) n (0)e

∞ 

π (r,l) n e = ρ,

(31a)

r=1 l=r n=l−r

+ ωn (0)e =

n=1

L  ∞  ∞ 

∞ 

ωn e = 1 − ρ − ν 0 e.

(31b)

n=0

Proof Let us differentiate Eqs. (15a)–(15c) with respect to s and then multiply these equations by e after setting s = 0. Now adding these multiplied equations and using Corollary 1 along with the facts that De = 0 and ν 0 [−D 0 ] = ω0 (0), one may formulate Eq. (31a). Similarly, from Eqs. (15d)–(15e), one may obtain (31b).  ∞ ∞ (r,l) Here one may note that L r=1 l=r n=l−r π n (0)e denotes the mean number of service completion in unit of time and multiplying this by E(B) will give ρ. On the  other hand, ∞ n=0 ωn (0)e represents the mean number of vacation termination per unit of time and multiplying this by E(V ) will give (1 − ρ − ν 0 e).

Remark 1 Using Eqs. (26a) in Corollary 2, one may compute σ = E(B)

 L r=1

ρ ∞ ∞ l=r

n=l−r

(r,l)+

πn

.

(32)

e

5 Performance Measures Various performance measures of a queueing system are often required for studying the behavior of the queueing system in detail. This section deals with the determination of different performance measures related to the model. Mean Queue-Length As the state probabilities at post-departure, arbitrary, and prearrival epochs are known, the corresponding mean queue-lengths can be easily obtained. For example, the mean queue-length can be obtained from the steady-state queue-length distribution at arbitrary epochs. The average number of customers in the queue can be given as Lq =

∞ 

npn e.

(33)

n=1

Similarly, the mean number the server is busy and on vacation  of customers when ∞ can be given by Lqb = ∞ n=1 nπ n e and Lqv = n=1 nω n e, respectively.

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Mean Waiting Time Applying Little’s law, the mean waiting time of an arbitrary customer in the queue in steady-state may be computed as Wq =

Lq . λ

(34)

Expected Length of a Busy and Idle Period Let the mean length of a busy period, i.e., the average time duration the server is busy to serve customers in a busy period is denoted by θb . Further, we define θi as the mean idle period, i.e., the average time period the server is on vacation or the server is dormant before starting a new busy period. From the definition of the carried load ρ (the fraction of time that the server is busy), one may have θb ρ . = θi 1−ρ

(35)

The mean number of service completion per unit of time is given by L ∞ ∞ (r,l) the mean number of busy period completion r=1 l=r n=l−r π n (0)e   and  ∞ ∞ (l,l) (L,l) ∞ L π (0)e + π (0)e . per unit of time is given by n n n=0 l=1 n=1 l=L+1 Hence, the mean number of customers served in a busy period can be calculated as L ∞ ∞ (r,l) r=1 l=r n=l−r π n (0)e . Therefore, the mean length of ∞ L   (l,l) (L,l) ∞ ∞ (0)e n=0 l=1 π n (0)e + n=1 l=L+1 π n a busy period is given by   (r,l)+ L ∞ ∞ E(B) e r=1 l=r n=l−r π n . (36) θ b = ∞ L  ∞ (l,l)+ (L,l)+ e+ ∞ e n=0 l=1 π n n=1 l=L+1 π n Now, using Eqs. (36) and (35), we may obtain   (r,l) + L ∞ ∞ $ % π e E(B) n r=1 l=r n=l−r 1−ρ θi = . · ∞  (l,l) + (L,l) + ∞ L ρ π e+ ∞ π e n=0

l=1

n

n=1

l=L+1

(37)

n

Hence, using the probabilities obtained in Sect. 3, one may compute θb and θi from Eqs. (36) and (37), respectively.

6 Computation Procedure of Queue-Length Distribution at Different Epochs In this section we have demonstrated how to compute queue-length distribution at different epochs, such as post-vacation-termination, post-service-completion, postdeparture, arbitrary epochs, and pre-arrival epochs.

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In order to compute all the probability vectors ω+ n (n ≥ 0), it is needed to calculate all the unknown joint probabilities involved in Eq. (11). Singh et al. [19] have used roots method for determination of the probability vectors at the post-departure epochs of MAP /R (a,b) /1 queueing system. For computation of the unknown joint probabilities involved in Eq. (11), we have used a similar kind of technique as described in [19]. We have considered service and vacation distributions having rational LST because distributions with rational LST cover a wide range of distributions that are applicable in queueing theory, see Botta et al. [4]. This implies that the LST of those distributions can be expressed in the form P (s) R(s) = Q(s) , where deg(P (s)) ≤ deg(Q(s)). For example, considering matrixexponential service-time distribution, the derivation of A(z) has been given by Singh et al. [19]. Following similar arguments for phase-type (P H -type) service distribution with representation (β, T ), where β is a row vector and T is a square matrix of same order, say ζ , one can derive A(z) = (I ζ ⊗ β)(−(D(z) ⊕ T ))−1 (I ζ ⊗ T 0 ),

(38)

where ⊗ and ⊕ represent the Kronecker product and Kronecker sum, respectively, and T 0 = −T eζ . Similar derivation can be done for V (z). Assuming service and vacation time distributions possess rational LST, let, dA(z) and dV (z) are the denominators of A(z) and V (z), respectively. Hence, each entry of the matrix [zL I − (A(z))L V (z)] is also a rational function with the denominator [dA(z)]LdV (z). Let the (i, j )-th (1 ≤ i, j ≤ m) entry of A(z) ai,j (z) vi,j (z) and V (z) is given by and , respectively. If the (i, k)-th entry of the dA(z) dV (z) (l) (z) ai,k matrix (A(z))l (1 ≤ l ≤ L) is denoted by , then the (i, j )-th entry of (dA(z))l [zL I − (A(z))L V (z)] is given by

 L

zL I − A(z) V (z)

i,j

=

ri,j (z) , [dA(z)]LdV (z)

where

ri,j (z) =

⎧ m  (L) ⎪ L L ⎪ ai,k (z)vk,i (z), i = j, ⎨z [dA(z)] dV (z) − m  ⎪ (L) ⎪ ai,k (z)vk,j (z), ⎩−

k=1

i = j.

(39)

k=1

Since W + (z) is a row vector order (1 × m), the j -th (1 ≤ j ≤ m) entry of ∞ + n can be given as Wj+ (z) = n=0 ωn,j z . Expanding both sides of Eq. (11) and then comparing component wise we get m equations in terms of m unknowns, viz. Wj+ (z). After canceling [dA(z)]LdV (z) from both sides of the expanded equation,

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a simplified system of linear equations may be written as W + (z)R(z) = M(z),

(40)

where R(z) is an m × m matrix whose (i, k)-th entry is given by rk,i (z) and M(z) is a 1 × m row vector whose j -th element is given by Mj (z) =

m 

zL

k=1

+

m 

L−1 m   l=1

  (l) + [ω+ a D ] + ω (z) [dA(z)]L−l l 0 l,i i,k 1,i

i=1

−1 [ω+ 0 (−D 0 ) ]1,i

L−1    l [D(z)]i,k − z [D l ]i,k

i=1

−

l=0

L−1  l=0

zl

m 

+ (L) ωl,i ai,k (z)

 vk,j (z).

i=1

Above system of Eqs. (40) can be solved for Wj+ (z) using Cramer’s rule and the solution can be given by Wj+ (z) =

|R j (z)| |R(z)|

1 ≤ j ≤ m,

(41)

where R j (z) is a square matrix whose (i, k)-th entries is given by (R j (z))i,k =

/ rk,i (z) Mi (z)

k = j. k = j, 1 ≤ i, j, k ≤ m.

For uniqueness of Wj+ (z), |R(z)| is considered to be a non-zero polynomial in z, i.e., |R(z)|= 0. Under steady-state conditions, it can be shown that |zL I − |R(z)| = 0 has exactly mL roots in |z|≤ 1 (including (A(z))L V (z)| = ([dA(z)]LdV (z))m multiplicity) which also includes the root at z = 1, see Gail et al. [8, Lemma 1]. The characteristic equation associated with queue-length distribution is defined as |zL I − (A(z))L V (z)|= 0, i.e., |R(z)|= 0. Let the roots of the characteristic equation whose value is less than 1 are denoted by z1 , z2 , · · · , zmL−1 and the root 1 is denoted by zmL . These roots can be used to determine the unknown joint probabilities of Eq. (11). Since Wj+ (z) (given by Eq. (41)) is convergent for |z|≤ 1, then zn (n = 1, 2, 3, · · · , mL) must be the zeros of the numerator of Eq. (41) and hence we can determine the unknown vectors by considering any one component of W + (z), say Wκ+ (z) (1 ≤ κ ≤ m). This implies that |R κ (zi )|= 0 1 ≤ i ≤ mL.

(42)

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The above system of homogeneous equation gives mL equations in terms of mL + unknowns, namely, ωn,j (0 ≤ n ≤ L, 1 ≤ j ≤ m), which leads to a trivial solution. Hence, a non-homogeneous system is needed to find a non of equation + (1) = 1 and |R(1)|= 0, from trivial solution. Using W + (1)e = 1, i.e., m W κ κ=1 Eq. (41), we have m 

|R "κ (1)|= |R " (1)|.

(43)

κ=1

Equations (42) for (1 ≤ i ≤ mL − 1) and (43) gives a non-homogeneous system + of equation. Solving them we get unknown joint probabilities ωn,j (0 ≤ n ≤ L − + 1, 1 ≤ j ≤ m) and probability vectors ωn (0 ≤ n ≤ L − 1) associated with Eq. (11). It is assumed that z1 , z2 , · · · , zmL−1 and zmL are distinct. If some roots of the characteristic equation, inside and on |z|= 1 are repeated, the above procedure needs some modifications to get the unknown joint probabilities, see Singh et al. [19]. Now after computing the probability vectors ω+ n (0 ≤ n ≤ L − 1), the rest probability vectors at post-vacation-termination epochs can be obtained from the V GF of ω+ n ’s, i.e., Eq. (11). After computing the queue-length probability vectors at post-vacationtermination epoch, the probability vectors at post-service-completion epoch can be obtained from Eqs. (2b) and (2c). The queue-length distribution at post-departure epoch can be computed from Eq. (12). Now, using the queue-length probability vectors at post-vacation-termination and post-service-completion epochs, the queue-length probability vectors at server’s vacation period can be obtained from Eq. (28). Thereafter, the queue-length probability vectors at server’s busy period can be obtained from Eq. (20) with the help of the previously determined queuelength probability vectors along with Eq. (26). Finally, the queue-length distribution at an arbitrary epoch can be computed from Eq. (24). At last, the queue-length distribution at pre-arrival epoch can be computed from Eq. (29).

7 Numerical Results and Discussion Based on the analytical results, a few numerical illustrations have been presented in this section. We have observed that if the parameters satisfy the stability criteria of the queueing system, then the numerical results are satisfactory. Considering phase-type (P H -type) service time and vacation time distribution, we have computed the queue-length distribution at various epochs. Although, the numerical computations were carried out with high precision, due to lack of place the results have been presented here up to 6 decimal places. For this queueing system, we have considered the service limit L as 3. The BMAP has been considered to have 3 phases, while the P H -distributions for the service and

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the vacation times have been considered to have 2 phases. The 3-state BMAP representation has been taken as ⎡

⎤ −0.95 0.08 0.05 D 0 = ⎣ 0.03 −0.87 0.04 ⎦ , 0.02 0.03 −0.73 ⎡ ⎤ 0.03 0.10 0.18 D 3 = ⎣ 0.07 0.11 0.05 ⎦ , 0.03 0.05 0.13 ⎡ ⎤ 0.0 0.0 0.0 D k = ⎣ 0.0 0.0 0.0 ⎦ , for k ≥ 4. 0.0 0.0 0.0

⎡

⎤ 0.02 0.15 0.07 D 1 = ⎣ 0.20 0.04 0.19 ⎦ , 0.06 0.17 0.01 ⎡ ⎤ 0.16 0.05 0.06 D 4 = ⎣ 0.08 0.09 0.07 ⎦ , 0.04 0.12 0.07

and

For this representation of BMAP , one may compute, π = [0.264842, 0.372648, 0.362510], λ = 1.982630, and λb = 0.761796. The P H -type representation of the service-time distribution has been taken as

−18.0 18.0 β 1 = [0.5, 0.5], T1 = . 0.0 −12.0 Hence the expected service time and the traffic intensity can be computed as E(B) = 0.111111 and ρ = λE(B) = 0.220292, respectively. The P H -type representation of the vacation time has been taken as

−20.0 20.0 . T2 = 0.5 −25.0

β 2 = [0.3, 0.7],

For this vacation time distribution the expected vacation time can be computed as E(V ) = 0.055. In Tables 1 and 2, we have presented the queue-length distribution at post-service-completion and post-vacation-termination epochs, respectively. Table 3 represents the distribution of the number of customers in the queue when the server is in vacation (ωn ). Whereas Table 4 describes the queue-length distribution at an arbitrary epoch.  From Table 4, it may be observed that ∞ π , which validates the n=0 p n ' correctness of the obtained numerical values. Further, it may be mentioned here that the value of σ can be computed from the Remark 1. Therefore, one may −1 compute ν 0 = σ ω+ 0 [−D 0 ] . Thus, for the given arrival, service time, and vacation time distributions, the ν 0 is computed as [0.186044, 0.271003, 0.270698], i.e., the probability that the server will remain ∞dormant is 0.727745. Moreover, Eq. (31b) has been verified, i.e., the value n=0 ωn e has been matched with the value (1 − ρ − ν 0 e). This may be also considered as a valid check of the numerical computations.

BMAP /R/1 − SV Queueing System Under G-Limited Service Discipline Table 1 Queue-length distribution at busy post-service-completion epochs

Table 2 Queue-length distribution at post-vacation-termination epochs

(3,3)+

(3,3)+

(3,3)+

125 (3,3)+

n

πn,1

πn,2

πn,3

πn

0 1 2 3 4 5 6 7 8 9 10 .. . 

0.010709 0.001793 0.000101 0.000790 0.001480 0.000192 0.000048 0.000108 0.000095 0.000013 0.000006 .. .

0.020170 0.002186 0.000101 0.001457 0.001940 0.000186 0.000079 0.000169 0.000096 0.000014 0.000010 .. .

0.027227 0.000933 0.000067 0.002060 0.001426 0.000129 0.000104 0.000168 0.000077 0.000012 0.000011 .. .

0.058106 0.004911 0.000269 0.004308 0.004846 0.000507 0.000231 0.000445 0.000268 0.000039 0.000027 .. .

n 0 1 2 3 4 5 6 7 8 9 10 .. . 

e

0.015351 0.026424 0.032228 0.074003

+ ωn,1

+ ωn,2

+ ωn,3

ω+ ne

0.055748 0.023214 0.004468 0.002893 0.005314 0.003059 0.000879 0.000522 0.000490 0.000283 0.000110 .. .

0.072681 0.026851 0.005142 0.004845 0.006787 0.003267 0.001031 0.000742 0.000621 0.000314 0.000131 .. .

0.060622 0.020785 0.003649 0.005853 0.006470 0.002763 0.000924 0.000749 0.000588 0.000278 0.000121 .. .

0.189052 0.070850 0.013258 0.013591 0.018571 0.009089 0.002834 0.002013 0.001699 0.000875 0.000362 .. .

0.097138 0.122609 0.102989 0.322737

126 Table 3 Queue-length distribution when the server on vacation

Table 4 Queue-length distribution at an arbitrary epoch

S. Ghosh et al.

n 0 1 2 3 4 5 6 7 8 9 10 .. . 

ωn,1 0.008986 0.003738 0.000719 0.000464 0.000852 0.000491 0.000141 0.000084 0.000078 0.000045 0.000018 .. .

n 0 1 2 3 4 5 6 7 8 9 10 .. . 

pn,1 0.214281 0.016514 0.013986 0.010658 0.003824 0.002436 0.001427 0.000779 0.000410 0.000240 0.000129 .. .

ωn,1 0.011714 0.004323 0.000827 0.000777 0.001089 0.000524 0.000165 0.000119 0.000099 0.000050 0.000021 .. .

ωn,1 0.009769 0.003347 0.000587 0.000939 0.001038 0.000444 0.000148 0.000120 0.000094 0.000045 0.000019 .. .

ωn e 0.030470 0.011408 0.002133 0.002180 0.002979 0.001459 0.000455 0.000322 0.000272 0.000140 0.000058 .. .

0.015642 0.019741 0.016580 0.051963

pn,1 0.307641 0.021810 0.018967 0.012586 0.004826 0.003004 0.001743 0.000928 0.000508 0.000289 0.000155 .. .

pn,1 0.301053 0.020889 0.018967 0.010718 0.004535 0.002814 0.001611 0.000857 0.000476 0.000268 0.000144 .. .

pn e 0.822974 0.059213 0.051921 0.033962 0.013185 0.008253 0.004782 0.002564 0.001393 0.000797 0.000428 .. .

0.264842 0.372648 0.362510 1.000000

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127

8 Conclusion and Future Scope This paper deals with detailed analysis of the infinite-buffer BMAP /R/1 − SV queueing system under G-limited service discipline. The analysis is based on the determination of the roots of the characteristic equation obtained at postvacation-completion and post-service-termination epochs. Using those roots, the queue-length distribution at post-service-completion and arbitrary epochs has been determined. Moreover, the detailed computational procedure for determining the queue-length probability vectors at different epochs has been described. Some performance measures of the system have been also discussed. In future, the BMAP /R/1 queueing system with G-limited service under adaptive vacations may be investigated. Further, one may be interested in the BMAP /R/1 queueing system under probabilistic-limited (P -limited) service discipline with the server’s vacation. Acknowledgments The authors are sincerely thankful to the anonymous referees for their valuable suggestions and constructive comments towards the significant restructure of the paper. The third author is partially supported by NSERC under the research grant number RGPIN-201406604.

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13. Kasahara, S., Takine, T., Takahashi, Y., Hasegawa, T.: MAP /G/1 queues under N-policy with and without vacations. J. Oper. Res. Soc. Jpn. 39(2), 188–212 (1996) 14. Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat. Stoch. Model. 7(1), 1–46 (1991) 15. Lucantoni, D.M., Meier-Hellstern, K.S., Neuts, M.F.: A single-server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Probab. 22(3), 676–705 (1990) 16. Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16(4), 764–779 (1979) 17. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The John Hopkins University Press, Baltimore (1981) 18. Saffer, Z., Telek, M.: Unified analysis of BMAP /G/1 cyclic polling models. Queueing Syst. 64(1), 69–102 (2010) 19. Singh, G., Gupta, U.C., Chaudhry, M.L.: Computational analysis of bulk service queue with Markovian arrival process: MAP /R (a,b)/1 queue. OPSEARCH 50(4), 582–603 (2013) 20. Singh, G., Gupta, U.C., Chaudhry, M.L.: Detailed computational analysis of queueing-time distributions of the BMAP /G/1 queue using roots. J. Appl. Probab. 53(4), 1078–1097 (2016) 21. Takagi, H.: Queueing Analysis Volume 1: Vacation and Priority Systems. Elsevier Science Pub. Co., Amsterdam (1991) 22. Tian, N., Zhang, Z.G.: Vacation Queueing Models: Theory and Applications, vol. 93. Springer, Berlin (2006) 23. Vishnevsky, V.M., Dudin, A.N., Semenova, O.V., Klimenok, V.I.: Performance analysis of the BMAP /G/1 queue with gated servicing and adaptive vacations. Perform. Eval. 68(5), 446– 462 (2011)

A Production Inventory System with Renewal and Retrial Demands G. Arivarignan, M. Keerthana, and B. Sivakumar

Abstract This paper presents a continuous review inventory system with make-tostock production facility. We assume that the arrival time points of demands form a renewal process and each demand requires only single item. The replenishment of stock is done by producing items one at a time. The production process is started when the inventory level drops to or below a prefixed inventory level, denoted by s(> 0), and is terminated when the maximum inventory level, namely S(> s), is reached. The inter-production time is assumed to be exponential. The customer, whose demand cannot be met during stock out period, enters an orbit of infinite size and from the orbit he sends signal to the inventory system to get his demand satisfied. The inter-retrial time between two successive retrials is assumed to follow exponential distribution. With suitable modeling process, we derived the joint probability distribution of number of customers in the orbit, status of the machine (producing or not) and the inventory level, using matrix geometric method. Keywords Renewal primary demand · Make-to-stock · Continuous review inventory system · Infinite orbit · Matrix geometric method

1 Introduction The literature on continuous review inventory systems mostly considered the replenishment of stock by ordering items (in lots) from a supplier or manufacturer. The most frequently used policies are “order up to,” equivalently (S − 1, S) policy (placing an order at every occurrence of a demand) or (r, Q) policy (placing an order for Q items when the inventory level drops at r). The performance analysis

G. Arivarignan () Department of Statistics, Manonmaniam Sundaranar University, Tirunelveli, India M. Keerthana · B. Sivakumar School of Mathematics, Madurai Kamaraj University, Madurai, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_9

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of these models have been carried out by [2, 3, 8, 12] and [1]. However, in certain cases the production of items in lieu of placing orders for them is preferred. This is necessitated to protect the trade mark rights of technology developed in-house, or to catch up with the changing trend or specifications in the manufacturing process, or to reduce the cost of stocking these items. Moreover, the production could be of two types: “make-to-order” and “make-to-stock”. In the former, the production is started whenever the demand arrives and in the latter, the production is started when the inventory is at a low level (already fixed) and run until the stock is accumulated to a desirable level. A detailed discussion on stochastic models for manufacturing system is given in [3]. He et al. [5, 6] considered a make-to-order production inventory system with arrivals of demands forming a Poisson process, production time having exponential distribution and zero lead time, and that the production can be initiated at any required time. He et al. [7] extended this work by having phase type distribution for the production time. They derived optimal replenishment policies by not only using the inventory level, but also the information on the number of outstanding orders. Krishnamoorthy and Narayanan [10] considered an inventory system with make-tostock production and assumed Markovian arrival process for demands and Markov production process for the production times. They derived the joint probability distribution of inventory level and the queue size and computed various measures of system performance in steady state. Kim [9] considered a two-station tandem production system with make-to-order production in one station and make-to-stock in another. The make-to-order facility processes the customer order with the option to accept or reject. They addressed the problem of coordinating the decisions by presenting a Markov decision model. In this paper, we considered an inventory system with one production machine for augmenting items to the stock. The inter-demand occurrence time points are assumed to have arbitrary distribution and customers are allowed to join an orbit in case of nonavailability of items in the inventory and are permitted to retry for their demands. A matrix geometric solution to the problem of finding limiting distribution of—number of customers in the system, status of production machine, and the inventory level—is provided.

2 Model Formulation Consider a continuous review system, in which the customers arrive to an inventory system demanding unit item. To augment items to the stock, the items are produced within the system. Production process starts only when the inventory level reaches a level (or below) a prefixed safety stock and is produced one-at-a time until the inventory level reaches the maximum of inventory. The customers who arrive during stock out periods are allowed to join an orbit and from there, he / she can retry to get their demands satisfied.

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The various random processes considered for the above model are listed below: • The inter-arrival times of customers are independent and identically distributed with arbitrary distribution F (·) , density function f (·), and mean m. Each demand requires only a single item. • The items are produced one at a time with random production time. • Production process is started when the inventory level drops to (or below) a prefixed quantity s and the production is terminated when the maximum inventory level S is reached (0 ≤ s < S). For Mathematical tractability, we assume that the production is switched on only at the time of demand points only. • The inter-production time is assumed to have exponential distribution with parameter α. • During the stock out period, the arriving customers are allowed to enter into an orbit of infinite size. Retrial by customers from the orbit are entertained. • The inter-retrial time is assumed to have exponential distribution with parameter θ . We assume that the orbiting customers follow first come first serve discipline.

2.1 Embedded MRP and Its Analysis Let X(t) denote the number of customers in the orbit, Y (t) denote the production status (1 for on and 0 for off), and L(t) denote the inventory level at time t > 0. The state space of these processes are, respectively, {0, 1, 2, . . .}, {0, 1}, and {0, 1, 2, . . . , S}. We consider the joint process (t) = (X(t), Y (t), L(t)) whose state space  is given by  = {(x, y, l) : x = 0, 1, 2, . . . ; if y = 0, then l = s + 1, s + 2, . . . , S − 1, S and x = 0, 1, 2, . . . ; if y = 1, then l = 0, 1, . . . , S − 1, S }. Let 0 = T0 < T1 < T2 < . . . be sequence of time points at which demand arrives to the system. Define Xn = X(Tn +), Yn = Y (Tn +), and Ln = L(Tn +). The state space " of the discrete time process {Xn , Yn , Ln } is given by " = {(x, y, l) : x = 0, 1, 2, . . . ; if y = 0, then l = s + 1, s + 2, . . . , S − 1 and x = 0, 1, 2, . . . ; if y = 1, then l = 0, 1, . . . , S − 1 .} This representation will help us to represent the state spaces in a block partitioned form and in turn we can write the associated matrices in block partitioned form.

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It can be shown that the discrete time stochastic process {Xn , Yn , Ln , Tn ; n = 0, 1, 2, . . .} with state space " × R + is a Markov renewal process satisfying the property, Pr{Xn+1 , Yn+1 , Ln+1 , Tn+1 − Tn ≤ t|X0 , · · · , Xn , Y0 , · · · , Yn , L0 , · · · , Ln , T0 , · · · , Tn } = Pr{Xn+1 , Yn+1 , Ln+1 , Tn+1 − Tn ≤ t|Xn , Yn , Ln } for all n = 0, 1, 2, . . . and t ∈ R + and (x, y, l), (x " , y " , l " ) ∈ " . Since this process is a time homogeneous one, we can write it as a function of duration between two successive demand time points, viz., Pr{Xn+1 = x " , Yn+1 = y " , Ln+1 = l " , Tn+1 − Tn ≤ t|Xn = x, Yn = y, Ln = l} = K(x,y,l)((x " , y " , l " ), t)

(1)

and the quantity in right-hand side of Eq. (1) is called semi-Markov kernel over " . It may be noted that the function Pr[(x, y, l), (x " , y " , l " )] = lim K(x,y,l)((x " , y " , l " ), t) t →∞

is the one-step transition probability function of the Markov chain {(Xn , Yn , Ln ), n = 0, 1, 2, . . .}. We introduce a set of blocks in the collection of states of the state space " as indicated below: ˆ 1, ˆ 2, ˆ . . .) " = (0, xˆ = ( x, 0, x, 1),

x = 0, 1, 2, . . .

x, 0 = ( (x, 0, s + 1), (x, 0, s + 2), . . . , (x, 0, S − 1)) x, 1 = ( (x, 1, 0), (x, 1, 1), . . . , (x, 1, S − 1)). We use the notation [D]ij as the (i, j )− th entry of a matrix D. The derivative of K(x,y,l)((x " , y " , l " ), t) is denoted by κ(x,y,l)(x " , y " , l " , t). We define the following sub-matrices: [B((x,y),(x ",y " )) (t)]ll " = κ(x,y,l)(x " , y " , l " , t) [A(x,x ") (t)]yy " = B((x,y),(x ",y " )) (t) [κ(t)]xx " = A(x,x " ) (t). It can be easily seen that the sub-matrices A(x,x ") (t) are zero matrices for x " > x + 1 and that these matrices do not depend on the individual values of x and x " , but on the difference namely, x − x " for x " ≤ x. For brevity, we write

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Ar (t) = A(x,x ") (t), where r = x − x " + 1 for x " ≤ x + 1 and x > 0. Thus we have ˆ ⎛ 0 0ˆ Cˆ 1 1ˆ ⎜ ⎜ Cˆ 2 ⎜ ˆ 2 κ(t) = ⎜ Cˆ 3 3ˆ ⎜ ⎝ Cˆ 4 .. .. . .

1ˆ Aˆ 00 Aˆ 1 Aˆ 2 Aˆ 3 .. .

2ˆ 0 Aˆ 0 Aˆ 1 Aˆ 2 .. .

3ˆ 0 0 Aˆ 0 Aˆ 1 .. .

4ˆ 0 0 0 Aˆ 0 .. .

... ⎞ ... ... ⎟ ⎟ ... ⎟ ⎟. ... ⎟ ⎠ .. .

The sub-matrices are defined below. 0 A˜ 00 = 1

0 Aˆ 0 = 1

$

$

0 1 % 0 0 0 Aˆ 11 00

0 0 0

1 % 0 . Aˆ 11 0

For k = 1, 2, . . . 0 $ ˆ 00 0 Ak Aˆ k = 1 Aˆ 10 k

1 % ˆ A01 k Aˆ 11

0 $ ˆ 00 Ck Cˆ 10

1 % Cˆ k01 . Cˆ k11

k

for k = 1, 2, . . .

0 Cˆ k = 1

k

We need the following notations for use in sequel: f (t) : pdf of interval time between two successive demands F (t) : distribution function associated with f (t) F¯ (t) : 1 − F (t) −αt )j pj (t) : e j(αt probability for production of j items in time t ! j  p¯ j (t) : 1 − pk (t) qj (t) :

k=0 e−θ t (θt )j j! j 

q¯j (t) : 1 −

k=0

probability for retrial of j customers in time t qk (t).

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With the above notation, we can write the following:



Aˆ 11 00

Aˆ 11 0



ij

= 

ij

=

f (t)p0 (t) i = 0, j = 0, 0 otherwise

f (t)p0 (t)q0 (t) i = 0, j = 0, 0 otherwise

for k = 1, 2, . . . Q − 1

Aˆ 10 k



ij

=

f (t)p¯j −i+k−1 (t)qk−1 (t) i = 0, 1, . . . , S − 1, j = S − k, . . . , S − 1, 0 otherwise

for k = Q, Q + 1, . . .

Aˆ 10 k



ij

=

f (t)p¯j −i+k−1 (t)qk−1 (t) i = 0, 1, . . . , S − 1, j = s + 1, . . . , S − 1, 0 otherwise

for k = 1, 2, . . . , S − 1

Aˆ 11 k

ij

⎧ ⎨ f (t)pj −i+k (t)qk−1 (t) i = k + 1, . . . , S − 1, j = i − k, . . . , S − 2, = f (t)pj −i+k (t)qk−1 (t) i = 0, 1, . . . , k, j = 0, . . . , S − 2, ⎩ 0 otherwise

for k = S, S + 1, . . .

Aˆ 11 k



ij

=

f (t)pj −i+k (t)qk−1 (t) i = 0, 1, . . . , S − 1, j = 0, . . . , S − 2, 0 otherwise

for k = 1, . . . , Q − 2

Aˆ 00 k



ij

=

f (t)qk−1 (t) i = s + k + 1, . . . , S − 1, j = i − k, 0 otherwise

for k = Q − 1, Q − 2 . . . , for k = 1, . . . , s

Aˆ 01 k



ij

=

Aˆ 00 k are zero matrices.

f (t)qk−1 (t) i = s + 1, . . . , s + k, j = i − k, 0 otherwise

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for k = s + 1, . . . , Q − 2

Aˆ 01 k

ij

⎧ ⎨ f (t)qk−1 (t) i = k + 1, . . . , s + k, j = i − k, = f (t)q¯k−2 (t) i = k j = i − k, ⎩ 0 otherwise

for k = Q − 1, . . . , S − 1

Aˆ 01 k

ij

⎧ ⎨ f (t)qk−1 (t) i = k + 1, . . . , S − 1, j = i − k, = f (t)q¯k−2 (t) i = k, j = i − k, ⎩ 0 otherwise Aˆ 01 k are zero matrices. For k = 1,

for k = S, S + 1, . . . ,

Cˆ k00



ij





Cˆ k10





ij

Cˆ k11

=

ij

=

Cˆ k01

f (t) i = s + 2, . . . , S − 1, j = i − 1, 0 otherwise 

ij

=

f (t) i = s + 1, j = i − 1, 0 otherwise

f (t)p¯j −i+k−1 (t) i = 0, 1, . . . , S − 1, j = S − k, . . . , S − 1, 0 otherwise

⎧ ⎨ f (t)pj −i+k (t) i = 1, . . . , S − 1, j = i − k, . . . , S − 2, = f (t)pj −i+k (t) i = 0, j = 0, . . . , S − 2, ⎩ 0 otherwise

for k = 2, 3, . . . Q − 1

Cˆ k10



ij

=

f (t)p¯j −i+k−1 (t)q¯k−2 (t) i = 0, 1, . . . , S − 1, j = S − k, . . . , S − 1, 0 otherwise

for k = Q, Q + 1, . . . 

f (t)p¯j −i+k−1 (t)q¯k−2 (t) i = 0, 1, . . . , S − 1, j = s + 1, . . . , S − 1, 10 ˆ Ck = ij 0 otherwise

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for k = 2, 3, . . . , S − 1

Cˆ k11

⎧ ⎨ f (t)pj −i+k (t)q¯k−2 (t) i = k, . . . , S − 1, j = i − k, . . . , S − 2, = f (t)pj −i+k (t)q¯k−2 (t) i = 0, 1, . . . , k − 1, j = 0, . . . , S − 2, ⎩ 0 otherwise

ij

for k = S, S + 1, . . . ,

Cˆ k11



ij

=

f (t)pj −i+k (t)q¯k−2 (t) i = 0, 1, . . . , S − 1, j = 0, . . . , S − 2, 0 otherwise

for k = 2, 3, . . . , Q − 2

Cˆ k00



ij

=

f (t)q¯k−2 (t) i = s + k + 1, . . . , S − 1, j = i − k, 0 otherwise

for k = Q − 1, . . . , for k = 1, 2, . . . , s

Cˆ k01



ij

=

Ck00 are zero matrices f (t)q¯k−2 (t) i = s + 1, . . . , s + k, j = i − k, 0 otherwise

for k = s + 1, . . . , Q − 2 

f (t)q¯k−2 (t) i = k, . . . , s + k, j = i − k, Cˆ k01 = ij 0 otherwise for k = Q − 1, . . . , S − 1

Cˆ k01



ij

for k = S, S + 1, . . . ,

=

f (t)q¯k−2 (t) i = k, . . . , S − 1, j = i − k, 0 otherwise

Cˆ k01 are zero matrices.

2.2 Steady State Analysis The transition probability matrix (tpm) P of the Markov chain {(Xn , Yn , Ln ), n = 0, 1, 2, . . .} is obtained from the Markov renewal kernel κ(t) by 

∞

P =

κ(t)dt, 0

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where the integration on right-hand side is performed element wise of the matrix. The matrix P can be written in block partitioned form ˆ ⎛ 0 ˆ0 C1 1ˆ ⎜ ⎜ C2 ⎜ P = 2ˆ ⎜ C3 ˆ3 ⎜ ⎝ C4 .. .. . .

1ˆ ˜ A0 A1 A2 A3 .. .

2ˆ 0 A0 A1 A2 .. .

3ˆ 0 0 A0 A1 .. .

4ˆ 0 0 0 A0 .. .

... ⎞ ... ... ⎟ ⎟ ... ⎟ ⎟, ... ⎟ ⎠ .. .

where the sub-matrices Ci and Ai ’s are respective integral values of Cˆ i and Aˆ i ’s. It may be that we have C1 e+ A˜ 0 e = e, Cn+1 +(An +. . .+A0 ) = e, n ≥ 1. noted ∞ Let A = i=0 Ai . Then A is a stochastic matrix. Following [11], we make the following statement. Proposition Since the Markov chain with tpm P is irreducible (and hence all states are positive recurrent) the matrix A is stochastic. The Markov chain is ergodic if and only if, 1 t|(X0 , Y0 , L0 ) = (x, y, l)] + Pr[(X(t), Y (t), L(t)) = (x " , y " , l), T1 ≤ t|(X0 , Y0 , L0 ) = (x, y, l)] = "(x,y,l) ((x " , y " , l " ), t) Pr[(X(t), Y (t), L(t)) = (x " , y " , l " ), t < T1 ≤ t + h|(X0 , Y0 , L0 ) = (x, y, l)] h↓0 h  t  = "(x,y,l) ((x " , y " , l " ), t) + lim + lim

h↓0 (x "" ,y "" ,l "" )∈" 0

×

Pr[(X1 , Y1 , L1 ) = (x "" , y "" , l "" ), w < T1 ≤ w + h|(X0 , Y0 , L0 ) = (x, y, l)] h

× Pr[(X(t − w), Y (t − w), L(t − w)) = (x " , y " , l)" |(X1 , Y1 , L1 ) = (x "" , y "" , l "" )]

= "(x,y,l) ((x " , y " , l " ), t) +

 (x "" ,y "" ,l "" )∈"

"

"



t

κ(x,y,l) ((x "" , y "" , l "" ), w)

0

"

×!(x "" ,y "" ,l "" ) ((x , y , l ), t − w)dw

where "(x,y,l)((x " , y " , l " ), t) = Pr[(X(t), Y (t), L(t)) = (x " , y " , l " ), T1 > t|(X0 , Y0 , L0 ) = (x, y, l)] and κ(x,y,l)((x " , y " , l " ), t) is the derivative of Markov renewal kernel. Hence we get the Markov renewal equation, !(x,y,l) ((x " , y " , l " ), t) = "(x,y,l) ((x " , y " , l " ), t) +

 (x,y,l)∈"

× !(u,v,k) ((x " , y " , l " ), t − w)dw.



∞

κ(x,y,l) ((u, v, k), w) 0

(7)

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We also have ˜ (x,y,l)(x " , y " , l, t), "(x,y,l)((x " , y " , l " ), t) = F¯ (t)" where ˜ (x,y,l)(x " , y " , l, t) = Pr[(X(t), Y (t), L(t)) = (x " , y " , l " ), |T1 > t, (X0 , Y0 , L0 ) " = (x, y, l)]. ˜ (x,y,l)(x " , y " , l, t) is obtained as Using r = x − x " , the function " ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p(l " −l) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p¯(l " −l) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p¯(x " −x−l−(S−l " )) (t)qr (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p¯(l " −l+r) (t)q(r) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p¯(l " −l+r) (t)q¯(r) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p¯(l " −l+r) (t)q(r) (t),

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p¯(l " −l+r) (t)q¯(r) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p(l " −l+r) (t)q¯(r) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (t)q(r) (t), p " ⎪ ⎪ (l −l+r) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p(l " −l+r) (t)q(r " ) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ q(r) (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0,

x " = x,

y " = y, l " = l,

x = 0,

y = 0, l = s + 1, . . . , S,

x"

y " = y, l " = l, . . . , S − 1,

= x,

x = 0,

y = 1, l = 0, 1, . . . , S − 1

x " = x,

y " = 0, l " = S,

x = 0,

y = 1, l = 0, . . . , S − 1,

x"

y " = 0, l " = 1, . . . , S,

= 1, . . . , x − S,

x > S,

y = 1, l = 1, . . . , S − 1,

x " = x − S + 1, . . . , x,

y " = 0, l " = S − r, . . . , S,

x > S,

y = 1, l = 1, . . . , S − 1,

x"

= 0,

y " = 0, l " = S − r, . . . , S,

x > S,

y = 1, l = 1, . . . , S − 1,

x " = 1, . . . , x,

y " = 0, l " = S − r, . . . , S,

x < S,

y = 1, l = 1, . . . , S − 1,

x"

y " = 0, l " = S − r, . . . , S,

= 0,

x < S,

y = 1, l = 1, . . . , S − 1,

x " = 0,

y " = 1, l " = l − x, . . . , S − 1,

x = 1, . . . , l − 1,

y = 1, l = 1, . . . , S − 1,

x"

y " = 1, l " = l − r, . . . , S − 1,

= 1, . . . , x,

x = 1, . . . , l − 1,

y = 1, l = 1, . . . , S − 1,

x " = 1, . . . , x;

y " = 1; l " = Max{1, l − r}, . . . , S − 1,

x = l, l + 1 . . . ,

y = 1, l = 1, . . . , S − 1,

x"

= x − l + s + 2, . . . , x, y " = 0, l " = l − r,

x = 1, . . . , l

y = 0, l = s + 2, . . . , S,

otherwise.

We shall use the following partition on  : ˜ 1, ˜ 2, ˜ . . .)  = (0, ˜i = ( )i, 0*, )i, 1*),

i = 0, 1, 2, . . .

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)i, 0* = ( (i, 0, s + 1), (i, 0, s + 2), . . . , (i, 0, S)) )i, 1* = ( (i, 1, 0), (i, 1, 1), . . . , (i, 1, S)) ˜ (x,y,l)(x " , y " , l, t) in a block partitioned matrix and collect the functions " 0˜ 0ˆ E1 (t) 1ˆ ⎜ ⎜ E2 (t) ⎜ ˆ ˜ 2 "(t) = ⎜ E3 (t) 3ˆ ⎜ ⎝ E4 (t) .. .. . . ⎛

1˜ 0 D1 (t) D2 (t) D3 (t) .. .

2˜ 0 0 D1 (t) D2 (t) .. .

3˜ 0 0 0 D1 (t) .. .

4˜ 0 0 0 0 .. .

... ⎞ ... ... ⎟ ⎟ ... ⎟ ⎟ ... ⎟ ⎠ .. .

. The Markov renewal equation (7) can be conveniently expressed as ! = " + κ , !,

(8)

where , represents the convolution of matrices κ and ! so that the entry at ((x, y, l), (x " , y " , l " ))th position in the resultant matrix has the second term of Eq. (7). Equation (8) is a generalization of renewal equation studied in renewal theory. Define the limiting probability distribution for , !(x " , y " , l " ) = lim !(x,y,l)((x " , y " , l " ), t),

(x, y, l) ∈ " (x " , y " , l " ) ∈ .

t →∞

By using the result of [4] we get, 

1

!(x " , y " , l " ) = 

(x,y,l)∈"



∞

×

π(x,y,l)m(x, y, l)

π(x,y,l)

(x,y,l)∈"

˜ (x,y,l)(x " , y " , l, t)dt, "

0

where m(x, y, l) is the mean recurrence time of the Markov chain (Xn , Yn , Ln ) in state (x, y, l). In this problemit is given by the mean time between two successive ∞ demand points, namely, m = 0 (1 − F (t))dt. Hence we get !(x " , y " , l " ) =

1 m

 (x,y,l)∈"



∞

π(x,y,l) 0

˜ (x,y,l)(x " , y " , l, t)dt. "

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3 Conclusion We presented in this article a stochastic model on inventory system maintained with a production facility in which the inter-demand times have arbitrary distribution and the production time is distributed as exponential. The customers whose demand cannot be satisfied due to want of stock are allowed to join an orbit of infinite size. These orbiting customers can retry for their demand according to first come first serve discipline and the inter-trial times are exponentially distributed. We used an embedded Markov renewal process and made a study on it. We used the matrix geometric solution approach provided by [11] and there by a wide range of algorithmic solution procedures can be used to get the invariant vector of the underlying Markov chain. Finally we derived a renewal equation which can be used to get the limiting probability distribution of the process under consideration. Acknowledgments The work of the author “G. Arivarignan” was supported by University Grants Commission, India, research award No. F.6-6/2017-18/EMERITUS-2017-18-OBC-9414/(SA-II). The work of the author “M. Keerthana” was supported by the UGC BSR Research Fellowship, University Grants Commission, India, F.25-1/2014-15 (BSR)/5-66/2007 (BSR).

References 1. Arivarignan, G., Sivakumar, B.: Inventory systems with renewal demands at service facilities. In: Srinivasan, S.K., Vijayakumar, A. (eds.) Stochastic Point Processes, pp. 108–123. Narosa Publishing House, New Delhi (2003) 2. Axsäter, S.: Inventory Control. Springer, Berlin (2006) 3. Buzacott, J.A., Shanthikumar, J.G.: Stochastic Models for Manufacturing Systems. Prentice Hall, New Jersey (1993) 4. Cinlar, E.: Introduction to Stochastic Processes. Prentice Hall Inc., Upper Saddle River (1975) 5. He, Q., Jewkes, E.M., Buzacott, J.A.: Analysis of the value of information used in inventory control of an inventory production system. In: ABS and ACORS Conference. Dalhousie University, Halifax (1999) 6. He, Q., Jewkes, E.M., Buzacott, J.A.: Performance measures of a make to order inventoryproduction system. IIE Trans. 32, 409–419 (2000) 7. He, Q., Jewkes, E.M., Buzacott, J.A.: The system of information used in inventory control of a make to order inventory-production system. IIE Trans. 34, 999–1013 (2002) 8. Kalpakam, S., Arivarignan, G.: Semi Markov models in inventory systems. Electron. J. Math. Phys. Sci. 18(5), 1–17 (1984–1985) 9. Kim, E.: On the admission control and demand management in a two-station tandem production system. J. Ind. Manag. Optim. 7(1), 1–8 (2011) 10. Krishnamoorthy, A., Narayanan, V.C.: Production inventory with service time and vacation to the server. IMA J. Manag. Math. 22, 33–45 (2011) 11. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover Publication Inc., New York (1994) 12. Zipkin, P.H.: Foundations of Inventory Management. McGraw-Hill, New York (2000)

A Queueing System with Batch Renewal Input and Negative Arrivals U. C. Gupta, Nitin Kumar, and F. P. Barbhuiya

Abstract This paper studies an infinite buffer single server queueing model with exponentially distributed service times and negative arrivals. The ordinary (positive) customers arrive in batches of random size according to renewal arrival process, and join the queue/server for service. The negative arrivals are characterized by two independent Poisson arrival processes, a negative customer which removes the positive customer undergoing service, if any, and a disaster which makes the system empty by simultaneously removing all the positive customers present in the system. Using the supplementary variable technique and difference equation method we obtain explicit formulae for the steady-state distribution of the number of positive customers in the system at pre-arrival and arbitrary epochs. Moreover, we discuss the results of some special models with or without negative arrivals along with their stability conditions. The results obtained throughout the analysis are computationally tractable as illustrated by few numerical examples. Furthermore, we discuss the impact of the negative arrivals on the performance of the system by means of some graphical representations. Keywords Batch arrival · Difference equation · Disasters · RCH · Renewal process · Negative customers

1 Introduction Since the pioneering work of Gelenbe [16] in the year 1989, queueing model with negative arrivals (also termed as G-networks) have gained considerable attention. A negative arrival causes the removal of one or more ordinary customer (also called positive customer) from the system, and prevents it from getting served. In the literature, negative arrivals are generally introduced by the name of “negative

U. C. Gupta () · N. Kumar · F. P. Barbhuiya Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_10

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customers” and/or “disasters.” The arrival of a negative customer removes one ordinary customer from the system, according to a definite killing strategy, i.e., RCH (Removal of Customer at the Head) or RCE (Removal of Customer at the End). Under RCH killing discipline, the customer who is undergoing service gets removed, while in case of RCE, the customer at the end of the queue is eliminated. Meanwhile, the occurrence of a disaster simultaneously removes all the present customers in the system thus making the system idle. Disasters are also known by the terms catastrophic events (Barbhuiya et al. [7]), mass exodus (Chen and Renshaw [12]) or queue flushing (Towsley and Tripathi [27]). Both, a negative customer and a disaster have no impact on the system when it is empty. For further references on different queueing models with negative arrivals the readers may refer to the bibliography by Van Do [28]. Initially, M/M/1 queueing model with positive and negative customers was studied by Harrison and Pitel [18]. They derived the Laplace transforms of the sojourn time density under both RCH and RCE killing discipline. They further extended their work to M/G/1 queue with negative arrivals and obtained the generating function of the queue length probability distribution (see Harrison and Pitel [19, 20]). Jain and Sigman [21] derived a Pollaczek–Khintchine formula for an M/G/1 queue with disasters using preemptive LIFO discipline, whereas Boxma et al. [8] considered the same model by assuming the disasters to occur in deterministic equidistant times or at random times. The M/M/1 queue with negative arrivals was first extended to the GI /M/1 queue by Yang and Chae [30], assuming the occurrence of negative customers (under RCE killing discipline) and disasters. Meanwhile, Abbas and Aïssani [1] investigated the strong stability conditions of the embedded Markov chain for GI /M/1 queue with negative customers. A discrete-time GI /G/1 queue with negative arrivals was considered by Zhou [31] where he derived the probability generating function of actual service time of ordinary customers. Recently, Chakravarthy [10] investigated a single server catastrophic queueing model assuming the arrival process to be versatile Markovian point process with phase type service time. All the work discussed till now was studied under steady-state condition. Kumar and Arivudainambi [23] and Kumar and Madheswari [24] obtained the transient solution of system size for the M/M/1 and M/M/2 queueing model with catastrophes, respectively. Following this, a time dependent solution for the system size of M/M/c queue with heterogeneous servers and catastrophes was considered by Dharmaraja and Kumar [13]. A survey on queueing models with interruptions due to various reasons such as catastrophes, server breakdowns, etc. can be found in Krishnamoorthy et al. [22]. The papers referred above, studies queueing models with negative arrivals of one form or the other, under the assumption of single arrival of positive customers. But in most of the real-world scenario, the request for service arrives in groups of random size. For example, transmission of messages to the service station occurs in the form of packets in batches, unfinished goods arrives in bulk into the production systems for further processing. This gives us a practical motivation to relax the assumption of single arrival and consider batch arrival of the positive customers into the system. We study a continuous-time GI X /M/1 queue which

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is influenced by negative customers (with RCH killing discipline) and disasters, occurring independently of one another according to Poisson process. The arrival of negative customers or disasters have no impact on the system when it is empty. We first formulate the model using the supplementary variable technique and then apply difference equation method to obtain the steady-state distribution of the number of positive customers in the system at different epochs. In the literature, most of the queueing models with negative arrivals are studied using the matrix geometric (matrix analytic) method or the embedded Markov chain technique. However, encouraged by some recent works (see Barbhuiya and Gupta [5, 6], Goswami and Mund [17]), we try to implement the methodology based on supplementary variable technique and difference equation method to study queueing model with negative arrivals. The whole procedure involved is analytically tractable and easy to implement, as we obtain explicit formulae of the system-content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equation and the corresponding constants. We discuss the stability conditions along with some special cases of the model. We also present some numerical results in order to illustrate the applicability of our theoretical work and study the influence of different parameters on the system performance. The queueing model described above may have possible use in computer communications and manufacturing systems (see Artalejo [3]). A real-world application can be experienced within a network of computers, where a message affected with virus often infects the whole system when it gets transferred from one node to another. A signal which immediately removes the message and prevents further transmission of it can be thought of as a negative customer. Moreover, a reset instruction in the computer database may be considered as a disaster as it clears all the stored files present in the system. In these systems, the stored files/data act as positive customers whereas clearing operation plays the role of the negative arrivals (see Wang et al. [29], Atencia and Moreno [4]). The remaining portion of the paper is organized as follows. In Sect. 2 we give a comprehensive description of the model under consideration. In Sect. 3 we perform the steady-state analysis of the model and discuss the stability condition. We deduce the results of some special cases of our model in Sect. 4 which is followed by some illustrative numerical examples in Sect. 5. Finally, we give the concluding remarks in Sect. 6.

2 Model Description We consider an infinite buffer queueing model wherein customers (positive customers) arrive into the system in batches and joins the queue. The arriving batch size is a random variable X with probability mass function P (X = i) = gi , i = 1, 2, . . .. For theoretical analysis and numerical implementation we assume that the maximum permissible size of the arriving batch is b, which also holds true in many real-world circumstances. Consequently, the mean arriving batch size

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Disaster which eliminates all the customers from the system Customers arrive into the system in batches

Negative customer removes only the customer undergoing service Customer leave the system after service completion

Departure Queue Customer in the service

Fig. 1 Pictorial representation of the GI X /M/1 queue with negative customer and disaster

b b i is g = i=1 igi and the probability generating function is G(z) = i=1 gi z . The inter-arrival times T between the batches are independent and identically distributed continuous random variables with probability density function (pdf) a(t), distribution function A(t), the Laplace–Stieltjes transform (L.S.T) A∗ (s), and the mean inter-arrival time λ−1 = a = −A∗(1)(0), where λ is the arrival rate of the batches and A∗(1)(0) is the derivative of A∗ (s) evaluated at s = 0. The customers are served individually by a single server and the service time follows exponential distribution with parameter μ. The system is affected by negative arrivals which is characterized by two independent Poisson arrival processes namely, negative customers and disasters with rate η and δ, respectively. The negative customer follows RCH killing discipline and removes only the customer undergoing service, while the occurrence of a disaster eliminates all the customers from the system. We further assume that the negative customer or disaster have no impact on the system when it is empty. The arrival process, service process, and the negative arrivals are independent of each other. The model described above may be mathematically denoted by GI X /M/1 queue with negative customers and disasters. One may refer to Fig. 1 for a pictorial representation of the model.

3 The Steady-State Analysis In this section we analyze the model described in Sect. 2 in steady state. We first formulate the governing equations of the system using supplementary variable technique (SVT) by considering the remaining inter-arrival time of the next batch as the supplementary variable. For this purpose, we denote the states of the system

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N(t) and U (t), respectively, as the number of customers in the system and the remaining inter-arrival time of the next batch, at time t. We further define qn (u, t)du = P [N(t) = n, u < U (t) ≤ u + du], n ≥ 0, u ≥ 0, and in steady state pn (u) = lim qn (u, t). t →∞

Relating the states of the system at two consecutive epochs t and t + $t and using the arguments of SVT we obtain the following difference-differential equations in steady state: ∞

 d p0 (u) = (μ + η)p1 (u) + δ − pk (u), du

(1)

k=1

−

d pn (u) = −(μ + η + δ)pn (u) du +a(u)

min{n,b} 

gi pn−i (0) + (μ + η)pn+1 (u), n ≥ 1.

(2)

i=1

Obtaining the steady-state solution directly from (1) and (2) is a rather difficult task. Therefore, for further analysis we take the transform for which we define pn∗ (s) =



∞ 0

e−su pn (u)du ⇒ pn = pn∗ (0) =



∞

pn (u)du, n ≥ 0.

0

Multiplying (1) and (2) by e−su , integrating with respect to u over 0 to ∞ and then separating Eq. (2) we obtain the transformed equations as − sp0∗ (s) = (μ + η)p1∗ (s) + δ

∞ 

pk∗ (s) − p0 (0),

(3)

k=1

(μ + η + δ − s)pn∗ (s) = A∗ (s)

n 

∗ gi pn−i (0) + (μ + η)pn+1 (s)

i=1

−pn (0), 1 ≤ n ≤ b − 1, (μ + η + δ − s)pn∗ (s) = A∗ (s)

b 

(4)

∗ gi pn−i (0) + (μ + η)pn+1 (s)

i=1

−pn (0), n ≥ b.

(5)

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Adding (3)–(5)  for all values of n, taking limit s → 1 and using the normalizing condition ∞ n=0 pn = 1 we have ∞ 

pn (0) =

n=0

1 = λ. a

(6)

The L.H.S of Eq. (6) denotes the mean number of arriving batch into the system per unit time such that the remaining inter-arrival time is 0, which is actually the arrival rate λ. We now define pn− as the probability that the number of positive customers in the system is n just before the  arrival of a batch, i.e., at pre-arrival epoch. Since − − pn− is proportional to pn (0) and ∞ n=0 pn = 1, we have the relation between pn and pn (0) as pn (0) pn (0) , n ≥ 0. pn− = ∞ = λ p (0) k=0 k

(7)

Based on the theory of difference equations we obtain the state probabilities at prearrival (pn− ) and arbitrary (pn ) epochs in the following section.

3.1 Steady-State System-Content Distributions We define the right shift operator D on the sequence of probabilities {pn (0)} and ∗ (s) for all n. Thus, (5) can be {pn∗ (s)} as Dpn (0) = pn+1 (0) and Dpn∗ (s) = pn+1 rewritten in the form * + b  ∗ ∗ b−i b gi D − D pn−b (0), n ≥ b.(8) (δ − s + (μ + η)(1 − D)) pn (s) = A (s) i=1

Substituting s = δ + (μ + η)(1 − D) in (8), we get the following homogeneous difference equation with constant coefficient:  ∗

A (δ + (μ + η)(1 − D))

b 

 gi D

b−i

−D

b

pn (0) = 0, n ≥ 0.

(9)

i=1

The corresponding characteristic equation (c.e.) is ∗

A (δ + (μ + η)(1 − z))

b  i=1

gi zb−i − zb = 0,

(10)

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which has exactly b roots, denoted by r1 , r2 , ..., rb , inside the unit circle |z| = 1. Thus the solution of (9) is of the form pn (0) =

b 

ci rin , n ≥ 0,

(11)

i=1

where c1 , c2 , ..., cb are the corresponding b arbitrary constants independent of n. Now using (11) in (8) we have the following non-homogeneous difference equation: (δ − s

+ (μ + η)(1 − D))pn∗ (s)

=

b  j =1

* ∗

cj A (s)

b 

+ gi rj−i

− 1 rjn , n ≥ b.

i=1

(12) The general solution of (12) is of the form pn∗ (s)

/ 4 $ % b A∗ (s)G(rj−1 ) − 1 δ−s n  =B 1+ + cj r n , n ≥ b, μ+η δ − s + (μ + η)(1 − rj ) j j =1

(13) where the first term in the R.H.S of (13) is the solution corresponding to the homogeneous equation of (12) for a fixed s, such that B is an arbitrary constant. Meanwhile, the second term in the R.H.S. is a particular solution of (12). Taking ∞ ∗ limit as s → 0 and summing over n from b to ∞ in (13), we have, n=b pn (0) = n ∞ ∞ δ−s tends to infinity as s → 0. Thus to n=b pn ≤ 1. However, B n=b 1 + μ+η ensure the convergence of the solution we must have B = 0 and thus (13) reduces to / 4 b  A∗ (s)G(rj−1 ) − 1 ∗ pn (s) = r n , n ≥ b. cj (14) δ − s + (μ + η)(1 − rj ) j j =1

We now find the conditions under which pn∗ (s) satisfies (14) for 1 ≤ n ≤ b − 1 as well. Thus substituting the respective values in (4) we obtain b  j =1

* cj

b 

+ gi rjn−i

= 0, 1 ≤ n ≤ b − 1,

i=n+1

which reduces to the following on using the condition gb = 0, b  j =1

cj rjn−b = 0, 1 ≤ n ≤ b − 1.

(15)

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Summing over n from 0 to ∞ in (11) and using relation (6) we obtain λ=

b  i=1

ci . 1 − ri

(16)

One may note that (15) and (16) together constitutes a system of b equations in b unknowns which can be solved to obtain the constants cj for j = 1, 2, . . . , b. Once cj ’s are obtained, the expression of pn (0) given in (11) becomes completely known and pn∗ (s) is given by pn∗ (s)

/

b 

=

cj

j =1

4

A∗ (s)G(rj−1 ) − 1

rjn , n ≥ 1.

δ − s + (μ + η)(1 − rj )

(17)

Now, using (7) and (17), the steady-state distribution of the number of positive customers in the system at pre-arrival and arbitrary epochs are given by pn− =

1 n ci ri , n ≥ 0, λ b

i=1

pn =

pn∗ (0)

=

b  j =1

p0 = 1 −

∞  n=1

(18)

/ cj

4

G(rj−1 ) − 1 δ + (μ + η)(1 − rj )

b  cj rj pn = 1 − 1 − rj j =1

/

rjn , n ≥ 1,

G(rj−1 ) − 1 δ + (μ + η)(1 − rj )

(19) 4 (20)

.

This completes the analysis of the model under consideration. It may be noted that the results derived so far are mainly expressed in terms of the roots of the c.e. (10) lying inside the unit circle. It can be proved that δ > 0 is a sufficient condition for the c.e. to have exactly b roots inside the unit circle (see Appendix), which ensures the stability of the system. Or in other words, due to the occurrence of disasters the system becomes empty and as a result the model under consideration always remains stable. Once the probability distributions are completely known, different characteristic measures determining the performance of the system can be easily established. For example, the average population size at  pre-arrival (L− ) and arbitrary (L) epochs ∞ − − are given by L = n=1 npn and L = ∞ n=1 npn , respectively. That is,  cj rj 1  ci ri L− = , L = λ (1 − ri )2 (1 − rj )2 b

b

i=1

j =1

/

G(rj−1 ) − 1 δ + (μ + η)(1 − rj )

4 .

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4 Special Cases In this section we discuss a few special cases of the model by considering some fixed values of the parameters. As a result our model reduces to some well-known classical queueing models with or without negative arrivals. Case 1: If η = 0 and δ = 0, i.e., negative customer or disaster does not occur or their occurrence have no impact on the system, then our model reduces to the classical GI X /M/1 queue. Consequently, the steady-state distributions of the number of customers in the system at pre-arrival and arbitrary epochs can be obtained directly from (18)–(20) by putting η = 0 and δ = 0, where rj ,  j = 1, 2, . . . , b are the roots of the c.e. zb − A∗ (μ − μz) bi=1 gi zb−i = 0 lying inside the unit circle, and then the corresponding arbitrary constants cj , j = 1, 2, . . . , b can be obtained by solving the system of Eqs. (15) and (16). Here it may be noted that λg < μ is the necessary and sufficient condition for the stability of the system. This particular queueing model has been extensively studied in the literature, both analytically and numerically, based on the use of embedded Markov chain technique and roots method (see Chaudhry and Templeton [11], Briere ` and Chaudhry [9], Easton et al. [14, 15]). However, the present paper provides an alternative procedure for the solution of the model which is theoretically tractable and easy to implement, as compared to the other approaches. Meanwhile, setting η = 0, δ = 0, g1 = 1, and gi = 0 for i ≥ 2 will give the steady-state solution for GI /M/1 queue. The c.e. will have a single root inside the unit circle (say r) under the condition λ < μ, and the corresponding arbitrary constant can be obtained from (16) as c = λ(1 − r). It is followed by the systemcontent distributions which can be obtained from (18)–(20). Case 2: If δ = 0, i.e., the disaster does not play any role and the only negative X /M/1 queue arrivals are the negative customers, then the model reduces to GI b ∗ with negative customers. The c.e. z − A ((μ + η) − (μ + η)z) bi=1 gi zb−i = 0 will have exactly b roots inside the unit circle under the necessary and sufficient condition λg < μ + η. Equations (15) and (16) can be solved for the arbitrary constants following which, the steady-state distributions of the number of positive customers in the system can be obtained from (18)–(20). As discussed in Case 1, the solution for GI /M/1 queue with negative customers (Yang and Chae [30]) can be further derived by assuming g1 = 1 and gi = 0 for i ≥ 2. Case 3: If η = 0, the system does not get affected by the negative customers and our model reduces to GI X /M/1 queue with disaster. Due to the impact of disasters, the system will always remain stable and hence the c.e. will have exactly b roots inside the unit circle under the sufficient condition δ > 0. The steady-state distributions can be derived from (18)–(20) after obtaining the constants from (15) and (16). Similarly as before, the solution for GI /M/1 queue with disasters (Park et al. [25]) can also be obtained.

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5 Numerical Observation In this section we demonstrate the analytical results obtained in Sect. 3 by some numerical examples, which are represented in tabular and graphical form. The results given in the table may be beneficial for other researchers who would like to compare their results using some other methods in the near future. Table 1 displays the steady-state distribution of the number of positive customers in the system for Poisson (M) and deterministic (D) arrival processes. The parameters chosen are λ = 10, μ = 10, η = 5, δ = 2, g1 = 0.2, g3 = 0.4, g6 = 0.3, and g10 = 0.1. The last row of the table depicts the average system content at various epochs. It is important to note that the system-content distributions in the 2nd and 3rd column are same due to the Poisson arrival process, which verifies the accuracy of our analytical results. Meanwhile, for deterministic inter-arrival time distribution, the L.S.T A∗ (s) is a transcendental function, which is approximated to a rational function using P ad e´(15, 15) approximation (see Akar and Arikan [2], Singh et al. [26]). Another interesting trend can be observed in the 4th and 7th column of the table. As n becomes larger, the ratio of the system-content distribution at pre-arrival epoch converges to a particular value which is the largest real root (say rb ) of the c.e. (10) lying inside the unit circle. This suggests that the limiting distributions at the pre-arrival epoch can be approximated by the unique largest root of the c.e. as pn− = λ1 cb rbn . Figure 2 investigates the influence of η on L for different values of δ. As η increase, L decreases for any value of δ, which is intuitive. Similarly, for a fixed η, L decreases with increasing δ. However, as δ becomes too large (δ = 10), L seems to attain a constant value irrespective of the values of η. A similar behavior can be experienced on plotting L against μ for different values of δ, and consequently it is omitted. Figure 3 depicts the impact of λ on L for different δ. Clearly, as λ increases L increases for any δ. However, when λ is kept fixed along with other parameters, L decreases significantly with the increase in δ. Finally, in Figs. 4 and 5 we, respectively, illustrate the impact of δ and η on L for different inter-arrival time distributions, namely, exponential (M), Erlang (E4 ), and deterministic (D). It may be observed in Fig. 4 that for each inter-arrival time distribution, L decreases as δ increases, which is obvious. However, for a fixed δ, L is equal for all the three distributions. A possible explanation for this phenomenon may be the frequent occurrence of disasters which removes all the customers including the batch which has just arrived. The effect of inter-arrival time distribution can be best understood from Fig. 5 as η increases. For higher values of η, L decreases significantly. However, for exponential inter-arrival time distribution L is greater, and decreases for Erlang followed by deterministic distribution. It may be mentioned that in all the numerical results generated throughout this section, the values of the parameters involved are not restricted to any condition except that δ > 0, as the system with disaster is always stable.

GI = M pn− 0.20533567 0.03093521 0.02830528 0.04682481 0.02575445 0.03186531 0.04637555 0.02567353 .. .

0.00000060 0.00000057 0.00000054 0.00000051 0.00000048 0.00000045 .. .

1.00000000 15.04001756

n 0 1 2 3 4 5 6 7 .. .

200 201 202 203 204 205 .. .

Sum Mean

1.00000000 15.04001756

0.00000060 0.00000057 0.00000054 0.00000051 0.00000048 0.00000045 .. .

pn 0.20533567 0.03093521 0.02830528 0.04682481 0.02575445 0.03186531 0.04637555 0.02567353 .. . 0.94509121 0.94509121 0.94509121 0.94509121 0.94509121 0.94509121 .. .

− pn+1 /pn− 0.15065676 0.91498603 1.65427828 0.55001705 1.23727398 1.45536181 0.55360053 1.05065616 .. .

1.00000000 12.39890533

0.00000009 0.00000008 0.00000008 0.00000007 0.00000007 0.00000006 .. .

GI = D pn− 0.23080160 0.03535630 0.03982161 0.04056222 0.03488259 0.04219365 0.03922244 0.02966955 .. .

1.00000000 14.40030123

0.00000010 0.00000009 0.00000009 0.00000008 0.00000008 0.00000007 .. .

pn 0.12004016 0.03653976 0.03420904 0.06060381 0.02886916 0.04113680 0.05948070 0.03095702 .. .

0.93533903 0.93533903 0.93533903 0.93533903 0.93533903 0.93533903 .. .

− pn+1 /pn− 0.15318913 1.12629474 1.01859822 0.85997718 1.20959058 0.92958177 0.75644328 1.07090478 .. .

Table 1 Steady-state distribution of the number of positive customers in the system at various epochs for different inter-arrival time distributions

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δ=1 δ=3 δ=8

35 30

δ=2 δ=5 δ=10

25

L

20 15 10 5 0

1

2

3

4

5

4

5

η

6

7

8

9

10

6

7

8

9

10

Fig. 2 Effect of η on L for various δ

40 δ=1 δ=3 δ=8

35 30

δ=2 δ=5 δ=10

25

L

20 15 10 5 0

1

2

3

λ

Fig. 3 Effect of λ on L for various δ

60 M

50

E4

L

40

D

30 20 10 0

1

3

5

7

10

δ

Fig. 4 Effect of δ on L for various inter-arrival distributions

13

15

17

20

A Queueing System with Batch Renewal Input and Negative Arrivals

155

20 18

M

16

E4

14

D

L

12 10 8 6 4 2 0

0

5

10

15

20

25

30

35

40

45

50

μ (or η)

Fig. 5 Effect of η on L for various inter-arrival distributions

6 Concluding Remarks In this paper, the steady-state analysis of a GI X /M/1 queueing model with negative customers and disasters has been presented. We have derived the explicit closedform expressions of the distribution of the number of positive customers in the system at pre-arrival and arbitrary epochs, in terms of roots of the associated characteristic equation. The results of some classical queueing models with or without negative arrivals have been discussed along with their stability conditions. Additionally, through some numerical examples, we have investigated the influence of negative customers and disasters on the performance characteristic of the system. The methodology used in this paper is based on supplementary variable technique and difference equation method which makes the analysis easily tractable, both theoretically and computationally. The procedure developed throughout the analysis can be utilized and further extended to study some more complicated models.

Appendix  Theorem 1 The c.e. A∗ (δ + (μ + η)(1 − z)) bi=1 gi zb−i − zb = 0 have exactly b roots inside the unit circle |z| = 1 subject to the condition δ > 0. b ∗ Proof (μ + η)(1 − b Let usb−iassume f1 (z) =∗ −z and f2 (z) =bA (δ + b−i is an analytic z)) i=1 gi z = K(z). Since A (δ + (μ + η)(1 − z)) g z i i=1  i function, it can be written in the form K(z) = ∞ i=1 ki z such that ki ≥ 0 for all i.

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Consider the circle |z| = 1 −  where  > 0 and is a sufficiently small quantity. Now |f1 (z)| = |zb | = (1 − )b = 1 − b + o() |f2 (z)| = |K(z)||

b 

gi zb−i | ≤ K(|z|)

i=1 ∗

b 

gi |z|b−i = K(1 − )

i=1 ∗

= A (δ) − {2A (δ)(b − g) − (μ + η)A

b 

gi (1 − )b−i

i=1 ∗(1)

(δ)} + o()

< 1 − b + o() under the sufficient condition δ > 0. Thus from Rouche’s ´ theorem we have exactly the same number of zeroes in f1 (z) and f1 (z) + f2 (z) inside the unit circle, and hence the theorem.



References 1. Abbas, K., Aïssani, D.: Strong stability of the embedded Markov chain in an GI /M/1 queue with negative customers. Appl. Math. Model. 34(10), 2806–2812 (2010) 2. Akar, N., Arikan, E.: A numerically efficient method for the MAP /D/1/K queue via rational approximations. Queueing Syst. 22(1), 97–120 (1996) 3. Artalejo, J.R.: G-networks: a versatile approach for work removal in queueing networks. Eur. J. Oper. Res. 126(2), 233–249 (2000) 4. Atencia, I., Moreno, P.: The discrete-time Geo/Geo/1 queue with negative customers and disasters. Comput. Oper. Res. 31(9), 1537–1548 (2004) 5. Barbhuiya, F.P., Gupta, U.C.: A difference equation approach for analysing a batch service queue with the batch renewal arrival process. J. Differ. Equ. Appl. 25(2), 233–242 (2019) 6. Barbhuiya, F.P., Gupta, U.C.: Discrete-time queue with batch renewal input and random serving capacity rule: GI X /GeoY /1. Queueing Syst. 91, 347–365 (2019) 7. Barbhuiya, F.P., Kumar, N., Gupta, U.C.: Batch renewal arrival process subject to geometric catastrophes. Methodol. Comput. Appl. Probab. 21(1), 69–83 (2019) 8. Boxma, O.J., Perry, D., Stadje, W.: Clearing models for M/G/1 queues. Queueing Syst. 38(3), 287–306 (2001) 9. Brière, G., Chaudhry, M.L.: Computational analysis of single-server bulk-arrival queues: GI X /M/1. Queueing Syst. 2(2), 173–185 (1987) 10. Chakravarthy, S.R.: A catastrophic queueing model with delayed action. Appl. Math. Model. 46, 631–649 (2017) 11. Chaudhry, M.L., James, G.C.: Templeton. In: First Course in Bulk Queues. Wiley, New York (1983) 12. Chen, A., Renshaw, E.: The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 34(1), 192–207 (1997) 13. Dharmaraja, S., Kumar, R.: Transient solution of a Markovian queuing model with heterogeneous servers and catastrophes. Opsearch 52(4), 810–826 (2015) 14. Easton, G., Chaudhry, M.L., Posner, M.J.M.: Some corrected results for the queue GI X /M/1. Eur. J. Oper. Res. 18(1), 131–132 (1984) 15. Easton, G., Chaudhry, M.L., Posner, M.J.M.: Some numerical results for the queuing system GI X /M/1. Eur. J. Oper. Res. 18(1), 133–135 (1984)

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16. Gelenbe, E.: Random neural networks with negative and positive signals and product form solution. Neural Comput. 1(4), 502–510 (1989) 17. Goswami, V., Mund, G.B.: Analysis of discrete-time batch service renewal input queue with multiple working vacations. Comput. Ind. Eng. 61(3), 629–636 (2011) 18. Harrison, P.G., Pitel, E.: Sojourn times in single-server queues by negative customers. J. Appl. Probab. 30(4), 943–963 (1993) 19. Harrison, P.G., Pitel, E.: M/G/1 queues with negative arrival: an iteration to solve a fredholm integral equation of the first kind. In: Proceedings of the Third International Workshop on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS’95)., pp. 423–426. IEEE, Piscataway (1995) 20. Harrison, P.G., Pitel, E.: The M/G/1 queue with negative customers. Adv. Appl. Probab. 28(2), 540–566 (1996) 21. Jain, G., Sigman, K.: A Pollaczek–Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33(4), 1191–1200 (1996) 22. Krishnamoorthy, A., Pramod, P.K., Chakravarthy, S.R.: Queues with interruptions: a survey. Top 22(1), 290–320 (2014) 23. Kumar, B.K., Arivudainambi, D.: Transient solution of an M/M/1 queue with catastrophes. Comput. Math. Appl. 40(10–11), 1233–1240 (2000) 24. Kumar, B.K., Madheswari, S.P.: Transient behaviour of the M/M/2 queue with catastrophes. Statistica 62(1), 129–136 (2002) 25. Park, H.M., Yang, W.S., Chae, K.C.: Analysis of the GI /Geo/1 queue with disasters. Stoch. Anal. Appl. 28(1), 44–53 (2009) 26. Singh, G., Gupta, U.C., Chaudhry, M.L.: Analysis of queueing-time distributions for MAP /DN /1 queue. Int. J. Comput. Math. 91, 1911–1930 (2014) 27. Towsley, D., Tripathi, S.K.: A single server priority queue with server failures and queue flushing. Oper. Res. Lett. 10(6), 353–362 (1991) 28. Van Do, T.: Bibliography on G-networks, negative customers and applications. Math. Comput. Model. 53(1–2), 205–212 (2011) 29. Wang, J., Huang, Y., Dai, Z.: A discrete-time on–off source queueing system with negative customers. Comput. Ind. Eng. 61(4), 1226–1232 (2011) 30. Yang, W.S., Chae, K.C.: A note on the GI /M/1 queue with Poisson negative arrivals. J. Appl. Probab. 38(4), 1081–1085 (2001) 31. Zhou, W.-H.: Performance analysis of discrete-time queue GI /G/1 with negative arrivals. Appl. Math. Comput. 170(2), 1349–1355 (2005)

Asymptotic Analysis Methods for Multi-Server Retrial Queueing Systems Ekaterina Fedorova

, Anatoly Nazarov

, and Alexander Moiseev

Abstract In this paper, we consider a multi-server retrial queueing system of type M/M/N. We propose the asymptotic methods for analysis of the system under long delay and heavy load conditions. Application areas of each method are defined and numerical examples are given. Keywords Retrial queues · Asymptotic analysis · Heavy load · Long delay

1 Introduction Queueing theory is widely used for solving different practical problems in real economic, technical, and social systems. There are two classes of queueing models: systems with queues and loss systems. However, in real systems, there are situations in which a queue is not identified explicitly, but also an arrival call is not lost if it comes when all service devices are unavailable. Often a primary call does not refuse to be serviced and performs repeated attempts after a random period of time. Examples of this can be found in telecommunication systems, cellular networks, and call centers [1, 21, 27, 32]. Thus, a new class of queueing system has appeared: systems with repeated calls or retrial queueing systems. The first papers regarding retrial queues were published in the middle of the twentieth century by Wilkinson, Cohen, Elldin, and Gosztony [9, 13, 18, 35]. Most were devoted to practical problems and the influence of repeated attempts on telephone traffic, communication systems, etc. A comprehensive description and detailed comparison of classical queueing systems and retrial queues was presented by Falin and Artalejo in [5, 6, 15]. Nowadays, there are many papers devoted to retrial queueing systems. Scientists from different countries have studied different types of retrial queues, developed methods of their investigation, and solved practical and theoretical problems in this E. Fedorova · A. Nazarov · A. Moiseev () Tomsk State University, Tomsk, Russian Federation © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_11

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area. However, the majority of the studies are performed by matrix methods [10, 12, 17, 19] and involve further numerical analysis or computer simulation [6–8, 22, 28, 31]. Analytical results are obtained only for the simplest models, e.g. retrial queues with a stationary Poisson arrival process and an exponential distribution of the service law (see [15]). In this paper, we use the asymptotic analysis method developed by the Tomsk scientific group for the study of different types of queueing systems and networks (e.g., in [24, 29]). The principle of the method is a derivation of asymptotic equations from systems of equations determining the model behavior and obtaining formulas for asymptotic functions under some limit condition. In the previous papers (e.g. [23, 26]), we applied the asymptotic analysis method for single-server retrial queueing systems under heavy load and long delay limit conditions. In addition, we proposed Gaussian, quasi-geometric, and gamma approximation methods for the single-server retrial queues [16, 25]. Thus, in this paper, we plan to generalize our results to a multi-server model. Asymptotic and approximate methods are also offered in [3, 11, 15, 30, 36], etc. The performance characteristics of retrial queues with Poisson arrival process under heavy and light loads and long delay conditions are studied by [2, 4, 14, 33]. In addition, the paper [34] is devoted to the “extreme” load of a retrial queue (when an intensity of primary calls tends to infinity or zero). The rest of the paper is organized as follows. In Sect. 2, the description of the mathematical model of the retrial queue M/M/N is described and the stochastic process of the system states is analyzed. In addition, we present the research directions. In Sect. 3, we determine a limit condition of a long delay and prove the theorem about the Gaussian form of the asymptotic characteristic function. In Sect. 4, the retrial queue is studied under a limit condition of a heavy load and we prove the theorem about the gamma distribution of the asymptotic characteristic function. Section 5 is devoted to the hyper-gamma approximation as an improvement of the gamma approximation. In Sect. 6, some numerical examples of the comparison of asymptotic distributions with exact ones (obtained via imitation modeling) are presented, and conclusions about each method application area are made.

2 Mathematical Model and Problem Statement Let us consider a multi-server retrial queueing system of type M/M/N. The system structure is presented in Fig. 1. The arrival process is Poisson with a rate λ. There are N servers with service times distributed exponentially with a rate μ. If a call arrives when there is a free server, this call occupies it for the service. Otherwise, the call goes to an orbit, where it stays for a random time distributed by the exponential law with a rate σ . After the delay the call attempts to obtain the service. If there is a free server, the call occupies it, otherwise the call instantly returns to the orbit.

Asymptotic Analysis Methods for Multi-Server Retrial Queueing Systems

161

σ

Fig. 1 Retrial queueing system M/M/N

σ σ

O

μ μ N

μ

Let i(t) be the random process of the number of calls in the orbit and n(t) be the random process which defines the servers block states as follows: ⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎨ 1, n(t) = 2, ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎩ N,

if all servers are free at the moment t, if 1 server is busy at the moment t, if 2 servers are busy at the moment t, if all servers are busy at the moment t.

The aim of the research is to find the probability distribution of the number of calls in the orbit. The process i(t) is not Markovian, therefore we consider the two-dimensional continuous-time Markov chain {n(t), i(t)}. Denote the stationary probability distribution of the system states {n(t), i(t)} by Pn (i) = P {n(t) = n, i(t) = i}, where n = 0 . . . N, i = 0 . . . ∞. The considered process is Markovian, thus, the following system of Kolmogorov equations for Pn (i) can be written ⎧ −(λ + iσ )P0 (i) + μP1 (i) = 0, ⎪ ⎪ ⎨ −(λ + iσ + nμ)Pn (i) + λPn−1 (i) + (i + 1)σ Pn−1 (i + 1) ⎪ +(n + 1)μPn+1 (i) = 0, for n = 1, . . . , N − 1, ⎪ ⎩ (i + 1)σ PN−1 (i + 1) − (λ + Nμ)PN (i) + λPN−1 (i) + λPN (i − 1) = 0. (1)

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Let us introduce the partial characteristic functions Hn (u) =

∞ 

ej ui Pn (i),

(2)

i=0

√ where j = −1 is an imaginary unit. Substituting functions (2) into Eqs. (1), the following system of equations is obtained: ⎧ j σ H0" (u) − λH0 (u) + μH1 (u) = 0, ⎪ ⎪ ⎪ ⎨ j σ H " (u) − j σ e−j u H " (u) − (λ + nμ)Hn (u) + λHn−1 (u) n n−1 (3) +(n + 1)μHn+1 (u)= 0, for n . . , N − 1, ⎪  = 1, .  ⎪ ⎪ ⎩ −j σ e−j u H " (u) − λ 1 − ej u + Nμ H (u) + λH (u) = 0. N

N−1

N−1

Analytical solution of systems (1) and (3) are unknown in the scientific literature. Therefore, we propose: – to obtain an asymptotic solution of system (3) under a long delay limit condition (σ → 0); – to obtain an asymptotic solution of system (3) under a heavy load limit condition (ρ → 1, where ρ = λ/(Nμ)); – to make conclusions about the application area of obtained asymptotic distributions using a comparison with computer simulation results.

3 Asymptotic Solution Under Long Delay Considering the long delay limit condition (σ → 0), we formulate the following theorem. Theorem 1 The asymptotic partial characteristic function of the probability distribution of the number of calls in the orbit for the retrial queueing system of M/M/N type under the long delay condition σ → 0 has the form of a Gaussian distribution 5 κ1 (j u)2 κ2 Hn (u) = Rn exp j u + , σ 2 σ 

(4)

where *

% N $  κ1 + λ n 1 R0 = μ n!

+−1

$ ,

Rn =

n=0

κ1 = λ

RN , 1 − RN

κ2 =

κ1 + λ μ

%n

1 n!

for n = 1, . . . , N,

λRN + (λ + κ1 )φN , 1 − RN − (λ + κ1 )gN

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φN and gN are defined by the following systems of equations, respectively, ⎧ ⎪ −(κ1 + λ)g0 + μg1 = R0 , ⎪ ⎪ ⎪ ⎪ −(κ1 + λ + nμ)gn + (κ1 + λ) gn−1 ⎪ ⎪ ⎪ ⎨ +(n + 1)μgn+1 = Rn − Rn−1 , n = 1, . . . , N − 1, −NμgN + (κ1 + λ) gN−1 = −RN−1 , ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ gn = 0, ⎪ ⎩ n=0

⎧ ⎪ −(κ1 + λ)φ0 + μφ1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −(κ1 + λ + nμ)φn + (κ1 + λ) φn−1 ⎪ ⎨ +(n + 1)μφn+1 = κ1 Rn−1 , n = 1, . . . , N − 1, −NμφN + (κ1 + λ) φN−1 = −λRn + κ1 RN−1 , ⎪ ⎪ ⎪ ⎪  ⎪ N ⎪ ⎪ φn = 0. ⎪ ⎩ n=0

To prove the theorem, we obtain the first- and the second-order asymptotic functions for the solution of system (3).

3.1 First-Order Asymptotics Denoting σ = ε, we use the substitutions u = εw,

Hn (u) = Fn (w, ε)

for n = 0, . . . , N.

Then system (3) can be rewritten as follows: ⎧ ∂F (w, ε) 0 ⎪ − λF0 (w, ε) + μF1 (w, ε) = 0, ⎪ ⎪j ⎪ ∂w ⎪ ⎪ ∂F (w, ε) ∂Fn−1 (w, ε) n ⎨ − j e −j εw − (λ + nμ)Fn (w, ε) + λFn−1 (w, ε) j ∂w ∂w ⎪ ⎪ +(n + 1)μFn+1 (w, ε) = 0, n = 1, . . . , N − 1, ⎪ ⎪     ⎪ ⎪ ⎩ −j e −j εw ∂FN −1 (w, ε) − λ 1 − ej εw + N μ F (w, ε) + λF N N −1 (w, ε) = 0. ∂w

(5) Denote Fn (w) = lim Fn (w, ε). From Eqs. (5), we derive the following system: ε→0

⎧ " j F (w) − λF0 (w) + μF1 (w) = 0, ⎪ ⎪ ⎨ 0" " (w) − (λ + nμ)Fn (w) + λFn−1 (w) j Fn (w) − j Fn−1 ⎪ +(n + 1)μF n = 1, . . . , N − 1 n+1 (w) = 0, ⎪ ⎩ " (w) − NμFN (w) + λFN−1 (w) = 0. −j FN−1

(6)

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The solution Fn (w) of system (6) is written as Fn (w) = Rn exp{j wκ1 },

(7)

where Rn is the asymptotic probability of n busy servers and κ1 is the asymptotic mean of the number of calls in the orbit. Substituting expression (7) into Eqs. (6), the system for Rn is obtained ⎧ ⎨ −(κ1 + λ)R0 + μR1 = 0, (λ + κ1 )Rn−1 − (κ1 + λ + nμ)Rn + (n + 1)μRn+1 = 0, ⎩ (λ + κ1 )RN−1 − NμRN = 0.

(8)

Obviously, the solution of system (8) has the form of discrete Erlang distribution $ Rn =

κ1 + λ μ

%n

1 , n!

*

% N $  κ1 + λ n 1 R0 = μ n!

+−1 .

(9)

n=0

Let us find the parameter κ1 . We sum all equations of system (5): ∂ j ∂w

*N−1 

+ Fn (w, ε) + λej w FN (w, ε) = 0.

n=0

Suppose ε → 0 and substituting expression (7), then the following equation is obtained: −κ1

N−1 

Rn + λRN = 0.

n=0

Taking into account the normalization requirement

N 

Rn = 1, we have

n=0

−κ1 (1 − RN ) + λRN = 0. Thus, κ1 is the solution of the following equation: κ1 = λ

RN , 1 − RN

where RN is the component of solution of system (8) that depends on κ1 .

(10)

Asymptotic Analysis Methods for Multi-Server Retrial Queueing Systems

Denoting x =

165

κ1 + λ , it is easy to show that x is the solution of the following μ

equation: N  μn − λ n=0

n!

x n = 0,

which has no more than one positive root x1 . Thus, κ1 is equal to κ1 = μx1 − λ.

(11)

In this way, we obtain the probabilities Rn and the mean κ1 for expression (7). The function Fn (w) is called the first-order asymptotic function.

3.2 Second-Order Asymptotics For more detailed analysis, the second-order asymptotics is derived. First, applying results of the first-order asymptotics, we use the following substitution:  κ  1 Hn (u) = Hn(2)(u) exp j u . σ

(12)

From system (3), the following system of equations for Hn2 (u) is obtained ⎧ " ⎪ j σ H0(2) (u) − (κ1 + λ)H0(2)(u) + μH1(2)(u) = 0, ⎪ ⎪ ⎪ ⎪ (2) " −j u (2) " (2) ⎪ ⎪ n (u) − j σ e ⎪ j σ H  Hn−1 (u) − (κ1 + λ + nμ)Hn (u) ⎨ (2) (2) + κ1 e−j u + λ Hn−1 (u) + (n + 1)μHn+1 (u) = 0, n = 1, . . . , N − 1,     ⎪ ⎪ −j u (2) " ju ⎪ −j σ e H (u) − λ 1 − e + Nμ HN(2)(u) ⎪ N−1  ⎪ ⎪  ⎪ ⎪ ⎩ + κ1 e−j u + λ H (2) (u) = 0. N−1 (13) We use the notation σ = ε2 ,

u = εw,

Hn(2) (u) = Fn (w, ε)

for n = 0, . . . , N.

(14)

Note here that the notation of the parameter ε and functions Fn (w, ε) differ from those in the previous subsection. We use the same symbols for brevity.

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Substituting (14) into system (13), we obtain ⎧ ∂F0 (w, ε) ⎪ ⎪ jε − (κ1 + λ)F0 (w, ε) + μF1 (w, ε) = 0, ⎪ ⎪ ∂w ⎪ ⎪ ∂Fn−1 (w, ε) ⎪ ∂Fn (w, ε) ⎪ ⎪ − j εe−j εw − (κ1 + λ + nμ)Fn (w, ε) jε ⎪ ⎨ ∂w   ∂w + κ1 e−j εw + λ Fn−1 (w, ε) + (n + 1)μFn+1 (w, ε) = 0, ⎪ ⎪     ⎪ ⎪ −j εw ∂FN−1 (w, ε) j εw ⎪ ⎪ −j εe + Nμ FN (w, ε) − λ 1 − e ⎪ ⎪ ∂w  ⎪ ⎪ ⎩ + κ e−j εw + λ F 1 N−1 (w, ε) = 0.

(15)

Let us solve system (15) in three steps. Step 1 Denote Fn (w) = lim Fn (w, ε). From Eqs. (15), we obtain the following ε→0

system for the functions Fn (w): ⎧ ⎨ −(κ1 + λ)F0 (w) + μF1 (w) = 0, −(κ1 + λ + nμ)Fn (w) + (κ1 + λ) Fn−1 (w) + (n + 1)μFn+1 (w) = 0, ⎩ −NμFN (w) + (κ1 + λ) FN−1 (w) = 0. Suppose the solution of this system has the form Fn (w) = Φ(w)Rn ,

(16)

where Rn has the same meaning as in the previous subsection and it can be calculated using expressions (9) and (11). Step 2 Consider the following expansion of the function Fn (w, ε) Fn (w, ε) = Φ(w) (Rn + j εwfn ) + O(ε2 ),

(17)

where fn is unknown. Let the function Φ(w) have the following form:  Φ(w) = exp

5 (j w)2 κ2 . 2

(18)

Substituting (17) and (18) into system (15), we derive the system for fn as follows: ⎧ −(κ1 + λ)f0 + μf1 = κ2 R0 , ⎪ ⎪ ⎨ −(κ1 + λ + nμ)fn + (κ1 + λ) fn−1 ⎪ +(n + 1)μfn+1 = κ1 Rn−1 + κ2 Rn − κ2 Rn−1 , ⎪ ⎩ −NμfN + (κ1 + λ) fN−1 = −λRn + κ1 RN−1 − κ2 RN−1 .

(19)

Asymptotic Analysis Methods for Multi-Server Retrial Queueing Systems

167

Note that the determinant of the system matrix is equal to zero and ranks of the system matrix and the extended matrix are equal. In this way, there are many solutions to system (19), and the general solution has the form fn = CRn + fn0 ,

(20)

where fn0 is a particular solution satisfying some additional condition, for example,  N n=0 fn = 0. Let us write the particular solution fn0 as follows: fn0 = κ2 gn + φn .

(21)

Substituting (21) into (19), we obtain the following system for gn : ⎧ ⎨ −(κ1 + λ)g0 + μg1 = R0 , −(κ1 + λ + nμ)gn + (κ1 + λ) gn−1 + (n + 1)μgn+1 = Rn − Rn−1 , ⎩ −NμgN + (κ1 + λ) gN−1 = −RN−1 ,

(22)

with the additional condition N 

gn = 0,

(23)

n=0

and the following system of equations for φn : ⎧ ⎨ −(κ1 + λ)φ0 + μφ1 = 0, −(κ1 + λ + nμ)φn + (κ1 + λ) φn−1 + (n + 1)μφn+1 = κ1 Rn−1 , ⎩ −NμφN + (κ1 + λ) φN−1 = −λRn + κ1 RN−1 ,

(24)

with the additional condition N 

φn = 0.

(25)

n=0

Obviously, systems (22)–(23) and (24)–(25) have unique solutions. Step 3 Summing up all equations of system (15), we obtain

κ1

N−1  n=0

Fn (w, ε) − λej εw FN (w, ε) − j ε

N−1  n=0

∂Fn (w, ε) = 0. ∂w

(26)

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Using expansions (17) and taking into account ej εw = 1 + j εw + O(ε2 ), formula (26) can be rewritten as κ1 j εw

N−1 

fn − λj εRN − λj εfN − j ε

n=0

Φ " (w) (1 − RN ) = O(ε2 ). Φ(w)

Substituting expression (18), we obtain κ1

N−1 

fn − λfN − λRN + κ2 (1 − RN ) = 0.

n=0

Taking into account formula (20), we have / C κ1

N−1 

4 Rn − λRN

+ κ1

n=0

N−1 

fn0 − λfN0 − λRN + κ2 (1 − RN ) = 0.

n=0

Using (10), we obtain the following expression: κ2 (1 − RN ) = λRN + λfN0 − κ1

N−1 

fn0 ,

n=0

or κ2 (1 − RN ) = λRN + (λ + κ1 )fN0 = λRN + (λ + κ1 ) (κ2 gN + φN ) , which does not depend on the constant C. Finally, we obtain κ2 as follows: κ2 =

λRN + (λ + κ1 )φN . 1 − RN − (λ + κ1 )gN

(27)

Thus, we define the function Fn (w), which is called the second-order asymptotic function. √ √ Taking into account the expressions ε = σ , w = u/ε = u/ σ and Eqs. (14) and (16), the approximation can be written as follows: 

Hn(2)(u)

5 (j u)2 κ2 = Rn exp . 2 σ

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Turning back to formula (12), we obtain the following asymptotic expression for the partial characteristic functions: 

5 κ1 (j u)2 κ2 Hn (u) = Rn exp j u + , σ 2 σ where κ1 and κ2 are defined by expressions (11) and (27), respectively, and the probability distribution Rn is defined by system (8). This completes the proof. Thus, we have proved that under the long delay condition the probability distribution P (i) of the number of calls in the orbit has a Gaussian approximation with parameters κ1 /σ and κ2 /σ . Finally, denote the probability function of the Gaussian distribution by G(x); then the discrete probability distribution P (i) of the process under study can be approximated as follows: P (i) = (G(i + 1) − G(i)) (1 − G(0))−1 .

4 Asymptotic Solution Under Heavy Load Denote the system load by ρ = λ/(Nμ). The stationary regime of the retrial queue exists if ρ < 1. Let us consider the system under the limit condition of the heavy load ρ → 1 or ε = 1 − ρ → 0. Theorem 2 The asymptotic characteristic function of the probability distribution of the number of calls in the orbit in the retrial queueing system of M/M/N type under the heavy load condition ρ → 1 has a gamma distribution of the form % $ j u −α h(u) = 1 − β

(28)

(μ + σ ) and the inverse scale parameter β = 1 − ρ. σ To prove the theorem, we introduce the following notation:

with the shape parameter α =

λ = (1 − ε)Nμ,

u = εw,

Hn (u) = εN−n Fn (w, ε)

for n = 0, . . . , N. (29)

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Substitute expressions (29) into system (3): ⎧ ∂F0 (w, ε) ⎪ ⎪ j σ εN−1 − (1 − ε)NμεN F0 (w, ε) + μεN−1 F1 (w, ε) = 0, ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ∂Fn (w, ε) ∂Fn−1 (w, ε) ⎪ ⎪ − j σ e−j εw εN−n j σ εN−n−1 ⎪ ⎪ ⎪ ∂w ∂w ⎨ − [(1 − ε)Nμ + nμ] εN−n Fn (w, ε) + (1 − ε)NμεN−n+1 Fn−1 (w, ε) ⎪ +(n + 1)μεN−n−1 Fn+1 (w, ε) = 0, for n = 1, . . . , N − 1, ⎪ ⎪ ⎪ ⎪ ∂F (w, ε) ⎪ ⎪ (1 − ε)NμεFN−1 (w, ε) − j σ e−j εw N−1 ⎪ ⎪ ∂w ⎪ 

 ⎪ ⎪ ⎩ − (1 − ε) 1 − ej u Nμ + Nμ FN (w, ε) = 0. (30) After some transformations, system (30) can be rewritten as follows: ⎧ ∂F (w, ε) 0 ⎪ jσ − (1 − ε)N μεF0 (w, ε) + μF1 (w, ε) = 0, ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ∂Fn−1 (w, ε) ∂Fn (w, ε) ⎪ ⎪ − j σ e −j εw ε − [(1 − ε)N μ + nμ] εFn (w, ε) ⎪ ⎨jσ ∂w ∂w 2 + (1 − ε)N με Fn−1 (w, ε) + (n + 1)μFn+1 (w, ε) = 0, for n = 1, . . . , N − 1, ⎪ ⎪ ⎪ ∂FN −1 (w, ε) ⎪ ⎪ (1 − ε)N μεFN −1 (w, ε) − j σ e −j εw ⎪ ⎪ ∂w ⎪  

⎪ ⎩ − (1 − ε) 1 − e j εw N μ + N μ FN (w, ε) = 0.

(31) Suppose the solution of system (31) has the form Fn (w, ε) = Fn (w) + εfn (w) + O(ε2 ),

(32)

where Fn (w) = lim Fn (w, ε). ε→0

Substituting expression (32) into system (31), and writing equalities for members with equal powers of ε, we obtain the following system for Fn (w): ⎧ " ⎨ j σ F0 (w) + μF1 (w) = 0, j σ Fn" (w) + (n + 1)μFn+1 (w) = 0, ⎩ " (w) + NμFN (w) = 0, j σ FN−1

n = 1, . . . , N − 1,

(33)

and the following system for fn (w): ⎧ " ⎨ j σf0 (w) + μf1 (w) = NμF0 (w), j σfn" (w) + (n + 1)μfn+1 (w) = NμFn (w), ⎩ " (w) + NμfN (w) = NμFN−1 (w). j σfN−1

n = 1, . . . , N − 1,

(34)

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Summing up all equations of system (31), we obtain the additional expression ∂ jσ ∂w

/

4 N−1 1  N−n ε Fn (w, ε) + (1 − ε)ej εw NμFN (w, ε) = 0. ε n=0

Take into account expression (32) and make some transformations. " " j σfN−1 (w) + j σ FN−2 (w) − (1 − j w)NμFN (w) + NμfN (w) = 0.

Using formulas (33) and (34), we obtain FN−1 (w) − (1 − j w)NFN (w) = 0.

(35)

Let us differentiate equation (35) and take into account the last equation of system (33). (μ + σ )FN (w) − j σ (1 − j w)NFN" (w) = 0. Clearly, the solution has the form FN (w) = C(1 − j w)−

(μ+σ ) σ

.

Using inverse expressions for substitutions (14), we obtain FN (w) = FN

u ε

$ = FN

u 1−ρ

%

$ % (μ+σ ) ju − σ . =C 1− 1−ρ

It is easy to show that C = 1 owing to the normalization requirement. Thus, the asymptotic characteristic function of the probability distribution of the number of calls in the orbit h(u) has the form of the gamma distribution characteristic function % $ j u −α h(u) = 1 − β μ+σ and the inverse scale parameter β = 1 − ρ. with the shape parameter α = σ This completes the proof. Denoting the probability distribution function of the gamma distribution by Γ (x), the discrete probability distribution of the number of calls in the orbit P (i) can be approximated as follows: P (i) = Γ (i + 1) − Γ (i).

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Moreover, during the study, we obtained the understandable conclusion that only the function FN (w) (it describes the situation when all servers are busy) is significant under the heavy load condition. This allows us to obtain the approximation of the probability distribution of the number of calls in the orbit for more general retrial queueing systems.

5 Hyper-Gamma Approximation During the proof of Theorem 2, we obtained the function FN−1 (w) (see formula (35)). Thus, we offer to apply the following approximation: h∗ (u) = FN (w) + εFN−1 (w) = FN

$

u 1−ρ

$

% + (1 − ρ)FN−1

u 1−ρ

% .

(36)

Consider functions FN (w) and FN−1 (w). In Sect. 4, it is shown that μ

FN (w) = C(1 − j w)− σ −1 . In addition, from Eq. (35) we have FN−1 (w) =

μ C (1 − j w)− σ . N

Substitute these expressions into (36) h(0) = FN (0) + (1 − ρ)FN−1 (0), and take into account the normalization equality h(0) ≡ 1 C + (1 − ρ) Therefore, we obtain C =

N . Then Eq. (36) can be rewritten as follows: 1−ρ+N

N h (u) = 1−ρ +N ∗

where α =

C = 1. N

$ % $ % 1−ρ j u −α j u −α+1 + , 1− 1− β 1−ρ+N β

μ + 1 and β = 1 − ρ. σ

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We use the notation % $ j u −α ΓN (u) = 1 − , β

% $ j u −α+1 ΓN−1 (u) = 1 − , β

and q =

N . 1−ρ +N

Thus, we have the final expression h∗ (u) = qΓN (u) + (1 − q)ΓN−1 (u).

(37)

Distribution (37) is called the hyper-gamma distribution.

6 Numerical Examples Let us consider some numerical examples to demonstrate the area of applicability of the proposed approximations. To do this, we provide simulations of the systems’ evolution and compare statistical results with analytical results derived in this paper. The comparison is performed by using the Kolmogorov distance [20] between respective cumulative distribution functions  i   6 7  d = max  p(l) ˜ − p(l) ,  i≥0  l=0

where p(l) is a probability distribution calculated using the approximation formulas (4), (28), or (37), and p(l) ˜ is an empiric distribution of the number of calls in the orbit based on the results of the simulation. For our purposes, we assume that values d ≤ 0.05 are sufficient for good accuracy of approximation. Let the number of servers N be equal to 10, the service rate of each server be μ = 1, and the arrival process be Poisson with the rate λ = Nμρ. Parameters ρ and σ will be variable in relation to the considering asymptotic conditions. First, we consider the long delay asymptotic condition. Let ρ = 0.8. The comparison of the Gaussian distributions 4 and empiric distributions is presented in Fig. 2 for various values of the parameter σ . Values of the Kolmogorov distance for this example are presented in Table 1. We note that the Gaussian approximation (4) becomes accurate enough for σ ≤ 0.1. Next, consider the asymptotic condition of the heavy load. Let σ = 1. Cumulative distribution functions of the gamma and hyper-gamma approximations (28) and (37), and the corresponding empiric distributions are shown in Fig. 3. Values of the Kolmogorov distance are presented in Table 2. Note that sufficient accuracy of approximation is achieved for ρ ≥ 0.97.

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1

1

1

0,8

0,8

0,8

0,6

0,6

0,6

0,4

0,4

a)

0,2

0,2

0

0,4

b)

0 0

5

10

15

20

c)

0,2 0

0

10

20

30

40

50

60

70

0

20

40

60

80

100 120

Fig. 2 Comparisons of the Gaussian approximation (dashed line) and the simulation results (solid line) for: (a) σ = 0.5; (b) σ = 0.1; (c) σ = 0.05 Table 1 Kolmogorov distances d for the Gaussian approximation for various values of the parameter σ

1 0.125

σ d

0.5 0.096

0.1 0.048

0.05 0.035

1

1

1

0,8

0,8

0,8

0,6

0,6

0,6

0,4

0,4

a)

b)

0

10

20

30

40

50

60

70

c)

0

0

0

0,4 0,2

0,2

0,2

0.01 0.018

0

20

40

60

80 100 120 140

0

50

100

150

200

Fig. 3 Comparisons of the gamma approximation (dashed line), the hyper-gamma approximation (dotted line, very close to gamma approximation), and the simulation results (solid line) for: (a) ρ = 0.9; (b) ρ = 0.95; (c) ρ = 0.97 Table 2 Kolmogorov distances dg and dh for the gamma and hyper-gamma approximations, respectively, for various values of the parameter ρ

ρ dg dh

0.9 0.218 0.215

0.95 0.095 0.093

0.96 0.077 0.076

0.97 0.049 0.048

0.98 0.022 0.022

In addition, we note that the hyper-gamma approximation is only slightly better than the gamma approximation in the example. However, the accuracy of the approximation and the difference between the hyper-gamma and gamma approximation results increase for fewer servers (e.g., for the system M/M/2, values of the Kolmogorov distance are presented in Table 3).

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Table 3 Kolmogorov distances dg and dh for the gamma and for hyper-gamma approximations, respectively, for the system M/M/2

ρ dg dh

0.9 0.118 0.100

0.95 0.058 0.046

0.97 0.033 0.025

7 Conclusions In this paper, the multi-server retrial queue M/M/N has been considered. We have proposed asymptotic methods for the study of the system under long delay and heavy load conditions. It has been proved that the asymptotic characteristic function of the number of calls in the orbit under the long delay condition is Gaussian (Sect. 3). Using the numerical comparison of the asymptotic and the empiric distributions, we have shown that the applicability area of this approximation is σ ≤ 0.1. In Sect. 4, the retrial queue has been studied under the limit condition of a heavy load, and the gamma form of the asymptotic characteristic function of the number of calls in the orbit has been proved. The numerical analysis allows to conclude that the asymptotic formulas can be applied for ρ ≥ 0.97. In addition, we have proposed an improvement of the asymptotic result in the form of the hyper-gamma approximation (Sect. 5), which is better than the gamma approximation for N < 10. In recent papers, we found that the formulas for asymptotic characteristic functions of the probability distribution of the number of calls in the orbit (under long delay and heavy load conditions) have the same form for single-server retrial queues with different arrival processes and service laws: M/M/1, M/GI /1, MMP P /M/1, and MMP P /GI /1. Thus, in future work, we plan to apply the proposed methods for multi-server retrial queueing systems with non-Poisson arrival processes.

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On the Application of Dynamic Screening Method to Resource Queueing System with Infinite Servers Michele Pagano

and Ekaterina Lisovskaya

Abstract Infinite-server queues are a widely used modelling tool thanks to their analytical tractability and their ability to provide conservative upper bounds for the corresponding multi-server queueing systems. A relatively new research field is represented by resource queues, in which every customer requires some volume of resources during her staying in the queue and frees it only at the end of the service. In a nutshell, in this paper the joint distribution of the processes describing the number of busy servers and the total volume of occupied resources is derived and the parameters of the corresponding bidimensional Gaussian distribution are explicitly calculated as a function of the arrival process characteristics and the service time and customers capacity distributions. The aim of this paper is twofold: on one side it summarizes in a ready-to-be-used way the main results for different arrival processes (namely, Poisson processes, renewal processes, MAP, and MMPP), on the other it provides a detailed description of the employed methodology, presenting the key ideas at the basis of powerful analysis tools (dynamic screening and asymptotic analysis methods), developed in the last two decades by Tomsk researchers. Keywords Resource queuing systems · Dynamic screening method · Asymptotic analysis method · Renewal processes · MMPP · MAP

M. Pagano () Department of Information Engineering, University of Pisa, Pisa, Italy e-mail: [email protected] E. Lisovskaya Tomsk State University, Tomsk, Russian Federation e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_12

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1 Introduction Infinite-server queues play a relevant role in queueing theory and in performance analysis. Indeed, different issues can be modelled in such a way, that the number of servers is really infinite or so big, that in practice there are always free servers. A typical example is represented by economical models, in which there is no reason to limit the number of contracts that can be signed between credit organizations and clients. Although in real systems physical resources are always finite, these models can be applied to the analysis of computing clusters and multi-core supercomputers, as well as to high-capacity routers (see [1] and references therein). Moreover, infinite-server queues have a higher analytical tractability than the corresponding multi-server systems: for several classes of arrival processes not only mean values of the performance indexes are available, but it is also possible to determine the corresponding probability distributions, at least under some asymptotic conditions. For instance, “heavy traffic” scenarios are often encountered in computer networks and the knowledge of the steady-state distribution of the number of busy servers can provide conservative upper bounds for the correct dimensioning of the system (e.g., output capacity of a router). Traditionally, in network modelling the service was associated to packets transmission or calls duration, but this assumption is getting less and less true in modern network architectures. Indeed, issues related to virtual machine allocation in cloud environments or performance of LTE (Long Term Evolution) networks require new queueing models, in which the customers ask for some resources (CPU/memory and radio resources, respectively) that are released at the end of the service. Such models are known in the literature as resource queueing systems. For instance, in [2] they are applied to the analysis of M2M traffic characteristics in a LTE network cell, while [3] presents an overview of the resource queuing systems used for modeling of a wide class of real systems with limited resources, focusing on wireless networks with exponentially distributed service time. Resource queues in connection with AQM (Active Queue Management) mechanisms are investigated in [4] under the processor sharing discipline, but the analysis is limited to Poisson arrivals. Finally, analytical results for systems with finite resources are given in [5], where M/M/n/m queues are considered and the service time is assumed to be proportional to the customer capacity. All the above-mentioned works deal with finite resource queueing systems, and analytical results are obtained under stringent condition for the arrival process and the service time distribution. However, the inadequacy of the Poisson process as arrival model is well-known in the literature [6, 7] and more realistic traffic models have been proposed in the literature, such as MAPs (Markov Arrival Processes) and MMPPs (Markov Modulated Poisson Processes). In case of infinite-server resource queues the previous limitations disappear and such models can be used to calculate conservative bounds on system performance under realistic traffic conditions and general distributions of the service time and the customer capacity.

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The aim of the paper is to present the principal results in the field of infiniteserver resource queueing systems, most of them derived and tested by the authors in the last few years and gathered in [8], where complete proofs and simulation results are also reported. However, to the best of our knowledge, this paper is the first attempt (at least in the English scientific literature) to collect the main results for such systems in a review work and provide for networking specialists the possibility of choosing the most suitable model and finding the relevant performance indexes. It is worth noticing that, above all in case of correlated arrivals, the derivation of the Gaussian approximation is quite cumbersome, so we mainly focus on the methodological elements and present a complete analysis only for Poisson arrivals. Indeed, we also aim to popularize powerful tools developed in the last decades by the “Tomsk queueing theory School,” such as the alternative description of MAPs, the dynamic screening method, and the asymptotic analysis method. The rest of the paper is organized as follows. In Sect. 2 we provide a thorough description of the analyzed queueing systems and recall some background results and definitions, while the following section details the application of the proposed methodology to the case of Poisson arrivals. Then, in Sect. 4 we generalize the analysis to renewal processes and MAPs, highlighting the differences with the Poisson case and summarizing the key results. Finally, the main contributions of the paper are pointed out in the Conclusions, together with future research directions.

2 Reference Model and Theoretical Background In this section we describe the system under analysis, introducing the notation and the mathematical apparatus used in the rest of the paper. In describing the different arrival processes we detail the definition of MAPs, since our notation is slightly different (although equivalent) from the one most widely used in the western literature. Finally, we briefly recall the dynamic screening method for the study of non-Markovian queueing systems.

2.1 Infinite-Server Resource Queueing System Let us consider an infinite-server queueing system with infinite resources (so no customer will be rejected) as shown in Fig. 1. An arriving customer can occupy any free server for a random service time ξ ≥ 0, characterized by a distribution function B(τ ) = P {ξ < τ } with finite first moment (roughly speaking, it is just required that the mean service time is finite and no assumptions are made on its variance). As already mentioned in the introduction, the customer requires during his service also some resource of random volume ν, described by a distribution function G(y) = P {ν < y} with finite first and second moments. When the service is completed, the customer leaves the system and frees the resource. Moreover, service times {τ }

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Fig. 1 Infinite-server resource queueing system

and customer capacities {ν} are assumed to be mutually independent and do not dependent on the epochs of customers’ arrivals. Let us fix an initial moment t0 (to get the steady-state regime it will be enough to consider t0 → −∞) and let the system be empty at time t0 . Denote by i(t) the number of customers in the system at time t; then, the total volume of occupied resources (i.e., the total customers capacity) is given by V (t) =

i(t ) 

νi

i=1

and the bidimensional process {i(t), V (t)} unambiguously characterizes the state of the considered queueing system. Due to the independence of the two components, it is easy to find a relation among them. Indeed, the characteristic function of the total customers capacity can be rewritten as    i   h(v) = M ej vV (t ) = M M ej v k=1 νk |i(t) = i =

∞ 

∞    i  i  M ej vν M ej v k=1 νk P {i(t) = i} = P {i(t) = i}

i=0

i=0

and, taking into account that    M ej vν =

∞

ej vy dG(y) = G∗ (v) ,

0

we get the link between traditional (the number of busy servers does not depend on the occupied resources) and resource queueing systems: h(v) =

∞  

i G∗ (v) P {i(t) = i} .

(1)

i=0

However, this elegant result does not solve our problem. Indeed, the distribution of the number of busy servers is known only for a limited set of systems (see

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Sect. 3.1 for Poisson arrivals); even in that case, quite often the analytical expression of the distribution of the total customers capacity is not available and only numerical approximations can be found. The process {i(t), V (t)} is, in general, non-Markovian; among the different approaches (for instance, the analysis of the embedded Markov chain and the method of supplementary variables) proposed in the literature, we consider the dynamic screening method that provides a unified framework for the analysis of infinite-server queueing systems (including tandem queues, queueing networks, and resource systems) and is described in Sect. 2.3.

2.2 Arrival Process The arrival process plays a major role in determining not only the queueing behavior, but also the analytical tractability of the system. Indeed, analytical results can be obtained only for Poisson arrivals, but it is well-known that the distribution of inter-arrival times is typically quite far from the exponential one and, above all, the arrivals are correlated [6]. To cope with these issues, we will consider two different classes of traffic models, widely used in the literature: renewal processes and MAPs (which include MMPPs as a special case). Unfortunately, for both classes closedform results are not available and only asymptotic approximations can be obtained under heavy traffic conditions. In this paper we introduce a scale parameter N → ∞ (high intensity parameter) and focus on the case of “infinitely growing arrival rate” (for the other regime, known in the literature as “infinitely growing service time,” see for instance [9]). In more detail, renewal processes are characterized by the sequence of interarrival times {ζn }, which are independent identically distributed random variables with common distribution A(z) = P {ζ < z}; in our analysis only the existence of finite mean and variance is assumed. Hence, the asymptotic condition simply corresponds to a scaled distribution A(zN) and the mean interarrival time goes to 0 as 1/N when N → ∞. As far as MAPs are concerned, we make use of the characterization developed by Tomsk researchers on the basis of the theory of doubly stochastic processes, which includes the following components [1]: – k(t): a continuous time ergodic Markov chain with K states and infinitesimal generator matrix Q = qkν , k, ν = 1, . . . , K – λk ≥ 0, k = 1, . . . , K: the conditional arrival rate for each state of the underlying Markov chain k(t), typically denoted through the diagonal matrix Λ = diag {λk } , k = 1, . . . , K – dkν k, ν = 1, . . . , K: the conditional probabilities that there is an arrival when the Markov chain k(t) changes its state from k to ν (it is assumed that dkk = 0), grouped in the matrix D = dkν , k, ν = 1, . . . , K

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In this way, unlike the classical notation [10], the model parameters have a clear physical interpretation and MMPPs can be easily obtained by setting D = 0 since state transitions of the underlying Markov chain just imply a rate change, but arrivals are not generated. Moreover, the asymptotic condition can be taken into account multiplying all the coefficient of the matrices Q and Λ by the high intensity parameter N → ∞. It is worth mentioning that this notation is equivalent to the classical one, based on matrices D0 and D1 . Indeed, it is possible to show that Λ + D ◦ Q] D0 = Q − [Λ D1 = Λ + D ◦ Q where ◦ denotes the Hadamard product (or entrywise product).

2.3 Dynamic Screening Method In a nutshell, the dynamic screening method is based on the construction of a suitable screened process and its markovization by the addition of a suitable component, depending in general on the arrival process. Let us consider two time axes (see Fig. 2): the first one displays the arrival times of all customers, while the other one corresponds to the screened customers. For any t ≥ t0 let us define a continuous function S(t) that assumes values in the interval [0, 1]; then, a customer arriving at time t is screened on the second axis (i.e., generates an event on it) with probability S(t). Since the screening probability depends on the arrival time t, the method is called dynamic. In more detail, for an infinite-server queue we assume that the system is empty at the initial time t0 , fix an arbitrary moment T > t0 and put S(t) = 1 − B(T − t)

Fig. 2 Screening of the customers’ arrivals

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i.e., S(t) represents the probability that a customer, arrived at time t < T , is not served by time T and so it still occupies some resources in the queue. Instead, with probability 1 − S(t) the customer has left the system and it is not screened on the second axis. Let us denote by {n(t)} and {W (t)} the counting process representing the number of screened event in the interval [t0 , t) and their total capacity, respectively. The process {n(t)} is, in general, non-Markovian (except the case of Poisson arrivals), but it can be markovized by adding a suitable component: – the residual time before the next arrival z(t) in case of renewal processes ⇒ the process {n(t), z(t)} is Markovian; – the state k(t) of the modulating Markov chain in case of MAPs ⇒ the process {n(t), k(t)} is Markovian; – the residual time z(t) and the state l(t) of the embedded Markov chain in case of semi-Markov processes ⇒ the process {n(t), z(t), l(t)} is Markovian. Moreover, the probability distributions of the number of customers in the system {i(t)} and the number of screened arrivals on the second axis {n(t)} coincide at time T: P {i(T ) = m} = P {n(T ) = m}

∀m = 0, 1, 2, . . .

(2)

The latter result, known as the fundamental equation of the dynamic screening method, can be easily verified starting from the equality of the corresponding conditional probabilities (given a sequence of L arrivals at times t1 , t2 , . . . tL ) P {i(T ) = m|t1 , t2 , . . . tL } = P {n(T ) = m|t1 , t2 , . . . tL } ∀m = 0, 1, 2, . . . for any number of arrivals L and any sequence of arrival times t1 , t2 , . . . tL , which is a direct consequence of the chosen S(t) as can be verified by direct calculation. Since the distributions of the multidimensional random variable (L, t1 , t2 , . . . tL ) are the same in the two cases, also the distributions of the random variables i(T ) and n(T ) (i.e., of the values of the processes {i(t)} and {n(t)} at time T ) coincide. It is easy to prove the same property for the extended process {i(t), V (t)}: P {i(T ) = m, V (T ) < z} = P {n(T ) = m, W (T ) < z} ∀m = 0, 1, 2, . . . and z ≥ 0

(3)

that, by analogy with (2), represents the fundamental equation of the dynamic screening method for resource queueing systems. To summarize, the essence of the dynamic screening method consists in the following steps: 1. Choose a suitable screening function S(t) and build the corresponding screened process {n(t)}; 2. Markovize the process {n(t), W (t)}, by adding the suitable component &(t);

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3. Determine the probabilistic characteristics of the extended process {n(t), W (t), &(t)}; 4. Derive the joint distribution of the process {n(t), W (t)} (and, in case, the marginal distributions if relevant); 5. Set t = T and, according to (3), get the distribution of the process {i(t), V (t)} at time t = T . Finally, note that T was chosen arbitrarily (the only condition is T > t0 ) and so we can calculate the probability distribution of the joint process at any time ; in particular, letting t0 → −∞, we can get the steady-state distribution, which is typically the parameter of interest in the study of queueing systems.

3 Analysis of Infinite-Server Resource Queueing System: Poisson Arrivals Let us assume that the arrival process is Poissonian with rate λ and denote by Mv /GI/∞ the corresponding resource queueing system to highlight that customers are characterized by their capacity v. Although in this special case the analysis can be carried out in different ways, we will take advantage of the analytical simplicity of the input process to better illustrate our general methodology. In more detail, at first in Sect. 3.1 we derive the Kolmogorov equation for the characteristic function of the bidimensional process {i(t), V (t)} and find the corresponding analytical solution that is possible thanks to the special structure of the arrival process. Then, in Sect. 3.2 we present the general approach that provides first- and second-order approximations of the characteristic function.

3.1 Direct Solution of Kolmogorov Equations Let us define the screened process as described in Sect. 2; thanks to the memoryless property of the exponential distribution, now the bidimensional stochastic process {n(t), W (t)} is Markovian and no additional component is required. To visually simplify the analysis, let us introduce the following notation: Δ

P {n(T ) = n, W (T ) < w} = P (n, w, t)

∀n = 0, 1, 2, . . . and w > 0 ,

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and assume that P (n, w, t) = 0 for negative values of n and w. According to the formula of total probability the following equality holds P (n, w, t + Δt) = P (n, w, t)(1 − λΔt) + P (n, w, t)λΔt (1 − S(t))  ∞ + λΔtS(t) P (n − 1, w − y, t)dG(y) + o(Δt) , 0

from which the set of Kolmogorov differential equations can be easily derived: ∂P (n, w, t) = λS(t) ∂t



∞

P (n − 1, w − y, t)dG(y) − P (n, w, t)

(4)

0

for n = 0, 1, 2, . . . and w > 0, with initial conditions / 1 n=w=0 P (n, w, t0 ) = 0 otherwise.

(5)

To solve the Kolmogorov differential equations, let us introduce the characteristic function Δ

h(u, v, t) = M {exp (j un(t) + j vW (t))} =

∞ 

 ej un

∞

ej vw P (n, dw, t).

0

n=0

(6) Taking into account that ∞ 

 e



∞

j un

e 0

n=0

= ej u

P (n − 1, d(w − y), t)dG(y)

0

∞ 





e 0



j vy

∞ 

w

P (n − 1, d(w − y), t)dG(y)

0

 e



w

j u(n−1)

e

j v(w−y)

P (n − 1, d(w − y), t) dG(y)

0

n=0 ∞

= ej u

 ej vy ej v(w−y)

0

∞

ju

∞

ej u(n−1)

n=0

= e

w

j vw



0

0

∗

= e G (v)h(u, v, t), ju

∞

ej vy h(u, v, t)dG(y) = ej u h(u, v, t)

where ∗

Δ



∞

G (v) = 0

ej vy dG(y) ,

ej vy dG(y)

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Eq. (4) can be rewritten as

∂h(u, v, t) = λS(t)h(u, v, t) ej u G∗ (v) − 1 ∂t

(7)

with the initial condition h(u, v, t0 ) = 1

(8)

and its solution is given by  5

 t h(u, v, t) = exp λ ej u G∗ (v) − 1 S(τ )dτ .

(9)

t0

For t = T and t0 → −∞, by virtue of (3) we obtain the characteristic function of the bidimensional process describing the number of busy servers and the total customers capacity in steady-state conditions: 

 h(u, v) = exp λb ej u G∗ (v) − 1 , (10) where Δ



∞

b =

(1 − B(τ )) dτ .

0

Putting v = 0 in (10), we get the characteristic function for the number of busy servers in steady-state conditions 

 Δ h(u) = h(u, v)|v=0 = exp λb ej u − 1

(11)

that coincides with the characteristic function of the Poisson distribution with parameter λb, in agreement with the well-known classical results for the M/GI/∞ queueing systems. In a similar way the characteristic function for the total customers capacity is  6 7 Δ h(v) = h(u, v)|u=0 = exp λb G∗ (v) − 1

(12)

in accordance with the results obtained by Oleg Tikhonenko [5] and with Eq. (1). Indeed, as shown by (11), in M/GI/∞ the number of busy servers has Poisson distribution with parameter λb and by direct substitution into (1) we get h(v) =

∞   i=0

= e

∞   ∗ i (λb)i −λb i e G∗ (v) P {i(t) = i} = G (v) i!

−λb λbG∗ (v)

e



6

i=0

∗

= exp λb G (v) − 1

7

.

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3.2 The Asymptotic Analysis Method The asymptotic analysis method in queueing systems aims at determining their characteristics under some limit condition [11]. In the following we will consider its application to the differential Eq. (7) in case of “infinitely growing arrival rate” and we look for its approximate solutions with different order of accuracy, namely “first-order asymptotic” h(u, v, t) ≈ h1 (u, v, t) and “second-order asymptotic” h(u, v, t) ≈ h1 (u, v, t)h2 (u, v, t), also known as Gaussian approximation. Note that it is possible to derive higher order asymptotics, but in that case the inversion of the characteristic function is, in general, possible only by numerical methods and, as stated in [1], at least for “traditional” queueing systems the gain is not significant in case of heavy traffic.

3.2.1 First-Order Asymptotic Analysis By performing the substitutions ε=

1 , u = εx, v = εy, h(u, v, t) = f1 (x, y, t, ε) λ

(13)

in Eq. (7), we obtain the following Cauchy problem: ⎧   ∂f (x, y, t, ε) ⎪ ⎨ε 1 = S(t)f1 (x, y, t, ε) ej εx G∗ (εy) − 1 ∂t ⎪ ⎩ f1 (x, y, t0 , ε) = 1.

(14)

For ε → 0, taking into account the first-order Taylor–Maclaurin expansion   ej εx = 1 + j εx + O ε2 the limit function f1 (x, y, t) = lim f1 (x, y, t, ε) satisfies the following differenε→0

tial equation: ∂f1 (x, y, t) = S(t)f1 (x, y, t) (j x + jya1) , ∂t where a1 is the average customer capacity, i.e.,  a1 =

∞

ydG (y) . 0

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Taking into account the initial condition f1 (x, y, t0 ) = 1, we get  5  t f1 (x, y, t) = exp (j x + jya1) S(τ )dτ t0

and, after performing the substitutions inverse to (13), the first-order approximation of h(u, v, t), i.e.,  5  t h(u, v, t) ≈ exp λ (j u + j va1 ) S(τ )dτ .

(15)

t0

3.2.2 Second-Order Asymptotic Analysis The second-order asymptotic provides the bidimensional Gaussian approximation of the process {i(t), V (t)}. Rewriting the corresponding characteristic function as  5  t S(τ )dτ h(u, v, t) = h2 (u, v, t) exp (j x + jya1) t0

the differential Kolmogorov equation (7) becomes   ∂h2 (u, v, t) + λ(j u + j va1 )S(t)h2 (u, v, t) = h2 (u, v, t)λS(t) ej εu G∗ (v) − 1 ∂t and, after performing the substitutions ε2 =

1 , u = εx, v = εy, h2 (u, v, t) = f2 (x, y, t, ε) , λ

(16)

we obtain the following differential equation: ε2

∂f2 (x, y, t, ε) + (j εx + j εya1)S(t)f2 (x, y, t, ε) ∂t   = S(t)f2 (x, y, t, ε) ej εx G∗ (εy) − 1

(17)

with the initial condition f2 (x, y, t0 , ε) = 1.

(18)

As before we consider the limit as ε → 0 and then use the second-order Taylor– Maclaurin expansion ej εx = 1 + j εx +

  (j εx)2 + O ε3 . 2

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Then, the limit function f2 (x, y, t) = lim f2 (x, y, t, ε) satisfies the following ε→0

differential equation: $ % ∂f2 (x, y, t) (j x)2 (jy)2 = S(t)f2 (x, y, t) + a2 + j xjya1 , ∂t 2 2

(19)

where a2 is the second moment of the random variable describing the customer capacity, i.e., 

∞

a2 =

y 2 dG (y) .

0

The solution of (19), with the initial condition f2 (x, y, t0 ) = 1, is $ f2 (x, y, t) = exp

(j x)2 (jy)2 + a2 + j xjya1 2 2

%

5

t

S(τ )dτ t0

and, performing the substitutions inverse to (16), we get the second-order approximation of h(u, v, t), i.e., % t 5  $ (j v)2 (j u)2 + a2 + j uj va1 S(τ )dτ . h(u, v, t) ≈ exp λ j u + j va1 + 2 2 t0 (20) Finally, for t = T and t0 → −∞, by virtue of (3) we obtain the second-order asymptotic for the characteristic function of the steady-state distribution of the bidimensional process {i(t), V (t)} 5  (j v)2 (j u)2 λb + λa2 b + j uj vλa1 b h(u, v) ≈ exp j uλb + j vλa1 b + 2 2 (21) that corresponds to the characteristic function of a bivariate Gaussian process with correlated components. This result has a much wider validity, not limited to Poisson arrival, as shown in the next section.

4 Asymptotic Analysis of Infinite-Server Resource Queueing System Dynamic screening and asymptotic analysis can be applied to a great variety of arrival processes and queueing systems. For instance, as shown in this section, the proposed methodology can be easily extended to renewal processes and MMPPs, a

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special case of MAPs widely used in teletraffic [10, 12]. For sake of brevity, we will just sketch the procedure, highlighting the additional complexity due to the change in the input process as well as the general validity of the Gaussian approximation and providing references with the detailed proof of the results.

4.1 The MMPP(ν) /GI/∞ Queue As already stated in Sect. 2.2, an MMPP is characterized by the two matrices Q and Λ and the evolution of the queue depends on the state of the modulating Markov chain k(t). Therefore, it is now necessary to work with the tridimensional Markovian process {k(t), n(t), W (t)}. Denoting the probability distribution of this process by P (k, n, w, t) = P {k(t) = k, n(t) = n, W (t) < w} , and applying the formula of total probability as in the Poisson case, we get P (k, n, w, t + Δt) = P (k, n, w, t)(1 − λk Δt)(1 + qkk Δt) + P (k, n, w, t)λk Δt (1 − S(t))  w + λk ΔtS(t) P (k, n − 1, w − y, t)dG(y) +



0

qνk ΔtP (ν, n, w, t) + o(Δt),

(22)

ν=k

for k = 1, . . . , K, n = 0, 1, 2, . . . and w > 0. From (22), we obtain the system of Kolmogorov differential equations  w

∂P (k, n, w, t) = λk S(t) P (k, n − 1, w − y, t)dG(y) − P (k, n, w, t) ∂t 0  qνk P (ν, n, w, t), (23) + ν

with initial conditions / P (k, n, w, t0 ) =

r(k) n = w = 0 0

otherwise,

where {r(k)}, k = 1, . . . , K are the stationary state probabilities of the modulating Markov chain k(t). Note that the first term on the right-hand side of (23) is similar to the one in (4), while the other one takes into account the state transitions in the modulating Markov chain.

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Introducing the partial characteristic function h(k, u, v, t) = M {exp (j un(t) + j vW (t))}  ∞ ∞  j un = e ej vw P (k, n, dw, t), 0

n=0

we can write the following system of equations:

 ∂h(k, u, v, t) = λk S(t)h(k, u, v, t) ej u G∗ (v) − 1 + h(ν, u, v, t)qνk ∂t ν with the initial condition h(k, u, v, t0 ) = r(k)

for k = 1, . . . , K ,

or in matrix form:

∂h(u, v, t) = h(u, v, t) S(t)(ej u G∗ (v) − 1) + Q , ∂t

(24)

with the initial condition h(u, v, t0 ) = r , where h(u, v, t) = [h(1, u, v, t), h(2, u, v, t), . . . , h(K, u, v, t)] and r = [r(1), r(2), . . . , r(K)] is the row-vector of the stationary distribution of the modulating Markov chain: 

rQ = 0 re = 1 ,

e being a column-vector with all entries equal to 1. To the matrix differential equation (24) we apply the asymptotic analysis method to get asymptotic results under the condition of “infinitely growing arrival rate.” Denoting by N the scaling parameter, we consider the family of MMPP processes ˜ and Q = N Q ˜ as N → ∞. Calculations are more cumbersome with = N since now we need to work with a matrix (and not scalar) equation, but, as shown in [13], the procedure is analogous to the Poisson case: the first- and second-order

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approximations are derived and then, by setting t = T and t0 → −∞, we obtain the characteristic function of the process {i(t), V (t)} at steady state:  (j u)2 (Nλb1 + Nκb2 ) h(u, v) ≈ exp Nλ(j u + j va1 )b1 + 2

5 (j v)2 2 (Nλa2 b1 + Na1 κb2 ) + j uj v(Nλa1 b1 + Nκa1 b2 ) , + 2

(25)

where a1 and a2 are the first and the second moments of the random variable describing the customer capacity, 

∞

b1 =

 (1 − B(τ ))dτ,

0

∞

b2 =

(1 − B(τ ))2 dτ

0

and ˜ λ = r e,

  ˜ − λI e, κ = 2g

where the row-vector g satisfies the linear matrix system ⎧   ⎨g Q ˜ = r λI − ˜ ⎩ge = 1. The form of the characteristic function (25) implies that the bidimensional process {i(t), V (t)} is asymptotically Gaussian with the vector of mathematical expectations a = N [λb1

λa1 b1 ]

and the covariance matrix

λb1 + κb2 λa1 b1 + κa1 b2 K=N . λa1 b1 + κa1 b2 λa2 b1 + κa12 b2 In the general case of MAPs [14], the procedure is exactly the same, only equality (22) slightly changes since a transition of the modulating Markov chain k(t) from state ν to state k (with k = ν) can now generate an arrival with probablity dνk . This corresponds to substitute the matrix Λ with Λ + Q ◦ D , leaving unchanged all the rest. Apart from the value of λ and κ, equality (25) still holds for the steady-state characteristic function and hence the previous considerations about Gaussianity can be extended to MAP(ν)/GI/∞ resource queues.

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4.2 The GI(ν) /GI/∞ Queue Let us consider as input flow a renewal process and assume that the inter-arrival time, characterized by the distribution A(z), has finite mean and variance, i.e., a=

1 = λ



∞



∞

and σ 2 =

(1 − A(z)) dz

0

(z − a) dA(z) .

0

In this case the memoryless property does not hold, hence it is necessary to take into account the residual time z(t) to obtain a Markovian process {z(t), n(t), W (t)}. Denoting its probability distribution by P (z, n, w, t) = P {z(t) < z, n(t) = n, W (t) < w} , the formula of total probability leads to the following equality (for n = 0, 1, 2, . . ., and z, w > 0): P (z, n, w, t + Δt) = [P (z + Δt, n, w, t) − P (Δt, n, w, t)] + P (Δt, n, w, t)(1 − S(t))A(z)  w P (Δt, n − 1, w − y, t)dG(y) + o(Δt), + A(z)S(t) 0

from which the Kolmogorov differential equation is easily derived: ∂P (z, n, w, t) ∂P (z, n, w, t) ∂P (0, n, w, t) = + (A(z) − 1) ∂t ∂z ∂z  w

∂P (0, n, w, t) ∂P (0, n − 1, w − y, t) dG(y) − + S(t)A(z) , ∂z ∂z 0 with initial condition / R(z)

P (z, n, w, t0 ) =

0

n=w=0 otherwise ,

where 1 R(z) = a



z 0

(1 − A(u))du

(26)

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is the stationary distribution of the renewal arrival process. Also in this case it is useful to rewrite the Kolmogorov equation in terms of the partial characteristic function h(z, u, v, t) = M {exp (j un(t) + j vW (t))}  ∞ ∞  j un = e ej vw P (z, n, dw, t) n=0

0

and we obtain the following equation: ∂h(z, u, v, t) ∂h(z, u, v, t) = ∂t ∂z   ∂h(0, u, v, t) A(z) − 1 + A(z)S(t) ej u G∗ (v) − 1 , + ∂z (27) with the initial condition h(z, u, v, t0 ) = R(z) .

(28)

Since the exact solution of (27) is, in general, not available, we apply the asymptotic analysis method under the condition of “infinitely growing arrival rate,” rewriting the distribution function as A(Nz) with N → ∞ as in Sect. 4.1. Following our usual approach, we get the second-order approximation of h(z, u, v, t) and, setting z → ∞, t = T , t0 → −∞, we obtain the characteristic function of the process {i(t), V (t)} in the steady-state regime (see [15] for the detailed proof):  (j u)2 (Nλb1 + Nκb2 ) h(u, v) ≈ exp Nλ(j u + j va1 )b1 + 2 +

5 (j v)2 (Nλa2 b1 + Na12 κb2 ) + j uj v(Nλa1 b1 + Nκa1 b2 ) , 2

(29)

where a1 , a2 , b1 , and b2 are the same as in (25), while the expression of κ has changed:   κ = λ3 σ 2 − a 2 . In complete analogy with the result in Sect. 4.1, the bidimensional process {i(t), V (t)} is asymptotically Gaussian with the vector of mathematical expectations a = N [λb1

λa1 b1 ]

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and the covariance matrix

λb1 + κb2 λa1 b1 + κa1b2 K=N λa1 b1 + κa1 b2 λa2 b1 + κa12b2



that have exactly the same expression (apart from the definition of κ and λ) as in the MMPP case.

5 Conclusions In this work we analyzed infinite-server resource queueing systems, collecting in a review paper the most relevant results we obtained in the last few years. To the best of our knowledge it is the first attempt in the English literature to describe a general analysis methodology for such systems and provide a list of readyto-be-used formulas for different arrival processes (namely, Poisson processes, renewal processes, MAP, and MMPP). The proposed approach is based on the application at first of the dynamic screening method (for markovization purposes) and then of the asymptotic analysis method (to find at least an asymptotic solution for the corresponding Kolmogorov equations). In a nutshell, the paper highlights that, under the condition of “infinitely growing arrival rate,” the joint distribution of the processes describing the number of busy servers and the total volume of occupied resources is bivariate Gaussian and provides analytical expressions for its parameters (mean vector and covariance matrix) as a function of the arrival process characteristics, the distribution of the service time and the first and second moments of the customers capacity distribution. Finally, it is worth mentioning that the proposed methodology is much more general and can be applied to other arrival processes (e.g., semi-Markov processes), heterogeneous customers/servers, multi-resource customers as well as to more complex resource systems, including tandem queues and queueing networks. Acknowledgments The publication has been prepared with the support of the University of Pisa PRA 2018–2019 Research Project “CONCEPT—COmmunication and Networking for vehicular CybEr-Physical sysTems.”

References 1. Nazarov, A., Moiseev, A.: Infinite-Server Queueing System and Networks (in Russian). Publishing House STL, Tomsk (2015) 2. Sopin, E.S., Ageev, K.A., Markova, E.V., Vikhrova, O.G., Gaidamaka, Y.V.: Performance analysis of M2M traffic in LTE network using queuing systems with random resource requirements. Autom. Control Comput. Sci. 52(5), 345–353 (2018)

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3. Gorbunova, A.V., Naumov, V.A., Gaidamaka, Y.V., Samouylov, K.E.: Resource queuing systems as models of wireless communication systems (in Russian). Inform. Primen 12(3), 48–55 (2018) 4. Tikhonenko, O., Kempa, W.: Queueing system with processor sharing and limited memory under control of the AQM mechanism. Autom. Remote Control 76(10), 1784–1796 (2015) 5. Tikhonenko, O., Kawecka, M.: Total volume distribution for multiserver queueing systems with random capacity demands. In: Kwiecie´n, A., Gaj, P., Stera, P. (eds.) Computer Networks, pp. 394–405. Springer, Berlin (2013) 6. Pagano, M., Rykov, V., Yuri, K.: Teletraffic Models (in Russian). Publishing House Infra-M, Moscow (2018) 7. Paxson, V., Floyd, S.: Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Netw. 3(3), 226–244 (1995) 8. Lisovskaya, E.: Asymptotic methods for the analysis of resource queueing systems with nonPoissonian arrival flows. Ph.D. Thesis, Tomsk State University (2018). Candidate of physical and mathematical Sciences 9. Lisovskaya, E., Moiseeva, S., Pagano, M., Potatueva, V.: Study of the MMPP/GI/∞ queueing system with random customers’ capacities. Inf. Appl. 11(4), 109–117 (2017) 10. Heffes, H., Lucantoni, D.M.: A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Sel. Areas Commun. 4, 856–868 (1986) 11. Nazarov, A., Moiseeva, S.: The Asymptotic Analysis Method in Queueing Theory (in Russian). Publishing House STL, Tomsk (2006) 12. Heyman, D.P., Lucantoni, D.: Modeling multiple IP traffic streams with rate limits. IEEE/ACM Trans. Netw. 11(6), 948–958 (2003) 13. Lisovskaya, E., Moiseeva, S., Pagano, M.: The total capacity of customers in the infinite-server queue with MMPP arrivals. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) Distributed Computer and Communication Networks, pp. 110–120. Springer, Cham (2016) 14. Kononov, I., Lisovskaya, E.: Analysis of infinite-server queues with arrivals of random volume (in Russian). In: Proceedings of the XV International Conference Named After A. F. Terpugov, vol. 1, pp. 67–71. Publishing House TSU, Tomsk (2016) 15. Lisovskaya, E., Moiseeva, S.: Asymptotic analysis of non-Markovian infinite-server queueing with renewal arrivals of random volume customers (in Russian). Tomsk State University J. Control Comput. Sci. 39, 30–38 (2017)

“Controlled” Versions of the Collatz–Wielandt and Donsker–Varadhan Formulae Aristotle Arapostathis

and Vivek S. Borkar

Abstract This is an overview of the work of the authors and their collaborators on the characterization of risk-sensitive costs and rewards in terms of an abstract Collatz–Wielandt formula and in case of rewards, also a controlled version of the Donsker–Varadhan formula. For the finite state and action case, this leads to useful linear and dynamic programming formulations for the reward maximization problem in the reducible case. Keywords Principal eigenvalue · Risk-sensitive control · Collatz–Wielandt formula · Donsker–Varadhan functional

1 Introduction This short article is an overview of the work of authors and their collaborators on a somewhat novel perspective of the risk-sensitive control problem on infinite time horizon that aims to optimize the asymptotic growth rate of a mean exponentiated total reward, resp., cost. The viewpoint taken here is based on the fact that the dynamic programming principle for this problem essentially reduces it to an eigenvalue problem seeking the principal eigenvalue and eigenvector for a monotone positively 1-homogeneous operator. This allows us to exploit the existing generalized Perron–Frobenius (or Krein–Rutman) theory which leads to some explicit expressions for the optimal growth rate. The first is the abstract Collatz– Wielandt formula (see [1]) which can be shown to hold for both cost minimization and reward maximization problems, though we have not exhausted all the cases

A. Arapostathis Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX, USA e-mail: [email protected] V. S. Borkar () Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_13

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in our work. The second is a variational formula for the principal eigenvalue that generalizes the Donsker–Varadhan formula for the same in the linear case. This seems workable only for the reward maximization problem. We first consider the discrete time case based on the results of [2] in the next two sections, followed by those for reflected diffusions in a bounded domain, based on [5], in Sect. 4. We then sketch, in Sect. 5, the very recent and highly nontrivial extensions to diffusions on the whole space developed in [3] and [6]. Finally, we recall in Sect. 6 some developments in the simple finite state-action setup from [10], where the aforementioned development allows us to derive the dynamic programming equations for risk-sensitive reward process in the reducible case. Section 7 concludes by highlighting some future directions.

2 Discrete Time Problems The celebrated Courant–Fischer formula for the principal eigenvalue of a positive definite symmetric matrix A ∈ Rd×d is λ = max

0=x∈Rd

x T Ax . x Tx

Consider an irreducible nonnegative matrix Q ∈ Rd×d . The Perron–Frobenius theorem guarantees a positive principal eigenvalue with an associated positive eigenvector for Q. Is there a counterpart of the Courant–Fischer formula for this eigenvalue? The answer is a resounding “YES”! It is the Collatz–Wielandt formula for the principal eigenvalue of an irreducible nonnegative matrix Q = [q(i, j )] ∈ Rd×d , stated as (see [17, Chapter 8]): $ λ =

sup

min

x=[x1 ,··· ,xd ]T , xi ≥0 ∀i i : xi >0

$ =

inf

max

x=[x1 ,··· ,xd ]T , xi >0 ∀i i : xi >0

(Qx)i xi (Qx)i xi

%

%

An alternative characterization can be given as follows. Write Q = ΓP ,

.

Controlled Version of the Donsker–Varadhan Formula

201

where κi :=



q(i, j ) ,

1≤i ≤d,

j

Γ := diag(κ1 , . . . , κd ) , κi > 0 , p(i, j ) :=

q(i,j )/κi

,

1 ≤ i, j ≤ d ,

P := [p(j | i)] , with P a stochastic matrix. In other words, we have pulled out the row sums {κi } of Q into a diagonal matrix Γ so that what is left is a stochastic matrix P . Also define  G0 := (π, P˜ ) : π is a stationary probability

 for the stochastic matrix P˜ = [p(j ˜ |i)] .

Then the following representation holds [12]: * log λ =

sup

(π,P˜ ) ∈ G0



+ 6  7 , π(i) κi − D p(· ˜ | i) p(· | i)

i

where D(·  ·) denotes the Kullback–Leibler divergence or relative entropy. This is the finite state counterpart of the Donsker–Varadhan formula [14] for the principal eigenvalue of a nonnegative matrix. As is well known, the infinite dimensional generalization of the Perron– Frobenius theorem is given by the Krein–Rutman theorem [13, 16]. There are also nonlinear variants of it. Let 1. B be a Banach space with a “positive cone” K such that K − K is dense in B, 2. T : B → B be a compact order preserving (i.e., f ≥ g ⇒ Tf ≥ T g), strictly increasing (i.e., f > g ⇒ Tf > T g), strongly positive (i.e., maps nonzero elements of K to its interior), positively 1-homogeneous (i.e., T (af ) = aTf for all a > 0) operator. A nonlinear variant of the Krein–Rutman theorem [18] then asserts that under some technical hypotheses, a unique positive principal eigenvalue and a corresponding unique (up to a scalar multiple) positive eigenvector for T exist. Our interest is in the following nonlinear scenario arising in risk-sensitive control: Consider – a controlled Markov chain {Xn } on a compact metric state space S; – an associated control process {Zn } in a compact metric control space U ; – a per stage reward function r : S × U × S → R such that r ∈ C(S × U × S);

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– a controlled transition kernel p(dy | x, u) with full support, such that for all Borel A ⊂ S, P (Xn+1 ∈ A | Xm , Zm , m ≤ n) = P (Xn+1 ∈ A | Xn , Zn ) = p(A | Xn , Zn ) .

(1)

This is called the controlled Markov property and the controls for which this holds are said to be admissible. The maps  (x, u) → f (y)p(dy | x, u), f ∈ C(S), f  ≤ 1, are assumed to be equicontinuous. The control problem is to maximize the asymptotic growth rate of the exponential reward, that is, to achieve  N−1

1  λ := sup sup lim inf log E e m=0 r(Xm ,Zm ,Xm+1 )  X0 = x . x∈S {Zm } N↑∞ N The second supremum in this definition is over all admissible controls. We allow relaxed (i.e., probability measure valued) controls {μn } taking values in P(S), in which case (1) gets replaced by P (Xn+1 ∈ A | Xm , μm , m ≤ n) = P (Xn+1 ∈ A | Xn , μn )  = p(A | Xn , z)μn (dz), n ≥ 0 . Define Tf (x) :=

 sup

φ:S→P (U ) measurable

p(dy | x, u)φ(du | x)er(x,u,y)f (y) .

This is a compact, order preserving, strictly increasing, strongly positive, positively 1-homogeneous operator. Using the nonlinear variant of the Krein–Rutman theorem stated above, this leads to an abstract Collatz–Wielandt formula [2]: Theorem 1 There exist ρ > 0, ψ ∈ int(C + (S)) such that T ψ = ρψ and  Tf dμ ρ = inf sup  f dμ f ∈ int(C + (S)) M+ (S)  Tf dμ  . inf = sup + f dμ f ∈ int(C + (S)) M (S) Also, log ρ is the optimal reward for the risk-sensitive control problem.

Controlled Version of the Donsker–Varadhan Formula

203

3 Variational Formula We now state a variational formula for the principal eigenvalue [2]. Let G denote the set of probability measures η(dx, du, dy) ∈ P(S × U × S) which disintegrate as η(dx, du, dy) = η0 (dx)η1 (du | x)η2(dy | x, u) , such that η0 is invariant under the transition kernel  η2 (dy | x, u)η1 (du | x) . U

These are the so-called “ergodic occupation measures” for discrete time control problems. Theorem 2 Under the above hypotheses, 

$  log ρ = sup η∈G

η0 (dx)η1(du | x)

r(x, u, y)η2 (dy | x, u)

%  − D η2 (dy | x, u) p(dy | x, u) . 

This can be viewed as a controlled version of the Donsker–Varadhan formula. The hypotheses above can be relaxed to 1. Range(r) = [−∞, ∞) with er ∈ C(S × U × S); 2. p(dy | x, u) need not have full support. The formula then is the same as before, the difference is that under the previous, stronger set of conditions, the supremum over x ∈ S in the definition of λ was redundant, it is no longer so. The extension proceeds via an approximation argument that approximates the given transition kernel by a sequence of transition kernels for which our original hypotheses hold. We thus have an equivalent concave maximization problem, in fact a linear program, as opposed to a “team” problem one would obtain from the usual “log transformation” as in, e.g., [15]. Furthermore, if ρ(ϕ) denotes the asymptotic growth rate for a randomized Markov control ϕ, then it can be shown that ρ = maxϕ ρ(ϕ), implying the sufficiency of randomized Markov controls. Some applications worth noting are [2]: 1. Growth rate of the number of directed paths in a graph. This requires −∞ as a possible reward to account for the absence of edges.

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2. Portfolio optimization in the framework of [8]. 3. Problem of minimizing the exit rate from a domain.

4 Reflected Diffusions Analogous results hold for reflected diffusions in a compact domain with smooth boundary. These are described by the stochastic differential equation     dX(t) = b Xt , Ut dt + σ Xt dWt − γ (Xt ) dξt , dξ(t) = 1{Xt ∈ ∂Q} dξt ,

(2)

for t ≥ 0. Here: 1. Q is an open connected and bounded set with C 3 boundary ∂Q; 2. {Wt }t ≥0 is a standard d-dimensional Wiener process; 3. the control {Ut }t ≥0 lives in a metrizable compact action space U and is nonanticipative, i.e., for t > s, W (t) − W (s) is independent of X0 ; Wy , Uy , y ≤ s; 4. b is continuous, and x → b(x, u) is Lipschitz uniformly in u; 5. σ is C 1,β0 and uniformly non-degenerate; 6. γi (x) = σ(x)σ(x)T η(x), where η(x) is the unit outward normal on ∂Q. In contrast to the preceding section, we first consider the cost minimization problem to highlight the differences with the reward maximization problem. Unlike the classical cost/reward criteria such as discounted and average cost/reward, the risk-sensitive cost and reward problems are not rendered equivalent by a mere sign flip, and the differences are stark. For cost minimization, the control problem is to minimize lim sup t ↑∞

t 1 log E e 0 r(Xs ,Us ) ds , t

where r is continuous. The corresponding “Nisio semigroup” is defined as follows. For t ≥ 0, let St f (x) :=

t inf Ex e 0 r(Xs ,Us ) ds f (Xt ) .

{Ut }t≥0

¯ → C(Q) ¯ is a semigroup of strongly continuous, bounded Lipschitz, Then St : C(Q) monotone, superadditive, positively 1-homogeneous, strongly positive operators with infinitesimal generator G defined by Gf (x) :=



 1  tr σ(x)σT (x)∇ 2 f (x) + min b(x, u) , ∇f (x) + r(x, u)f (x) . u∈U 2 (3)

Controlled Version of the Donsker–Varadhan Formula

205

Let   ¯ := f : Q ¯ → ¯ ∇f (x), γ (x) = 0 for x ∈ ∂Q . Cγ2,+ (Q)  [0, ∞) : f ∈ C 2 (Q), As in the discrete case, the nonlinear Krein–Rutman theorem then leads to the ¯ satisfying following conclusions. There exists a unique pair (ρ, ϕ) ∈ R × Cγ2,+ (Q), ϕ(0) = 1, such that St ϕ = eρt ϕ . This solves Gϕ(x) = ρϕ(x) , x ∈ Q,

and ∇ϕ(x), γ (x) = 0 , x ∈ ∂Q .

The abstract Collatz–Wielandt formula for this problem is  ρ = =

inf

sup

¯ f ∈Cγ2,+ (Q),f >0

¯ ν∈P (Q)

sup

inf

¯ f ∈Cγ2,+ (Q),f >0

¯ ν∈P (Q)

Q¯

Gf dν f

Q¯

Gf dν . f



In the uncontrolled case, the first formula above is the convex dual of the Donsker– Varadhan formula for the principal eigenvalue of G: %

$ ρ =

sup

¯ ν∈P (Q)

Q¯

r(x)ν(dx) − I (ν)

,

where  $ I (ν) := −

inf

¯ f ∈Cγ2,+ (Q),f >0

Q¯

Lf f

% dν ,

with Lf (x) :=

 1  tr σ(x)σT (x)∇ 2 f (x) + b(x) , ∇f (x) . 2

For the risk-sensitive reward problem, the same abstract Collatz–Wielandt formula holds, except that the definition of the operator G now has a “max” in place of the “min.” But as in the discrete time case, one can go a step further and have a variational formulation. Let 1 R(x, u, w) := r(x, u) − |σT (x)w|2 , 2

(x, u, w) ∈ Q¯ × U × Rd ,

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and  M := μ ∈ P(Q¯ × U × Rd ) :  ¯ Q×U ×Rd

5 ¯ , Af (x, u, w)μ(dx, du, dw) = 0 ∀ f ∈ C 2 (Q) ∩ Cγ (Q)

with Af (x, u, w) :=

 8 9 1  tr σ(x)σT (x)∇ 2 f (x) + b(x, u) + σ(x)σT (x)w, ∇f (x) 2 (4)

¯ Recall the definition of an “ergodic occupation measure” for f ∈ C 2 (Q) ∩ C(Q). [4]. For a stochastic differential equation as in (2), but with the drift b replaced with b(x, u) + σ(x)σT (x)w, and w taking values in some compact metrizable space, this measure  is the time-t marginal of a stationary state-control process Xt , v(Xt ), w(Xt ) , perforce independent of t. Thus, in the case when the parameter w lives in a compact space, by a standard characterization of ergodic occupation measures (ibid.), M is precisely the set thereof for controlled diffusions whose (controlled) extended generator is A. This, however, is not necessarily the case if w lives in Rd . An example to keep in mind is the one-dimensional stochastic differential equation  2  √ X dXt = e t /2 − Xt dt + 2 dWt . It is straightforward to verify that the standard Gaussian density satisfies the Fokker– Planck equation. However, the diffusion is not even regular, so it does not have an invariant probability measure. Therefore, we refer to M as the set of infinitesimal ergodic occupation measures. The variational formula for this model is  ρ = sup

¯ ×Rd μ∈M Q×U

R(x, u, w)μ(dx, du, dw) .

This result is from [6]. An analogous abstract Collatz–Wielandt formula for the risk-sensitive cost minimization problem was derived in [5]. We have not derived a corresponding variational formula. Even if one were to do so, it is clear that it will be a “sup-inf/infsup” formula rather than a pure maximization problem. This is already known through a different route: it forms the basis of the approach initiated by Fleming and McEneaney [15] and followed by many, in which the Hamilton–Jacobi– Bellman equation for the risk-sensitive cost minimization problem is converted to an Isaacs equation for an ergodic payoff zero sum stochastic differential game. The aforementioned expression then is simply the value of this game. Going by pure

Controlled Version of the Donsker–Varadhan Formula

207

analogy, for the reward maximization problem, one would expect this route to yield a stochastic team problem wherein the two agents seek to maximize a common payoff, but non-cooperatively, i.e., without either of them having knowledge of the other person’s decision. What this translates into is that under the corresponding ergodic occupation measure, the two control actions are conditionally independent given the state. The set of such measures is non-convex. What we have achieved instead is a single concave programming problem, which is a significant simplification from the point of view of developing computational schemes for the problem. This also brings to the fore the difference between reward maximization and cost minimization in risk-sensitive control.

5 Diffusions on the Whole Space Here we consider a controlled diffusion in Rd of the form dXt = b(Xt , Ut ) dt + σ(Xt ) dWt , where 1. W is a standard d-dimensional Brownian motion; 2. the control Ut lives in a metrizable compact action space U and is nonanticipative, i.e., for t > s, W (t) − W (s) is independent of X0 ; Wy , Uy , y ≤ s; 3. b(x, u) is continuous and locally Lipschitz continuous in x uniformly in u ∈ U; 4. σ is locally Lipschitz continuous and locally non-degenerate; 5. b and σ have at most affine growth in x. Without loss of generality, we may take Ut to be adapted to the increasing σ -fields generated by {Xt , t ≥ 0}. Then these hypotheses guarantee the existence of a unique weak solution for any admissible control {Ut }t ≥0 ([4, Chapter 2]). As before, we let r(x, u) be a continuous running reward function, which is locally Lipschitz in x uniformly in u, and is also bounded from above in Rd . We define the optimal risk-sensitive value J ∗ by J ∗ := sup lim inf {Ut }t≥0 T →∞

T 1 log E e 0 r(Xt ,Ut ) dt , T

where the supremum is over all admissible controls. Consider the extremal operator 8

  9 . (x) := 1 trace a(x)∇ 2 f (x) + max b(x, u), ∇f (x) + r(x, u)f (x) Gf 2 u∈U

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. is defined by for f ∈ C 2 (Rd ). The generalized principal eigenvalue of G   . := inf λ ∈ R : ∃ φ ∈ W2,d (Rd ), ϕ > 0, Gφ . − λφ ≤ 0 a.e. in Rd , λ∗ (G) loc (5) d d where W2,d loc (R ) denotes the local Sobolev space of functions on R whose d d generalized derivatives up to order 2 are in Lloc (R ), equipped with its natural semi-norms. We assume that r − λ∗ is negative and bounded from above away from zero on the complement of some compact set. This is always satisfied if −r is an inf-compact function, that is the sublevel sets {−r ≤ c} are compact (or empty) in Rd × U for each c ∈ R, or if r is a positive function vanishing at infinity and the process {Xt }t ≥0 is recurrent under some stationary Markov control. Then there exists a unique positive Φ∗ ∈ C 2 (Rd ) normalized as Φ∗ (0) = 1 which . ∗ = λ∗ Φ∗ . In other words, the eigenvalue λ∗ = λ∗ (G) . is simple. Let solves GΦ ϕ∗ := log Φ∗ . As shown in [6], the function

H(x) :=

2 1  T σ (x)∇ϕ∗ (x) , 2

x ∈ Rd

is an infinitesimal relative entropy rate. We let Z := Rd × U × Rd , and use the single variable z = (x, u, w) ∈ Z. Let P(Z) denote the set of probability measures on the Borel σ -algebra of Z, and MA denote the set of infinitesimal ergodic occupation measures for the operator A in (4) defined for f ∈ C 2 (Rd ), which here can be written as  MA :=

 μ ∈ P(Z) :

Z

Af (z) μ(dz) = 0

5 ∀ f ∈ Cc2 (Rd ) ,

where Cc2 (Rd ) is the class of functions in C 2 (Rd ) which have compact support. Recall the definition R(x, u, w) := r(x, u) − 12 |σT (x)w|2 in Sect. 4. We also define  5  P∗ (Z) := μ ∈ P(Z) : H(x) μ(dx, du, dw) < ∞ , Z

 5  R(z) μ(dz) > −∞ . P◦ (Z) := μ ∈ P(Z) : Z

The following is a summary of the main results in [6, Section 4].

Controlled Version of the Donsker–Varadhan Formula

209

Theorem 3 We have . = J ∗ = λ∗ (G)

 sup

inf

Z

μ∈P∗ (Z ) g∈Cc2 (Rd )



=

max

μ∈MA-∩P∗ (Z )

Z

  Ag(z) + R(z) μ(dz)

R(z) μ(dz) .

Suppose that the diffusion matrix a is bounded and uniformly elliptic, and either |b|2 −r is inf-compact, or b, x− has subquadratic growth, or 1+|r| is bounded. Then MA ∩ P◦ (Z) ⊂ P∗ (Z), and P∗ (Z) may be replaced by P(Z) in the variational H is bounded, then formula above. If, in addition, 1+|ϕ | ∗

. = J ∗ = λ∗ (G)

 inf

sup

g∈Cc2 (Rd ) μ∈P (Z )

 Z

 Ag(z) + R(z) μ(dz) .

We continue with the Collatz–Wielandt formula in Rd for the risk-sensitive cost minimization problem. This is studied in [3]. Here, we have a running cost r(x, u) which is bounded from below in Rd × U, and is locally Lipschitz in x uniformly in u. The assumptions on b and σ are as stated in the beginning of the section, except that we may replace the affine growth assumption with the more general condition   sup b(x, u), x+ + σ(x)2 ≤ C0 1 + |x|2

∀ x ∈ Rd ,

u∈U

for some constant C0 > 0. The risk-sensitive optimal value Λ∗ is defined by Λ∗ :=

inf

{Ut }t≥0

lim sup T →∞

T

1 log E e 0 r(Xs ,Us ) ds . T

The operator G here is as in (3) but for f ∈ C 2 (Rd ), and we let the generalized principal eigenvalue λ∗ (G) be defined as in (5). The running cost does not have any structural properties that penalize unstable behavior such as near-monotonicity or inf-compactness, so uniform ergodicity for the controlled process needs to be assumed. Let Lf (x, u) :=

 8 9 1  tr σ(x)σT (x)∇ 2 f (x) + b(x, u), ∇f (x) . 2

We consider the following hypothesis.

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Assumption 1 The following hold. (i) There exists an inf-compact function  ∈ C(Rd ), and a positive function V ∈ d W2,d loc (R ), satisfying infRd V > 0, such that sup LV(x, u) ≤ κ1 1K (x) − (x)V(x) ∀ x ∈ Rd ,

(6)

u∈U

for some constant κ1 and a compact set K. (ii) The function x → β(x) − maxu∈U r(x, u) is inf-compact for some β ∈ (0, 1). As noted in [7], the Foster–Lyapunov equation in (6) cannot in general be satisfied for diffusions with bounded a and b. Therefore, to treat this case, we consider an alternate set of conditions. Assumption 2 The following hold. d (i) There exists a positive function V ∈ W2,d loc (R ), satisfying infRd V > 0, constants κ1 and γ > 0, and a compact set K such that

sup LV(x, u) ≤ κ1 1K (x) − γ V(x)

∀ x ∈ Rd .

u∈U

(ii) r − ∞ + lim sup|x|→∞ maxu∈U r(x, u) < γ . is,

Let o(V) denote the class of continuous functions f that grow slower than V, that |f (x)| → 0 as |x| → ∞. We quote the following result from [7]. V(x)

Theorem 4 Grant either Assumption 1, or 2. Then Λ∗ = λ∗ (G) =

 sup

inf

d f ∈C 2,+ (Rd )∩o(V) μ∈P (R )



=

inf

sup

f ∈C 2,+ (Rd ) μ∈P (Rd )

Rd

Rd

Gf dμ f

Gf dμ , f

(7)

where C 2,+ (Rd ) denotes the set of positive functions in C 2 (Rd ). We should remark here that the class of test functions f in the first representation formula in (7) cannot, in general, be enlarged to C 2,+ (Rd ). It is also interesting to consider the substitution f = eψ . Then (7) transforms to  λ∗ (G) =

sup

inf

d ψ∈C 2,+ (Rd )∩o(log V) μ∈P (R )



=

inf

sup

ψ∈C 2,+ (Rd ) μ∈P (Rd )

Rd

Rd

F [ψ](x) μ(dx)

F [ψ](x) μ(dx) ,

Controlled Version of the Donsker–Varadhan Formula

211

with F [ψ](x) := inf sup

u∈U w∈Rd

6 7 Aψ(x, u, w) + R(x, u, w) .

This underscores the discussion in the last paragraph of Sect. 4.

6 Finite State and Action Space For discrete time problems with finite state and action spaces (i.e., |S|, |U | < ∞ in Sects. 2 and 3), one can go significantly further for the reward maximization problem. We recall below some results in this context from [10]. Consider a controlled Markov chain {Yn } on S with state-dependent action space at state i given by U˜ i := ∪u∈U ({u} × Vi,u ) , where Vi,u :=

5   q(j | i, u) = 1 . q(· | i, u) : q(· | i, u) ≥ 0, j

This is isomorphic to P(S). Let K := ∪i∈S ({i} × U˜ i ) . The (controlled) transition probabilities of {Yn } are   p˜ j | i, (u, q(· | i, u)) := q(j | i, u) . Define the per stage reward r˜ : K × S → R by     r˜ i, (u, q(· | i, u)), j := r(i, u, j ) − D q(· | i, u) p(· | i, u) . Let {(Zn , Qn ), n ≥ 0} denote the U˜ Yn -valued control process. Consider the problem: Maximize the long run average reward lim inf N↑∞

N−1 7 1  6 E r˜ (Yn , (Zn , Qn ), Yn+1 ) . N n=0

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Define the corresponding ergodic occupation measure γ ∈ P(K × S) by   γ (i, (u, dq), j ) := γ1 (i)γ2 (u, dq | i)γ3 j | i, (u, q) , where γ1 is an invariant probability distribution (not necessarily unique) under the transition kernel    γˇ (j | i) = γ2 (u, dq | i)γ3 j | i, (u, q) . u

Vi,u

Let E denote the set of such γ . The above average reward control problem is equivalent to the linear program: P0

Maximize 

γ (i, (u, dq), j )˜r (i, (u, q), j )

i,j,u

over E. Recall that E is specified by linear constraints and its extreme points correspond to stationary Markov policies ([9, Chapter V]). The maximum will be attained at an extreme point of E corresponding to a stationary Markov policy. This LP can be simplified as Maximize    γ " (i, u, j ) r(i, u, j ) − D q(· | i, u) p(· | i, u) i,j

over E˜ :=



γ " ∈ P(S × U × S) : γ " (i, u, j ) = γ1 (i)ϕ(u | i)q(j | i, u), where γ1 (·)   is invariant under the transition kernel γ˘ (j | i) := u ϕ(u | i)q(j | i, u) . The dual LP is: Minimize λ˘ subject to λ˘ ≥ λ(i) ,    λ(i) + V (i) ≥ q(j | i, u) r˜ (i, (u, q(· | i, u)), j ) + V (j ) , j

λ(i) ≥



q(j | i, u)λ(j ) , ∀ i ∈ S, (u, q(· | i, u)) ∈ U˜ i .

j

The proof goes through finite approximations. Note that the LP has infinitely many constraints. However, it does pave the way for the corresponding dynamic

Controlled Version of the Donsker–Varadhan Formula

213

programming principle. The dynamic programming formulation equivalent to the above LP turns out to be as follows: λ∗ = max λ(i) , i

λ(i) + V (i) = λ(i) =

max



(u,q(· | i,u))∈Bi

max

(u,q(· | i,u))∈Bi

  q(j | i, u) V (j ) + r˜ (i, (u, q(· | i, u), j )) ,

j



q(j | i, u)λ(j ) ,

(†)

j

∀i ∈S, where Bi is the Argmax in (†). Once again, the proof goes through finite approximations.

7 Future Directions There are several directions left uncharted in this broad problem area. Some of them are listed below. 1. There are some in-between cases that need to be analyzed, e.g., controlled Markov chains with countably infinite state space. Under the strong “Doeblin condition,” the abstract Collatz–Wielandt formula has been derived for these in [11]. This needs to be extended to more general cases. 2. The counterpart of the dynamic programming equations derived for reducible risk-sensitive reward processes can also be expected to hold for risk-sensitive cost problems and is yet to be established. 3. Concrete computational schemes based on approximate concave maximization problems is another direction worth pursuing. Acknowledgments The work of A.A. was supported in part by the National Science Foundation through grant DMS-1715210, and in part the Army Research Office through grant W911NF-171-001. The work of V.S.B. was supported by a J. C. Bose Fellowship from the Government of India.

References 1. Akian, M., Gaubert, S., Nussbaum, R.: A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones (2011). arXiv1112.5968 2. Anantharam, V., Borkar, V.S.: A variational formula for risk-sensitive reward. SIAM J. Control Optim. 55(2), 961–988 (2017). https://doi.org/10.1137/151002630

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3. Arapostathis, A., Biswas, A.: A variational formula for risk-sensitive control of diffusions in Rd . SIAM J. Control Optim. 58(1), 85–103 (2020). https://doi.org/10.1137/18M1218704 4. Arapostathis, A., Borkar, V.S., Ghosh, M.K.: Ergodic Control of Diffusion Processes. Encyclopedia of Mathematics and its Applications, vol. 143. Cambridge University Press, Cambridge (2012) 5. Arapostathis, A., Borkar, V.S., Kumar, K.S.: Risk-sensitive control and an abstract CollatzWielandt formula. J. Theor. Probab. 29(4), 1458–1484 (2016). https://doi.org/10.1007/s10959015-0616-x 6. Arapostathis, A., Biswas, A., Borkar, V.S., Suresh Kumar, K.: A variational characterization of the risk-sensitive average reward for controlled diffusions in Rd (2019). arXiv1903.08346 7. Arapostathis, A., Biswas, A., Saha, S.: Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control. J. Math. Pures Appl. 124, 169–219 (2019). https:// doi.org/10.1016/j.matpur.2018.05.008 8. Bielecki, T., Hernández-Hernández, D., Pliska, S.R.: Risk sensitive control of finite state Markov chains in discrete time, with applications to portfolio management. Math. Methods Oper. Res. 50(2), 167–188 (1999). https://doi.org/10.1007/s001860050094 9. Borkar, V.S.: Topics in Controlled Markov Chains. Pitman Research Notes in Mathematics, vol. 240. Longman Scientific and Technical, Harlow (1991). https://doi.org/10.1007/978-14612-5320-4 10. Borkar, V.S.: Linear and dynamic programming approaches to degenerate risk-sensitive reward processes. In: 56th IEEE Annual Conference on Decision and Control (CDC). pp. 3714–3718 (2017). https://doi.org/10.1109/CDC.2017.8264204 11. Cavazos-Cadena, R.: Characterization of the optimal risk-sensitive average cost in denumerable Markov decision chains. Math. Oper. Res. 43(3), 1025–1050 (2018). https://doi.org/10. 1287/moor.2017.0893 12. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Applications of Mathematics, vol. 38, 2nd edn. Springer, New York (1998). https://doi.org/10.1007/978-14612-5320-4 13. de Pagter, B.: Irreducible compact operators. Math. Z. 192(1), 149–153 (1986). https://doi.org/ 10.1007/BF01162028 14. Donsker, M.D., Varadhan, S.R.S.: On a variational formula for the principal eigenvalue for operators with maximum principle. Proc. Nat. Acad. Sci. U.S.A. 72, 780–783 (1975). https:// doi.org/10.1073/pnas.72.3.780 15. Fleming, W.H., McEneaney, W.M.: Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33(6), 1881–1915 (1995). https://doi.org/10.1137/S0363012993258720 16. Kre˘ın, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat. Nauk. 3(1(23)), 3–95 (1948) 17. Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). https://doi.org/10.1137/1.9780898719512 18. Ogiwara, T.: Nonlinear Perron-Frobenius problem on an ordered Banach space. Jpn. J. Math. 21(1), 42–103 (1995). https://doi.org/10.4099/math1924.21.43 19. Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York (1994)

An (s, S) Production Inventory System with State Dependant Production Rate and Lost Sales S. Malini and Dhanya Shajin

Abstract In this paper, the system under study is a production inventory system that follows (s, S) replenishment policy and having state dependent production rate. The system considered has infinite capacity where customers arrive according to Poisson process. The service time follows exponential distribution. Further in the system, when the inventory level depletes to s, the production process is switched on and is kept on till the inventory level reaches its maximum capacity S. The production time follows exponential distribution with parameter θi , where i represents number of items in the inventory and 0 ≤ i ≤ S − 1. It is assumed that no new customers join the queue when there is void inventory. This yields an explicit product form solution for the steady state probability vector of the system, though there exists a dependence relationship between number of customers joining the queue and time interval for which the production process is turned on. Long run performance measures are computed and lost sales of the system is analysed. A comparison chart that points out the reduction of lost sales with state dependent production rate is also provided along with numerical illustrations for the performance measures. An expected cost function is constructed to numerically investigate the optimal (s, S) pair. Keywords Production inventory system · (s,S) reordering policy · State dependent production rate · Stochastic decomposition

1 Introduction Queueing theory is the branch of Mathematics that models and analyses queues or waiting lines. The aim of queueing theory is to develop mathematical models that can predict system behaviour. The system under consideration is those that

S. Malini () · D. Shajin Department of Mathematics, Amrita School of Arts and Sciences, Amrita Vishwa Vidyapeetham, Kochi, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_14

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provide service for arbitrarily rising demands. Queueing systems with inventory control are one of the areas under focus for the past decade. By inventory, we mean any physical material stored, under process or waiting for processing in a system like raw materials, goods, etc. or any item served to the customer after his service period. Queueing system with inventory is studied under different heads as—Single Server Queueing Systems with Inventory, Queueing Inventory System with Stochastic Environment, Queueing Inventory System with Substitution Flexibility, Queueing System with Production-Inventory, Queueing System with Service Inventory, Continuous Review Inventory Systems with Server Vacation, Queueing Inventory System with Postponed Demands/Customers, Queueing Systems with New Inventory Models [4]. Here we consider an (s, S) production inventory system where the production mode is also taken into consideration along with inventory management. The reordering policy under study is (s, S) policy where s is the reorder level and S is the maximum inventory level. Turning the pages of development, a primary study on inventory with positive service time was done by Sigman and Simchi-Levi [10], in which a FIFO M/G/1 queue is considered with customer arrival that follows Poisson process and time taken for service having an arbitrary distribution and each customer requires one item of inventory. A maximum capacity level and a minimum reorder level are also considered in the system. Followed by this was the study made by Berman et al. [2] in which a model was developed for an inventory management system. The service mode of the system under consideration was to serve single unit of inventory on the completion of each service. Also, it was assumed that there exists a deterministic nature for arrival and service processes and the formation of queue happens only when inventories are out of stock. An optimization for order quantity minimizing the cost factor was also developed through the paper. Schwarz and Daduna [9] in their work determined stationary distribution of the system whose reorder point is taken to be zero. Various reordering policies like (r, S), (r, Q) and some general randomized order policies are considered for study. They derived product form solutions assuming no customer joining happens when the inventory level is zero. Later Saffari and Haji[8] studied M/M/1 queueing inventory system under (r, Q) policy whose lost sales, stationary distributions of arising demands in the system and inventory level in the system considering lead times as stochastic variables were derived. Several performance measures were also derived along with cost analysis. Related works in production industry include work by He and Jewkes [3] which examines a production system in which demands follow Poisson process which are processed as per FCFS principle. Using Markovian decision process approach an optimal replenishment policy is arrived at. A recent development in this area is the work by Baek and Moon [1] that evaluates a production inventory system as an M/M/1 queue. The customer arrival is presumed to occur as a Poisson process with a single server to render service that follows exponential distribution. The replenishment of items happens either from an external agent who follows (r, Q) policy or from an internal production system, where production process is supposed to follow Poisson process. The product form solution in terms of joint probability

An (s, S) Production Inventory System with State Dependant Production Rate. . .

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is arrived at and applying the same, performance measures and cost model are developed. In our study, the work by Krishnamoorthy et al. [7] requires special mention. A queueing inventory system under (r, S) and (s, S) policy is analysed to obtain joint probability distribution of number of customers and inventory level. Krishnamoorthy and Viswanath [5] published the first work in the case of a production inventory system. The paper considered a production inventory system with Markovian arrival process, Markovian production times, service time following phase type distribution, under (s, S) reordering policy. Under stability condition, state distributions and various performance measures are evaluated. Current study in this paper finds its motivation from the paper—Stochastic decomposition in production inventory with service time by Krishnamoorthy and Viswanath [6] where demands follow Poisson process, service rates and production times follows exponential distribution under a constant rate. The current paper develops a comparative study with the former one in the sense that, here we assume state dependent production rates. Remaining paper is developed in the following manner. Section 2 provides a model description, followed by system analysis in Sect. 3. In Sect. 4 performance measures of the system are evaluated and in Sect. 5 the production cycle is analysed. Following it, in Sect. 6, an expected cost function per unit time for the system under consideration is developed for which numerical examples are provided. An optimal value of s and S is arrived at from the illustrations.

2 Model Description Here we consider an (s, S) production inventory system with single server and infinite capacity. The customer arrival occurs according to Poisson process with rate, λ. The services follow exponential distribution with rate, μ. The production is switched on when the inventory level diminishes to s and is switched off as soon as it reaches to S. The switch on mode is denoted by 1 and switch off mode by 0. Thus, when the inventory level is between s + 1 and S − 1, the production mode can be either 0 or 1. It is assumed that each production is of 1 unit and the production times follow the exponential distribution with parameter, θi where i represents the number of items in the inventory. Also, 0 ≤ i ≤ S − 1. Basic assumptions chosen for this model are as follows: • No customer joins the queue when the inventory level is zero. • The product takes negligible time to reach the retail shop from the production unit.

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2.1 Mathematical Model The above conditions are modelled using  = {N(t), I (t), C(t); t ≥ 0} which is a Continuous Time Markov Chain (CTMC). Here N(t) represents the number of customers, I (t) represents the number of items in the inventory at the production centre and C(t) represents the production mode as on or off. Also, C(t) = 1 for 0 ≤ I (t) ≤ s = 0 for I (t) = S = 0 or 1 for s + 1 ≤ I (t) ≤ S − 1 The state space of the above process is given by {(n, i, 1); n ≥ 0, 0 ≤ i ≤ S − 1} ∪ {(n, i, 0), n ≥ 0, s + 1 ≤ i ≤ S} When N(t) = n, known as the level of the system, there are 2S − s states in the level.

2.2 Notations In the following model explanation, below mentioned notations are used: • • • • • • •

Im → Identity matrix of order m en → Column vector of 1’s of order n × 1 e → Column vector of 1’s of respective order 0 → Zero matrix of respective order i → Number of customers (0 to ∞) j → Number of inventory (0 to S) k → Production mode (0 or 1)

2.3 Transitions For the CTMC  = {(N(t), I (t), C(t); t ≥ 0)} is given by • Customer Arrival (rate λ) (i, j, 1) → (i + 1, j, 1) for i ≥ 0; 1 ≤ j ≤ S − 1 (i, j, 0) → (i + 1, j, 0) for i ≥ 0; s + 1 ≤ j ≤ S

An (s, S) Production Inventory System with State Dependant Production Rate. . .

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• Service Process (rate μ) (i, j, 1) → (i − 1, j − 1, 1) for i ≥ 1; 1 ≤ j ≤ S − 1 (i, s + 1, 0) → (i − 1, s, 1) for i ≥ 1 (i, j, 0) → (i − 1, j − 1, 0) for i ≥ 1; s + 2 ≤ j ≤ S • Production Process (rate θi ) (i, j, 1) → (i, j + 1, 1) for i ≥ 0; 0 ≤ j ≤ S − 2(i, S − 1, 1) → (i, S, 0) for i ≥ 0 • For the remaining transitions we have the rate zero.

2.4 Infinitesimal Generator The infinitesimal generator for the CTMC  applying the transitions described is given by ⎛

⎞ Q R0 ⎜ R2 R1 R0 ⎟ ⎜ ⎟ ⎜ ⎟ R2 R1 R0 G=⎜ ⎟ ⎜ ⎟ R2 R1 R0 ⎝ ⎠ .. .. .. . . .

(1)

Here, • • • •

R0 is the arrival matrix that represents the transition rates of customer arrival R2 is the service matrix that represents the transition rates of service times and R1 represents the system stay in same state. R00 = Q is given by ⎛

⎞

⎜ ⎜ ⎜ Q=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−θ0 θ0 ⎜ 0 −(λ + θ1 ) (θ1 ) ⎜ 0 −(λ + θ2 ) ⎜ .. .. ⎜ . . ⎜ ..

.

..

. −(λ + θs−1 )

θs −(λ + θs ) N " 0 Jl Nl 0 Jl .. .. . . ..

. N" 0

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with 7 6 N " = 0 θs ,

0 N "" = , θs−1

0 0 Nl = for s + 1 ≤ l ≤ S − 1, 0 θl

−(λ) 0 Jl = for s + 1 ≤ l ≤ S − 1, 0 −(λ + θl )

0 0 R0 = 0 λIm−1 R1 = Q −

μ R0 λ

⎛ ⎜μ 0 ⎜ ⎜ μ0 ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ R2 = ⎜ μ 0 ⎜ ⎜ H1 0 ⎜ ⎜ H2 0 ⎜ ⎜ Jl ⎜ ⎜ .. .. ⎜ . . ⎝ .. . H3 with



6 7 μ μ0 H1 = , H2 = , H3 = μ 0 μ 0μ

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0

An (s, S) Production Inventory System with State Dependant Production Rate. . .

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3 System Analysis Here we analyse the state of the system.

3.1 Stability Condition For the same, we first establish the stability condition by defining D = R0 +R1 +R2 as the generator matrix. Then D has a steady state probability vector. Let it be !. Then ! can be decomposed as (φ0 , φ1 , φ2 , . . . , φs , φ(s+1)0, φ(s+1)1, . . . , φS0 ). The system under study is similar to a LIQBD (Level Independent Quasi Birth Death Process). Hence the condition for stability is given by ! R0 e < ! R2 e ⇒ λ < μ

3.2 Steady State Probability Vector For evaluating the steady state vector of the process , we assume that the production inventory system performs with negligible service time and there are no backlog ˜ = {I (t), C(t); t ≥ 0} of demands. The respective Markov Chain is given by  where I (t) is the inventory level and C(t) is the production mode. The state space of the above state process is given by s i=0

{i}

S i=s=1

{((i, 0), (i, 1))}



S

˜ = {I (t), C(t); t ≥ 0} is The infinitesimal generator for the above state process  given by ⎛

⎞

⎜ ⎜ ⎜ G˜ = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−θ0 θ0 ⎜ λ −(λ + θ1 ) θ1 ⎜ 0 λ −(λ + θ2 ) ⎜ .. .. ⎜ . . ⎜ ..

.

..

. −(λ + θs−1 ) θs−1 0 λ −(λ + θs ) 0 H˜1 0

N" Jl Nl 0 H˜2 Jl H2 0 .. .. . . Jl N" H˜3 −λ

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λ λ λ H1 , H˜2 = H2 , H˜3 = H3 and all other matrices are as described μ μ μ for the process . ˜ let π be the steady state probability vector. For the process ,

where H˜1 =

⇒ π = (π0 , π1 , π2 , . . . , πs , π(s+1)0, π(s+1)1, . . . , πS ). Then π satisfies the equations ˜ =0 πG πe = 1 On solving, the various components of π can be obtained as πi0 = πS0 for s + 1 ≤ i ≤ S − 1. λS−i S−s j :s+(j −1) 1 πi1 = λ πS0 for 0 ≤ i ≤ s − 1 j =1 k=i θi θk S−i :i+(j −1) 1 πi1 = λj πS0 for s ≤ i ≤ S − 1 j =1 k=i θk Applying the normalizing condition, πS0

⎧ ⎡ s−1 S−s ⎨ λS−i  j ⎣ = λ ⎩ θi i=0

j =1m

s+(j :−1) k=i

⎤ 1 ⎦+ θk

S−1  i=s

⎫−1 ⎡ ⎤ i+(j S−i ⎬  :−1) 1 ⎣ ⎦ + (S − s) λj ⎭ θk j =1

k=i

The steady state probability vector for the original system under study is computed using π. Let x be the steady state probability vector of the original system. Then x must satisfy the equations xG = 0 and xe = 1. Now x can be partitioned as (x0 , x1 , . . . , xS ) corresponding to the levels. Here each xi can be partitioned into • P (number of customers in the system) • P (number of items in the inventory at the production centre and the production mode k) xi (j, k) = pi πj (k) Let xi = Kρ i π, i ≥ 0.

An (s, S) Production Inventory System with State Dependant Production Rate. . .

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Here ρ = μλ and K is a constant to be determined. On computation the value of K is found to be 1 − ρ, for: Considering the equations, xG = 0 and xe = 1. xG = 0 on simplification gives x0 Q + x1 R2 = 0

(2)

xi−1 R0 + xi R1 + xi+1 R2 = 0, otherwise

(3)

Now we aim to check if this system of equations are satisfied: We have from Eq. (2) x0 Q + x1 R2

λ = Kπ Q + R2 μ But from structure of matrices we have Q+

λ ˜ and π G ˜ =0 R2 = G μ

⇒ x0 Q + x1 R2 = 0 Also, from Eq. (3) xi−1 R0 + xi R1 + xi+1 R2   % $ λ λ 2 i = Kρ π R0 + R1 + R2 μ deμ   % $ λ λ 2 μ  i R2 = Kρ π R0 + Q − R0 + μ λ deμ

λ i−1 = Kρ π Q + R2 μ ˜ = Kρ i−1 π G =0 Hence, the condition xG = 0 is satisfied. Now, applying the normalizing condition xe = 1 we have   K 1 + ρ + ρ2 + · · · = 1 ⇒ K = 1 − ρ

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Thus, the steady state probability vector x is given by xi = (1 − ρ) ρ i π

where ρ =

λ μ

Theorem Under necessary and sufficient condition, λ < μ, the steady state probability vector of the system under consideration has a product form decomposition and can be written as the product of probability of number of customers in the system and probability of number of items in the inventory. xi = (1 − ρ) ρ i π

where ρ =

λ μ

with π = (π0 , π1 , π2 , . . . , πs , π(s+1)0, π(s+1)1, . . . , πS ) and πi0 = πS0 for s + 1 ≤ i ≤ S − 1. λS−i S−s j :s+(j −1) 1 λ πS0 for 0 ≤ i ≤ s − 1 j =1 k=i θi θk S−i :i+(j −1) 1 = λj πS0 for s ≤ i ≤ S − 1 j =1 k=i θk

πi1 = πi1 πS0 =





S−1 S−i j :i+(j−1) 1 λS−i S−s j :s+(j−1) 1 λ λ + i=0 j=1m k=i i=s j=1 k=i θi θk θk 5−1 + (S − s) s−1



4 System Performance Measures • Mean number of customers in the system, Ecust =

λ μ−λ

• Expected number of items in the inventory in the system, Einvent =

s  i=0

i · πi1 +

S−1  s+1

i · (πi1 + πi0 ) + SπS0

An (s, S) Production Inventory System with State Dependant Production Rate. . .

225

• Expected rate of production, Epro.rat e =

s 

S−1 

θ i · πi +

i=0

θi · πi1

i=s+1

• Expected loss of customers, Ec.loss = λ · π01 • Expected production switch on rate, Eson = λ · πS

5 Production Cycle By production cycle we mean the time period during which the production process is switched on. Let the production process be switched on at the time epoch T0 when there are s + 1 inventories in the system and a service has been completed. Till the epoch T0 the production mode is kept off. Once the production process is turned on, it is kept in on mode till the inventory level reaches to the maximum capacity, S. Let T1 be that time epoch. Thus, the length of the production cycle will be T1 − T0 .

5.1 Mathematical Model The production cycle can be modelled as the time until absorption for the Markov Chain " = {(N(t), I (t)) ; t ≥ 0}. Here, N(t) is the number of customers and I (t) is the inventory level in the system. The state space of the above process is given ∞ > > by, {(i, j ) |0 ≤ j ≤ S − 1} {$} where $ represents the absorbing state where i=0

we switch off the production. For the Markov chain " all the transitions happen in the same manner as for  except for the absorbing state $. The infinitesimal generator for the above process " is given as C=

A −Ae 0 0



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where

⎡ A=

T00 T0 ⎢ T2 T1 ⎢ 0 T2 ⎢ . ⎢ .. ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

T0 T1 T0 .. .. . . .. .. .. . . .

in which

⎛

T00

⎛ ⎜ ⎜ ⎜ ⎜ T1 = ⎜ ⎜ ⎜ ⎝

⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

−θ0

−θ0

θ0 −(λ + θ1 ) ..

.

⎞ θ1 −(λ + θ2 ) .. . ..

.

..

. −(λ + θs−1 )

θ0 −(λ + μ + θ1 ) ..

.

θS−2 −(λ + θS−1 )

⎞ θ1 −(λ + μ + θ2 ) .. . ..

.

..

. −(λ + μ + θS−2 )

⎡ T2 =

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

θS−2 −(λ + μ + θS−1 )

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎤

0 0 ⎢μ 0 ⎥ ⎢ .. .. ⎥ ⎢0 . . ⎥ , T0 ⎢ ⎥ .. .. ⎣ ⎦ . . 0 0 0 0 μ 0

=



0 0 0 λ · IS−1

5.2 Steady State Analysis Let zi (j ) denote the expected time until absorption of the process " from the state (i, j ). Define the row vector zT such that   zT = z0 T , z1 T , . . . . where each zi is the column vector with S elements. Also, let τi (j ) be the probability of switching on the production process. At this point, there are ‘i’ customers and ‘j ’ inventories in the system. Define τ to be the probability vector such that τ = (τ0 , τ1 , . . . .)

An (s, S) Production Inventory System with State Dependant Production Rate. . .

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Here each τi is a vector with dimension S × 1. It is to be noted that when j = s, τi (j ) = 0. Thus, τi (s) can be calculated using the values of x. τi (s) = (1 − ρ) ρ i , for i ≥ 0. Now, the expected length of production cycle denoted by EP C is given by EP C = τ · z =

∞ i=0

· ρ · ρ i · zi (s)

To find the vector z, we apply the concept that z satisfies the equations: Cz = −e ⇒ T00 · z0 + T0 · z1 = −e T2 · zi−1 + T1 · zi + T0 · zi+1 = −e for i ≥ 1

(4)

The above system of equations are solved by assuming that the production system is having instantaneous service without any backlogs. Let the expected length of production cycle under this assumption be denoted by EP˜ C . ˜ = {(Y (t)) ; t ≥ 0} whose absorbing To solve for EP˜ C , consider the CTMC " state is ∇. Here, Y (t) represents the inventory level during the course of production ˜ is given by cycle at time t. The state space of the above Markov Chain " C=

J −Je 0 0



where ⎛

θ0 −θ0 ⎜ λ −(λ + θ ) θ1 1 ⎜ ⎜ λ −(λ + θ2 ) ⎜ 0 ⎜ .. .. ⎜ . . J =⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ ⎝ λ −(λ + θS−2 ) θS−2 λ −(λ + θS−1 )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Let −J −1 e = (Y0 , Y1 , . . . , Ys−1 ) be the column vector whose (s + 1)th entry is EP˜ C . By the relation J (−J −1 e) = −e we arrive at the following set of equations: −θ0 · Y0 + θ0 · Y1 = −1 λ · Yi−1 + (λ + θi · Yi + θi · Yi+1 ) = −1, for 1 ≤ i ≤ S − 2 λ · YS−2 + (λ + θS−1 ) · YS−1 = −1

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S. Malini and D. Shajin

On computation, the above set of equations yields i

Yi − Yi+1 =

j =0

S−1

YS−1 =

j =0

λi−j

1 ,0 ≤ i ≤ S − 2 k=j θk :S−1  1 

:i

λS−1−j

θk

k=j

A solution to the above set of equations gives Ys =

S−2 l j =0

l=s

l−j

λ

 1  S−1 :S−1  1  + λS−1−j k=j θk j =0 k=j θk

:l

This gives the expected length EP˜ C . Now, from Eq. (4) we have $ % λ  λ 2

∞ $ λ % i λ T1 + T2 zi+1 T00 + · T2 z0 + T0 + i=0 μ μ μ μ ∞  λ i =− ·e i+0 μ

(5)

From the structure of matrices it can be observed that λ T2 = J μ λ λ T1 = T00 T0 + μ μ λ  λ 2 λ T1 + J T0 + T2 = μ μ μ T00 +

Applying the above relation to Eq. (5), it reduces to J

∞  λ i 1 ·e · zi = i=0 μ K ∞  λ i ⇒ K · · zi = −(J −1 )e i=0 μ

(6)

From Eqs. (4) and (6) it follows that the expected length of production cycle and the expected length of production cycle with instantaneous service are equal. Thus, EP C =

S−2 l l=s

j =0

λl−j

:l k=j

$

1 θk

%

+

S−1 j =0

λS−1−j

:S−1 $ 1 % k=j θk

An (s, S) Production Inventory System with State Dependant Production Rate. . .

229

6 Analysis of Expected Cost Function per Unit Time Based on the above performance measures, an expected cost function per unit time is created and the optimal values of s and S are evaluated. The cost function is given by Fc = Einvent ∗ Cinvent + Epro.rat e ∗ Cpro.rat e + Ec.loss ∗ Cc.loss + M ∗ Eson where • • • •

Cinvent is the holding cost per inventory per unit time. Cpro.rat e is the cost for producing unit inventory per unit time Cc.loss is the cost occurred due to loss of customers and M is the fixed cost for starting the production.

Note: In all the below mentioned cases we have followed an assumption that the variable production rates follow a relation with each other as θi = (S − (i − 1)) · θ for 1 ≤ i ≤ S.

6.1 Effect of Maximum Inventory Level S The effect of maximum inventory level S on cost function and various performance measures discussed above is tabulated as per the tables. Here, the values of s, λ, μ, θ and other basic costs are assumed to be constant throughout the calculations. Case 1: When θ = 6 An optimal value for the cost function is obtained and is indicated in bold letters as in Table 1. From the table it can be concluded that as the value of S increases, the value of Einvent also increases. Coming to production rates, as the maximum inventory level increases, the expected production rate also increases. Also it can be seen that as the maximum inventory level increases, the expected loss is found Table 1 Effect of max inventory level S for θ = 6, λ = 5, μ = 12, s = 3, Cinvent = 50; Cpro.rate = 200; Cc.loss = 400; M = 2000

S Einvent 6 4.65957 7 5.28534 9 6.476668 10 7.052521 13 8.731847 20 12.501280

Epro.rat e 7.665957 8.111103 10.131786 11.317054 15.841013 26.930208

Ec.loss 0.001216912 0.000357524 0.000061156 0.000031314 0.000000651 0.000000584

EP C 937 484.3 × 101 122.030 × 103 610.311 × 103 762.939 × 105 506.046 × 1015

Es.on 0.400705 0.206020 0.206020 0.061866 0.030666 0.010971

FC 2568.066 2298.671 2519.370 2739.782 3666.131 6033.048

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Table 2 Effect of max inventory level S for θ = 2, λ = 5, μ = 12, s = 3, Cinvent = 50; Cpro.rate = 200; Cc.loss = 400; M = 2000

S 7 9 10 13 15 20

Einvent 4.250082 5.904106 6.709735 9.031418 10.510924 14.028628

Epro.rat e 5.124119 3.654156 3.271005 3.021602 3.371441 4.774770

Ec.loss 0.059368 0.008299 0.003417 0.000533 0.000235 0.000050

EP C 4843 1220.30 × 102 6103.11 × 102 7629.39 × 104 1907.35 × 106 5960.46 × 109

Es.on 0.797907 0.267705 0.157480 0.058634 0.041093 0.022298

FC 2856.891 1564.768 1306.016 1173.374 1282.116 1701.004

to follow a decreasing pattern. This is because there is less chance for backlogs of demands when there is a large capacity level. As the maximum inventory level increases, the length of production cycle also increases, i.e., the time for which the system must be kept on to reach maximum level will increase with S. Another observation is that the expected rate at which production is switched on decreases with increasing value of S, because there is less chance that the inventory level falls beyond the replenishment level. Case 2: When θ = 2 An optimal value for the cost function and expected production rate is obtained and is indicated in bold letters as in Table 2. In this case also, as the value of S increases, the value of Einvent also increases. Also, as the maximum inventory level increases, the expected loss is found to follow a decreasing pattern. As the maximum inventory level increases, the length of production cycle also increases, i.e. the time for which the system must be kept on to reach maximum level will increase with S. The expected rate at which production is switched on decreases with increasing value of S.

6.1.1 Comparison Chart for Customer Loss The following Table 3 provides a comparison for the expected customer loss in comparison with a system following fixed production rate: Table 3 Effect of max inventory level S for θ = 2.5, λ = 5, s = 10, Cinvent = 50; Cpro.rate = 200; Cc.loss = 400; M = 2000

S 12 13 15 16

Var pro rate 1.9082 × 10−10 3.2634 × 10−11 8.790 × 10−12 6.685 × 10−12

Fixed pro rate 0.033 0.030 0.024 0.022

An (s, S) Production Inventory System with State Dependant Production Rate. . .

231

6.2 Effect of Minimum Inventory Level s The effect of minimum inventory level s on cost function and various performance measures discussed above is tabulated as per the tables. Here, the values of S, λ, μ, θ and other basic costs are assumed to be constant throughout the calculations. Case 1: When θ = 2 The cost function is obtained and increases with increase in value of s as in Table 4. Meanwhile, as the value of s increases, the value of Einvent also increases, because the stock is being replenished more frequently. Also, as the minimum inventory level increases, the expected production rate increases. As the minimum inventory level increases, the expected loss is found to follow a decreasing pattern as inventories are added to the system in s shorter interval of time (i.e., smaller the value of S − s, more frequent is the addition of inventory). Also, as the minimum inventory level increases, the length of production cycle decreases, i.e., the time for which the system must be kept on to reach maximum level will decrease with s. The expected rate at which production is switched on increases with increasing value of s. This is because as s approaches to S, there will be more chance of switching on the system. Case 2: When θ = 10 The cost function is obtained and decreases with increase in value of s (Table 5). Other conclusions obtained include: As the value of s increases, the value of Einvent also increases. But as the minimum inventory level increases, the expected production rate decreases. As the minimum inventory level increases, the expected loss is found to follow a decreasing pattern. Also, as the minimum inventory level increases, the length of production cycle decreases, i.e., the time for which the system must be kept on to reach maximum level will decrease with s. Similarly, the expected rate at which production is switched on increases with increasing value of s.

Table 4 Effect of min inventory level s for θ = 2, λ = 5, μ = 12, S = 13, Cinvent = 50; Cpro.rate = 200; Cc.loss = 400; M = 2000

s 5 6 7 8 9 10

Einvent 9.498185 9.708196 9.900285 10.072336 10.221815 10.345635

Epro.rat e 3.000086 3.274726 3.663149 4.437888 5.182550 5.881182

Ec.loss 0.0000068 0.0000031 0.0000018 0.0000013 0.0000011 0.000001

EP C 76,292,967 76,289,061 76,269,530 76,171,874 75,683,593 73,242,187

Es.on 0.102597 0.157450 0.267563 0.473054 0.793881 1.152411

FC 1280.147 1455.268 1762.778 2337.309 3135.367 3998.346

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S. Malini and D. Shajin

Table 5 Effect of min inventory level s for θ = 10, λ = 5, μ = 12, S = 13, Cinvent = 50; Cpro.rate = 200; Cc.loss = 400; M = 2000

s 3 5 6 7 9

Einvent 8.662487 9.588038 10.043674 10.492828 11.36246

Epro.rat e 23.42065 18.911789 16.491345 14.631114 10.944421

Table 6 Effect of min inventory level s for θ = 2.5, λ = 5, S = 15, Cinvent = 50; Cpro.rate = 200; Cc.loss = 400; M = 2000

Ec.loss 0.000001645105 0.000000007112 0.000000000602 0.000000000082 0.000000000030

EP C 76,293,904 76,292,967 76,289,061 76,269,530 75,683,593 s 2 3 4 6

Es.on 0.022583 0.035036 0.045657 0.0621868 0.1456481

Var pro rate 2.4680 × 10−5 1.722 × 10−6 1.3978 × 10−9 1.2074 × 10−9

FC 5162.4225 4331.833 3891.766 3575.238 3048.303

Fixed pro rate 0.088 0.073 0.061 0.044

6.2.1 Comparison Chart for Customer Loss The following Table 6 provides a comparison for the expected customer loss in comparison with a system following fixed production rate:

7 Conclusion This paper analyses an (s, S) production inventory system with state dependent production rate and lost sales. The steady state analysis is performed assuming the system is having negligible service time and no backlogs of demand, and a product form solution is developed. Further, the expected length of production cycle is formulated in the paper. An expected cost function per unit time is also developed with which the optimal values of s and S are calculated numerically. From the numerical exemplars it is obvious that, the loss rate can be considerably decreased with state dependent production rates along with an optimal value for expected cost function. It is to be noted that the expected customer loss in the system reduces considerably with increase in production rate. One can extend the study to derive an optimal analytical expression for dependency between number of items in the inventory and production rates.

An (s, S) Production Inventory System with State Dependant Production Rate. . .

233

References 1. Baek, J.W., Moon, S.K.: The M/M/1 queue with a production-inventory system and lost sales. Appl. Math. Comput. 233, 534–544 (2014) 2. Berman, O., Kaplan, E.H., Shevishak, D.G.: Deterministic approximations for inventory management at service facilities. IIE Trans. 25(5), 98–104 (1993) 3. He, Q.-M., Jewkes, E., Buzacott, J.: Optimal and near-optimal inventory control policies for a make-to-order inventory-production system. Eur. J. Oper. Res. 141(1), 113–132 (2002) 4. Karthikeyan, K., Sudhesh, R.: Recent review article on queueing inventory systems. Res. J. Pharm. Technol. 9, 2056 (2016) 5. Krishnamoorthy, A., Narayanan, V.C.: Production inventory with service time and vacation to the server. IMA J. Manag. Math. 22(1), 33–45 (2011) 6. Krishnamoorthy, A., Viswanath, N.C.: Stochastic decomposition in production inventory with service time. Eur. J. Oper. Res. 228(2), 358–366 (2013) 7. Krishnamoorthy, A., Manikandan, R., Lakshmy, B.: A revisit to queueing-inventory system with positive service time. Ann. Oper. Res. 233(1), 221–236 (2015) 8. Saffari, M., Asmussen, S., Haji, R.: The M/M/1 queue with inventory, lost sale, and general lead times. Queueing Syst. 75(1), 65–77 (2013) 9. Schwarz, M., Sauer, C., Daduna, H., Kulik, R., Szekli, R.: M/M/1 queueing systems with inventory. Queueing Syst. 54(1), 55–78 (2006) 10. Sigman, K., Simchi-Levi, D.: Light traffic heuristic for an M/G/1 queue with limited inventory. Ann. Oper. Res. 40, 371–380 (1992)

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent Phase-Type Claims and a Constant Barrier A. S. Dibu

and M. J. Jacob

Abstract Inspired by the problems with random income feature, this paper focuses on an insurance risk model with MAP inter-arrival time for premiums as well as claims. We study the model for a convex combination of two types of interdependent Phase-type claims, where the probability of claim switching is directly associated with the inter-arrival time of claims. Furthermore, the surplus process of this model is assumed to be restricted by a horizontal barrier “b” above the initial surplus “u”. The transient analysis of the corresponding Markovian fluid flow model is considered to develop the integral equations governing the Gerber–Shiu function and the expected discounted dividends paid until ruin. The closed-form solutions for these integral equations are obtained in terms of Lundberg roots. When the premium sizes are Phase-type distributed, the solutions are explicit at “u = b”. For “u ≤ b”, the solutions are explicit when the premium sizes are distributed exponentially. Finally, to validate and present the tractability of these solution expressions, some numerical illustrations are provided in individual cases. Keywords Markovian arrival process · Random incomes · Phase-type claims · Inter-dependent claims

1 Introduction The insurance risk model considered in this paper realises a surplus process with random incomes. The complexity of the premium process structures in various financial and insurance markets is investigated in [24]. Instead of a constant premium rate, Boikov [7] linked the stochastic premium process in a surplus process to characterise more realistic cash inflows. Temnov [28] estimated the metric distances between the ruin probabilities of risk models corresponding to

A. S. Dibu () · M. J. Jacob Department of Mathematics, National Institute of Technology Calicut, Kozhikode, Kerala, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_15

235

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A. S. Dibu and M. J. Jacob

both stochastic and constant premium processes. Bao and Ye [6] and Labbé and Sendova [19] dealt with the Gerber–Shiu analysis in the compound Poisson risk model with compound Poisson premiums. Further exercises in risk processes with random incomes were done by Hao and Yang [17] and Jieming et al. [18] in delayed claim strategies, Labbé et al. [20] in a model with amount sizes to take positive as well as negative values, Gao and Wu [16] in a model with two classes of delayed claims and Shija and Jacob [26] in the Markov-modulated model. In recent years, the risk processes with multi-phase arrivals have got much attention in the literature. Several strategic surplus processes are modelled in a multi-phase environment with the aid of a general class of arrival process so-called the Markovian arrival processes (MAP) for claim arrivals. The versatile MAP is introduced by Neuts [25] and Latouche and Ramaswami [22] developed the matrixanalytic methods to analyse the Markovian fluid flow process. The Gerber–Shiu analysis of the risk processes under Markovian set-up is done in [1, 29] under absolute ruin with debit interest and heavy-tailed claim sizes, Dong and Liu [15] in the latent tax model. Furthermore, the generalised versions of the Gerber–Shiu function have been investigated by Cheung and Landriault [13] using a rewardbased measure and Cheung and Feng [11] using a cost-utility measure. Meanwhile, Landriault and Shi [21] analysed the occupation times and Li et al. [23] analysed the probability function of the number of claims up to ruin in models having the MAP inter-arrival time for claims. Apart from all these papers, we consider a multi-phase cash inflow by modelling the premium arrivals into the MAP version. Generally, the Phase-type (PH) amount sizes are fused to the MAP interarrival times, since both have multi-phase structure and the matrix-analytic methods are developed to analyse the Markovian fluid flow models. These methodologies are inspired in [4] to analyse a risk process using corresponding fluid queues. Technically, this kind of fusing will establish correlation between claims and interarrival times (see [1, 5] and [2], and references therein for further details). Risk processes with barrier strategy are proposed by De Finetti [14]. Under some assumptions, he observed that the maximum dividend is availed to the shareholders while implementing the barrier strategy. Some of the recent problems handled in the MAP risk model under dividend strategy are as follows: with perturbation, Cheung et al. [12] studied a barrier problem and Cheng and Wang [10] studied the threshold dividend problem. Meanwhile, in the non-perturbed model, Ahn et al. [2] studied a horizontal barrier problem and Zhang et al. [30] studied an Erlangised dividend problem. The main quantities of interest in all the recent works are the expected discounted dividends paid out until ruin and the Gerber–Shiu function. The remaining sections in this paper are organised as follows. The next section introduces the model assumptions and notations. In Sect. 3, the governing renewal equations satisfying the two measures of interest, the expected discounted dividends until ruin and the Gerber–Shiu function, are established. The closed-form solutions are obtained for the two measures by taking barrier as initial surplus level u = b, in Sect. 4. For exponential premiums, the closed-form solutions at the general initial surplus level (u ≤ b) are then obtained in Sect. 5. The numerical examples for a

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

237

scalar and two-state versions are illustrated in Sect. 6. The concluding statements are remarked in Sect. 7.

2 The MAP Premium Model with Horizontal Barrier Strategy In this paper, we consider a random income surplus process given by Np (t )

U (t) = u +

 i=1

Xi −

N c (t ) 

Yj

(1)

j =1

which initiates at U (0) = u. The process (1) will and negative  have positive Np (t ) jumps due to the aggregate processes of premium i=1 Xi and that of claim   Nc (t ) j =1 Yj respectively. The inter-arrival times of premium and claim, and the amount sizes of premium and claim are all assumed to be mutually independent. In all the aforementioned papers with stochastic income, the premium is assumed to have a stochastic behaviour, but all the cash inflows are realised through a single phase/channel. Instead of assuming a single phase, a multi-phase cash inflow to the surplus process is realised in this model. For developing the multi-phase model, the inter-arrival time of premiums, Tp , is assumed to follow a MAP. A MAP having n ≥ 1 transient phases with representation MAPn (β, E0 , E1 ) is a two-dimensional Markov process {(Np (t), Jp (t))} having state space N × {1, 2, . . . , n} for t ≥ 0. Here, the counting process Np (t) denotes the number of premium arrivals and Jp (t) is the state of the underlying continuous time Markov chain (CTMC) of premium arrivals at time t ≥ 0. The surplus process U (t) is thus capable of realising the cash inflows through countably infinite number of territories (regions), offices or the financial products of the insurance company. Furthermore, the claim arrival is governed by the MAP interarrival time, Tc with the representation MAPm (α, D0 , D1 ) having m ≥ 1 transient phases. The two-dimensional Markov process associated with the MAP of claims is denoted by {(Nc (t), Jc (t))} having state space N × {1, 2, . . . , m} for t ≥ 0 in which Nc (t) is the number of claim arrivals and Jc (t) is the state of underlying CTMC of claim arrivals at time t ≥ 0 (see [25] and [22] for further details about MAP). The process defined by Eq. (1) will either have a positive jump due to a premium arrival or have a negative jump due to a claim arrival at the first renewal time τ 1 . To bring out what happens at τ 1 , define τ = min(Tp , Tc ) which will again follow a MAP with representation MAPmn ([α ⊗β], F0 , F1 ) having mn ≥ 1 transient phases. Here, F0 = D0 ⊕ E0 = D0 ⊗ I n + E0 ⊗ I m F1 = D1 ⊕ E1 = D1 ⊗ I n + E1 ⊗ I m

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A. S. Dibu and M. J. Jacob

in which I m and I n denote identity matrices of order m and n, respectively, ⊕ is the Kronecker sum and ⊗ is the Kronecker product. Then for t ≥ 0, the inter-arrival times τ is a two-dimensional Markov process {(N(t), J (t))} having state space, N × E where E = {1, 2, . . . , mn}. Then, N(t) denotes the number of renewals and J (t) is the state of underlying CTMC of renewals at time t ≥ 0 . Due to the matrix structure of the MAP premium inter-arrival time, the premium sizes, {Xi }i∈N+ , are assumed to follow a PH distribution with representation PHm1 (γ , G) where the transition between the transient phases of the CTMC is given by m1 -dimensional square matrix G and the initial probability vector is given by the m1 -dimensional row vector γ . For the minimum of two MAP arrivals, τ , the transitions of related fluid queue process will be governed by an irreducible CTMC with transition rate matrix  

T11mn×mn T12mn×mnm1 F0 γ ⊗ F1 T= = (2) T21mnm1 ×mn T22mnm1 ×mnm1 g ⊗ I mn G ⊗ I mn where I mn denotes the identity matrix of order mn and g = −Ge1 m1 in which denotes the m -dimensional column vector of ones. The PH claims that e1 1 m1 disturb the surplus process (1) are of two classes—one kind with representation PHn1 (γ 1 , G1 ) and the another kind with representation PHn2 (γ 2 , G2 ). The related fluid process thus comprises of an irreducible CTMC which generates two transition rate matrices—one for the first kind which is given by  "

T =

T"11mn×mn

T"21mnn

1 ×mn



T"12mn×mnn

1

T"22mnn

1 ×mnn1



F0 γ 1 ⊗ F1 = g1 ⊗ I mn G1 ⊗ I mn



and the another kind which is given by  ""

T =

T""11mn×mn

T""21mnn

2 ×mn

T""12mn×mnn

T""22mnn



=

2

2 ×mnn2

γ 2 ⊗ F1 F0 g2 ⊗ I mn G2 ⊗ I mn



1 1 1 where g1 = −G1 e1 n1 and g2 = −G2 en2 in which en1 and en2 denote, respectively, the n1 -dimensional and n2 -dimensional column vectors of ones. Then the probability density functions of premiums and the two classes of claims satisfy the condition

 F1 =

∞ x=0

 T12 eT22 x T21 dx =

∞ x=0

"

T"12 eT22 x T"21 dx =



∞ x=0

""

T""12 eT22 x T""21 dx.

During the temporary time that the CTMC stays in the states governed by the matrices T11 , T"11 and T""11 , the fluid level stays idle with no increase or decrease. On the other hand, the fluid process characterised by transition rates of positive jumps in the states governed by T22 and negative jumps in the states governed by T"22 and T""22 . The fluid queue process and the surplus process are connected by

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

239

relating the duration of the idle fluid level with the duration of the idle surplus level and, by freezing the time evolution upon a transition from idle level to increasing (decreasing), the increasing (decreasing) fluid flow with the premium (claim) size random variables (see [1, 4, 5] for further details). In this paper, the inter-dependent claims structure proposed by Boudreault et al. [8] is generalised to the MAP/PH set-up which satisfies 



∞

F1 =

y=0



+

" γ 1 ⊗ F1 e[Λ⊗I n ]t eT22 y T"21 dy

∞



y=0

"" γ 2 ⊗ F1 (I mn − e[Λ⊗I n ]t ) eT22 y T""21 dy

where Λ = diag[−λi ] for i = 1, 2, . . . , m. This inter-dependency implies that the probability of a claim from first (second) kind is an exponentially decreasing (increasing) function of the time separating this event from the last claim arrival time and λ"i s denote the Bayesian estimate of the rate of exponential decrease (increase) in the ith claim arriving phases. Remark 1 The inter-dependent strategy of two classes of claim sizes can be adapted to the premium sizes also. We restrict the inter-dependence assumption for the claim sizes to avoid redundancy in the analysis. The security loading factor, θ = (average cash inflow/average cash outflow)−1, is assumed to be positive where the average cash inflow is given by 6  7−1 E {X} = π p [−γ ⊗ E1 ] [G ⊗ I n ]−1 e1 E Tp nm1 and the average cash outflow is given by [E {Tc }]−1 E {Y } = π c



∞ t =0



66 7 7 π c ⊗ D1 eΛt e[D0 ⊗I m ]t −γ 1 ⊗ D1 ⊗ I m

6 7−1 1 × G1 ⊗ I m2 em2 n dt 1  ∞   π c ⊗ D1 I m − eΛt e[D0 ⊗I m ]t + πc ×

66

t =0

7 76 7−1 1 −γ 2 ⊗ D1 ⊗ I m G2 ⊗ I m2 em2 n dt 1

where π p and π c are the stationary probability row vectors of the CTMC, Jp = {Jp (t); t ≥ 0} and Jc = {Jc (t); t ≥ 0 , respectively. Furthermore, e1 and m2 n 1

denote the m2 n1 -dimensional and m2 n2 -dimensional column vectors of ones, e1 m2 n 2

respectively, and I m2 denote the identity matrix of order m2 .

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A. S. Dibu and M. J. Jacob

The surplus process (1) is further restricted under the horizontal barrier, b ≥ u, satisfying the equation

dUb (t) =

⎧ N (t ) p N c (t )   ⎪ ⎪ ⎪ ⎪d Xi − d Yj , Ub (t) ≤ b ⎪ ⎨ j =1

i=1

N c (t ) ⎪  ⎪ ⎪ ⎪ −d Yj , ⎪ ⎩

Ub (t) > b.

j =1

Under the barrier-restricted strategy, the insurance company can undertake the economic interest of the shareholders by paying out the extra surplus above the constant barrier b as dividends. For expressing the revised risk process Ub (t) formally (as of Fig. 1), let χ(t) = sup{U (v) | 0 ≤ v ≤ t} be the running maximum of surplus process U (t). Denoting Db (t) = max{χ(t) − b, 0} as the aggregate dividends paid by the company up to time t, the revised risk process Ub (t) can be thus expressed as Ub (t) = U (t) − Db (t)

(3)

with the understanding that the aggregate dividends paid by the company up to time t ≥ 0 is zero for Ub (t) < b. For the amended process given by Eq. (3), we analyse the expected discounted dividends paid until ruin (EDDR) and the Gerber–Shiu function (GSF). For this U b (t)

Db (

2)

b

u U b (T − ) T

0 1

2

t

3

|U b (T )|

Fig. 1 A realisation of the surplus process Ub (t)

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

241

analysis, we denote the classical ultimate ruin time as T = inf{t ≥ 0 : Ub (t) < 0} (T = ∞ if the set is empty). Then for the discounting factor δ ≥ 0,  Dδ,b (T ) =

T t =0

e−δt dDb (t)

defines the discounted value of the aggregate dividends paid out until the time of ruin T . The EDDR for the initial surplus level u is defined as 6 7 ν δ,b (u) = E Dδ,b (T ) | Ub (0) = u = [α ⊗ β] V 1 δ,b (u), 0 ≤ u ≤ b.

(4)

Here, V 1 δ,b (u) denotes the mn-dimensional column vector for which

V1 δ,b (u)

i

6 7 = E Dδ,b (T ) | Ub (0) = u, J (0) = i

(5)

Equation (5) determines the EDDR for the initial capital u and the initial phase i ∈ E. Furthermore, let |Ub (T )| and Ub (T − ) be the deficit at ruin and the surplus immediately before ruin, respectively. Then for δ ≥ 0, the mn-dimensional column vector Φ 1 δ,b (u) defined as

Φ1 δ,b (u)

i



= E e−δT w(Ub (T − ), |Ub (T )|) 1(T < ∞) | Ub (0) = u, J (0) = i

which determines the GSF with the initial surplus level u and the initial phase i ∈ E. The function w(x, y) is the penalty insurer abide to settle whenever the surplus process confirms a ruin time (T < ∞). As a consequence of the arguments from [1], the Gerber–Shiu function is given by φ δ,b (u) = [α ⊗ β] Φ 1 δ,b (u), 0 ≤ u ≤ b.

(6)

3 The Governing Renewal Equations for the Expected Discounted Dividends Paid Out Up To Ruin and the Gerber–Shiu Function The section focuses on developing two renewal equations that govern the process (3): one which satisfies the EDDR and the other which satisfies the GSF with the initial surplus level Ub (0) = u. For this development, Theorems 1 and 2 will provide the existence of integral equations that satisfy the mn-dimensional vectors 1 V1 δ,b (u) and Φ δ,b (u), respectively.

242

A. S. Dibu and M. J. Jacob

Theorem 1 For 0 ≤ u ≤ b, the measure V 1 δ,b (u) satisfies the integral equation F0 V 1 δ,b (u) = F



b−u x=0

T12 eT22 x T21 V 1 δ,b (u + x) dx +



× (x + u

− b) e1 mn

+V1 δ,b (b)



∞

T12 eT22 x T21

x=b−u

dx + C1 δ,b (u)

(7)

in which e1 mn denotes the mn-dimensional column vector of ones. Here, F = [E0 ⊗ I m ] [δI mn − F0 ]−1 , F" = [D0 ⊗ I n ] [δI mn − ([Λ ⊗ I n ] + F0 )]−1 and F"" = [D0 ⊗ I n ] [δI mn − F0 ]−1 are mn-dimensional square matrices. Furthermore, 

C1 δ,b (u)

u

=



y=0

  " "" F" T"12 eT22 y T"21 − T""12 eT22 y T""21

"" + F"" T""12 eT22 y T""21 V 1 δ,b (u − y) dy

is an mn-dimensional column vector which furnishes the renewal of EDDR process when a claim arrives at τ 1 . Proof In the range of initial surplus 0 ≤ u < b, the first renewal on the process (3) may be due to an arrival of either a claim or a premium. Conditioning on τ 1 , the time of first renewal (claim or premium arrival time), it follows V1 δ,b (u) =



∞

t =0



6  7 e−δt β ⊗ P r τ 1 = t, τ 1 = Tp

b−u x=0



+

×

t =0  u y=0

 +

∞ t =0 u

 ×





1 eT22 x T21 (x + u − b)e1 + V (b) dx dt mn δ,b

∞

x=b−u ∞ −δt [Λ⊗I n ]t

 +

eT22 x T21 V 1 δ,b (u + x) dx

y=0

e

e

[α ⊗ P r (τ 1 = t, τ 1 = Tc )]

"

eT22 y T"21 V 1 δ,b (u − y) dy dt   e−δt I mn − e[Λ⊗I n ]t [α ⊗ P r (τ 1 = t, τ 1 = Tc )] ""

eT22 y T""21 V 1 δ,b (u − y) dy dt.

(8)

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243

where      P r τ 1 = t, τ 1 = Tp = P rij τ 1 = t, τ 1 = Tp i,j =1,2,...,mn   P r (τ 1 = t, τ 1 = Tc ) = P rij (τ 1 = t, τ 1 = Tc ) i,j =1,2,...,mn .   Here, P rij τ 1 = t, τ 1 = Tp and P rij (τ 1 = t, τ 1 = Tc ) are the probabilities that the first shock on the process happens due to a claim arrival and a premium arrival, respectively, along with a transition from ith state to j th state at time t ≥ 0 for i, j ∈ E. Then, P r (τ 1 = t, τ 1 = Tc ) = F−1 0 [E0 ⊗ I m ]   P r τ 1 = t, τ 1 = Tp = F−1 0 [D0 ⊗ I n ] . As a consequence of Bayes’ theorem, it follows that F0 t P r (τ 1 > t | τ 1 = Tc ) = F−1 [−F0 ]−1 F1 0 [E0 ⊗ I m ] e   F0 t P r τ 1 > t | τ 1 = Tp = F−1 [−F0 ]−1 F1 . 0 [D0 ⊗ I n ] e

(9) (10)

After applying these probabilities and integrating over time in Eq. (8), after some can rearrange the obtained equality to the integral equation (7). Now without any further arguments, V 1 δ,b (b) satisfies V1 δ,b (b) =



∞ t =0



∞



x=0 ∞

× +

t =0

 



1 eT22 x T21 x e1 + V (b) dx dt mn δ,b e−δt e[Λ⊗I n ]t [α ⊗ P r (τ 1 = t, τ 1 = Tc )]

b

× +

6  7 e−δt β ⊗ P r τ 1 = t, τ 1 = Tp

"

eT22 y T21 " V 1 δ,b (b − y) dy dt

y=0 ∞

t =0



  e−δt I mn − e[Λ⊗I n ]t [α ⊗ P r (τ 1 = t, τ 1 = Tc )]

b

×

y=0

""

eT22 y T21 "" V 1 δ,b (b − y) dy dt.

Hence at u = b, Eq. (7) can be alternatively written as F0 V 1 δ,b (b) = F



∞ x=0

1 xT12 eT22 x T21 e1 dx + F V (b) + C1 1 δ,b mn δ,b (b).

(11)

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A. S. Dibu and M. J. Jacob

Theorem 2 For 0 ≤ u ≤ b, the measure Φ 1 δ,b (u) satisfies the integral equation F0 Φ 1 δ,b (u) = F



b−u

T12 eT22 x T21 Φ 1 δ,b (u + x) dx

x=0



+

∞

x=b−u

1 T12 eT22 x T21 Φ 1 (b) dx + D1 δ,b δ,b (u) + Wδ,b (u). (12)

Here, D1 δ,b (u)





u

=

y=0

  " "" F" T"12 eT22 y T"21 − T""12 eT22 y T""21

"" + F"" T""12 eT22 y T""21 Φ 1 δ,b (u − y) dy,

is an mn-dimensional column vector which furnishes the renewal of GSF process when a claim arrives at τ 1 and 6 1 7 " 1 "" 1 W1 δ,b (u) = F w 1 (u) − w 2 (u) + F w 2 (u),  ∞ " T" y "  ∞ "" T"" y "" 22 t w(u, y − u) dy and w 1 (u) = 22 t w where, w1 1 (u) = y=u T12 e 2 2 2 y=u T12 e " " 1 "" "" 1 (u, y − u) dy for which t2mnn ×1 = −T22 emnn1 and t2mnn ×1 = −T22 emnn2 . Here, 1

2

1 e1 mnn1 and emnn2 denote mnn1 -dimensional and mnn2 -dimensional column vectors of ones, respectively.

Proof Using the same arguments of Theorem 1, it follows Φ1 δ,b (u) =



∞ t =0



+ +

6  7 e−δt β ⊗ P r τ 1 = t, τ 1 = Tp ∞

e T21 × x=b−u  ∞ −δt [Λ⊗I n ]t t =0

T22 x

e



×

y=0

 +

∞ t =0

x=0

(x + u −

b) e1 mn

eT22 x T21 Φ 1 δ,b (u + x) dx

+ Φ1 δ,b (b)



dx dt

"

eT22 y T"21 Φ 1 δ,b (u − y) dy +



∞ y=u

" eT22 y t"2 w(u, y − u) dy dt

  e−δt I mn − e[Λ⊗I n ]t × [α ⊗ P r (τ 1 = t, τ 1 = Tc )]



×

b−u

[α ⊗ P r (τ 1 = t, τ 1 = Tc )]

e

u





u

e y=0

T""22 y

T""21 Φ 1 δ,b (u

 − y) dy +

∞ y=u

"" eT22 y t""2 w(u, y

− u) dy dt. (13)

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

245

After substituting the probabilities from Eqs. (9) and (10), integrate out the time in Eq. (13) to furnish the integral equation (12) and then at u = b, Eq. (12) reduces to the form 1 1 1 F0 Φ 1 δ,b (b) = F F1 Φ δ,b (b) + Dδ,b (b) + Wδ,b (b).

(14)

4 The Closed-Form Solutions of V  (b) and Φ  (b) δ,b δ,b For analysing the solution of the governing equations (7) and (12), the expressions 1 of V 1 δ,b (u) and Φ δ,b (u) at the horizontal barrier b have to be determined. This 1 section focuses to derive the expressions of V 1 δ,b (b) and Φ δ,b (b) by applying analytical Laplace inversion on the Laplace transformed version of the implicit integral equations (11) and (14),  ∞ respectively. For applying these transformations to the equations, let f˜(sx ) = x=0 e−sx f (x) dx be the Laplace transform from the real domain of x to the complex domain of s of an integrable function f (.). The proposition below provides the Laplace transform of Eqs. (11) and (14).  ∞ −sy " T" y " Proposition 1 Let M˜ 1 (sy ) = T12 e 22 T21 dy and M˜ 2 (sy ) = y=0 e  ∞ −sy "" T"" y "" 1 ˜1 T12 e 22 T21 dy. Then the Laplace transforms V˜ δ,b (sb ) and Φ δ,b (sb ) of y=0 e 1 1 the measures V δ,b (b) and Φ δ,b (b), respectively, will satisfy the following equations: 1 ˜1 Lb (sb )V˜ δ,b (sb ) = A V (sb ) 1

1

˜ Φ (sb ) ˜ δ,b (sb ) = A Lb (sb )Φ

(15) (16)



˜1 where Lb (sb ) = F0 − F F1 − F" M˜ 1 (sb ) − M˜ 2 (sb ) − F"" M˜ 2 (sb ), A V (sb ) = ∞ 1 1 T x 1 1 22 T e ˜ Φ (sb ) = W ˜ (sb ), the Laplace transform of 21 mn dx and A δ,b sb F x=0 xT12 e 1 Wδ,b (b). Proof Taking the Laplace transform from the domain of b to s, Eqs. (11) and (14) yield  ∞

1 1 1 T22 x 1 ˜ 1 (sb ) (17) ˜ ˜ F0 V δ,b (sb ) = F xT12 e T21 emn dx + F1 V δ,b (sb ) + C δ,b sb x=0 ˜1 ˜1 ˜1 ˜1 F0 Φ δ,b (sb ) = F F1 Φ δ,b (sb ) + Dδ,b (sb ) + Wδ,b (sb )

(18)

246

A. S. Dibu and M. J. Jacob

respectively. Here

˜ 1 (sb ) = F" M˜ 1 (sb ) − M˜ 2 (sb ) + F"" M˜ 2 (sb ) V˜ 1 (sb ) C δ,b δ,b

1 ˜ 1 (sb ) = F" M˜ 1 (sb ) − M˜ 2 (sb ) + F"" M˜ 2 (sb ) Φ ˜ δ,b (sb ). D δ,b 1

1

˜ δ,b (sb ) to the left-hand side, Now collecting the coefficients of V˜ δ,b (sb ) and Φ respectively, of Eqs. (17) and (18) will provide the Laplace transform of the integral equations (11) and (14). In Eqs. (15) and (16), the elements of mn-dimensional square matrix Lb (sb ) are rational which can be represented in the form Lb (sb ) = [Lb (sb )]i,j =1,2,...,mn

pij (sb ) ⊗ I n for i, j = 1, 2, . . . , m = qij (sb ) =

7 1 6 pij (sb ) ⊗ I n for i, j = 1, 2, . . . , m Qb (sb )

where Qb (sb ) = [qb (sb )]nκ for κ ∈ {1, 2, . . . , m} in which qb (sb ) is the n1 + n2 degree unique

denominator polynomial qij (s) that occurs in κ number of columns pij (sb ) adj of . Denoting the Lb (sb ) and Ldet b (sb ), respectively, as the adjoint and qij (sb ) determinant of Lb (sb ), (15) and (16) can be alternatively settled as adj

1 V˜ δ,b (sb ) =

Lb (sb )

˜1 Φ δ,b (sb ) =

Lb (sb )

Ldet b (sb ) adj

Ldet b (sb )

˜1 A V (sb )

(19)

˜1 A Φ (sb )

(20) adj

where Ldet b (sb ) and the elements of the mn-dimensional square matrix Lb (sb ) 1 adj , are rational. Then Ldet b (sb ) and Lb (sb ) can be represented as multiple of Qb (sb ) which is followed due to the MAP/PH structure of the model. Now, the Laplace transform inversion of Eqs. (19) and (20) with respect to sb will determine the 1 expression for V 1 δ,b (b) and Φ δ,b (b), respectively, which is explained by Theorem 3 given below. adj

adj

det Theorem 3 Let Qb (sb ) = Qb (sb )Lb (sb ), Qdet b (sb ) = Qb (sb )Lb (sb ) and det denote ρ b:l for l = 1, 2, . . . , κ (n1 + n2 ) as the distinct roots of Qb (sb ) = 0, having algebraic multiplicity n, occur in the left half of the complex plane. Then for κ(n +n )  an arbitrary ωb ∈ / ρ b:l l=11 2 and using the generalised Hermite’s interpolating

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

247

polynomial, the closed-form expressions for the solution of equations (11) and (14) are given by 

1

V1 δ,b (b) =

Qdet b (ωb ) 1

Φ1 δ,b (b) =

Qdet b (ωb )

b x=0



b x=0

ξ ωb (x)A1 V dx

(21)

ξ ωb (x)A1 Φ (b − x)dx

(22)

where ξ z (b) =

κ(n n 1 +n2 )  j =1

l=1

A1 V = F



∞

x=0

clj (z, b)

d j −1 j −1

dsb

adj

Qb (sb )|sb =ρ b:l

xT12 eT22 x T21 e1 mn dx

1 1 = −FT12 T−1 22 (em1 ⊗ I mn )emn 1 A1 Φ (b) = Wδ,b (b)

in which  n n−j z − ρ b:l  b(n−j −i) eρ b:l b g (i) l (ρ b:l ) clj (z, b) = P l (z) Γ (j ) Γ (n − j − i + 1)Γ (i + 1) 6

7−1

i=0

P l (z) = g l (z) ⎤n ⎡ κ(n: 1 +n2 )   z − ρ b:l ⎦ . =⎣ l=1

Proof Multiplying and dividing by Qb (sb ) in the right-hand side of Eqs. (19) and (20), it follows adj

1 V˜ δ,b (sb ) =

Qb (sb )

˜1 Φ δ,b (sb ) =

Qb (sb )

Qdet b (sb ) adj

Qdet b (sb )

˜1 A V (sb )

(23)

˜1 A Φ (sb ).

(24)

248

A. S. Dibu and M. J. Jacob

κ(n +n )  Now for an arbitrary ωb ∈ / ρ b:l l=11 2 and using the generalised Hermite’s interpolating polynomial (see [27]), Eqs. (23) and (24) can be alternatively written as ξ˜ ωb (sb )

1 V˜ δ,b (sb ) =

Qdet b (ωb ) ξ˜ ωb (sb )

˜1 Φ δ,b (sb ) = where ξ˜ z (sb ) =

Qdet b (ωb )

˜1 A V (sb )

(25)

˜1 A Φ (sb )

(26)

κ(n1 +n2 ) n

adj d j−1 j =1 c˜ lj (z, sb ) ds j−1 Qb (sb )|sb =ρ b:l

l=1

in which the

b

coefficient function clj (z, sb ) is given by

 (j −1) n−j    z − ρ b:l i (i) z − ρ b:l g (ρ b:l )R li (z, sb ) c˜ lj (z, sb ) = P l (z) Γ (j ) Γ (i + 1) l

(27)

i=0

for which R li (z, sb ) is a weight function fixed to neutralise the degree of polynomiadj als comprised in the parent matrix, Qb (u). A suitable divided difference form for this weight function is given by

z − ρ b:l R li (z, sb ) = sb − ρ b:l

n−(j −1)−i .

(28)

1 This form will admit the closed-form solutions of V 1 δ,b (b) and Φ δ,b (b) in terms of Lundberg roots. Substituting the divided difference weight function R li (z, sb ) given by Eq. (28) in Eq. (27), the particular form of coefficient function c˜ lj (z, sb ) can be represented by the equation



z − ρ b:l c˜ lj (z, sb ) = P l (z) Γ (j )

n

n−j  i=0

g (i) l (ρ b:l ) (n−(j −1)−i) Γ (i + 1) (sb − ρ b:l )

Now, inverting Eqs. (25) and (26) with respect to sb will yield the expressions for EDDR (21) and GSF (22), respectively, with barrier b, being the initial surplus level.

5 The Closed-Form Solutions of V  (u) and Φ  (u) δ,b δ,b for Exponential Premiums 1 In this section, we work on obtaining the expressions for V 1 δ,b (u) and Φ δ,b (u), for 0 ≤ u ≤ b, as the solution of defective renewal equations (7) and (12). For obtaining the solution, the method of analytical Laplace inversion is applied on the expressions

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

249

equivalent to the Laplace transform of the differential equations derived from the governing renewal equations (7) and (12). The existence of the differential equations is detailed in Theorem 4 below. For proceeding the analysis, the generalised inverse of mn × mnm1 dimensional component matrix F T12 of matrix T given by Eq. (2) is required. We observe that it is difficult to provide such an explicit generalised matrix suitable for the analysis with the PH premiums unless the columns of F T12 are linearly independent. Hence through Theorem 4, we restrict our analysis by taking exponential premiums so that m1 = 1 and thus the matrix F T12 reduces to a square matrix of order mn and will be invertible if the columns of F1 are linearly independent. In Theorem 4, the integrodifferential equations that satisfy EDDR and GSF for the initial capital “u ≤ b” are obtained. 1 1 Theorem 4 Taking Z1 δ,b (u) = Bδ,b (u)+Wδ,b (u) and assuming that the columns of 1 F1 are linearly independent, the measures V 1 δ,b (u) and Φ δ,b (u), respectively, satisfy the non-homogeneous first order integro-differential equations

$ %

d + F T12 T22 M−1 F0 + F T12 T21 V 1 I mn δ,b (u) du $ %  d −1 1 A1 + F T12 T22 M (u) + C (u) = I mn δ,b δ,b du $ %

d I mn + F T12 T22 M−1 F0 + F T12 T21 Φ 1 δ,b (u) du $ %  d 1 + F T12 T22 M−1 Z1 = I mn δ,b (u) + Dδ,b (u) du

(29)

(30)

where M−1 is the inverse of the mn order square matrix F T12 and A1 δ,b (u) = F



∞

z=b

1 T12 eT22 (z−u) T21 (z − b)e1 mn + V δ,b (b) dz

⎧

⎨−FT eT22 (b−u) T−1 (e1 ⊗ I )e1 + T V 1 (b) , 12 mn 21 m mn δ,b 22 1 = ⎩0, B1 δ,b (u) = F =



∞

z=b

T12 eT22 (z−u) T21 Φ 1 δ,b (b) dz

/ 1 −FT12 eT22 (b−u) T−1 22 T21 Φ δ,b (b), u ≤ b 0,

u > b.

u≤b u>b

250

A. S. Dibu and M. J. Jacob

Proof Using change of variables, Eqs. (7) and (12) can be written as F0 V 1 δ,b (u)

 =F

F0 Φ 1 δ,b (u) = F



b z=u b z=u

1 1 T12 eT22 (z−u) T21 V 1 δ,b (z)dz + Aδ,b (u) + Cδ,b (u) 1 1 T12 eT22 (z−u) T21 Φ 1 δ,b (z)dz + Zδ,b (u) + Dδ,b (u).

Now, the left product operation with M−1 throughout in the above equations yields M−1 F0 V 1 δ,b (u) =



b z=u

  −1 1 1 A eT22 (z−u) T21 V 1 (z)dz + M (u) + C (u) δ,b δ,b δ,b (31)

M−1 F0 Φ 1 δ,b (u) =



b z=u

  −1 1 Z1 eT22 (z−u) T21 Φ 1 δ,b (z)dz + M δ,b (u) + Dδ,b (u) . (32)

Differentiating Eqs. (31) and (32) with respect to initial capital u will then deliver,  d 1 d  1 V δ,b (u) = M−1 Aδ,b (u) + C1 (u) δ,b du du     1 −1 1 + T22 M−1 A1 δ,b (u) + Cδ,b (u) − T22 M F0 + T21 V δ,b (u)

M−1 F0

 d 1 d  1 Φ δ,b (u) = M−1 Zδ,b (u) + D1 δ,b (u) du du     1 −1 1 + T22 M−1 Z1 δ,b (u) + Dδ,b (u) − T22 M F0 + T21 Φ δ,b (u)

M−1 F0

which can be alternatively written as $ %

d −1 I mn + T22 M F0 + T21 V 1 δ,b (u) du $ %   d 1 + T22 M−1 A1 = I mn δ,b (u) + Cδ,b (u) du $ %

d + T22 M−1 F0 + T21 Φ 1 I mn δ,b (u) du % $   d 1 + T22 M−1 Z1 (u) + D (u) . = I mn δ,b δ,b du Then the left product operation with F T12 on the above equations provides the required equations (29) and (30) and hence the theorem holds. Now, applying the Laplace transform on Eqs. (29) and (30), we have the following corollary.

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

251

Corollary 1 The Laplace transforms with respect to the initial capital “u ≤ b” of 1 the measures V 1 δ,b (u) and Φ δ,b (u) satisfy the equations 1 1 Lu (su )V˜ δ,b (su ) = B˜ V (su ) 1

1

˜ δ,b (su ) = B˜ Φ (su ) Lu (su )Φ

(33) (34)



  = su I mn + FT12 T22 M−1 F0 − F" M˜ 1 (su ) − M˜ 2 (su ) Lu (su )  −F"" M˜ 2 (su ) + FT12 T21 and

where,

  −1 ˜ 1 1 ˜1 B (s ) = s I + FT T M Aδ,b (su ) − A1 u u mn 12 22 V δ,b (0) + F0 V δ,b (0)   1 1 1 B˜ Φ (su ) = su I mn + FT12 T22 M−1 Z˜ 1 δ,b (su ) − Zδ,b (0) + F0 Φ δ,b (0).

(35) (36)

Proof Taking the Laplace transform on Eqs. (29) and (30) and collecting the 1 ˜1 coefficients of V˜ δ,b (su ) and Φ δ,b (su ) on the left-hand side of the transformed equations will bring out Eqs. (33) and (34) and hence completes the proof. 1 The expression for V 1 δ,b (u) and Φ δ,b (u) will be explicit if the solution of the 1 1 unknown quantities V δ,b (0) and Φ δ,b (0) in Eqs. (35) and (36), respectively, is available. To follow up, we use Lemma 1 to determine the solutions at zero initial capital. 6 Lemma 1 Let Δρ δ = diag(ρ δ,1 , ρ δ,2 , . . . , ρ δ,mn ) and Γ δ = γ δ,1 , γ δ,2 , . . . , 7T be the eigenvalues matrix and the left eigenvectors matrix of Pδ = γ δ,mn Γ −1 Δ Γ ρ δ δ , respectively. Then ρ δ,i ’s are determined by the roots of the equation δ det Lu (s) = 0 which must be in the right-half complex plane and γ δ,i can be obtained by solving the equations γ δ,i Lu (ρ δ,i ) = 0; i = 1, 2, . . . , mn. 1 Proposition 2 The exact solutions for F0 V 1 δ,b (0) and F0 Φ δ,b (0) are given by 1 F0 V 1 δ,b (0) = Aδ,b (0) − Pδ

 −

∞ u=0

−

∞ u=0

respectively.

∞ u=0

e−Δρ δ u A1 δ,b (u)du

e−Pδ u FT12 T22 M−1 A1 δ,b (u) du

1 F0 Φ 1 δ,b (0) = Zδ,b (0) − Pδ







∞ u=0

(37)

e−Δρδ u Z1 δ,b (u)du

e−Pδ u FT12 T22 M−1 Z1 δ,b (u) du

(38)

252

A. S. Dibu and M. J. Jacob

Proof For the proposed model in this paper, equation (17) in [29] which is derived using Theorems 1 and 2 of [9], is revised to obtain the suitable form given by 

Pδ + FT12 T22 M−1



F0 − F" M˜ 1 (Pδ ) − M˜ 2 (Pδ )  −F"" M˜ 2 (Fδ ) + FT12 T21 = 0

(39)

By Lemma 1, it is easy to show that Eq. (39) is equivalent to γ δ,i Lu (ρ δ,i ) = 0 for i = 1, 2, . . . , mn. Setting su to ρ δ,i in Eqs. (33) and (34) yields   1 −1 ˜ 1 γ δ,i F0 V 1 ρ Aδ,b (ρ δ,i ) (0) = γ A (0) − γ I + FT T M 12 22 δ,i mn δ,i δ,b δ,i δ,b  ∞ = γ δ,i A1 (0) − ρ γ e−ρ δ,i u A1 δ,i δ,i δ,b δ,b (u) du  −

u=0

∞ u=0

e−ρ δ,i u γ δ,i FT12 T22 M−1 A1 δ,b (u) du

(40)

  1 −1 ˜ 1 γ δ,i F0 Φ 1 Zδ,b (ρ δ,i ) δ,b (0) = γ δ,i Zδ,b (0) − γ δ,i ρ δ,i I mn + FT12 T22 M  ∞ = γ δ,i Z1 e−ρ δ,i u Z1 δ,b (0) − ρ δ,i γ δ,i δ,b (u) du  −

u=0

∞ u=0

e−ρ δ,i u γ δ,i FT12 T22 M−1 Z1 δ,b (u) du

(41)

respectively for i = 1, 2, . . . , mn. The matrix form of Eqs. (40) and (41) is given by Γ δ F0 V 1 δ,b (0)

=

Γ δ A1 δ,b (0) − Δδ Γ δ  −

∞ u=0

Γ δ F0 Φ 1 δ,b (0)

=



∞ u=0

∞ u=0

e−Δδ u A1 δ,b (u)du

e−Δδ u Γ δ FT12 T22 M−1 A1 δ,b (u) du

Γ δ Z1 δ,b (0) − −





∞

Δδ Γ δ u=0

(42)

e−Δδ u Z1 δ,b (u)du

e−Δδ u Γ δ FT12 T22 M−1 Z1 δ,b (u) du

(43)

respectively. Multiplying throughout by Γ −1 δ in Eqs. (42) and (43) will result in Eqs. (37) and (38), respectively, and hence the proof is completed. Remark 2 By the Perron–Frobenius theorem [3], the eigenvalue of Pδ with the minimum real part, say ρ δ,1 , is real and strictly less than the real part of all other eigenvalues. Let γ δ,1 = γ δ be the associated left eigenvector normalised by γ δ e1 mn = 1. Then all components of γ δ are real and non-negative. In particular for

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253

δ = 0, we have the minimum non-negative root of Ldet u (su ) = 0, ρ 0,1 = 0, and the associated left eigenvector, γ 0,1 = π , the stationary probability row vector of the CTMC, {J (t)}t ≥0. In Eqs. (33) and (34), the elements of mn-dimensional square matrix Lu (su ) are rational which can be represented in the form Lu (su ) = [Lu (su )]i,j =1,2,...,mn

tij (su ) = ⊗ I n for i, j = 1, 2, . . . , m rij (su ) =

7 1 6 tij (su ) ⊗ I n for i, j = 1, 2, . . . , m Qu (su )

where Qu (su ) = [qu (su )]nκ for κ ∈ {1, 2, . . . , m} in which qu (su ) is the m+n1 +n2 degree rij (s

denominator polynomial

u ) that occurs in κ number of columns unique pij (sb ) tij (su ) adj which is same as in . Now denoting Lu (su ) and Ldet of u (su ) rij (su ) qij (sb ) as the adjoint and determinant of Lu (su ), Eqs. (33) and (34) can be alternatively settled as adj

Lu (su ) ˜ 1 1 V˜ δ,b (su ) = det B V (su ) Lu (su )

(44)

adj

Lu (su ) ˜ 1 ˜1 Φ B Φ (su ) δ,b (su ) = Ldet u (su )

(45) adj

where Ldet u (su ) and the elements of the mn-dimensional square matrix Lu (su ) 1 adj are rational. Then Ldet u (su ) and Lu (su ) can be represented as multiple of Qu (su ) which is followed again due to the MAP/PH structure. Now, inverting Eqs. (44) 1 and (45) with respect to su will give the expression for V 1 δ,b (u) and Φ δ,b (u), respectively, which is explained by adopting the similar arguments from Theorem 3 and modified as given below. adj

adj

det Theorem 5 Let Qu (su ) = Qu (su )Lu (su ) and Qdet u (su ) = Qu (su )Lu (su ) det where Qu (su ) is the denominator polynomial of Lu (su ). For l = 1, 2, . . . , κ(m + n1 + n2 ), let ρ u:l denote the distinct roots of Qdet u (su ) = 0 for which “κm” number of roots are in the right half, κ(n1 + n2 ) number of roots are in the left half of the complex plane, with all roots having algebraic multiplicity n. Then  κ(m+n +n ) for an arbitrary ωb ∈ / ρ u:l l=1 1 2 , the closed-form expressions for the

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solution of equations (11) and (14) are obtained by using the generalised Hermite’s interpolating polynomial which are given by V1 δ,b (u) = Φ1 δ,b (u) = where, ηz (u) =

1 Qdet u (ωu ) 1 Qdet u (ωb )

 

u

x=0 u x=0

η ωu (x)B 1 V (u − x)dx

(46)

η ωu (x)B 1 Φ (u − x)dx

(47)

κ(m+n1 +n2 ) n

adj d j−1 j =1 d lj (z, u) ds j−1 Qu (su )|su =ρ u:l

l=1

and

u

% $ d −1 1 B1 + FT A1 (u) = I T M mn 12 22 V δ,b (u) + F0 V δ,b (0)δ d (u) du % $ d −1 1 + FT Z1 (u) = I T M B1 mn 12 22 Φ δ,b (u) + F0 Φ δ,b (0)δ d (u) du in which n n−j  ) z − ρ u:l  u(n−j −i) eρ u:l u f (t l (ρ u:l ) d lj (z, u) = K l (z) Γ (j ) Γ (n − j − i + 1)Γ (i + 1) i=0

6 7−1 K l (z) = f l (z) ⎡ ⎤n κ(m+n :1 +n2 )   =⎣ z − ρ u:l ⎦ l=1

and δ d (u) is the Dirac delta function. Proof Operating left product and division by Qu (su ) in the right-hand side of Eqs. (44) and (45), we have adj

Qu (su ) ˜ 1 1 V˜ δ,b (su ) = det B V (su ) Qu (su )

(48)

adj

Qu (su ) ˜ 1 ˜1 Φ B Φ (su ). δ,b (su ) = Qdet u (su )

(49)

κ(m+n +n )  / ρ u:l l=1 1 2 and using the generalised Hermite’s Now for an arbitrary ωu ∈ interpolating polynomial, Eqs. (48) and (49) can be alternatively written as 1 V˜ δ,b (su ) =

ξ˜ ωu (su ) Qdet u (ωu )

1 B˜ V (su )

(50)

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

˜1 Φ δ,b (su ) = where ξ˜ z (su ) =

ξ˜ ωu (su ) Qdet u (ωu )

1 B˜ Φ (su )

(51)

κ(m+n1 +n2 ) n

adj d j−1 ˜ j =1 d lj (z, su ) ds j−1 Qu (su )|su =ρ u:l

l=1

in which the

u

coefficient function d˜ lj (z, su ) is given by d˜ lj (z, su ) = K l (z)

255

 (j −1) n−j    z − ρ u:l i (i) z − ρ u:l f (ρ u:l )S li (z, su ) Γ (j ) Γ (i + 1) l i=0

for which S li (z, su ) is given by

z − ρ u:l S li (z, su ) = su − ρ u:l

n−(j −1)−i .

Inverting Eqs. (50) and (51) with respect to su will yield the expressions for EDDR and GSF with initial surplus level 0 ≤ u ≤ b.

6 Numerical Examples An illustration on a scalar prototype is done, to validate expressions in the previous section, using Example 1. Two other numerical examples are illustrated in a twostate model through Examples 2 and 3 to show the tractability of expressions in the multi-phase environment. The computational codes are compiled in Mathematica, and the GUI of MATLAB is utilised to plot the graphs. Example 1 Before moving to the multi-phase model, the expressions derived in Sect. 6 are to be essentially validated. A single phase model is considered in this example. We use the expression of GSF φ δ,b (u) given by Eq. (6) and obtained the outputs of Laplace transform of ruin time ψ δ,b (u) by taking unit penalty function. Furthermore, the other parameters are set as follows: δ = 0.6 as the discounting factor and b = 10 as the horizontal barrier. The corresponding plots of ψ δ,b (u) against the initial capital u resemble Fig. 1(b) of example (4.1) in [31] for different values of λ. While increasing the Bayesian parameter λ, the probability of switching from type I claims to the relatively lower intensity type II claims increases. In the example, the intensity of claim sizes of type II claims is less than that of type I claims. From the graph (See Fig. 2) and the table values (see Table 1), it is evident that the values of Laplace transform of ruin time decrease by an increase in the parameter λ and converges to zero for unbounded initial capital.

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0.5 =1 =2 =3 =4 =5

0.4

0.3

0.2

0.1

0 0

1

2

3

4

5

6

7

8

9

10

Fig. 2 Single-state model with barrier, b = 10—Laplace transform of ruin time vs initial capital Table 1 Single-state model with barrier, b = 10—ruin probabilities vs initial capital

λ 1 2 3 4 5

ψ 0.6,10(0) 0.4944 0.4860 0.4800 0.4757 0.4724

ψ 0.6,10 (1) 0.2754 0.2523 0.2371 0.2254 0.2165

ψ 0.6,10(3) 0.0911 0.07615 0.0662 0.0590 0.0536

ψ 0.6,10 (7) 0.0115 0.0073 0.0063 0.0051 0.0043

ψ 0.6,10 (9) 0.0073 0.0023 0.0038 0.0030 0.0027

Example 2 In this example, we consider a two-state model having exponential premiums with rate G0 = [0.7]. Along with δ = 1 and b = 10, the following parameters are further taken into account: α = [1, 0] ,

β = [0.8, 0.2]





−0.9 0.1 0.7 0.1 , D1 = , D0 = 0 −0.8 0.1 0.7 E0 =

γ 1 = [0.9, 0.1] , G1 =



−1 0.3 , 0 −0.7

E1 =



0.4 0.3 , 0.3 0.4





−0.9 0.9 −1.8 1.8 , γ 2 = [0.6, 0.4] and G2 = . 0 −4 0 −1.6

The Bayesian parameter matrix is taken as Λ = diag [−1, −0.9] so that the positive averagecash flow is assured with a security loading factor, θ = 0.14. The values of EDDR ν 1,10 (u) , and by using unit penalty function in the expression of GSF, the

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

257

  values of Laplace transform of ruin time ψ 1,10 (u) are obtained  for 0 ≤ u ≤ b. Furthermore, the expected discounted deficit at ruin φ 1,10 (u) is also obtained by taking the penalty function as w(x, y) = e−ηy in the expression of GSF. In Example 2, we expect an increasing function of ν 1,10 (u) which is expected to coincide with ν 1,10 (10) at u = b = 10 (see Table 2a and Figs. 3a, b and 4). The plots and table values support the intuitions we have. The values of ψ 1,10 (u) and φ 1,10(u) also report the same coinciding property. Furthermore, we had an independent expression for all the performance measures given by (37) and (38) for zero initial capital. Our final expressions given by (46) and (47) limit to the values of these expressions at zero capital. This also validates the accuracy of the closed-form solutions, (46) and (47). Example 3 For u = b, a two-state model having Phase-type premiums is illustrated below with the following parameters along with δ = 1. α = [1, 0] ,



−0.9 0.9 D0 = , 0 −0.8

β = [0.8, 0.2]

−1 1 E0 = , 0 −0.9



0 0 D1 = , 0.8 0

0 0 E1 = , 0.9 0



−1 1 G0 = γ 0 = [0.7, 0.3] , , γ 1 = [0.9, 0.1] , 0 −0.9



−0.9 0.9 −1.8 1.8 G1 = , γ 2 = [0.6, 0.4] and G2 = . 0 −0.8 0 −1.6 The Bayesian parameter matrix is taken as Λ = diag [−1, −0.9] so that the positive average cash flow is assured with a security loading factor θ = 0.457. unit penalty functions, the values of The values of EDDR ν 1,b (b) , and by using  Laplace transform of ruin time ψ (b) and expected discounted deficit at ruin 1,b   φ 1,b (b) are obtained for 0 ≤ b ≤ ∞ (see Table 2b, Figs. 5a, b and 6). On observing the output values of EDDR and the GSF under unit penalty function and under discounted penalty function that depends only on deficit at ruin, it is noted that as b tends to ∞, the EDDR ν 1,b (b) increases and converges to a constant while the values of Laplace transform of ruin time and the discounted deficit at ruin settles down to zero. The outputs are evident to support our expectations.

u 0 1 3 5 7 9

ν 1,10 (u) 0.0075394 0.0157673 0.0438987 0.1092820 0.2674270 0.6498320

ψ 1,10 (u) 0.383013 0.232532 0.0765822 0.028174 0.0128726 0.0072313

(a) With exponential premiums and for u ≤ b

Table 2 Two-state model

φ 1,10 (u) 0.44138800 0.24320700 0.08110710 0.03001510 0.01369940 0.00767374 b 0 1 3 5 10 ∞

ν 1,b (b) 0.508040 0.562532 0.632340 0.659250 0.674023 0.675333

ψ 1,b (b) 0.24771900 0.16703100 0.06366140 0.02381490 0.00194031 0.00000000

(b) With Phase-type premiums and for u = b φ 1,b (b) 0.4152220 0.2515440 0.09363030 0.03446420 0.00277501 0.00000000

258 A. S. Dibu and M. J. Jacob

1.2

Laplace transform of ruin time

Expected discounted dividends paid until run

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent. . .

=1 and b=10

1 0.8 0.6 0.4 0.2 0 0

1

2

3

4 5 6 Initial capital

7

8

9

10

259

0.25 0.2 0.15 0.1 0.05 0

0

1

2

3

4 5 6 7 Initial capital, u=b

(a)

8

9

10

(b)

Fig. 3 Two-state model with exponential premiums. (a) Expected discounted dividends paid until ruin vs initial capital. (b) Laplace transform of ruin time vs initial capital

Expected discounted deficit at ruin

0.45

± =1 and b=10

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

1

2

3

4

5

6

7

8

9

10

Initial capital Fig. 4 Two-state model with exponential premiums—expected discounted deficit at ruin vs initial capital

7 Conclusion The paper introduces the random income feature in a MAP risk model. The interarrival time of random income is considered to follow a MAP to realise the cash inflows through multiple phases in an insurance company. Two types of claims are assumed to follow Phase-type distribution for which the claims are inter-dependent according to an exponential rate matrix in which the rates may be dependent on prior data. Furthermore, the surplus process is restricted by a horizontal barrier, b ≥ u. For u = b, the closed-form expressions of the expected discounted dividends

A. S. Dibu and M. J. Jacob 0.68 0.66

Laplace transform of ruin time

Expected discounted dividend paid until ruin

260

=1

0.64 0.62 0.6 0.58 0.56 0.54 0.52 0.5

0

1

2

3

4

5

6

7

8

9

10

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

Initial capital, u=b

3

4

5

6

7

8

9

10

Initial capital

(a)

(b)

Fig. 5 Two-state model with Phase-type premiums. (a) Expected discounted dividend paid until ruin vs initial capital, u = b. (b) Laplace transform of ruin time vs initial capital

Expected discounted deficit at ruin

0.45

± =1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

1

2

3

4

5

6

7

8

9

10

Initial capital, u=b Fig. 6 Two-state model with Phase-type premiums—expected discounted deficit at ruin vs initial capital, u = b

paid out until ruin, and the Gerber–Shiu function is obtained using the method of Laplace transforms when the premiums are Phase-type distributed. While for a general initial capital u ≤ b, the closed-form expressions of the expected discounted dividends paid out until ruin and the Gerber–Shiu function are obtained using the method of Laplace transforms when the premiums are exponentially distributed. The expressions are validated in the scalar prototype by mimicking the results of example (4.1) in [31] and examples for two-state models are illustrated under both exponential and Phase-type premiums.

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Remark 3 An important observation on implementing the Lundberg’s roots method for a multi-phase random income model is that the algebraic multiplicity of the Lundberg’s roots is nothing but the number of phases for the MAP inter-arrival times of premium.

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22. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999) 23. Li, J., Dickson, D.C., Li, S.: Analysis of some ruin-related quantities in a Markov-modulated risk model. Stoch. Model. 32(3), 351–365 (2016) 24. Melnikov, A.: Risk Analysis in Finance and Insurance. Chapman and Hall/CRC, Boca Raton (2011) 25. Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16(4), 764–779 (1979) 26. Shija, G., Jacob, M.: Gerber-Shiu function of Markov modulated delayed by-claimtype risk model with random incomes. J. Math. Finance 6(4), 489 (2016) 27. Spitzbart, A.: A generalization of Hermite’s interpolation formula. Am. Math. Mon. 67(1), 42–46 (1960) 28. Temnov, G.: Risk process with random income. J. Math. Sci. 123(1), 3780–3794 (2004) 29. Zhang, Z., Yang, H., Yang, H.: On the absolute ruin in a MAP risk model with debit interest. Adv. Appl. Probab. 43(1), 77–96 (2011) 30. Zhang, Z., Eric, C., Cheung, K.: The Markov additive risk process under an erlangized dividend barrier strategy. Methodol. Comput. Appl. Probab. 18(2), 275 (2016) 31. Zou, W., Gao, J.w., Xie, J.h.: On the expected discounted penalty function and optimal dividend strategy for a risk model with random incomes and interclaim-dependent claim sizes. J. Comput. Appl. Math. 255, 270–281 (2014)

A PH Distributed Production Inventory Model with Different Modes of Service and MAP Arrivals Salini S. Nair and K. P. Jose

Abstract This paper studies a production inventory model with retrial of customers under (s, S) policy. The arrival of customers is according to a Markovian Arrival Process with representation (D0 , D1 ) and service times follow an exponential distribution. The production process follows a phase-type distribution. When the inventory level reduces to a pre-assigned level s due to demands, production starts and service is given at a reduced rate. This reduced rate continuous up to the zero level of inventory. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level. An arriving customer, who notices the buffer full, proceeds to an orbit of infinite capacity with some probability and decides to leave the system with the complementary probability. An orbiting customer may retry from the orbit and inter-retrial times are exponentially distributed with linear rate. Various system performance measures of the model are defined. A suitable cost function is constructed and analyzed algorithmically. The optimum (s, S) pair is obtained. The effect of correlation between two successive inter-arrival times is also analyzed. Keywords Production inventory · Retrial of customers · Markovian arrival process · Phase-type distribution

1 Introduction Many researchers are interested in queuing-inventory systems with the production of inventory items for the last few decades. Nowadays, manufacturing companies produce inventory in response to the actual demand. Production inventory under (s, S) policy can be used to model these types of systems efficiently. Different notions such as retrial of customers, impatience of customers, server vacations,

S. S. Nair () · K. P. Jose PG and Research Department of Mathematics, St. Peter’s College, Kolenchery, Kerala, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_16

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interruptions of the service as well as the production process, etc. are being investigated in different production inventory systems. Artalejo et al. [1] were the first to study inventory policies with positive lead time along with the retrial of unsatisfied customers. They obtained solutions algorithmically. Krishnamoorthy and Jose [8] analyzed three production inventory systems with positive service time and retrial of customers. They assumed all the underlying distributions to be exponential. They find out that the model with buffer size equal to the current inventory level is the best profitable model. Benjaafar et al. [3] analyzed a production inventory system by considering customer impatience. They obtained that the optimal policy can be described by a production base-stock level and an admission threshold. They also characterized analytically the sensitivity of these thresholds to operating parameters. Krishnamoorthy and Viswanath [9] considered a production inventory system with positive service time. The time for producing each item was assumed to follow a Markovian production scheme. The customer arrival process followed a Markovian arrival process. The service time to each customer followed a phasetype distribution. They obtained the effect of the control variables s and S on the fraction of time the system goes out of inventory and on the expected loss rate of customers. He and Zhang [6] considered an inventory-production system consisting of a warehouse and a production facility. They obtained explicit solutions and developed computational methods for analyzing system performance measures. Yu and Dong [17] considered a production lot size problem as a renewal process and used a numerical approach to find out the optimal solution to the problem. Wensing and Kuhn [16] analyzed periodic replenishment processes that exhibited order crossover. It was compared with the existing concept of outstanding orders. Based on this, formulas were developed to give an exact analysis of three essential performance indicators of a periodic-review order-up-to inventory system with independent stochastic lead times. Krishnamoorthy et al. [10] considered an (s, S) production inventory system with positive service time, interruption to both service and production process. Production time and service time followed Erlang distribution and other random variables were exponentially distributed. They obtained an explicit expression for the necessary and sufficient condition for the stability of the system under study. Several system performance measures were derived, and their dependence on the system parameters was studied numerically. Anoop and Jacob [11] studied a multi-server Markovian queuing system by considering the servers as a standard production inventory and they obtained the condition for checking ergodicity and the steady state solutions. Beak and Moon [2] studied an (s, S) production inventory system. They analyzed the model using a regenerative process and obtained the result that the queue size and inventory level processes were independent in steady state. Zare et al. [18] analyzed a production inventory system consisting of a warehouse and a distribution center. The time taken to transport the inventory between these two centers was generally distributed. They obtained the optimal reorder point at the distribution

Production Inventory Model with Different Modes of Service

265

center and the optimal base-stock level in the warehouse. Chan et al. [4] studied a production inventory model with non-stop production. They assumed deterioration during deliveries. They optimized the cost for the system in which some of the cost parameters were production rate dependent. The primary concern in most of the manufacturing companies is to meet customer’s demands and to make maximum profit. Therefore, manufacturers have to increase or decrease production and sales of the items according to the demand of customers. Thus the effects of variations in production rate and service rate on inventory systems need further study. Jose and Salini [7] studied two production inventory systems with positive service time and retrial of customers. They assumed different rates of production depending on the inventory level and constructed a suitable cost function for an algorithmic solution using Matrix Analytic Method. Dhanya et al. [14] introduced two modes of service in an inventory system with positive service time. They assumed positive lead time for replenishment and obtained the stability condition. Product form solution for system state distribution is established. The optimal value of the service rate in a lower mode of service is also obtained. This paper describes a production inventory system with the retrial of customers and varying service rates. The arrival of customers is according to a Markovian arrival process with representation (D0 , D1 ) of order m1 . The service times follow an exponential distribution with rate μ. When the inventory level reduces to s, production starts and service is given at a reduced rate. The service time distribution has rate αμ, 0 < α < 1 when the inventory level lies between 0 and s. The production process is switched off when the inventory reaches the maximum level S. The production process follows phase-type distribution with representation (β, T ) of order m2 . The arriving customers are directed to a buffer of finite capacity equal to the current inventory level. An arriving customer, who notices the buffer full, proceeds to an orbit of infinite capacity with probability γ or decides to leave the system with probability (1 − γ ). An orbiting customer may retry from there and inter-retrial times are exponentially distributed with parameter iθ when there are i customers in the orbit. A retrial customer, who finds the buffer full returns back to the orbit with probability δ or decides to leave the system with probability (1 − δ). This paper is organized as follows. Section 2 contains the mathematical description of the model. Section 3 deals with the system stability. Performance measures are included in Sect. 4. Numerical results are provided in Sect. 5 and a related optimization problem is discussed in Sect. 6. Section 7 describes the Correlation analysis. Finally, concluding remarks are included in Sect. 8.

2 Mathematical Description of the Model To analyze the model mathematically, we use the following notations. • I (t): Inventory level at time t.

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• N(t): Number of customers in the orbit at time t. • M(t): Number of customers in the buffer at time t. / 0, if the production is in OFF mode • F (t)= 1, if the production is in ON mode • J1 (t): Phase of the arrival process at time t. • J2 (t): Phase of the production process at time t. Now, {X(t), t ≥ 0}, where X(t) = (N(t), F (t), I (t), M(t), J1 (t), J2 (t)) is a level dependent quasi birth-and-death process on the state space {(i, 0, j, h, k) : i ≥ 0; j = s + 1, . . . , S; h = 0, . . . , j ; k = 1, . . . , m1 } ∪ {(i, 1, j, h, k, l) : i ≥ 0; j = 0, . . . , S − 1; h = 0, . . . , j ; k = 1, . . . , m1 ; l = 1, . . . , m2 }. Now, we describe the transitions in the Markov chain as follows. (a) Transitions due to arrival of customers From (i, 0, j, h, k) to (i, 0, j, h + 1, k) is given by D1 , j = s + 1, . . . , S; h = 0, . . . , j − 1 From (i, 1, j, h, k, l) to (i, 1, j, h + 1, k, l) is given by D1 ⊗ Im2 , j = 0, . . . , S − 1; h = 0, . . . , j − 1 From (i, 0, j, j, k) to (i + 1, 0, j, j, k) is given by γ D1 , j = s + 1, . . . , S From (i, 1, j, j, k, l) to (i + 1, 1, j, j, k, l) is given by γ D1 ⊗ Im2 , j = 0, . . . , S − 1 (b) Transitions due to service completion From (i, 0, s + 1, h, k) to (i, 1, s, h − 1, k, l) is given by Im1 ⊗ μβ, h = 1, . . . , j From (i, 0, j, h, k) to (i, 0, j − 1, h − 1, k) is given by μIm1 , j = s + 2, . . . , S; h = 1, . . . , j From (i, 1, j, h, k, l) to (i, 1, j − 1, h − 1, k, l) is given by αμIm1 m2 , j = 1, . . . , s; h = 1, . . . , j

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From (i, 1, j, h, k, l) to (i, 1, j − 1, h − 1, k, l) is given by μIm1 m2 , j = s + 1, . . . , S − 1; h = 1, . . . , j (c) Transitions due to completion of production of an item From (i, 1, j, h, k, l) to (i, 1, j + 1, h, k, l) is given by Im1 ⊗ T 0 β, j = 0, . . . , S − 2; h = 0, . . . , j From (i, 1, S − 1, h, k, l) to (i, 0, S, h, k) is given by Im1 ⊗ T 0 , h = 0, . . . , S − 1 (d) Transitions due to retrial of customers from the orbit From (i, 0, j, h, k) to (i − 1, 0, j, h + 1, k) is given by iθ Im1 , j = s + 1, . . . , S; h = 0, . . . , j − 1 From (i, 0, j, j, k) to (i − 1, 0, j, j, k) is given by iθ (1 − δ)Im1 , j = s + 1, . . . , S From (i, 1, j, h, k, l) to (i − 1, 1, j, h + 1, k, l) is given by iθ Im1 m2 , j = 1, . . . , S − 1; h = 0, . . . , j − 1 From (i, 1, j, j, k, l) to (i − 1, 1, j, j, k, l) is given by iθ (1 − δ)Im1 m2 , j = 0, . . . , S − 1 (e) Transitions that leave the first four coordinates fixed From (i, 0, j, 0) to (i, 0, j, 0) is given by D0 − iθ Im1 , j = s + 1, . . . , S From (i, 0, j, h) to (i, 0, j, h) is given by D0 − μIm1 − iθ Im1 , j = s + 1, . . . , S; h = 1, . . . , j − 1 From (i, 0, j, j ) to (i, 0, j, j ) is given by D0 + (1 − γ )D1 − μIm1 − iθ (1 − δ)Im1 , j = s + 1, . . . , S

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From (i, 1, 0, 0) to (i, 1, 0, 0) is given by (D0 + (1 − γ )D1 ) ⊕ T − iθ (1 − δ)Im1 m2 From (i, 1, j, 0) to (i, 1, j, 0) is given by D0 ⊕ T − iθ Im1 m2 , j = 1, . . . , S − 1 From (i, 1, j, h) to (i, 1, j, h) is given by D0 ⊕ T − αμIm1 m2 − iθ Im1 m2 , j = 1, . . . , s; h = 1, . . . , j − 1 From (i, 1, j, j ) to (i, 1, j, j ) is given by (D0 + (1 − γ )D1 ) ⊕ T − αμIm1 m2 − iθ (1 − δ)Im1 m2 , j = 1, . . . , s From (i, 1, j, h) to (i, 1, j, h) is given by D0 ⊕ T − μIm1 m2 − iθ Im1 m2 , j = s + 1, . . . , S − 1; h = 1, . . . , j − 1 From (i, 1, j, j ) to (i, 1, j, j ) is given by (D0 + (1 − γ )D1 ) ⊕ T − μIm1 m2 − iθ (1 − δ)Im1 m2 , j = s + 1, . . . , S − 1 The generator matrix of the Markov chain is given by ⎤ ⎡ A1,0 A0 ⎥ ⎢A2,1 A1,1 A0 ⎥ ⎢ ⎥ ⎢ A A A 2,2 1,2 0 Q=⎢ ⎥ ⎢ A2,3 A1,3 A0 ⎥ ⎦ ⎣ .. .. .. . . .

(1)

where A1,i , i ≥ 0 governs transitions from i to i; A0 , transitions from i to i + 1; A2,i , i ≥ 1, transitions from i to i − 1. Neuts–Rao [13] truncation method is used to modify the infinitesimal generator Q to the form, where A1,i = A1 and A2,i = A2 for i ≥ N.

3 System Stability In order to find the stability of the system, we take Lyapunov test function (Falin and Templeton [5]), define φ(r) = i, if r is a state in the level i. The mean drift yr ,

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269

for any r belonging to the level i ≥ 1, is given by yr =



qrp (φ(p) − φ(r))

p=r

=



qru (φ(u) − φ(r)) +

u



qrv (φ(v) − φ(r)) +

v



qrw (φ(w) − φ(r))

w

where u, v, and w vary over the states belonging to the levels (i − 1), i and (i + 1), respectively. Then, by using the definition of φ, we can define φ(u) = i − 1, φ(v) = i and φ(w) = i + 1. yr = −

 u

qru +



qrw

w

⎧ ⎪ ⎪−iθ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −iθ(1 − δ) + γ (D1 e)k , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = −iθ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −iθ(1 − δ) + (γ (D1 e) ⊗ em2 )(k−1)m1 +l , ⎪ ⎪ ⎪ ⎪ ⎩

r = (i, 0, j, h, k), j = s + 1, . . . , S; h = 1, . . . , j − 1; k = 1, . . . , m1 r = (i, 0, j, h, k), j = s + 1, . . . , S; h = j ; k = 1, . . . , m1 r = (i, 1, j, h, k), j = 0, . . . , S − 1; h = 1, . . . , j − 1; k = 1, . . . , m1 ; l = 1, . . . , m2 r = (i, 1, j, h, k), j = 0, . . . , S − 1; h = j ; k = 1, . . . , m1 ; l = 1, . . . , m2

Since (1 − δ) > 0, for any  > 0, we can find N " large enough so that yr < − for any r belonging to the level i ≥ N " . Hence, by Tweedi’s result [15] the system under consideration is stable.

3.1 Rate Matrix R and Truncation Level N We use iterative method to find R. Denote the sequence of R by {Rn (N)} and is defined by R0 (N) = 0 and Rn+1 (N) = (−Rn2 (N)A2 (N) − A0 (N))A−1 1 (N). The value of N must be chosen such that |η(N) − η(N + 1)| < , where  is an arbitrarily small value and η(N), the spectral radius of R(N). For detailed discussion of selection of the value of N, one can refer to Neuts [12].

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4 Performance Measures of the Model The (i + 1)th component of the steady state probability vector x = (x0 , x1 , x2 , . . . , xN−1 , xN , . . .) is given by xi = (yi,0,j , yi,1,j ) where yi,0,j = (yi,0,j,0,1 , . . . , yi,0,j,0,m1 , . . . , yi,0,j,j,1 , . . . , yi,0,j,j,m1 ), j = s + 1, . . . , S; yi,1,j = (yi,1,j,0,1,1, . . . , yi,1,j,0,1,m2 , . . . , yi,1,j,0,m1 ,1 , . . . , yi,1,j,0,m1 ,m2 , . . . , yi,1,j,j,1,1 , . . . , yi,1,j,j,1,m2 , . . . , yi,1,j,j,m1 ,1 , . . . , yi,1,j,j,m1 ,m2 ), j = 0, . . . , S − 1. Then, 1. Expected Inventory level, EI , in the system is given by EI =

j  j  m1 m1  m2 ∞  ∞  S  S−1    jyi,0,j,h,k + jyi,1,j,h,k,l i=0 j =s+1 h=0 k=1

i=0 j =0 h=0 k=1 l=1

2. Expected number of customers, EO, in the orbit is given by  EO =

∞ 

 ixi e

i=1

3. Expected number of customers, EB, in the buffer is given by EB =

j  m1 S ∞   

hyi,0,j,h,k +

i=0 j =s+1 h=0 k=1

j  m1  m2 S−1  ∞   i=0 j =0 h=0 k=1 l=1

4. Expected switching rate, ESR, is given by ESR =

m1 ∞  s+1   i=0 h=1 k=1

μ yi,0,s+1,h,k

hyi,1,j,h,k,l

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5. Expected number of departures, EDS, after completing service is given by EDS =

j  m1 ∞  S  

μ yi,0,j,h,k +

i=0 j =s+1 h=1 k=1

+

∞ 

j  m1  m2 ∞  s  

αμ yi,1,j,h,k,l

i=0 j =1 h=1 k=1 l=1

j  m1  m2 S−1  

μ yi,1,j,h,k,l

i=0 j =s+1 h=1 k=1 l=1

6. Expected number of external customers lost, EL1 , before entering the orbit per unit time is EL1 =

m1 ∞  S   yi,0,j,j,k ((1 − γ )D1 e) i=0 j =s+1 k=1

+

m1  m2 ∞ S−1  

yi,1,j,j,k,l ((1 − γ )D1 ⊗ em2 )

i=0 j =0 k=1 l=1

7. Expected number of customers lost, EL2 , due to retrials per unit time is EL2 = θ (1 − δ)

m1 ∞  S  

iyi,0,j,j,k em1

i=0 j =s+1 k=1

+ θ (1 − δ)

m1  m2 S−1  ∞  

iyi,1,j,j,k,l em1 m2

i=0 j =0 k=1 l=1

8. Overall rate of retrials, ORR, is given by ORR = θ

∞ 

 ix i e

i=1

9. Successful rate of retrials, SRR, is given by j −1  m1 ∞  S   SRR = iθ yi,0,j,h,k em1 i=0 j =s+1 h=0 k=1

+

j −1  m1  m2 ∞  S−1   i=0 j =0 h=0 k=1 l=1

iθ yi,1,j,h,k,l em1 m2

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10. Expected number of crossovers, ECC, in one cycle is ECC =

m1  m2 s−1  ∞  

yi,1,s−1,h,k,l (em1 ⊗ T 0 )

i=0 h=0 k=1 l=1

+

m1  m2 ∞  s+1  

μ yi,1,s+1,h,k,l

i=0 h=1 k=1 l=1

5 Numerical Results We analyze the model by considering the performance measures overall and successful rate of retrials (ORR and SRR) and expected number of crossovers in one cycle (ECC). The values of ORR, SRR, and ECC by varying the parameters α, γ , δ, and θ are given in the following tables. Consider the following parameter values.



−2.1 1.0 0.1 1.0 m1 = 2, m2 = 2, D0 = , D1 = 1.0 −3.1 1.0 1.1 β = (0.5, 0.5), T =





2 −6 3 , T0 = 2 1 −4

Then the average arrival rate = 1.6 and correlation between two successive interarrival times = −0.0067. In Table 1, overall rate of retrials (ORR) decrease and successful rate of retrials (SRR) increases. As the service rate increases, the number of customers in the orbit decreases and hence ORR decreases. But increase in service rate increases the rate of successful retrials. The increase in service rate decreases ECC as in Table 1 Variations in α (S = 7; s = 2; γ = 0.6; N = 50; θ = 1.5; δ = 0.8; μ = 5)

α 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ORR 13.2152 13.2145 13.2138 13.2132 13.2127 13.2122 13.2118 13.2114 13.2111

SRR 1.3473 1.3485 1.3494 1.3502 1.3509 1.3514 1.3518 1.3522 1.3525

ECC 1.1766 1.1651 1.1192 1.0626 1.0059 0.9533 0.9059 0.8639 0.8268

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Table 1. In Tables 2 and 3, values of all the performance measures increase. As the probability of primary customers going to orbit and probability of retrial customers going to orbit increase, the number of customers in the orbit increases. This leads to the increase of ORR, SRR, and ECC. In Table 4, values of all the performance measures except ECC increase with the increase in retrial rate. Table 2 Variations in γ (S = 7; s = 2; α = 0.6; N = 50; θ = 1.5; δ = 0.8; μ = 3)

γ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ORR 0.5192 1.1318 1.8466 2.6704 3.6066 4.6542 5.8068 7.0526 8.3742

SRR 0.0561 0.1185 0.1874 0.2627 0.3440 0.4305 0.5211 0.6141 0.7080

ECC 0.4147 0.4350 0.4569 0.4801 0.5046 0.5300 0.5559 0.5818 0.6073

Table 3 Variations in δ (S = 7; s = 2; α = 0.6; γ = 0.6; N = 50; θ = 1.5; μ = 3)

δ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ORR 1.1416 1.2838 1.4657 1.7060 2.0373 2.5208 3.2854 4.6542 7.7400

SRR 0.1368 0.1507 0.1680 0.1902 0.2198 0.2615 0.3245 0.4305 0.6355

ECC 0.4376 0.4421 0.4476 0.4547 0.4641 0.4773 0.4971 0.5300 0.5903

ORR 10.5773 11.3201 12.0443 12.7515 13.4432 14.1205 14.7846 15.4364 16.0769

SRR 1.2258 1.2323 1.2372 1.2408 1.2436 1.2456 1.2471 1.2483 1.2492

ECC 0.8380 0.8365 0.8343 0.8316 0.8285 0.8252 0.8218 0.8182 0.8146

Table 4 Variations in θ (S = 7; s = 2; α = 0.6; γ = 0.6; N = 50; δ = 0.8; μ = 3)

θ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

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6 Optimization Problem Consider the following costs. • • • • • • • •

C = the fixed cost c1 = the procurement cost/unit/unit time c2 = the holding cost of inventory/unit/unit time c3 = the holding cost of customers in the orbit/unit/unit time c4 = the holding cost of customers in the buffer/unit/unit time c5 = the cost due to loss of primary customers/unit/unit time c6 = the cost due to loss of retrial customers/unit/unit time c7 = the cost due to service/unit/unit time.

We define the expected total cost/unit time as ET C = (C +(S −s)c1 )ESR +c2 EI +c3 EO +c4EB +c5 EL1 +c6 EL2 +c7 EDS

6.1 Numerical Illustrations Here, we find out the optimum values of the parameters α, γ , δ, and θ corresponding to the expected minimum total cost. We calculate the minimum expected total cost, by varying one parameter and keeping all others fixed. In Fig. 1, the optimum

Fig. 1 ET C versus α. S = 7; s = 2; γ = 0.6; N = 50; θ = 1.5; μ = 3; δ = 0.8; C = 20; c1 = 0.8; c2 = 0.1; c3 = 1; c4 = 0.99; c5 = 1.02; c6 = 1.02; c7 = 16.45

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Fig. 2 ET C versus γ . S = 7; s = 2; N = 50; θ = 1.5; μ = 3; δ = 0.8; α = 0.6; C = 20; c1 = 1; c2 = 12.9; c3 = 1; c4 = 1; c5 = 1; c6 = 1; c7 = 1

value of α which minimizes the expected total cost is obtained. The optimum value of α is 0.3 and the minimum ET C is 33.3947. The minimum ET C is 51.2581 at γ = 0.5 in Fig. 2. Figure 3 shows that the minimum ET C is 328.1441 at δ = 0.8. From Fig. 4, the optimum retrial rate θ is 1.3 and the minimum ET C is 30.5237.

6.2 Optimum (s, S) Pair We find out the optimum (s, S) pair, by fixing the parameter values and cost values. The optimum value of s, for each value of S, is obtained as in Table 5. The optimum value of s is 6 for all values 12, 13, 14, 15, 16 and 17 of S considered. The optimum (s, S) pair, which minimizes ET C is (6, 15) and the minimum value of ET C is 17.1538.

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Fig. 3 ET C versus δ. S = 7; s = 2; γ = 0.6; N = 50; θ = 1.5; μ = 3; α = 0.6; C = 20; c1 = 7; c2 = 88.5; c3 = 10.1; c4 = 0.9; c5 = 1; c6 = 0.8; c7 = 0.8

Fig. 4 ET C versus θ. S = 7; s = 2; γ = 0.6; N = 50; μ = 3; δ = 0.8; α = 0.6; C = 20; c1 = 1.5; c2 = 6.8; c3 = 1; c4 = 0.1; c5 = 1; c6 = 1; c7 = 1

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Table 5 Optimum (s, S) pair (γ = 0.6; N = 50; θ = 1.5; δ = 0.8; μ = 3; α = 0.6; C = 20;. c1 = 1; c2 = 1; c3 = 4; c4 = 1; c5 = 2; c6 = 2; c7 = 1)

s 4 5 6 7 8 9

S 12 17.9286 17.6202 17.5974 17.8814 18.5415 19.7641

13 17.7367 17.4054 17.3196 17.4791 17.9069 18.6733

14 17.6398 17.3028 17.1854 17.2771 17.5801 18.1196

15 17.6188 17.2854 17.1538 17.2081 17.4406 17.8564

16 17.6599 17.3350 17.1991 17.2333 17.4248 17.7688

17 17.7529 17.4391 17.3044 17.3286 17.4959 17.7965

7 Correlation Analysis 7.1 MAP with Negative Correlation Let ⎡

⎤ ⎡ ⎤ −2.0 2.0 0 0 0 0 D0 = ⎣ 0 −1.1 0 ⎦ , D1 = ⎣ 0.1 0 1.0⎦ 0 0 −13.2 12.0 0 1.2 Then r = −0.3690. That is, the arrival process has negative correlated arrivals with correlation between two successive inter-arrival times is −0.3690.

7.2 MAP with Positive Correlation Let ⎡

⎡ ⎤ ⎤ −2.0 2.0 0 0 0 0 D0 = ⎣ 0 −1.1 0 ⎦ , D1 = ⎣1.0 0 0.1 ⎦ 0 0 −13.2 1.2 0 12.0 Then r = 0.3690. That is, the arrival process has positive correlation with value 0.3690.

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Table 6 Effect of correlation on performance measures (S = 15; s = 6; γ = 0.6; N = 50; θ = 1.5; δ = 0.8; μ = 3; α = 0.6;)

r −0.3690 0.3690

ESR 0.0504 0.0537

EI 7.8710 8.0844

EO 0.2736 2.4005

EB 1.0079 1.7592

EL1 0.0020 0.0437

r −0.3690 0.3690

EL2 0.0617 0.5351

EDS 1.2839 1.2210

ORR 0.4105 3.6007

SRR 0.1389 1.1226

ERC 0.2970 0.2838

7.3 Effect of Correlation on Performance Measures From Table 6, when correlation between two successive inter-arrival times is positive, all performance measures except EDS and ERC have higher values.

8 Concluding Remarks In this paper, we analyzed a production inventory system with different service rates and retrial of customers. We investigated the stability of the system and derived various performance measures of the system in the steady state. A suitable cost function is constructed and analyzed. The optimum value of α is obtained graphically. The optimum values of other parameters are also obtained. The optimum (s, S) pair is also computed. The effect of correlation between two successive inter-arrival times on different performance measures is also analyzed. The analyzed model can be extended further by assuming Batch Markovian Arrival Process (BMAP). Acknowledgments Salini S. Nair acknowledges the financial support of University Grants Commission of India under Faculty Development Programme F.No. FIP/12th Plan/KLMG045 TF07/2015.

References 1. Artalejo, J.R., Krishnamoorthy, A., Lopez-Herrero, M.J.: Numerical analysis of (s, s) inventory systems with repeated attempts. Ann. Oper. Res. 141(1), 67–83 (2006) 2. Baek, J.W., Moon, S.K.: A production–inventory system with a Markovian service queue and lost sales. J. Korean Stat. Soc. 45(1), 14–24 (2016) 3. Benjaafar, S., Gayon, J.P., Tepe, S.: Optimal control of a production–inventory system with customer impatience. Oper. Res. Lett. 38(4), 267–272 (2010)

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On a Generalized Lifetime Model Using DUS Transformation P. Kavya and M. Manoharan

Abstract In this paper, we propose a new lifetime distribution based on the generalized DUS transformation by using Weibull distribution as the baseline distribution. This new distribution exhibits various behaviour of hazard function like increasing, decreasing and inverse bathtub. Here we try to study the characteristics of the new distribution and also analyse a real data set to illustrate the flexibility of the model. Keywords DUS transformation · Inverse bathtub · Weibull distribution

1 Introduction For analysing the lifetime data, there are number of models available in the literature. Earlier, only the constant, increasing and decreasing hazard rates received serious consideration. But, in real-life problem, a situation does arise when the hazard rate is expected to be non-monotone, for example, human life. To model such problems, several non-monotone hazard rate distributions were introduced. Mudholkar and Srivastava [7], Xie and Lai [10], Gupta et al. [4] and Xie et al. [11] are notable research works related to the non-monotone hazard rate data. There is a growing interest in the study of inverse bathtub hazard rates nowadays. A study of head and neck cancer data, [2] showed inverse bathtub hazard rates, in which the hazard rate initially increased, attained a maximum, and then decreased before it finally stabilized due to therapy. Log-normal, log-logistic, Burr Type XII, Burr Type III, log-Burr Type XII and the inverse Weibull distributions are some of the statistical distributions that show inverse bathtub hazard rates.

P. Kavya () · M. Manoharan Department of Statistics, University of Calicut, Malappuram, Kerala, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_17

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In the present study, we used a transformation called DUS transformation proposed by Kumar et al. [5]. If F (x) is the cdf of some baseline distribution, then the cdf G(x) of new distribution is given by, G(x) =

eF (x) − 1 . e−1

Maurya et al. [6] introduced a new class of distribution by using the generalization of DUS transformation. The cdf of Generalized DUS (GDUS) transformation is e(F

G(x) =

α (x))

−1 , e−1

where F (x) be the cdf of some baseline distribution. In both these papers, they used exponential distribution as the baseline distribution. Our main objective in this paper is to introduce a new class of distribution which includes all types of failure rates for suitable choice of parameter. By using GDUS transformation, the obtained distribution is expected to possess both monotone and non-monotone failure rates depending on the values of the parameter. The Weibull distribution has wide application in reliability and survival analysis. Depending on the shape parameter, Weibull models show different types of observed failures of components. Therefore here we consider Weibull distribution with parameters λ and k as the baseline distribution in GDUS transformation. The cdf and pdf of Weibull x k x k distribution are, respectively, G(x) = 1 − e−( λ ) and g(x) = ( λk )( xλ )k−1 e−( λ ) , x > 0, λ, k > 0. Using GDUS and Weibull distribution, the cdf and pdf of the new distribution, i.e., GDUS Weibull Distribution (GDUSWD) can be obtained as

F (x) =

f (x) =

αkx k−1 e

−( λx )k

e

$ %α −( x )k 1−e λ

−1

e−1

  x k α−1 1 − e−( λ ) e

;

(1)

$ % x k α 1−e−( λ )

, x > 0, α, λ, k > 0.

λk (e − 1)

(2) The hazard function of the distribution is,

h(x; α, λ, k) =

x k αkx k−1 e−( λ )



⎛

1−e

λk ⎝e − e

−( λx )k

$

α−1 e

$ % ⎞ x k α 1−e−( λ )

⎠

x k

1−e−( λ )

%α

.

(3)

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The paper deals with the selected topics as follows: In Sect. 2, we plot the pdf and hazard rates for different values of parameters of the GDUSWD. Various statistical characteristics of proposed distribution like moments, quantile function order statistic and Re´nyi entropy are included in Sect. 3. The parametric estimation for the new distribution is discussed in Sect. 4. In Sect. 5 we illustrate the flexibility of the proposed model for a real data set by using AIC (Akaike information criterion) and BIC (Bayesian information criterion), and finally recapitulate the conclusions in Sect. 6.

2 Shape of the pdf and Hazard Function

0.0

0.2

0.4

f(x)

0.6

0.8

1.0

The distribution function may seem complicated, so we plot it to gain a better understanding of the nature of the distribution. Using Eq. (2), the plots of pdf for various values of the parameters α, λ and k are given in Fig. 1.

0

2

4

6

8

10

x Fig. 1 The probability density plot of GDUSWD. (Red) α = 0.5, k = 1.5, λ = 1.5; (blue) α = 1.5, k = 2, λ = 1.5; (green) α = 0.8, k = 2, λ = 1.5

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(x) The term η(x) = −f f (x) , where f (x) is the density function of the distribution " and f (x) is the first derivative of f (x) with respect to x, is defined by Glaser [3] for the study of the shapes of the hazard rate. He stated the following theorem:

Theorem 1 "

1. If η (x) > 0 for all x > 0, then the distribution has increasing failure rate (IFR). " 2. If η (x) < 0 for all x > 0, then the distribution has decreasing failure rate (DFR). " " 3. Suppose there exists x0 > 0 such that η (x) > 0 for all x ∈ (0, x0 ), η (x0 ) = 0, " and η (x) < 0 for all x > x0 and  = limx→0 f (x) exists. Then if (i)  = 0, the distribution has inverse bathtub failure rate. (ii)  = ∞, the distribution has DFR. In GDUSWD,   x k −(k − 1) kx k−1 (α − 1)e−( λ ) −( λx )k α−1 −( xλ )k η(x) = + − α(1 − e ) e 1− x k x λk 1 − e−( λ ) and "

η (x) =

k − 1 k(k − 1)x k−2 + x2 λk $ % k(α − 1) k 2(k−1) −( λx )k −( λx )k k−2 − )e − kx (1 − e (k − 1)x λk λ

x k k − k x 2(k−1)e−2( λ ) λ

x k x k x k x k kα − k x k−1 e−( λ ) (1 − e−( λ ) )α−2 (α − 1)e−( λ ) + (1 − e−( λ ) ) . λ (4) "

The obtained expression of η (x) is too complicated, so we have used the software MATHEMATICA for checking the conditions mentioned in Theorem 1. Here we have observed that "

– When α ≤ 0.5, we have η (x) < 0 for all x > 0, hence the distribution has DFR " – When α ≥ 1, we have η (x) > 0 for all x > 0, hence the distribution has IFR " – When 0.5 < α < 1, there exists a x0 such that η (x) > 0 when x ∈ (0, x0 ), " " η (x0 ) = 0 and η (x) < 0 for all x > x0 , where x0 depends on the value of α, λ and k. From Eq. (2), we can easily verify that limx→0 f (x) = 0, hence the distribution has inverse bathtub shaped failure rate. Figure 2 shows the different shapes of hazard rates.

285

0

2

4

h(x)

6

8

10

On a Generalized Lifetime Model Using DUS Transformation

0.0

0.2

0.4

0.6

0.8

1.0

x

0.4

h(x)

0.3 0.0

0.00

0.1

0.05

0.2

0.10

h(x)

0.15

0.20

0.5

0.25

0.6

0.7

0.30

(a)

0

2

4

6

8

10

0.0

0.2

0.4

0.6

0.8

1.0

x

x

(c)

(b)

Fig. 2 The hazard rate plots of GDUSWD. (a) (Blue) α = 0.3, k = 2, λ = 0.1, (red) α = 0.4, k = 2, λ = 0.3. (b) (Blue) α = 0.8, k = 2, λ = 4, (red) α = 0.6, k = 2, λ = 3. (c) (Green) α = 1.5, k = 2, λ = 1.5, (blue) α = 2, k = 2, λ = 1.5

3 Some Analytical Characteristics Different statistical characteristics like moments, quantile function, order statistic and Re´nyi entropy of our proposed distribution are discussed below.

3.1 Moments The moments are used to understand the various characteristics of the proposed distribution. The rth raw moment of the GDUSWD is αk E(X ) = k λ (e − 1)



r

∞

x 0

r+k−1 −( λx )k

e

  x k α−1 1 − e−( λ ) e

$ % x k α 1−e−( λ )

dx

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Expanding the exponential term ex =

E(Xr ) =

αk λk (e − 1)



∞ 0

∞

xi i=0 i! ,

we get

 α −( λx )k ∞   1 − e  α−1 x k x k dx. x r+k−1 e−( λ ) 1 − e−( λ ) ! =0

Here the summation is absolutely convergent, we can interchange the summation and integral. ∞

 1 αk E(X ) = k λ (e − 1) !



r

=0

∞

x k

x r+k−1 e−( λ )

0

Using the expansion of series, (1 − y)b = E(Xr ) =

    x k α−1 x k α 1 − e−( λ ) 1 − e−( λ ) dx.

∞

i=0 (−1)

 i b yi i

and simplifying

$ % ∞  ∞  α (−1)m α + α − 1 Γ ( kr + 1) . r λk (e − 1) ! m k +1 ( m+1 k ) =0 m=0

λ

This expression of rth raw moment gives variance and other higher order central moments.

3.2 Quantile Function The pth quantile function Q(p) of our proposed distribution is obtained from the equation, F (Q(p)) = p

1 1 k Q(p) = λ −log(1 − log(1 + p(e − 1)) α ) .

(5)

3.3 Order Statistic Order statistics are sample values placed in ascending order. The study of order statistics deals with the applications of these ordered values and their functions. Let X1 , X2 , . . . , Xn be a random sample of size n from the GDUSWD distribution and X(1) , X(2) , . . . , X(n) denote the corresponding order statistics. The pdf and cdf of the rth order statistics fr (x) and Fr (x) are given by fr (x) =

n! F r−1 (x)[1 − F (x)]n−r f (x) (r − 1)!(n − r)!

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and Fr (x) =

n $ %  n j F (x)[1 − F (x)]n−j . j j =r

The pdf fr (x) and cdf Fr (x) of rth order statistic of our proposed distribution are obtained by using Eqs. (1) and (2) as, ⎛ fr (x) =

⎜e n! ⎜ (r − 1)!(n − r)! ⎝

α( λk )( xλ )k−1 e

−( xλ )k

⎞r−1 ⎛

$ % x k α 1−e−( λ )

− 1⎟ ⎟ ⎠ e−1

  x k α−1 e 1 − e−( λ )

$ % x k α 1−e−( λ )

⎜e − e ⎜ ⎝ e−1

⎞n−r ⎟ ⎟ ⎠

$ % x k α 1−e−( λ )

(6)

e−1 and

Fr (x) =

n $ 

%

⎛

n ⎜ ⎜e j ⎝

j =r

$ % x k α 1−e−( λ )

⎞j ⎛

$ % x k α 1−e−( λ )

⎜ − 1⎟ ⎟ ⎜e − e ⎠ ⎝ e−1 e−1

⎞n−j ⎟ ⎟ ⎠

.

(7)

The pdf of smallest and largest order statistics X(1) and X(n) is obtained by putting r = 1 and r = n, respectively, in Eq. (6). The cdf of X(1) and X(n) is obtained by putting r = 1 and r = n, respectively, in Eq. (7).

3.4 Entropy Entropy is interpreted as the degree of disorder or randomness in the system. Re´nyi entropy [8] is one of the well-known entropy measures. If random variable X has the pdf f (x), then the Re´nyi entropy is defined as, 1 log JR (γ ) = 1−γ



γ

(8)

f (x)dx ,

where γ > 0 and γ = 1. From Eq. (2), we get,  ∞ 0

f γ (x)dx =

 ∞$ 0

$

%γ ( x )k   x k x k γ (α−1) γ 1−e λ αk γ (k−1) −γ ( ) ( ) λ λ 1−e x e e λk (e − 1)

%α

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after some algebra, which is used in Sect. 3.1 

∞ 0

$ % ∞ ∞   γ i αi + γ α − γ (−1)j . f (x)dx =  γ −1 j (γ + j ) λk (e − 1)γ i=0 j =0 i! α γ k γ −1

γ

With this, Eq. (7) becomes, $ % $ % γ α k JR (γ ) = log − log 1−γ e−1 λk ⎡ ⎤ ∞  ∞ i $αi + γ α − γ % (−1)j  γ 1 ⎦. log ⎣ + 1−γ i! (γ + j ) j

(9)

i=0 j =0

4 Estimation Maximizing the logarithm of likelihood function is the most common method for finding the estimates—called MLE (Maximum Likelihood Estimates)—of the parameters involved in the given distribution. In this section, we use this method for obtaining the maximum likelihood estimates of the parameters α, λ and k of the proposed distribution. The log-likelihood function is, log L(x; λ) =

n :

f (xi ; λ)

i=1

log L(x; α, λ, k) =n logα + n log

$ % n  k logxi + n(k − 1)logλ + (k − 1) λ i=1

−

n    xi k

λ

i=1

+

n  

+ (α − 1) xi

 k α

xi k

log(1 − e−( λ ) )

i=1

1 − e−( λ )

i=1

n 

− n log(e − 1).

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Differentiating this function with respect to the parameters we get,    xi k  xi k α xi k  n  ∂log L 1 − e−( λ ) log 1 − e−( λ ) + log 1 − e−( λ ) , = + ∂α α n

n

i=1

i=1

n

xi k n  x k k  ∂log L −n n(k − 1) k i=1 xik e−( λ ) i = + + − (α − 1) x i k −( λ ) ∂λ λ λ λk+1 λ λ 1−e

i=1

−α

n  

xi

1 − e−( λ )

 k α−1

i=1

xi k  xi k k , e−( λ ) λ λ

and x     xi k ∂log L n i = + n logλ + log(xi ) − log ∂k k λ λ n

n

i=1

+ (α − 1)

n  i=1

+α

n   i=1

i=1

e

x −( λi )k

1−e

x −( λi )k

xi k

1 − e−( λ )

α−1

 x k i

λ xi k

e−( λ )

log

x  i

λ

 x k i

λ

log

x  i

λ

.

Equating these partial derivatives to zero yields three non-linear equations, and their solutions provide the maximum likelihood estimate of the parameters α, λ and k. Newton–Raphson method can be used to solve these equations with the help of the available statistical packages.

5 Application In this section, we have checked the flexibility of the proposed distribution and compared it with some well-known distributions namely, Inv. Lindley, Inv. Exponential, Gn. Inv. Lindley, Inv. Weibull, Inv. Gamma, Inv. Gaussian and Gn. Inv. Exponential, see Vikas et al. [9]. For the purpose of comparison, we have considered a set of real data of flood levels [1], 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.324, 0.296, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265. Many authors have used this data for checking the flexibility of their proposed distributions. We have used AIC (Akaike information criterion) and BIC (Bayesian

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Table 1 AIC, BIC and ranks [1(best) to 8( worst)] of the fitted models

Model Inv. Lindley (θ ) Inv. Exponential(λ) Gn. Inv. Lindley (α, θ ) Inv. Weibull(α, λ) Inv. Gamma(α, β) Inv. Gaussian(μ, λ) Gn. Inv. Exponential(α, λ) GDUSWD(α, λ, k)

AIC

BIC

0.8291 7.4806 −28.2950 −28.1947 −28.2833 −27.7040 −26.8224 −482.4123

0.1302 6.7817 −29.6930 −29.5927 −29.6812 −29.1020 −28.2203 −482.4209

Rank 7 8 2 4 3 5 6 1

information criterion) for comparing our model with other models. AIC and BIC are defined as, AIC = 2 × k − 2 × Log L, and BIC = k × Log(n) − 2 × Log L, where n is the sample size, k is the number of parameters and L is the maximum value of the likelihood function for the considered distribution. A minimum value of AIC and BIC is a sign of better fit of distributions. From Table 1, it can be seen that the proposed distribution gives the lowest AIC and BIC values. So we can conclude that GDUSWD provides the best fit for the data set compared to the other distributions given in this study.

6 Conclusion In the present study, we have introduced a new lifetime distribution exhibiting increasing, decreasing and inverse bathtub failure rates. We then derived its moments, quantile function, order statistic and Re´nyi entropy. We have considered a real dataset and compared our proposed distribution with some other well-known distributions. It is seen that among the distributions considered, the one proposed here fits best with the data. Thus, we can say that the GDUSWD is more flexible compared to the distributions mentioned in this study.

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References 1. Dumonceaux, R., Antle, C.: Discrimination between the lognormal and the Weibull distributions. Technometrics 15, 923–926 (1973) 2. Efron, B.: Logistic regression, survival analysis, and the Kaplan-Meier curve. J. Am. Stat. Assoc. 83, 414–425 (1988) 3. Glaser, R.E.: Bathtub and related failure rate characterizations. J. Am. Stat. Assoc. 75, 667–672 (1980) 4. Gupta, R.C., Gupta, P.L., Gupta, R.D.: Modeling failure time data by Lehmann alternatives. Commun. Stat. Theory Methods 27, 887–904 (1998) 5. Kumar, D., Singh, U., Singh, S.K.: A method of proposing new distribution and its application to bladder cancer patient data. J. Stat. Appl. Probab. Lett. 2, 235–245 (2015) 6. Maurya, S.K., Kaushik, A., Singh, S.K., Singh, U.: A new class of distribution having decreasing, increasing, and bathtub-shaped failure rate. Commun. Stat. Theory Methods 46(20), 10359–10372 (2017) 7. Mudholkar, G., Srivastava, D.: Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliab. 42, 299–302 (1993) 8. Renyi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 547–561. University of California Press, Berkeley (1961) 9. Sharma, V.K., Singh, S.K., Singh, U., Merovci, F.: The generalized inverse Lindley distribution: a new inverse statistical model for the study of upside-down bathtub data. Commun. Stat. Theory Methods 45(19), 5709–5729 (2016) 10. Xie, M., Lai, C.: Reliability analysis using an additive Weibull model with bathtub shaped failure rate function. Reliab. Eng. Syst. Saf. 52, 87–93 (1996) 11. Xie, M., Goh, T., Tang, Y.: A modified Weibull extension with bathtub-shaped failure rate function. Reliab. Eng. Syst. Saf. 76, 279–285 (2002)

Analysis of Inventory Control Model for Items Having General Deterioration Rate V. P. Praveen and M. Manoharan

Abstract Deterministic inventory control models for stochastic deteriorating items have been extensively studied in the past. However, there is not much work reported to model situations where different phases of deterioration rate are prevalent. In this paper, we develop a deterministic inventory control model with stochastic deterioration incorporated through additive Weibull distribution. In this study, an elegant approach is proposed to consider a time-dependent demand in the planning process and we consider that the holding cost totally depends on time and shortages are allowed for this model. The objective is to minimize the total inventory cost of the proposed model. Finally, the formulated model is illustrated through numerical examples to determine the effectiveness of the proposed model. Keywords Inventory control · Stochastic deterioration · Additive Weibull distribution · Time-dependent demand · Shortage · Optimization

1 Introduction It is reasonable to note that a product may be understood to have a lifetime which ends when utility reaches zero. Looking through the inventory models with deteriorating items extensively studied by researchers in the past shows that the deteriorating rate is considered constant or treated as some real valued functions. However, in real life situations, combinations of various factors would form the basis for modeling the inventory problem for deteriorating items. Deterioration is defined as decay, damage, or spoilage such that items cannot be used for intended purpose. Research in this direction began with the work of Whitin [23] who considered fashion goods deteriorating at the end of a prescribed storage period. Derman and

V. P. Praveen () · M. Manoharan Department of Statistics, University of Calicut, Malappuram, Kerala, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_18

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Klien [3] have studied the problem of choosing the order of issue of items from a stock pit materials under varying changes with time. Ghare and Schrader [5] studied the inventory model in which the deterioration is proportional to the inventory at the beginning of the time period. The economic order quantity under the condition of constant demand and exponential decay was derived. Covert and Phillip [1] and Phillip [18] developed an EOQ formula where the rate of deterioration is treated as random variable with time of deterioration following Weibull distribution. Donaldson [4] gave the fundamental result in the development of economic order quantity models with time-varying demand patterns. He established the classical no shortage inventory model with a linear trend in demand over a known and finite horizon. Dave and Patel [2] studied a deteriorating inventory model where the demand rate is a linear increasing function of time and with an assumption that shortages are not allowed. According to Steven Nahmias [14], certain type inventories undergo change in storage so that in time they may become partially or entirely unfit for consumption. Hence deterioration of stocks taken place overtime and the quantity that deteriorates can be modeled as a linear function of time and quantity. Mandan and Phaujdhari [12] developed a single item deterministic order model for deteriorating items with uniform rate of production and stock dependent consumption rate is presented. Goswamy and Choudhuri [6] developed an inventory replenishment policy over a deteriorating item having deterministic demand pattern with a linear trend and shortages. Raafat [19] reviewed inventory models with deteriorating items. Hariga [8] has studied optimum inventory lot sizing model for deterioration items with general continuous time-varying demand over a finite planning horizon. Krishnamurthy and Varghese [10] considered a continuous review deterioration of items with shortages. Goyal and Giri [7] gave recent trends of modeling in deteriorating items inventory. They classified inventory models on the basis of demand variations and various other conditions or constraints. Ouyang et al. [15] developed an inventory model for deteriorating items with exponential declining demand and partial backlogging. Ajanta Roy [20] developed a deterministic inventory model when the deterioration rate is time proportional, demand rate is a function of selling price, and holding cost is time dependent. Mandal [11] gave an EOQ inventory model for Weibull distributed deteriorating items under ramp type demand and shortages. Hung [9] gave an inventory model with generalized type demand, deterioration, and back order rates. Shah et al. [21] integrated time-varying deterioration and holding cost rates in the inventory model where shortages were not prohibited. Mishra et al. [13] gave an inventory model for deteriorating items with time-dependent demand and time-varying holding cost under partial backlogging. Tripathi and Pandey [22] presented an inventory model for deteriorating items with Weibull distributed deterioration and time-dependent demand under trade-credit policy. Yadav and Vats [25] proposed an inventory model with constant holding cost under partial backlogging and inflation. Pervin et al.[16] presented an inventory model in a declining demand for deteriorating items under trade-credit policy. Pervin et al.[17] proposed an inventory model with shortage under time-dependent demand and timevarying holding cost including stochastic deterioration, etc.

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The main motivation of recommending the new model is among the various consideration of demands in EOQ models; the time-dependent demand approach is very sensible to represent a perfect realistic situation. In order to address more realistic circumstances, we considered non-instantaneous deteriorating items and shortages are allowed to occur for displaying the real condition. Many inventory models were presented for variable deterioration rate with deterioration rate follows exponential distribution, Weibull distribution, etc. Most of these works were proposed to model any of the three parts in a deterioration rate. However, there is not much work reported to model situations where different phases of deterioration rate are prevalent. In this paper, we considered all the phases of deterioration rate by using additive Weibull model based on the simple idea of combining the failure rates of two Weibull distributions proposed by Xie and Lai [24]. The proposed model is validated with the help of four illustrative numerical examples. With the help of examples, we discussed the effectiveness of the proposed model. On these lines, different sections in this paper are developed and presented.

2 Model Description 2.1 Notations and Assumptions The following notations and assumptions are made in order to formulate the problem and EOQ model. A p T δ s I (t) I0 I1 (t) I2 (t) TC D(t) k θ (t)

Ordering cost per order; Unit purchasing cost per item; Length of cycle time; Backlogging rate, 0 ≤ δ ≤ 1; Lost sale cost per unit; Inventory level at time t, t ≥ 0; Maximum inventory level during [0, T ]; Inventory level that changes with time t during the production period; Inventory level that changes with time t during the non-production period; Total average cost; The demand rate D(t) at time t is a linearly increasing function of t; i.e., D(t) = x + yt, 0 ≤ t ≤ T , where x and y are nonnegative constants; Replenishment rate which is always finite; The distribution of the time to deterioration of an item follows the additive Weibull distribution with density function, (abt b−1 + cdt d−1 )exp(−at b − ct d ), t ≥ 0, where a > 0 and c > 0 are scale parameters and b > d > 0 or (b < d < 0) are shape parameters. The inventory level will change at a changing rate. Hence, to present differential model we use the deterioration rate function for additive Weibull distribution θ (t). The term θ (t)=abt b−1 +

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cdt d−1 , t ≥ 0, where 0 ≤ a, c ≤ 1 and b, d > 0, gives the on hand inventory deteriorates per unit time; The stock level reached in the cycle at the end of production period will be used in non-production period; Shortage cost per unit time, i.e., shortages are allowed to occur; Holding cost per item per time unit is time dependent and is assumed as h(t) = h + γ t, t ≥ 0, where γ > 0 and h > 0.

2.2 Methodology In the proposed inventory model, depending on the above assumptions, the inventory system can be considered as follows. At the beginning of each inventory cycle with zero stock level, k units of products arrive at the system. Up to time t1 due to replenishment, the inventory level meets the demand in the market. Replenishment is occurring during the production period only and at time t = t1 production stops and inventory level reaches the level Q which is the stock level that will be used in the non-production period. The inventory level in stock during the non-production period is diminishing due to those reasons of market demand and deterioration of items during the time interval [t1 , T ] and shortages begin to be accumulated which are partially backlogged. Next, the inventory level is declining to its lowest position at time t = T . Just after the cycle period the process repeats itself with k units of products arrived at the system, so replenishment is instantaneous and lead time is zero. Figure 1 depicts the proposed inventory system. Now I (t) denote the inventory position at time t, t ≥ 0, then the differential equations during the interval[0,T ] that describes the instantaneous state I (t), where all the sudden states of the inventory level are involved, are given by dI1 (t) + θ (t)I1 (t) = k − D(t) with I1 (0) = 0, 0 ≤ t ≤ t1 dt

(1)

and dI2 (t) + θ (t)I2 (t) = −D(t) with I2 (T ) = 0, I1 (t1 ) = I2 (t1 ) = Q, t1 ≤ t ≤ T dt (2) The solution of (1) using the boundary condition is I1 (t) = (k − x) t − −y

ab b+1 cd t t d+1 − (b + 1) (d + 1)



ab cd t2 − t b+2 − t d+2 , 0 ≤ t ≤ t1 2 2(b + 2) 2(d + 2)

(3)

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Fig. 1 Graphical representation of our proposed model

and the solution of (2) using the boundary condition is I2 (t) = x

     t1 − t + at b + ct d t − t1   a  b+1 c  d+1 t1 − t b+1 + t1 − t d+1 (b + 1) (d + 1) % $ % $ 2  t1 t2 ct d  2 at b − + + t − t12 +y 2 2 2 2



+

  a  b+2 c  d+2 b+2 d+2 , t1 ≤ t ≤ T (4) −t −t + + t t (b + 2) 1 (d + 2) 1

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Using the boundary condition I2 (t1 ) = Q, the equation becomes

    I2 (t) = Q 1 + a t1b − t b + c t1d − t d +x



    t1 − t + at b + ct d t − t1

  c  d+1 a  b+1 b+1 d+1 t t + −t −t + (b + 1) 1 (d + 1) 1 $ 2 % $ b %  t1 t2 ct d  2 at − + +y + t − t12 2 2 2 2

   c  d+2 a b+2 b+2 d+2 t t + + −t −t , t1 ≤ t ≤ T (b + 2) 1 (d + 2) 1

(5)

The maximum inventory level during [0,T ] is given by 

t1

I0 =

I1 (t)dt 0



t12 ab cd − t b+2 − t d+2 2 (b + 1)(b + 2) 1 (d + 1)(d + 2) 1 3

t ab cd t1b+3 − t1d+3 −y 1 − 6 2(b + 2)(b + 3) 2(d + 2)(d + 3)



= (k − x)

(6)

For the proposed model, a cost structure is imposed and it is analyzed by the criteria of minimization of the total expected cost per unit time. Hence for obtaining the total inventory cost, we calculate the following terms: – Annual ordering cost, OC = A

(7)

. – Total annual stock holding cost, HC, during time span [0,t1] is defined as follows:  t1 HC = h(t)I1 (t)dt 0



t2 ab cd t1b+2 − t d+2 H C = h(k − x) 1 − 2 (b + 1)(b + 2) (d + 1)(d + 2) 1 3

t1 ab cd b+3 d+3 − t t −hy − 6 2(b + 2)(b + 3) 1 2(d + 2)(d + 3) 1



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299

t13 ab cd − t1b+3 − t d+3 3 (b + 1)(b + 3) (d + 1)(d + 3) 1

4 t cd ab −γ y 1 − t1b+4 − t1d+4 . 8 2(b + 2)(b + 4) 2(d + 2)(d + 4)



+γ (k − x)

(8)

– P urchase cost, PC, during time span [t1 ,T]  P C = p I0 +



T

δD(t)dt t1



t2 ab cd t b+2 − t d+2 P C = p(k − x) 1 − 2 (b + 1)(b + 2) 1 (d + 1)(d + 2) 1 3

t1 ab cd b+3 d+3 − t t −py − 6 2(b + 2)(b + 3) 1 2(d + 2)(d + 3) 1 1 +pδx(T − t1 ) + pδy(T 2 − t12 ). 2



(9)

– Deteriorating cost, DC 

t1

DC = p

k − (x + yt)dt

0

*

t2 DC = p (k − x)t1 − b 1 2

+ .

(10)

– Shortage cost, SC, during time span [t1 ,T] is expressed as:  SC = p2

T

I2 (t)dt t1

$ % $ %

T b+1 + bt1b+1 T d+1 + dt1d+1 SC = Qp2 (T − t1 ) + a T t1b − + c T t1d − b+1 d +1 2T t1 − T 2 − t12 a + (T b+2 − T b+1 t1 + T t1b+1 − t1b+2 ) +xp2 2 b+1

c + (T d+2 − T d+1 t1 + T t1d+1 − t1d+2 ) d +1 $ % 3T t12 − T 3 − 2t13 T b+1 t12 − t1b+3 a T b+3 − t1b+3 +yp2 + − 6 2 b+3 b+1

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$ %

T d+1 t12 − t1d+3 c T d+3 − t1d+3 − 2 d+3 d +1

T b+3 − t1b+3 a + t1b+2 (T − t1 ) − b+2 b+3

T d+3 − t1d+3 c d+2 t . (T − t1 ) − + d +2 1 d+3 +

(11)

– Lost sale cost, Not all customers are willing to wait for the next lot size to arrive during the shortage period [t1 ,T], which may cause some loss in profit. Hence Lost sale cost, LSC  LSC = s

T

(1 − δ)D(t)dt

t1

 T 2 − t12 . LSC = s(1 − δ) x(T − t1 ) + y 2 

(12)

Hence, the total average cost of the system per time unit denoted by TC is defined as TC =

1 [OC + H C + P C + DC + SC + LSC]. T

(13)

where the component costs are as given in equation numbers from (7) to (12).

2.3 Solution Procedure The total average cost given by (13) is a highly nonlinear equation in T and t1 and our problem is to determine the optimal values of T and t1 that minimize the total average cost TC. The optimum values of T and t1 are obtained by equating to zero the first order partial derivatives of total average cost (T C) with respect to T and t1 as follows: ∂T C =0 ∂t1

(14)

∂T C =0 ∂T

(15)

and

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These two partial derivatives yield a minimizer (T ∗ , t1∗ ) provided that the following second order sufficient conditions are satisfied at that point. *

∂ 2T C ∂t12

+$

∂ 2T C ∂T 2

%

$ −

∂ 2T C ∂t1 ∂T

$

%

%2 >0

(16)

and *

∂ 2T C ∂t12

+ > 0,

∂ 2T C ∂T 2

>0

(17)

Equation (13) is our objective function which needs to be minimized. For this, we use the classical optimization techniques. Equations (14) and (15) obtained thereafter and are highly nonlinear in the variable T and t1 . However, if we give particular values to the discrete variables, our objective function becomes the function of two variables T and t1 . We have used the mathematical software MATHEMATICA to arrive at the solution of the system in consideration. We can obtain the optimal values of different values of the time with the help of this software. With the use of these optimal values, Eq. (13) provides minimum total average cost per unit time of the system in consideration. We can also show the solution procedure step by step as given below: Algorithm Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:

Initialize the value of the variables A, h, p, k, x, y, a, b, c, d, γ , δ, Q, p2 and s. Find t1∗ which satisfying (14) Find T ∗ which satisfying (15) If such t1∗ and T ∗ are found, then check if that t1∗ and T ∗ are also satisfying (16) and (17) If every condition is satisfied, then calculate T C from (13) The optimal solution is (T ∗ , t1∗ , T C ∗ ), where t1∗ and T ∗ are the associated values of t1 and T , respectively, and T C ∗ is the associated value of T C.

3 Numerical Examples The proposed model is illustrated below by considering the examples, where all associated parameters are taken in proper unit. The optimal solution of the inventory system is calculated with the help of Mathematica software. Example 1 Consider an inventory system with parameters A = 2000, h = 0.8, p = 20, k = 35, x = 20, y = 40, a = 0, b = 1.6, c = 0, d = 1.1, γ = 0.95, δ = 0.7, Q = 25, p2 = 6, s = 8. Then the optimal solution is t1∗ = 1.4321 and T ∗ = 6.11341 and the minimum total average inventory cost T C ∗ = 1132.41. The

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Fig. 2 Total average cost according to various choices of parameters with t1 , T , and T C along the x-axis, the y-axis, and the z-axis, respectively

graphical representation of the total average cost in Example 1 for the proposed model is shown in Fig. 2. Example 2 Consider an inventory system with parameters A = 2000, h = 0.8, p = 20, k = 35, x = 20, y = 40, a = 0.5, b = 0.8, c = 0.5, d = 0.4, γ = 0.95, δ = 0.7, Q = 25, p2 = 6, s = 8. Then the optimal solution is t1∗ = 1.6143 and T ∗ = 6.4638 and the minimum total average inventory cost T C ∗ = 1867.85. The graphical representation of the total average cost in Example 2 for the proposed model is shown in Fig. 3. Example 3 Consider an inventory system with parameters A = 2000, h = 0.8, p = 20, k = 35, x = 20, y = 40, a = 0.5, b = 1.6, c = 0.5, d = 0.7, γ = 0.95, δ = 0.7, Q = 25, p2 = 6, s = 8. Then the optimal solution is t1∗ = 1.6472 and T ∗ = 6.25146 and the minimum total average inventory cost T C ∗ = 5139.46. The

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Fig. 3 Total average cost according to various choices of parameters with t1 , T , and T C along the x-axis, the y-axis, and the z-axis, respectively

graphical representation of the total average cost in Example 3 for the proposed model is shown in Fig. 4. Example 4 Consider an inventory system with parameters A = 2000, h = 0.8, p = 20, k = 35, x = 20, y = 40, a = 0.5, b = 1.6, c = 0.5, d = 1.1, γ = 0.95, δ = 0.7, Q = 25, p2 = 6, s = 8. Then the optimal solution is t1∗ = 1.93214 and T ∗ = 6.14792 and the minimum total average inventory cost T C ∗ = 5076.92. The graphical representation of the total average cost in Example 4 for the proposed model is shown in Fig. 5. From the above four numerical examples, if we are not considering deterioration of the item (see Example 1), the minimum total average inventory cost is less than that of the proposed inventory system with additive Weibull distribution deterioration. Example 3 deals with constant rate of deterioration and that has the higher total average inventory cost than the other two examples. So this tells us

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Fig. 4 Total average cost according to various choices of parameters with t1 , T , and T C along the x-axis, the y-axis, and the z-axis, respectively

that when we are dealing with general deterioration rate our proposed model with additive Weibull distribution deterioration is more appropriate.

4 Conclusion The proposed model incorporates some realistic and practical features viz. the demand function and holding costs being totally time dependent, the inventory deteriorates at a variable rate over time and assumptions like shortages are allowed and are completely backlogged. The new model is introduced with additive Weibull deterioration rate to model different phases of deterioration rate. Four numerical assessments of the theoretical model have been done to illustrate the theory. The variations in the system statistics with a variation in system parameters have also been illustrated graphically. The solution obtained has also been revealed that the model is found to be quite suitable and stable. All these facts together make this study very unique and matter-of-fact and we examined total inventory cost at each of these situations.

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Fig. 5 Total average cost according to various choices of parameters with t1 , T , and T C along the x-axis, the y-axis, and the z-axis, respectively

Acknowledgement The authors wish to thank the anonymous referees for their constructive comments and suggestions on the earlier version of the paper.

References 1. Covert, R.P., Philiips, G.C.: An EOQ model for items with Weibull distribution deterioration. AIIE. Trans. 323–326 (1973) 2. Dave, U., Patel, L.K.: (T , Sj ) policy inventory model for deteriorating items with time proportional demand. J. Oper. Res. Soc. 32, 137–142 (1981) 3. Derman, C., Klein, M.: Inventory depletion. Manage. Sci. 4, 450–456 (1958) 4. Donaldson, W.A.: Inventory replenishment policy for a linear trend in demand-an analytical solution. Oper. Res. Q. 28(3), 663–670 (1977) 5. Ghare, P.M., Schrader, G.P.: A model for an exponentially decaying inventory. J. Ind. Eng. 14(5), 238–243 (1963) 6. Goswami, A., Choudhuri, K.S.: An EOQ model for deteriorating items with shortages and a linear trend demand. J. Oper. Res. Soc. 42, 1105–1110 (1991) 7. Goyal, S.K., Giri, B.C.: Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 134, 1–16 (2001)

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8. Hariga, M.A, Benkherouf, L.: Optimal and heuristic inventory replenishment models for deteriorating items with exponential time-varying demand. Eur. J. Oper. Res. 79, 123–137 (1994) 9. Hung, K.C.: An inventory model with generalized type demand, deterioration and backorder rates. Eur. J. Oper. Res. 208(3), 239–242 (2011) 10. Krishna Moorthy, A., Varghese, T.V.: Inventory with disaster. Optimization 35, 83–93 (1995) 11. Mandal, B.: An EOQ inventory model for Weibull distributed deteriorating items under ramp type demand and shortages. OPSEARCH 47(2), 158–165 (2010) 12. Mandan, B.N, Phaujdhari, S.: An inventory model for deteriorating items and stock depended consumption rate. J. O. R. Soc. 40, 483–488 (1989) 13. Mishra, V.K., Singh, L.S., Kumar, R.: An inventory model for deteriorating items with time dependent demand and time-varying holding cost under partial backlogging. J. Ind. Eng. Int. 9, (2013). Article number: 4 14. Nahmias, S.: Perishable inventory theory- a review. Oper. Res. 30, 680–708 (1982) 15. Ouyang, L.Y., Wu, K.S., Cheng, M.C.: An inventory model for deteriorating items with exponential declining demand and partial backlogging. Yugoslav J. Oper. Res. 15, 277–288 (2005) 16. Pervin, M., Mahata, G.C., Roy, S.K.: An inventory model with demand declining market for deteriorating items under trade credit policy. Int. J. Manage. Sci. Eng. Manage. 11, 243–251 (2015) 17. Pervin, M., Weber, G.W., Roy, S.K.: Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration. Ann. Oper. Res. 260 437–460 (2016) 18. Phillip, G.C.: A generalized EOQ model for items with Weibull distribution deterioration. AIIE Trans. 6(2), 159 (1974) 19. Raafat, F., Wolfe, P.M., Eidin, H.K.: An inventory model for deteriorating items. Comput. Ind. Eng. 20, 89–94 (1991) 20. Roy, A.: An inventory model for deteriorating items with price dependent demand and timevarying holding cost. Adv. Modell. Optim. 10(1), 25–37 (2008) 21. Shah, N.H., Soni, H.N., Patel, K.A.: Optimizing inventory and marketing policy for noninstantaneous deteriorating items with generalized type deterioration and holding cost rates. Omega 41(2), 421–430 (2013) 22. Tripathi, R.P., Pandey, H.S.: An EOQ model for deteriorating item with Weibull time dependent demand rate under trade credits. Int. J. Inf. Manage. Sci. 24, 329–347 (2013) 23. Whitin, T.M.: The Theory of Inventory Management. Princeton University Press, Princeton (1957) 24. Xie, M., Lai, C.D.: Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Safe. 52, 87–93 (1995) 25. Yadav, R.K.,Vats, A.K.: A deteriorating inventory model for quadratic demand and constant holding cost with partial backlogging and inflation. IOSR J. Math. 10(3), 47–52 (2014)

A Two-Server Queueing System with Processing of Service Items by a Server A. Krishnamoorthy and Divya V.

Abstract We consider a two-server (S1 and S2 ) queueing system in which the customers arrive according to Markovian arrival process. Each customer is to be provided with a processed item (inventory) at the end of his service. S1 provides service alone, whereas S2 provides service and also processes the items required to serve the customers. The maximum number of processed item permitted is L. The processing time follows phase type distribution. When the inventory level hits L, S2 starts serving customers if any waiting; else stays idle. S1 is dedicated to service only. Service is rendered only if there are processed items. Also, when a customer arrives to the system when both servers are idle, S1 provides him service and S2 continuously remains idle even if it has completed the processing of L items. The duration of service time given by both servers follows phase type distributions of same order, but S1 provides service at a slower rate than S2 . If the inventory level drops to a predetermined level s due to a service completion by S2 , then he starts processing items. If the inventory level drops to level s due to a service completion by S1 , then the customer served by S2 is shifted to S1 to provide him the residual service; S2 starts processing items. The arrival process is independent of the inventory processing and service process. The long run behavior of the system is analyzed under condition for stability. We derive some important distributions

A. Krishnamoorthy: Emeritus Fellow (EMERITUS-2017-18 GEN 10822(SA-II)), University Grants commission, India. The author’s “Divya V.” research was supported by the University Grants Commission, Govt. of India, under Faculty Development Programme (Grant No.F.No.FIP/12th Plan/KLKE008 TF 04) in Department of Mathematics, Cochin University of Science and Technology, Cochin-22. A. Krishnamoorthy () Department of Mathematics, CMS College, Kottayam, Kerala, India Divya V. Department of Mathematics, N.S.S. College, Cherthala, Kerala, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_19

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associated with the model. Numerical investigation of the optimal values of L and s is provided. Keywords Two-server queue · Additional item for service · Control policy · Level crossing problem

1 Introduction This paper is concerned with a two-server queueing system in which Server 1 (S1 ) provides service alone, whereas Server 2 (S2 ) provides service and also processes the item required (we call this “additional item”) to serve the customers. Each customer requires exactly one additional item for his service. In the absence of this additional item service cannot be provided. Therefore, S2 keeps processing the item until it hits a threshold value L. At this epoch he switches to serve customers, if any waiting. However, when the additional item level reduces to s, S2 returns to processing items. His service rate is higher than that of S1 ; both servers provide service according to phase type distributed random variable. Processing of each additional item requires a Phase distributed type amount of time, independent of the arrival and service processes. This work is an extension of those discussed in Kazimirsky [5], Hanukov et al. [4], Divya et al. [3], Baek et al. [1], and Dhanya et al. [2] to the two-server case. In classical queueing theory, it is implicitly assumed that if the server is ready to serve and customers are available to receive service then the service process proceeds. Either availability of “additional” items required to provide service is not taken into consideration/ignored or its abundance is taken for granted. In the latter case, the holding cost incurred is completely ignored. Sometimes the item(s) required for service may not be available. In such cases, service cannot be provided even when server is readily available and customer(s) are waiting. Typical example in medical case is operation theater. In the absence of “organ” for a patient in need of it, surgery cannot be performed. In a vehicle repair shop, a vehicle requiring a specific part replacement cannot be serviced if spares are not available. Thus, in several cases, availability of both customers and servers alone cannot guarantee service. This naturally leads to the investigation of availability of additional item(s) required to provide service. Then some control problems also arise—how much of additional item(s) to be held, time required to procure such items, and so on. This leads to the consideration of holding cost, shortage cost, and associated revenue loss. Kazimirisky [5] was the first to introduce “additional items needed for service.” He considered a BMAP/G/1 queue, in which the server proceeds to produce additional items whenever no customer is found at a departure epoch. Exactly one processed item is needed for each customer. Service time distribution of customers depends on whether processed item is available or not. Thus, there are two distinct service time distributions.

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Baek et al. [1] considered MMAP of customers of two types—type I (high priority) and type II (low priority). Both type of customers require a certain minimum number of additional items to start their service. Type I customers do not have a waiting space. If a type I customer is in service while another type I customer arrives, the latter leaves the system. On the other hand, if a type II customer is in service, the former is pushed out of the system by the type I arrival provided the number of additional items available is at least equal to the minimum number required to start its service. Else, it leaves the system without changing the status. Type II customers have an infinite capacity waiting space. Additional items arrive to the system according to MAP. They investigate system stability and analyze its performance. Dhanya et al. [2] extend the above to retrial queueing setup. Hanukov et al. [4] analyze a simple queueing system where again additional items are needed for service of a customer (one item for each customer). The arrival process is Poisson and service time is exponentially distributed. Divya et al. [3] considered a single server queue in which customers arrive according to Markovian arrival process. Whenever customers are not waiting, the server goes for vacation and produces inventory for future use during this period. The maximum inventory level permitted is L. The processing time is phase type distributed. The server returns from vacation when there are N customers in the system. The service time follows two distinct phase type distributions depending on whether there is processed items or no processed item available at service commencement epoch. The rest of the paper is arranged as follows. The model description and mathematical formulation are given in Sect. 2. Section 3 provides steady-state analysis of the model. Section 4 contains some level crossing problems. Some important performance measures are provided in Sect. 5 and a related cost function is analyzed in Sect. 6. Some numerical experiments to find the optimal values of L and s are discussed in Sect. 7. Notations and abbreviations used in the sequel: e(a): Column vector of 1" s of order a e: Column vector of 1" s of appropriate order. Ia : identity matrix of order a. ea (b): column vector of order b with 1 in the ath position and the remaining entries zero. – CT MC: Continuous time Markov chain – MAP : Markovian arrival process – LI QBD: Level independent quasi-birth and -death

– – – –

2 Model Description and Mathematical Formulation We consider a two-server queueing system in which the customers arrive according to Markovian arrival process with representation (D0 , D1 ) of order n. Each customer is to be provided with a processed item at the end of his service. S1 is

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always available to the customers provided processed item is available, whereas S2 produces items for service (inventory) for future use whenever the inventory level drops to a threshold s. Until the inventory level reaches L (the maximum permitted level) he does not provide service to customers. The inventory processing time follows phase type distribution PH(α, T ) of order m1 . After processing L items, S2 starts serving customers if any waiting; else stays idle. S1 is dedicated to service only. Servers provide service only if there are processed items. Also, when a customer arrives to an empty system, S1 provides him service and S2 remains idle even he is not engaged in processing the inventory. The service time at S2 follows phase type distribution P H (β, S) of order m2 and that at S1 follows phase type distribution P H (β, θ S) of order m2 , 0 < θ < 1. If the inventory level drops to a predetermined level s after a service completion by S2 , then he starts processing items. If the inventory level drops to a predetermined level s after a service completion by S1 , then the customer served by S2 is shifted to S1 for the remaining service and S2 goes for processing items. The arrival process is independent of the inventory processing and service process.

2.1 The QBD Process The model described in Sect. 2 can be studied as a LIQBD process. First, we introduce the following notations: At time t: N(t): the number of customers in the system, I (t): the number of processed items in the inventory, ⎧ ⎨ 0, when S2 is idle J (t) : status of S2 = 1, when S2 is processing items ⎩ 2, when S2 is serving a customer ⎧ ⎨ processing/service phase of S2 K1 (t) = 0, when S2 is idle ⎩ ⎧ ⎨ service phase of S1 K2 (t) = 0, when S1 is idle ⎩ M(t): the phase of arrival of the customer.

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311

It is easy to verify that {(N(t), I (t), J (t), K1 (t), K2 (t), M(t)) : t ≥ 0} is a LIQBD with state space: (i) no customer in the system l(0) = {(0, i, 1, k1 , 0, p) : 0 ≤ i ≤ L − 1, 1 ≤ k1 ≤ m1 , 1 ≤ p ≤ n} ∪ {(0, i, 0, 0, 0, p) : s + 1 ≤ i ≤ L, 1 ≤ p ≤ n} (ii) when there is 1 customer in the system l(1) = {(1, 0, 1, k1, 0, p) : 1 ≤ k1 ≤ m1 ; 1 ≤ p ≤ n} ∪ {(1, i, 1, k1 , k2 , p) : 1 ≤ i ≤ L − 1; 1 ≤ k1 ≤ m1 ; 1 ≤ k2 ≤ m2 ; 1 ≤ p ≤ n} ∪ {(1, i, 0, 0, k2 , p) : s + 1 ≤ i ≤ L; 1 ≤ k2 ≤ m2 ; 1 ≤ p ≤ n} ∪ {(1, i, 2, k1 , 0, p) : s + 1 ≤ i ≤ L − 1; 1 ≤ k1 ≤ m2 ; 1 ≤ p ≤ n} (iii) when there are h customers in the system, h ≥ 2: l(h) = {(h, 0, 1, k1 , 0, p) : 1 ≤ k1 ≤ m1 ; 1 ≤ p ≤ n} ∪ {(h, i, 1, k1 , k2 , p) : 1 ≤ i ≤ L − 1; 1 ≤ k1 ≤ m1 ; 1 ≤ k2 ≤ m2 ; 1 ≤ p ≤ n} ∪ {(h, i, 2, k1 , k2 , p) : s + 1 ≤ i ≤ L; 1 ≤ k1 , k2 ≤ m2 ; 1 ≤ p ≤ n} The infinitesimal generator of this CTMC is ⎤ A00 A01 ⎥ ⎢ A10 A11 A12 ⎥ ⎢ ⎥ ⎢ A21 A1 A0 Q¯ = ⎢ ⎥ ⎥ ⎢ A2 A1 A0 ⎦ ⎣ .. .. .. . . . ⎡

where A00 contains transitions within level 0; A01 , A10 , A11 , A12 , A21 represent transitions from level 0 to level 1, from level 1 to level 0, within level 1, from level 1 to level 2, from level 2 to level 1, respectively; A0 represents transitions from level h to level h + 1 for h ≥ 2, A1 represents transitions within the level h for h ≥ 2 and A2 represents transitions from level h to h − 1 for h ≥ 3. The boundary blocks A00 , A01 , A10 , A11 , A12 , A21 are of orders (s + 1)m1 n + (L − s − 1)(1 + m1 )n + n, ((s + 1)m1n + (L − s − 1)(1 + m1)n + n) × (m1n + sm1 m2 n + (L − s − 1)(2m2 + m1 m2 )n+m2 n),(m1 n+sm1 m2 n+(L−s−1)(2m2 +m1 m2 )n+m2 n)×((s+1)m1 n+ (L − s − 1)(1 + m1 )n + n), m1 n + sm1 m2 n + (L − s − 1)(2m2 + m1 m2 )n + m2 n, (m1 n + sm1 m2 n + (L − s − 1)(2m2 + m1 m2 )n + m2n) × (m1 n + sm1 m2 n + (L − s − 1)(m1 m2 n+m2 2 n)+m22 n), (m1 n+sm1 m2 n+(L−s−1)(m1m2 n+m2 2 n)+m22 n)× (m1 n + sm1 m2 n + (L − s − 1)(2m2 + m1m2 )n + m2 n), respectively. A0 , A1 , A2 are square matrices of order m1 n + sm1 m2 n + (L − s − 1)(m1 m2 n + m2 2 n) + m2 2 n. (h ,i ,j ,k ,l ) Define the entries of Apq2(h12,i12,j1 ,k2 1 ,l21 ) as transition submatrices which contains transitions of the form (p, h1 , i1 , j1 , k1 , l1 ) → (q, h2 , i2 , j2 , k2 , l2 ), where q = 0 or 1, when p = 0; q = 0, 1 or 2, when p = 1 and q = 1, when p = 2. (h ,i ,j ,k ,l ) (h ,i ,j ,k ,l ) (h ,i ,j ,k ,l ) Define the entries of A0(h2 ,i2 ,j2 ,k 2,l 2) , A1(h2 ,i2 ,j2 ,k 2,l 2) , A2(h2 ,i2 ,j2 ,k 2,l 2) as transition 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 submatrices which contains transitions of the form (g, h1 , i1 , j1 , k1 , l1 ) → (g + 1, h2 , i2 , j2 , k2 , l2 ), where g ≥ 2, (g, h1 , i1 , j1 , k1 , l1 ) → (g, h2 , i2 , j2 , k2 , l2 ), where g ≥ 2, (g, h1 , i1 , j1 , k1 , l1 ) → (g −1, h2 , i2 , j2 , k2 , l2 ), where g ≥ 3 respectively. Since none or one event alone could take place in a short interval of time

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with positive probability, in general, a transition such as (g1 , h1 , i1 , j1 , k1 , l1 ) → (g2 , h2 , i2 , j2 , k2 , l2 ) has positive rate only for exactly one of g2 , h2 , i2 , j2 , k2 , l2 different from g1 , h1 , i1 , j1 , k1 , l1 .

(i2 ,j2 ,k1" ,k2" ,l2 )

A00(i

1 ,j1 ,k1 ,k2 ,l1 )

=

(i2 ,j2 ,k1" ,k2" ,l2 )

A01

A

(i1 ,j1 ,k1 ,k2 ,l1 )

(i2 ,j2 ,k1" ,k2" ,l2 ) 10(i1 ,j1 ,k1 ,k2 ,l1 )

⎧ 0 ⎪ ⎪ T α ⊗ In ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T 0 ⊗ In ⎪ ⎪ ⎨ ⎪ T ⊕ D0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D0 ⎪ ⎪ ⎩

=

i2 = i1 + 1, 0 ≤ i1 ≤ L − 2; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = L − 1, i2 = L; j1 = j2 = 1; 1 ≤ k1 ≤ m1 , k1" = 0; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = i2 , 0 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = i2 , s + 1 ≤ i1 ≤ L; j1 = j2 = 0; k1 = k1" = 0; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n

⎧ Im ⊗ D1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Im1 ⊗ (β ⊗ D1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β ⊗ D1 ⎪ ⎪ ⎩

⎧ ⎪ Im1 ⊗ (θS 0 ⊗ In ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θS 0 α ⊗ I ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ θS 0 ⊗ I n = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S 0 α ⊗ In ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S 0 ⊗ In ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

i1 = i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = i2 , 1 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = 0; 1 ≤ k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = i2 , s + 1 ≤ i1 ≤ L; j1 = j2 = 0; k1 = k1" = 0; k2 = 0; 1 ≤ k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

i2 = i1 − 1, 1 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 , 1 ≤ k2 ≤ m2 ; k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = s + 1, i2 = s; j1 = 0, j2 = 1; k1 = 0, 1 ≤ k1" ≤ m1 ; 1 ≤ k2 ≤ m2 ; k2" = 0; 1 ≤ l1 , l2 ≤ n

i2 = i1 − 1, s + 2 ≤ i1 ≤ L; j1 = j2 = 0; k1 = k1" = 0; 1 ≤ k2 ≤ m2 ; k2" = 0; 1 ≤ l1 , l2 ≤ n

i1 = s + 1, i2 = s; j1 = 2, j2 = 1; 1 ≤ k1 ≤ m2 , 1 ≤ k1" ≤ m1 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i2 = i1 − 1, s + 2 ≤ i1 ≤ L − 1; j1 = 2, j2 = 0; 1 ≤ k1 ≤ m2 , k1" = 0; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n

A Two-Server Queueing System with Processing of Service Items by a Server

(i2 ,j2 ,k " ,k " ,l2 ) A11(i ,j ,k1 ,k2 ,l ) 1 1 1 2 1

(i2 ,j2 ,k1" ,k2" ,l2 )

A12(i

1 ,j1 ,k1 ,k2 ,l1 )

(i ,j2 ,k1" ,k2 ,l2 )

A212(i

1 ,j1 ,k1 ,k2 ,l1 )

=

⎧ 0 T (α ⊗ β) ⊗ In ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ T α ⊗ Im2 n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T 0 ⊗ Im2 n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T ⊕ D0 ⎪ ⎪ ⎪ ⎪ θS ⊕ D0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S ⊕ D0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ⊕ θS ⊕ D0 ⎪ ⎪ ⎩

⎧ ⎪ ⎪ Im1 ⊗ D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Im1 m2 ⊗ D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

313

i1 = 0, i2 = 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = 0,

1 ≤ k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

1 ≤ i1 ≤ L − 2, i2 = i1 + 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

i1 = L − 1, i2 = L; j1 = 1, j2 = 0; 1 ≤ k1 ≤ m1 , k1" = 0;

1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

i1 = i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0

1 ≤ l1 , l2 ≤ n i1 = i2 , s + 1 ≤ i1 ≤ L; j1 = j2 = 0; k1 = k1" = 0;

1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

i1 = i2 , s + 1 ≤ i1 ≤ L − 1; j1 = j2 = 2; 1 ≤ k1 , k1" ≤ m2 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = i2 , 1 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ;

1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

i1 = i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = i2 , 1 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n = β ⊗ (Im2 ⊗ D1 ) i1 = i2 , s + 1 ≤ i1 ≤ L; j1 = 0, j2 = 2; k1 = 0, ⎪ ⎪ ⎪ ⎪ 1 ≤ k1" ≤ m2 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n ⎪ ⎪ ⎪ ⎪ ⎪ Im2 ⊗ (β ⊗ D1 ) i1 = i2 , s + 1 ≤ i1 ≤ L − 1; j1 = j2 = 2; ⎪ ⎪ ⎪ ⎪ 1 ≤ k1 , k1" ≤ m2 ; k2 = 0, 1 ≤ k2" ≤ m2 ; ⎪ ⎪ ⎪ ⎩ 1 ≤ l1 , l2 ≤ n

=

⎧ ⎪ Im1 ⊗ (θS 0 ⊗ In ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Im1 ⊗ (θS 0 β ⊗ In ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ θS ⊗ Im2 n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S 0 ⊗ I m2 n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S 0 α ⊗ I m2 n + B ⎪ ⎪ ⎪ ⎩

i1 = 1, i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ,

1 ≤ k2 ≤ m2 , k2" = 0; 1 ≤ l1 , l2 ≤ n

i2 = i1 − 1, 2 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i2 = i1 − 1, s + 2 ≤ i1 ≤ L; j1 = j2 = 2; 1 ≤ k1 , k1" ≤ m2 ; 1 ≤ k2 ≤ m2 ; k2" = 0; 1 ≤ l1 , l2 ≤ n i2 = i1 − 1, s + 2 ≤ i1 ≤ L; j1 = 2, j2 = 0; 1 ≤ k1 ≤ m2 , k1" = 0; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = s + 1, i2 = s; j1 = 2, j2 = 1; 1 ≤ k1 ≤ m2 ; 1 ≤ k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

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A. Krishnamoorthy and Divya V.

where ⎡ ⎢ ⎢ B=⎢ ⎣

α ⊗ B1 α ⊗ B2 .. .

⎤ ⎥ ⎥ ⎥ ⎦

α ⊗ Bm2 where 6 7 Bmi = 0 · · · θ S 0 ⊗ In · · · 0 , where θ S 0 ⊗ In is in the ith position

(i2 ,j2 ,k1" ,k2" ,l2 )

A0(i

1 ,j1 ,k1 ,k2 ,l1 )

(i2 ,j2 ,k1" ,k2" ,l2 )

A1(i

1 ,j1 ,k1 ,k2 ,l1 )

⎧ ⎪ ⎪ Im1 ⊗ D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨I m1 m2 ⊗ D1 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Im2 ⊗ D1 ⎪ ⎪ 2 ⎪ ⎩

=

i1 = i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0; 1 ≤ l1 , l2 ≤ n i1 = i2 , 1 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = i2 , s + 1 ≤ i1 ≤ L; j1 = j2 = 2; 1 ≤ k1 , k1" ≤ m2 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

⎧ 0 ⎪ ⎪ T (α ⊗ β) ⊗ In ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T 0 α ⊗ Im2 n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T 0β ⊗ I ⎪ m2 n ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ T ⊕ D0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ⊕ θS ⊕ D0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S ⊕ θS ⊕ D0 ⎪ ⎪ ⎪ ⎩

i1 = 0, i2 = 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = 0, 1 ≤ k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n 1 ≤ i1 ≤ L − 2, i2 = i1 + 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = L − 1, i2 = L; j1 = 1, j2 = 2; 1 ≤ k1 ≤ m1 , 1 ≤ k1" ≤ m2 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; k2 = k2" = 0 1 ≤ l1 , l2 ≤ n i1 = i2 , 1 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = i2 , s + 1 ≤ i1 ≤ L; j1 = j2 = 2; 1 ≤ k1 , k1" ≤ m2 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

A Two-Server Queueing System with Processing of Service Items by a Server

(i2 ,j2 ,k1" ,k2" ,l2 )

A2(i

1 ,j1 ,k1 ,k2 ,l1 )

=

⎧ Im1 ⊗ (θS 0 ⊗ In ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Im1 ⊗ (θS 0 β ⊗ In ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ S 0 α ⊗ Im2 n + B ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I ⊗ (θS 0 β ⊗ I ) + S 0 β ⊗ I ⎪ n m2 n ⎪ ⎪ m1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

315

i1 = 1, i2 = 0; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 , 1 ≤ k2 ≤ m2 , k2" = 0; 1 ≤ l1 , l2 ≤ n i2 = i1 − 1, 2 ≤ i1 ≤ L − 1; j1 = j2 = 1; 1 ≤ k1 , k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i1 = s + 1, i2 = s; j1 = 2, j2 = 1; 1 ≤ k1 ≤ m2 ; 1 ≤ k1" ≤ m1 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n i2 = i1 − 1, s + 2 ≤ i1 ≤ L; j1 = j2 = 2; 1 ≤ k1 , k1" ≤ m2 ; 1 ≤ k2 , k2" ≤ m2 ; 1 ≤ l1 , l2 ≤ n

3 Steady-State Analysis Let π = (π0 , π1 , . . . , πL ) ⎡ denote the steady-state probability vector of the⎤generator F0 F1 ⎥ ⎢F F F ⎥ ⎢ 2 3 4 ⎥ ⎢ F5 F3 F4 ⎥ ⎢ ⎥ ⎢ .. .. .. ⎥ ⎢ . . . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ F F F 5 3 4 ⎥ ⎢ ⎥ ⎢ F5 F3 F6 ⎥. ⎢ A = A0 + A1 + A2 = ⎢ ⎥ F7 F8 F9 ⎥ ⎢ ⎥ ⎢ F10 F8 F9 ⎥ ⎢ ⎥ ⎢ .. .. .. ⎥ ⎢ . . . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ F10 F8 F9 ⎥ ⎢ ⎣ F10 F8 F11 ⎦ F12 F13 Then π satisfies π A = 0, π e = 1.

(1)

The LI QBD description of the model indicates that the queueing system is stable (see Neuts [6]) if and only if the left drift exceeds that of right drift. That is, π A0 e < π A2 e

(2)

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A. Krishnamoorthy and Divya V.

The vector π cannot be obtained directly in terms of the parameters of the model. From (1), we get πi = πi−1 Ui−1 , 1 ≤ i ≤ L

(3)

where ⎧ −F1 (F3 + U1 F5 )−1 ⎪ ⎪ ⎪ ⎪ ⎪ −F4 (F3 + Ui+1 F5 )−1 ⎪ ⎪ ⎪ ⎪ ⎨ −F4 (F3 + Us F7 )−1 Ui = −F6 (F8 + Us+1 F10 )−1 ⎪ ⎪ ⎪ −F9 (F8 + Ui+1 F10 )−1 ⎪ ⎪ ⎪ ⎪ −F9 (F8 + UL−1 F12 ) ⎪ ⎪ ⎩ −F11 (F13 )−1

f or i = 0 f or 1 ≤ i ≤ s − 2 f or i = s − 1 f or i = s f or s + 1 ≤ i ≤ L − 3 f or i = L − 2 f or i = L − 1

From the normalizing condition πe = 1, we have ⎛ π0 ⎝

j L−1 :

⎞ Ui + I ⎠ e = 1

(4)

j =0 i=0

We get π0 by solving (1) and (4). Substituting (3) and (4) in (2) gives the stability condition as ⎡ j j s : L−1   : ⎣ Ui (Im1 m2 ⊗ D1 )ee + Ui (Im1 m2 ⊗ D1 )ee π0 (Im1 ⊗ D1 )ee + j =1 i=0

+

L−1 :



j =s+1 i=0

Ui (Im2 2 ⊗ D1 )ee
1 −  →0 level s + 1. Here, M() is chosen in such a way that P h=0 h for every  > 0. The infinitesimal generator of the process is given by ⎡

⎤ 0 0 0 ··· ⎢ E0 B C ⎥ ⎢ ⎥ ⎢ 0 ⎥ B C ⎢E ⎥ ⎢ . ⎥ U =⎢ . .. .. ⎥. . . ⎢ . ⎥ ⎢ 0 ⎥ ⎢E ⎥ B C ⎣ ⎦ .. .. .. . . . where ⎡

⎤ F1 G1 ⎢H F G ⎥ ⎢ 1 2 2 ⎥ ⎢ ⎥ H2 F2 G2 ⎢ ⎥ ⎢ ⎥ B=⎢ .. .. .. ⎥ . . . ⎢ ⎥ ⎢ ⎥ ⎣ H2 F2 G2 ⎦ F3 with ⎡

T ⊕ D0 T 0 α ⊗ In ⎢ . .. ⎢ .. . F1 = ⎢ ⎢ ⎣ T ⊕ D0 T 0 α ⊗ In T ⊕ D0

H1 =

⎤   ⎥ ⎥ ⎥ , G1 = Im1 ⊗ D1 , ⎥ Is ⊗ (Im1 ⊗ (β ⊗ D1 )) ⎦

0 0 , ⊗ (θ S 0 ⊗ In )) 0

Is ⊗ (Im1 ⎡ T ⊕ D0 T 0 (α ⊗ β) ⊗ In ⎢ T ⊕ θ S ⊕ D0 T 0 α ⊗ Im2 n ⎢ ⎢ .. .. F2 = ⎢ . . ⎢ ⎣ T ⊕ θ S ⊕ D0 T 0 α ⊗ Im2 n T ⊕ θ S ⊕ D0

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

A Two-Server Queueing System with Processing of Service Items by a Server

⎡ G2 = Im1 +sm1 m2 ⊗D1 , H2 = ⎣ Im1 ⎡ ⎢ ⎢ ⎢ ⎢ F3 = ⎢ ⎢ ⎢ ⎣

T ⊕ D0 − Im1 ⊗ Δ

319

⎤ 0 0 ⎦, ⊗ (θ S 0 ⊗ In ) 0 0 0 Is−1 ⊗ (Im1 ⊗ (θ S β ⊗ In )) ⎤

T 0 (α ⊗ β) ⊗ In T ⊕ θ S ⊕ D0 − Im1 m2 ⊗ Δ

T 0 α ⊗ Im2 n

..

..

. T ⊕ θ S ⊕ D0 − Im1 m2 ⊗ Δ T 0 α ⊗ Im2 n T ⊕ θ S ⊕ D0 − Im1 m2 ⊗ Δ

.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

with ⎡ ⎢ Δ=⎣

⎤

δ1 ..

⎥ ⎦.

. δn

⎡

⎤ 0 ···0··· 0 ⎢ .. .. .. ⎥ ⎢ . .⎥ C=⎢. ⎥ , where, ⎣0 ···0··· 0⎦ 0 · · · C" · · · 0 ⎡ ⎤ 0 0 ⎦ C " = ⎣ Im1 ⊗ (θ S 0 ⊗ In ) 0 0 0 Is−1 ⊗ (Im1 ⊗ (θ S β ⊗ In )) E0 =

E10 e(M) ⊗ E20



with ⎡ ⎢ E10 = ⎣

0 .. . T 0 ⊗ e (n)

⎤ ⎥ 0 ⎦ , E2 =



0 T 0 ⊗ e (m2 n)



Let yk , k = 0, 1, · · · be the probability that the number of downcrossings from inventory level s to s − 1 is k. Then yk is the probability that the absorption occurs from the level k for the process χ1 . Hence, yk are given by y0 = γ1 (−B)−1 E 0 and for k = 1, 2, 3, . . . yk = γ1 ((−B)−1 C)k (−B)−1 E 0

320

A. Krishnamoorthy and Divya V.

where, γ1 = (1/d)(xx 0,0,1,1,0,1, · · · , x 0,s,1,m1,0,n , · · · , x M,0,1,1,0,1, · · · , x M,s,1,m1 ,m2 ,n ) with d=

m1  n s  

x 0,i,1,k1 ,0,p +

i=0 k1 =1 p=1

m1  m2  s  n M  

x h,i,1,k1 ,k2 ,p

h=1 i=0 k1 =1 k2 =1 p=1

Thus, we arrive at the lemma. Lemma 1 The expected number of downcrossings from inventory level s to s − 1 before hitting s + 1 is E(i) =

∞ 

kyk

k=0

4.2 Distribution of Number of Upcrossings of Inventory Level from s to s + 1 Before Hitting s − 1 To find this distribution, first we find the distribution of duration of time till upcrossing from s to s + 1 occurs before hitting s − 1. This again can be studied as the time until absorption in a CTMC, χ2 = {(N1 (t), N2 (t), I (t), J (t), K1 (t), K2 (t), K3 (t))}, where N1 (t) denotes the number of upcrossings from s to s + 1, N2 (t), the number of customers in the system, I (t), number of processed items, J (t), status of S2 , K1 (t), processing/service phase of S2 , K2 (t), the service phase of S1 , K3 (t), the arrival phase at time t. The state space of the process is {(h, 0, j, 1, k1 , 0, l) : h ≥ 0, s ≤ j ≤ L−1, 1 ≤ k1 ≤ m1 , 1 ≤ l ≤ n} ∪ {(h, 0, j, 0, 0, 0, l) : h ≥ 0, s + 1 ≤ j ≤ L, 1 ≤ l ≤ n} ∪ {(h, i, j, 1, k1 , k2 , l) : h ≥ 0, 1 ≤ i ≤ M, s ≤ j ≤ L − 1, 1 ≤ k1 ≤ m1 , 1 ≤ k2 ≤ m2 , 1 ≤ l ≤ n} ∪ {(h, 1, j, 0, 0, k2, l) : s + 1 ≤ j ≤ L, 1 ≤ k2 ≤ m2 , 1 ≤ l ≤ n} ∪ {(h, 1, j, 2, k1 , 0, l) : h ≥ 0, s + 1 ≤ j ≤ L − 1, 1 ≤ k1 ≤ m2 , 1 ≤ l ≤ n} ∪ {(h, i, j, 2, k1 , k2 , l) : h ≥ 0, 2 ≤ i ≤ M, s + 1 ≤ j ≤ L, 1 ≤ k1 , k2 ≤ m2 , 1 ≤ l ≤ n} ∪ {∗}, where ∗ denotes the absorbing state  of  indicating the hitting M() x e > 1 −  →0 level s + 1. Here, M() is chosen in such a way that P h=0 h for every  > 0. Let zk , k = 0, 1, · · · be the probability that the number of upcrossings from inventory level s to s + 1 is k. Then zk is the probability that the absorption occurs from the level k for the process χ2 . Proceeding on similar lines as in the proof of Lemma 1, we arrive at lemma.

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Lemma 2 The expected number of upcrossings from inventory level s to s+1 before hitting s − 1 is E(i) =

∞ 

kzk

k=0

5 Performance Measures In this section, we use the notations ηk for absorption rate from phase k in P H (α, T ), σk for absorption rate from phase k in P H (β, S), θ σk for absorption (1) rate from phase k in P H (β, θ S), and dpp" to denote pp" th entry of D1 .  1. Expected number of customers in the system, Es = ∞ h=1 hxhe 2. Expected number of processed items in the inventory, Eit =

m1 L−1 

n 

ix0,i,1,k1 ,0,p +

i=1 k1 =1 p=1

L  n 

m1  m2  ∞ L−1 n  

m2  n 

m2  L  n 

ixh,i,1,k1 ,k2 ,p +

h=1 i=1 k1 =1 k2 =1 p=1 L−1 

ix0,i,0,0,0,p +

i=s+1 p=1

x1,i,0,0,k2,p +

i=s+1 k2 =1 p=1

ix1,i,2,k1 ,0,p +

i=s+1 k1 =1 p=1

m2  m2  ∞  L  n 

ixh,i,2,k1 ,k2 ,p

h=2 i=s+1 k1 =1 k2 =1 p=1

3. Expected rate at which the inventory processing is switched on, Ripo =

m2  n 

σk1 x1,s+1,2,k1,0,p +

k1 =1 p=1

m2  n 

θ σk2 x1,s+1,0,0,k2,p +

k2 =1 p=1 m2  m2  ∞  n 

(θ σk2 + σk1 )xh,s+1,2,k1,k2 ,p

(10)

h=2 k1 =1 k2 =1 p=1

4. Expected rate of switching of S2 to service mode, Rsn =

m2  L  n  n  k2 =1 i=s+1 p=1 p " =1

(1)

dpp" x1,i,0,0,k2 ,p + m1  m2  ∞  n  h=2 k1 =1 k2 =1 p=1

ηk1 xh,L−1,1,k1,k2 ,p

(11)

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6 Analysis of a Cost Function We construct a cost function based on the above performance measures. Let c1 : Unit time cost for switching on inventory processing c2 : Unit time cost for switching of S2 to service mode h1 : Unit time cost for holding a customer h2 : Unit time cost for holding an item in inventory Then the expected cost per unit time, C = c1 Ripo + c2 Rsn + h1 Es + h2 Eit

7 Numerical Experiments We find optimal s and optimal L by using the above cost function.

6 7 6 7 −4 4 −3 3 We fix α = 0.9 0.1 , T = , β = 0.8 0.2 , S = , 0 −4 0 −3 θ = 0.6, c1 = 100, c2 = 5, h1 = 30 and h2 = 1. For the arrival process of type II customers, we consider the following five set of matrices for D0 and D1 1. Exponential (EXP) D0 = (−1), D1 = (1) 2. Erlang (ERA) ⎡

⎤ ⎡ ⎤ −3 3 0 000 D0 = ⎣ 0 −3 3 ⎦ , D1 = ⎣ 0 0 0 ⎦ 0 0 −3 300 3. Hyperexponential (HEXP) D0 =





−3.4000 0 0.6800 2.7200 , D1 = 0 −0.8500 0.1700 0.6800

4. MAP with negative correlation (MNA) ⎡ ⎤ ⎤ 0 0 0 −0.8101 0.8101 0 ⎦ , D1 = ⎣ 0.0810 0 1.2687 ⎦ D0 = ⎣ 0 −1.3497 0 38.0761 0 2.4304 0 0 −40.5065 ⎡

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5. MAP with positive correlation (MPA) ⎡

⎤ ⎡ ⎤ −0.8101 0.8101 0 0 0 0 ⎦ , D1 = ⎣ 1.2687 0 0.0810 ⎦ D0 = ⎣ 0 −1.3497 0 0 0 −40.5065 2.4304 0 38.0761 These two MAP processes are normalized so as to have an arrival rate of 1. The arrival process labeled MNA has correlated arrivals with correlation between two successive interarrival times given by −0.4211 and the arrival process corresponding to the one labeled MPA has a positive correlation with value 0.4211. Tables 1, 2, 3, 4, 5 indicate the effect of the parameter s on various performance measures and the cost function corresponding to different arrival processes when L is fixed. In the following, we summarize the observations based on these tables. Table 1 Effect of s: Fix L = 20 and arrival process as EXP

s 2 3 4 5 6 7 8 9 10 Ripo 0.035 0.037 0.039 0.042 0.045 0.049 0.053 0.058 0.064 Rsn 0.178 0.180 0.181 0.182 0.184 0.186 0.188 0.191 0.194 Es 1.984 1.952 1.923 1.894 1.866 1.837 1.808 1.779 1.750 Eit 10.467 10.976 11.485 11.994 12.500 13.005 13.507 14.005 14.497 C 74.336 74.105 73.990 73.918 73.883 73.894 73.963 74.110 74.340 Bold values represent the optimal value for the cost function Table 2 Effect of s: Fix L = 20 and arrival process as ERA

s 2 3 4 5 6 7 8 9 10 Ripo 0.034 0.036 0.038 0.041 0.045 0.048 0.053 0.058 0.064 Rsn 0.199 0.201 0.202 0.204 0.206 0.209 0.212 0.215 0.219 Es 1.553 1.527 1.501 1.475 1.448 1.421 1.393 1.365 1.336 Eit 10.487 11.001 11.516 12.031 12.546 13.061 13.576 14.089 14.602 C 61.475 61.420 61.414 61.432 61.475 61.553 61.680 61.872 62.155 Bold values represent the optimal value for the cost function Table 3 Effect of s: Fix L = 20 and arrival process as HEXP

s 2 3 4 5 6 7 8 9 10 Ripo 0.035 0.037 0.039 0.042 0.045 0.049 0.053 0.058 0.064 Rsn 0.171 0.172 0.173 0.175 0.176 0.178 0.180 0.183 0.186 Es 2.152 2.119 2.090 2.060 2.032 2.003 1.975 1.947 1.920 Eit 10.457 10.963 11.469 11.975 12.478 12.978 13.474 13.966 14.451 C 79.319 79.051 78.923 78.848 78.815 78.831 78.909 79.068 79.333 Bold values represent the optimal value for the cost function

3

0.035 0.078 16.645 11.122 514.38

2

0.033 0.076 16.697 10.644 515.27

0.036 0.079 16.631 11.605 514.69

4

Bold values represent the optimal value for the cost function

s Ripo Rsn Es Eit C

Table 4 Effect of s: Fix L = 20 and arrival process as MPA

0.040 0.081 16.629 12.090 515.37

5 0.043 0.083 16.630 12.573 516.19

6 0.046 0.085 16.633 13.054 517.08

7

0.050 0.088 16.635 13.533 518.03

8

0.055 0.092 16.637 14.009 519.04

9

10 0.060 0.095 16.639 14.480 520.14

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Table 5 Effect of s: Fix L = 20 and arrival process as MNA

s 2 3 4 5 6 7 8 9 10 Ripo 0.035 0.037 0.040 0.042 0.046 0.049 0.054 0.059 0.065 Rsn 0.208 0.209 0.211 0.212 0.214 0.216 0.219 0.222 0.225 Es 2.100 2.068 2.037 2.008 1.9778 1.949 1.918 1.890 1.858 Eit 10.418 10.918 11.427 11.924 12.430 12.918 13.419 13.892 14.381 C 77.951 77.702 77.546 77.460 77.380 77.383 77.399 77.542 77.729 Bold values represent the optimal value for the cost function

We see that Ripo increases when s increases. This happens because when s increases, the inventory level reaches s more rapidly from above. Rsn also increases as s increases. This is due to the fact that when s increases, S2 is switched on to processing at a faster rate and hence the inventory level reaches to maximum value L at a faster rate and as a result S2 switched on to service mode if customers are waiting. Es decreases as s increases. This happens since when s increases both Ripo and Rsn increase and as a result customers get service at a faster rate. Eit increases as s increases. This is because when s increases, S2 is switched on to processing mode at a faster rate. The cost function first decreases reaches a minimum value and then increases for all arrival processes. The optimal cost varies for different arrival processes (see Fig 1). It is the highest for MPA. This shows the effect of positive correlation. Tables 6, 7, 8, 9, 10 indicate the effect of the parameter L on various performance measures and the cost function when s is fixed. We summarize the observations based on these tables below. Ripo decreases as L increases. This is due to the fact that the level s is attained at a slower rate. Rsn also decreases as L increases. This happens since L is attained at a slower rate. Es increases as L increases. This happens since when L increases both Ripo and Rsn decrease and as a result customers get service at a slower rate. Eit increases as L increases since more items are processed at a stretch. The cost function first decreases reaches a minimum value and then increases for all arrival processes. The optimal cost varies for different arrival processes (see Fig 2). It is the highest for MPA. This shows the effect of positive correlation.

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74.4

ERA

EXP

62

C

C

74.2 61.8

74 61.6 61.4

73.8 2

4

6

8

2

10

4

6

8

10

s

s 79.4 HEXP

C

79.2

79

78.8

2

4

6

8

10

s 78

520

MPA

MNA

77.8

C

C

518

77.6

516

77.4

514 2

4

6

8

s Fig. 1 Effect of s on C when L = 20

10

2

4

6

s

8

10

9

0.109 0.216 1.641 5.597 66.80

8

0.131 0.227 1.617 5.090 67.86

10 0.093 0.208 1.667 6.096 66.43

11 0.081 0.202 1.695 6.591 66.52

Bold values represent the optimal value for the cost function

L Ripo Rsn Es Eit C

Table 6 Effect of L: Fix s = 3 and arrival process as EXP

12 0.072 0.197 1.723 7.083 66.90

13 0.064 0.194 1.751 7.573 67.48

14 0.058 0.191 1.780 8.062 68.21

15 0.053 0.188 1.809 8.549 69.04

16 0.049 0.186 1.838 9.035 69.96

17 0.045 0.184 1.867 9.521 70.93

18 0.042 0.182 1.895 10.006 71.96

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9

0.111 0.244 1.216 5.694 54.46

8

0.134 0.255 1.187 5.208 55.51

10 0.094 0.235 1.247 6.178 54.12

11 0.081 0.229 1.277 6.660 54.22

Bold values represent the optimal value for the cost function

L Ripo Rsn Es Eit C

Table 7 Effect of L: Fix s = 3 and arrival process as ERA

12 0.072 0.223 1.307 7.142 54.61

13 0.064 0.219 1.336 7.624 55.19

14 0.058 0.215 1.365 8.106 55.90

15 0.053 0.212 1.393 8.588 56.70

16 0.048 0.209 1.421 9.070 57.57

17 0.045 0.206 1.448 9.552 58.49

18 0.041 0.204 1.475 10.035 59.44

328 A. Krishnamoorthy and Divya V.

9

0.108 0.208 1.817 5.559 71.93

8

0.130 0.219 1.795 5.048 73.02

10 0.092 0.200 1.842 6.063 71.54

11 0.081 0.194 1.867 6.561 71.59

Bold values represent the optimal value for the cost function

L Ripo Rsn Es Eit C

Table 8 Effect of L: Fix s = 3 and arrival process as HEXP

12 0.071 0.189 1.894 7.056 71.94

13 0.064 0.186 1.921 7.549 72.49

14 0.058 0.183 1.949 8.040 73.20

15 0.053 0.180 1.977 8.529 74.01

16 0.049 0.178 2.005 9.017 74.92

17 0.045 0.176 2.033 9.504 75.88

18 0.042 0.175 2.062 9.991 76.90

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9

0.101 0.124 16.70 5.551 517.3

8

0.122 0.139 16.71 5.033 519.3

10 0.087 0.114 16.69 6.063 516.1

11 0.076 0.106 16.69 6.573 515.3

Bold values represent the optimal value for the cost function

L Ripo Rsn Es Eit C

Table 9 Effect of L: Fix s = 3 and arrival process as MPA

12 0.067 0.100 16.68 7.080 514.7

13 0.060 0.096 16.68 7.587 514.4

14 0.055 0.092 16.67 8.093 514.2

15 0.050 0.088 16.67 8.599 514.1

16 0.046 0.085 16.66 9.104 514.0

17 0.043 0.083 16.66 9.609 514.0

18 0.040 0.081 16.65 10.113 514.1

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9

0.111 0.250 1.745 5.488 70.14

8

0.132 0.264 1.723 4.940 71.12

10 0.094 0.242 1.776 5.979 69.81

11 0.082 0.235 1.800 6.505 69.90

Bold values represent the optimal value for the cost function

L Ripo Rsn Es Eit C

Table 10 Effect of L: Fix s = 3 and arrival process as MNA

12 0.072 0.230 1.833 6.987 70.32

13 0.065 0.225 1.860 7.500 70.89

14 0.059 0.222 1.891 7.980 71.67

15 0.054 0.219 1.919 8.483 72.50

16 0.049 0.216 1.950 8.964 73.46

17 0.046 0.214 1.979 9.461 74.44

18 0.042 0.212 2.009 9.942 75.51

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72

ERA

EXP

58

C

C

70

56

68

66

54 10

15

20

10

15

20

15

20

L

L 78 HEXP

C

76

74

72 10

15

20

L 76

520

MNA

MPA

74

C

C

518

72

516

514

70 10

15

L Fig. 2 Effect of L on C when s = 3

20

10

L

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8 Conclusion We considered a MAP/(PH,PH)/2 queue with processing of service items by a server. We analyzed the model in steady state by matrix analytic method and also derived some important distributions. Also we provided some numerical experiments to find the optimal values of L and s.

References 1. Baek, J., Dudina, O., Kim, C.: A Queueing system with heterogeneous impatient customers and consumable additional items. Int. J. Math. Comput. Sci. 27(2), 367–384 (2017) 2. Dhanya, S., Dudin, A.N., Olga, D., Krishnamoorthy, A.: A two-priority single server retrial queue with additional items. J. Ind. Manag. Optim. https://doi.org/10.3934/jimo.2019085 3. Divya, V., Krishnamoorthy, A., Vishnevsky, V.M.: On a queueing system with processing of service items under vacation and N-policy. In: DCCN 2018, CCIS 919, pp. 43–57 (2018) 4. Hanukov, G., Avinadav, T., Chernonog, T., Spiegal, U., Yechiali, U.: A queueing system with decomposed service and inventoried preliminary services. Appl. Math. Model. 47, 276–293 (2017) 5. Kazimirsky, A.V.: Analysis of BMAP/G/1 queue with reservation of service. Stoch. Anal. Appl. 24(4), 703–718 (2006) 6. Neuts, M.F.: Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD (1981)

A Two-Stage Tandem Queue with Specialist Servers T. S. Sinu Lal, A. Krishnamoorthy, V. C. Joshua, and Vladimir Vishnevsky

Abstract The queueing system considered consist of two multi-server stations in series. Customers arrive according to a Markovian Arrival Process to an infinite capacity queue at the first station. There are c servers who provide identical exponentially distributed service at the first station. A customer at the head of the queue can enter into service if any one of the servers at the first stage is idle. At the second station there are N identical servers called specialist servers . The service time distribution of specialist severs is phase type. There is a finite buffer in between the two stations. On completion of service at first stage, a customer needs service at the second station with probability p or leaves the system with probability 1 − p. In the former case, the customer joins the second station for service in case the waiting room is not full, else he is lost to the system. A customer in the finite buffer can enter into service if at least one of these servers is free. Stability of the system is established and stationary distribution is obtained using Matrix Analytic Methods. We compute distribution of waiting time of customers in the first queue, the mean number of customers lost due to capacity restriction of the waiting space of the second station and the mean waiting time of customers who get into service at the second station. An optimization problem on the capacity of second waiting station is also analyzed. Keywords Tandem queue · Specialist server · Matrix analytic method · Phase type distribution · Markovian arrival process

T. S. Sinu Lal · A. Krishnamoorthy · V. C. Joshua () Department of Mathematics, CMS College Kottayam, Kottayam, Kerala, India e-mail: [email protected];[email protected];[email protected] http://www.cmscollege.ac.in V. Vishnevsky Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_20

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1 Introduction Tandem queues receive much attention in the area of mathematical modeling for its wide range of applications to fit into a great deal of queueing situations that arise in common life as well as in various fields of science, industry, and technology. In a tandem queue, an arriving customer is offered multistage service facilities with various strategies. Numerous examples of such models can be found in our everyday life itself. A very observable example is a hospital situation where an arriving patient is first examined at the casualty clinic and after the preliminary examination at the casualty clinic the patient’s illness is sometimes cured or otherwise he may be sent to the clinic of a specialist for further treatment. The situation perfectly exemplifies a tandem queue of two stations, first is the casualty clinic and the second, the clinic of the specialist doctor. The problem discussed in this paper is motivated by this example. Analogous situations can also be found in manufacturing systems. The production process of many commodities are completed at service facilities arranged sequentially. Manufacturing of motor vehicles, air crafts, etc. are best examples for this. The mathematical model playing behind all these queueing systems are also applicable in the design and control of various communication networks. A message transmission system used for security checking of transmitting messages best illustrates this. In this case messages (data packets) are initially examined at the first stage and are transmitted if no threat is found otherwise it is passed on to the experts for further examination. Likewise tandem queues are observable in many realms of life. Phase type distributions were introduced as a generalized version of exponential distribution by M F Neuts (1975). The set of all phase type distributions is a dense subset of set of all distributions defined on the nonnegative real line. This means given any arbitrary probability distribution defined on the nonnegative real line, it can be approximated using a sequence of phase type distributions converging to it. This is the main reason why phase type distributions are widely used in stochastic modeling. The inadequacy of the stationary Poisson process in modeling arrivals to a system is its inability to address correlated arrival flows. This is best overcome by the introduction of Markovian arrival process (MAP) Neuts [19] and more results on MAP by Lucantony [17]. The set of Markovian arrival process is a versatile class which includes PH renewal process, Markov Modulated Process, etc. Moreover any stochastic counting process can be approximated with a sequence of Markovian arrival process with the desired degree of accuracy. The use of phase type distributions for service times and Markovian arrival process for customer arrivals increases the complexity of models and hence analysis becomes extremely tedious. Matrix analytic method, [20] is the most advantageous tool to compute the stationary distribution of the process of the system and hence evaluating the performance metrics. Latouche and Ramaswamy [16] is a emphasis on the applications of matrix analytic method on various queueing models. Chakravarthy et al. [4] study a MAP/PH/c queueing system with retrials and search. Gomez-Corral et al. [7, 9] provide detailed study on performance evaluation

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of a two stage tandem queue. Reference [9] describes a two-stage tandem queue with blocking, which occurs according to a mechanism called blocking after service. In this model, there is a finite buffer between stations and when the capacity exceeds, the customers are forced to be blocked at the first station, if they want to proceed to the next station. A tandem queue with MAP and blocking is studied in [7]. In the model studied in [7], there is no intermediate buffer between the stations. A MAP/PH/1/1→./PH/1/k+1 queue with blocking and retrial is described in [8]. This model has no waiting space before the first station and which leads to customer loss at the early stage of service. In [10, 11] Kim et al. analyze tandem queueing network of two stations. The model in [11] has a finite and an infinite intermediate buffer between stations. A model for interactive voice response of call centers is investigated in [13]. The paper [12] studies a MMAP/PH/N queueing system with an optimal strategy of control by the number of active servers in a multi-server queue. The model studied in [2] is of the type MAP /P H /c1 → ./P H /c2 /c2 +k2 , this is a loss system due to capacity restriction of queue at first service station. Reference [1] is an excellent survey of queueing networks with finite capacity queues. For wide range of details of MAP [3] is referred. References [6, 14, 15, 18] study models with finite buffers and use matrix analytic methods to compute stationary distribution of the system process. A detailed description of performance analysis of queueing networks can be seen in [21]. For fundamentals of stochastic process [5] is referred. The coming section of the paper is arranged as follows. In Sect. 2 mathematical model of the system is described and analysis is carried out. Also stability of the system is characterized in this section. In Sect. 3 the steady state distributions of the process are obtained. Section 4 includes the waiting time distribution of a customer in the intermediate buffer. Performance characteristics of the system are defined in Sect. 5. A constraint on system revenue is defined as a cost function in Sect. 6. The model is numerically illustrated in Sect. 7. The study concludes with Sect. 8.

2 Model Description The queueing system under study consists of two multi-server stations operating in tandem. Customers arrive to the first station according to a Markovian Arrival Process (MAP). The MAP is directed by an underlying random process νt , t ≥ 0, which itself is an irreducible continuous time Markov Chain on a finite state space{1,2. . . ,w}. The transition intensities of the process ν t are defined by the square matrices D0 and D1 , each having dimension w × w. The matrix D0 corresponds to the chain transitions without generating any arrival, whereas D1 corresponds to transitions generating an arrival of a customer. The matrix D = D0 + D1 is the infinitesimal generator of the process νt , , t ≥ 0. The stationary state distribution η of this chain is the unique solution to the system ηD = 0, ηe = 1, where 0 is a zero row vector and e is the column vector of 1’s having appropriate dimension. The fundamental arrival rate λ is given by λ = ηD1 e. The coefficient 2 2 of variation of successive arrivals Cvar is given by Cvar = 2λη(−D0 )−1 e − 1

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and the correlation coefficient Ccor of the successive inter arrivals is given by 2 . At the first station there are c identical Ccor = λη(−D0 )−1 D1 (−D0 )−1 e −1)/Cvar exponential servers, each having service rate μ , 0 < μ < ∞ arranged in parallel. An arriving customer can directly enter into one of these servers if at least one of them is idle or otherwise is required to wait in an infinite queue in front of the first station. For a customer coming out of the first station, a probabilistic decision making is carried out to determine whether to quit the system or to proceed towards the second station. A customer proceeds to the second station with a probability p or otherwise leaves the system forever with the complimentary probability 1 − p. The second station has N identical servers each with phase type distributed service time. This phase type distribution of the service time of these servers has the irreducible representation (α, S). The process of the system is modeled as the irreducible continuous time Markov chain X(t) = {(q1(t), q2 (t), ζ 1 (t), ζ 2 (t), . . . .ζ r (t), a(t)), t ≥ 0}, which is a quasi birth–death (QBD) process. The first component q1 (t) is the number of customers up to the first station. Those who receiving service from the first station together with those in the infinite waiting line constitute q1 (t) and it can take nonnegative integer values. q2 (t) is the number of customers in the second stage of service, those customers waiting in the finite buffer and receiving service from the second station. q2 (t) takes values from the set {0, 1, 2, . . . k + N}. For i = 1, 2, . . . , r, ζ i (t) is the number of customers in the second station with service phase i, so that 0 ≤ ζ 1 (t) + ζ 2 (t) + · · · + ζ r (t) ≤ N, r is the total number of phases of service. This method of representing the phases of service was put forward by Ramaswamy and Lucantoni (1985). Taking the components ζ i (t) has a great advantage of reducing the dimension of the state space considerably small. If we count phase of each of the N servers separately, for larger values of N, the number of states in each level shows huge hike due to the number of phases of the service process and dimension of the MAP. Computation of performance characteristics become extremely tedious when the state space grow indefinitely. a(t) is the phase of arrival process which can take values in {1, 2, . . . , w}. The state space S = S1 ∪ S2 . S1 = {(i, j, k1, k2 , . . . , kr , l), i ∈ Z + , 0 ≤ j ≤ N + k, 1 ≤ km ≤ r, 1 ≤ l ≤ w}, and S2 = {(i, 0, l), i ∈ Z + , 1 ≤ l ≤ w}. The states in S1 correspond to nonzero values of q2 and states of S2 correspond to q2 = 0. The levels in S are determined by the values of i, and in the ith level the number of states is given by li = (1 + N−1 i=1 di + kdN )w states, where di is the number of combinations of nonnegative integers(i1, i2 , . . . , ir ) such that 0 ≤ ij ≤ N and rj =1 ij = i.

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The infinitesimal generator of the QBD process X(t) is of the form ⎛

⎞ B00 B0 ⎜B B B ⎟ ⎜ 10 11 0 ⎟ ⎜ ⎟ B20 B21 B0 ⎜ ⎟ ⎜ ⎟ Q=⎜ .. .. .. ⎟. . . . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ B B B 2 1 0 ⎝ ⎠ .. .. .. . . . The blocks of the generator matrix are obtained as follows. B0 = diag(D1 , Id1 ⊗ D1 , Id2 ⊗ D1 , . . . , IdN−1 ⊗ D1 , IdN ⊗ D1 , . . . IdN ⊗ D1 ) Bi1 = B00 − iμIli B1 = B00 − cμIlN , ⎛

B00

B 00 ⎜ B 10 B 11 ⎜ ⎜ B 20 B 21 ⎜ ⎜ .. .. ⎜ . . =⎜ ⎜ N0 ⎜ B B N1 ⎜ ⎜ .. ⎝ .

⎞

..

. B N+k0 B N+k+11

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ,⎟ ⎟ ⎟ ⎟ ⎟ ⎠

where each of the subblocks B ij is described below. B 00 = D 0 B i1 = [Mlk ] + Idi ⊗ D 0 , i = 1, 2, . . . N B N+i1 = B N1 for i=1,2. . . ,k, where [Mlk ] is a block matrix, in which each Mlk is a matrix of order w for 1 ≤ l ≤ di ,1 ≤ k ≤ di . The blocks Mlk are defined below.

Ml,k

⎧ ⎪ |ζ i (t1 ) − ζ i (t2 )| > 1for at least one i ⎪ ⎪ 0w×w ⎪ ⎪ ⎪ ⎪ ⎨   = diag( ζ i (t1 )Sii , . . . , ζ i (t1 )Sii )w ζ i (t1 ) = ζ i (t2 )∀i = 1, 2 . . . , r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ diag(ζ i (t1 )Si0 αj , . . . , ζ i (t1 )Si0 αj ))w |ζ i (t1 ) − ζ j (t2 )| = 1for exactly one pair (i,j),

where 0w×w is a zero square matrix of order w. B j o = [Tlk ], 1 ≤ j ≤ N − 1,

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where [Tlk ] is a block matrix, in which each Tlk where1 ≤ l ≤ di and1 ≤ k ≤ di is a square block of order w each Tlk are defined by

Tlk =

⎧ 0w×w ⎪ ⎪ ⎨

|ζ i (t1 ) − ζ i (t2 )| > 1 for at least one i

⎪ diag(ζ i (t)i Si0 , . . . , ζ i (t)i Si0 )w |ζ i (t1 ) − ζ i (t2 )| = 1∀i = 1, 2 . . . , r, ⎪ ⎩

,

where 0w×w is a zero square matrix of order w. For j ≥ N, B j 0 is defined by B j o = [Vlk ], where [Vlk ] is a block matrix, in which each Vlk where 1 ≤ l ≤ di and 1 ≤ k ≤ di is a square block of order w each Vlk are defined by

Vlk =

⎧ 0w×w ⎪ ⎪ ⎨

|ζ i (t1 ) − ζ i (t2 )| > 1for at least one i

⎪ diag(ζii Si0 αi , . . . , ζii Si0 αi )w |ζ i (t1 ) − ζ i (t2 )| = 1∀i = 1, 2 . . . , r ⎪ ⎩

For i=1,2. . . c-1 Bi0 is defined as follows. Bi0 = B∗∗ − iμIwdi ,B2 = B∗∗ − cμIwdi ⎛ qμIw pμα ⊗ Iw ⎜ qμIwdi pμIwdi ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ qμI pμI ⎜ wdi wdi B∗∗ = ⎜ qμIwdN pμIwdN ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ ⎝ qμIwdN pμInwdN μIwdN Theorem 1 The Markov chain described above is stable if and only if π0 D1 +

N−1  i=1

πi Idi ⊗ D1 +

N+k  i=N

πi IdN ⊗ D1 < cμ

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ .⎟ . ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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Proof Let B = B0 + B1 + B2 ,then B takes the form ⎞ C0 C0# ⎟ ⎜ C$ C C# ⎟ ⎜ 0 1 1 ⎟ ⎜ C1$ C2 C2# ⎟ ⎜ ⎟ ⎜ .. .. .. ⎟ ⎜ . . . ,⎟. B=⎜ ⎟ ⎜ $ # ⎟ ⎜ CN CN CN ⎟ ⎜ ⎟ ⎜ .. .. # ⎠ ⎝ . . CN $ CN CN ⎛

where the subblocks are defined as below. Cj# = D 0 , C1# = D 1 + cpμα ⊗ Ia , Cj# = B j i + jpμα ⊗ Ia for2 ≤ j ≤ N C1 = B 00 , Cj = B j 1 + cpμIdj for j 1 for at least one i ⎪ ⎪[0]a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨diag( ζ i (t )S 0 α · · ·  ζ i (t )S 0 α )ζ i (t ) = ζ i (t )∀i = 1, 2 . . . , r 1 i i 1 i i a 1 2 Δ2 (l, k) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |ζ i (t1 ) − ζ j (t2 )| = 1 for exactly one pair diag(ζ i (t1 )Si0 αj . . . ζ i (t1 )Si0 αj )a ⎪ ⎪ ⎪ ⎩

(i,j).

The waiting time of a tagged customer in the buffer is the time until the Markov chain W (t) enters the absorbing state. The waiting time of rth customer follows the phase type distribution with the irreducible representation (γ , Ω). Here γ is the initial probability vector whose mth component as 1 and all other components are

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zero, which means that transition always starts from the level r. The distribution function Fm of the waiting time is given by Fm (t) = 1 − γ (exp(t)e). In the above, γ (exp(t)e) is the mth entry of the column vector exp(t)e. m is given Expected waiting time of a customer who joins as the mth customer, Ew by the formula, m Ew = −γ −1 e.

Waiting time of an arbitrary customer in the buffer Ew =

∞  k 

m yiN+m Ew ,

i=0 m=1

where yiN+m = ξiN+m edN w

5 Performance Measures 1. Expected number of customers in first stage Ec1 =

∞ 

iξi eli

i=o

 li = (1 + N−1 combinations of i=1 di + kdN )w where di is the number of  nonnegative integers(i1, i2 , . . . , ir ) such that 0 ≤ ij ≤ N and rj =1 ij = i. 2. Expected number of customers in the queue Eq1 =

∞ 

(j − c)ξj elj .

j =c+1

3. Expected number of busy servers in the first station Eb1 = Ec1 − Eq1 . 4. Expected number of customers in the finite buffer EBF =

∞ N+k   j =0 i=0

iξj i .

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5. Expected number of busy severs in the second station Es2 =

∞ N−1  

i(ξij edi w ) + N

j =0 i=0

∞ N+k  

ξij edN w .

j =0 i=N

6. Expected number of customers in the system= Expected number of customers in Stage 1+Expected number of customers in stage 2 Ecs = Ec1 + Es2 . 7. Average intensity of flow of customers from first stage Avf = Ec1 μ. 8. Probability that the system is empty P e = ξ00 ew . 9. Expected time spend by a customer in the first queue Ec1 Et1 = Avf . 10. Expected number of customers leaving the system after the first stage of Service Ln1 =

c 

∞ 

iqμξi e(N+k+1)di w +

i=1

cqξi e(N+k+1)di w .

i=c+1

 11. Probability that the buffer is full, P bf = ∞ i=0 ξ iN+k e(N+k+1)diw . 12. Expected waiting time of a customer in the system

r Ew =

⎧ ⎨ Et1 ((1 − p) + p ∗ P bf ), if the customer is leaving after the first stage ⎩

Et1 1 + p ∗ (1 − P bf ) ∗ Ew , if the customer proceeds to second station.

13. Expected departure rate from the first station 1 Erat e = cμ

∞ 

ξ i e(N+k+1)di w + μ

i=c

c−1 

iξ i e(N+k+1)diw .

i=0

1 )q. 14. Rate of losing customers after the first stage,RL = P bf (Erat e 15. Rate at which customers proceeds to second station from first 1 RP 2 = Erat e (1 − P bf )p.

16. Expected departure rate from the second station 2 Erat e =

dj dN  r r N  ∞     (ξ i Si0 ) + (ξ i S10 αi ). j =1 m=1 i=1

j =N+1 m=1 i=1

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17. Probability that all servers at the first station are busy ∗ = Pbusy

∞ N+k  

ξj i e.

j =0 i=N

18. Probability that at least one server remains idle at first station 1 Pidle =

c N+k  

ξj i e.

j =0 i=N

19. Probability that there are at least one server in first station is busy 1 Pbusy =

c 

ξj e(N+k+1)di w .

j =1

20. Probability that there are at least one server in second station is busy 2 Pbusy =

∞ N+k  

ξj i e.

j =0 i=1

6 Optimal Control on System Parameters A revenue function involving parameters governing the system process is defined as follows: Φc = Ln1 ∗ C1 + Es2 ∗ C2 C1 represents the service cost per unit time for a server in the first station and C2 per unit time for a specialist server. Then Φc is actually a function of system parameters. Optimal values of a parameter can be obtained by changing the values of that parameter over the given range and keeping all other parameters constant. The values of Φc are plotted against the values of parameters and the graphs obtained are convex in nature and hence values at which system revenue reaches its maximum over the given range can be determined. It is to be noted that the values assumed by Φc depend on the specific values assigned to the costs C1 and C2 .

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7 Numerical Experiments 7.1 Example 1 In this example, we consider a tandem queue of two stations where servers at both stations have exponentially distributed service times with parameters μ1 and μ2 , respectively, and customers arrive according to a Poisson process with parameter λ. We fix λ = 0.5, μ2 =3, w=2, and N=2. The experiment is carried out by varying different parameters. In Tables 1 and 2 the service rate μ1 of the initial servers is varied and the corresponding variations in system performance measures are calculated. From Table 1, the length of the queue in front of the first station decreases with the increase in μ1 and this is pictorially represented in Fig. 1. Also the number of customers in Table 1 Variation in system performance measures with respect to μ1

μ1 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

Ec1 36.8151 10.7943 5.4412 3.4959 2.5578 2.0220 1.6792 1.4416 1.2668 1.1324

Eq1 34.9402 9.1091 3.9398 2.1422 1.3226 0.8844 0.6245 0.4587 0.3472 0.2691

Eb1 1.8749 1.6852 1.5014 1.3537 1.2352 1.1375 1.0547 0.9829 0.9196 0.8632

EBF 0.3125 0.6594 0.9325 1.1479 1.3346 1.5105 1.6834 1.8551 2.0251 2.1918

Ec2 3.1034 5.8427 8.2159 10.2280 12.1234 14.0596 16.0913 18.2108 20.3836 22.5693

Ecs 39.9184 16.6370 13.6570 13.7239 14.6812 16.0816 17.7705 19.6524 21.6504 23.7017

Avf 0.5625 0.5056 0.4504 0.4061 0.3706 0.3413 0.3164 0.2949 0.2759 0.2590

Table 2 Variation in system performance measures with respect to μ1

μ1

Pe

Et 1

Ln1

1 Pbusy

1 Erat e

Pbf

Es2

0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

0.0134 0.0372 0.0600 0.0814 0.1027 0.1251 0.1486 0.1729 0.1976 0.2222

122.7169 35.9809 18.1372 11.6531 8.5259 6.7399 5.5973 4.8052 4.2226 3.7746

116.467 30.363 13.131 7.1406 4.4086 2.948 2.0816 1.529 1.1573 0.897

0.9257 0.8086 0.6944 0.6007 0.5244 0.4612 0.4079 0.3624 0.3233 0.2895

93.9660 86.7843 79.8980 74.5461 70.3697 66.9616 64.0387 61.4292 59.0349 56.8015

0.0038 0.0088 0.0115 0.0122 0.0116 0.0105 0.0093 0.0081 0.0070 0.0061

0.9798 0.9293 0.8740 0.8293 0.7934 0.7629 0.7357 0.7103 0.6862 0.6629

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40 35 30 25 Eq1

c=1

20 15 10

c=3

5

c=5

0 0.1

0.2

0.3

0.4

0.5

μ1

0.6

0.7

0.8

0.9

1

Fig. 1 Variation in queue length with service rate of initial servers for different values of c 2.5

2 c=7

EBF

1.5 c=5

1 c=3

0.5

0 0.1

0.2

0.3

0.4

0.5

P

0.6

0.7

0.8

0.9

1

Fig. 2 Variation in number of customers accumulated in intermediate buffer with the increase in p

Fig. 3 Variation in number of customers accumulated in intermediate buffer with μ2

the intermediate buffer is directly proportional to μ1 and is represented in Fig. 2. With the increase in μ2 , the mean number of customers in the intermediate buffer decreases and it is represented graphically in Fig. 3. Figure 4a shows variation in EBF(along Z axis) with respect to μ1 and p. In Fig. 4b variation in EBF(along Z axis) with c and μ2 is depicted. Figure 5 shows that the system revenue increases with increase in buffer size(K) and reaches its maximum in the given range of values.

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1.5

4 3.5

1

EBF

EBF

3 2.5

0.5

2 1.5

0 8

1 2

0.5

1.5

0.1

0.4

4

0.2 0

1

0.6

0.3

0.5

μ

0.8

6

0.4

1

c

p

(a)

2

0.2 0

μ1

(b)

Fig. 4 Variation in EBF with respect to the different parameters

22.5 22.45 22.4

c

Cost(Φ )

22.35 22.3 22.25 22.2 22.15 22.1 22.05

0

500

1000 K

Fig. 5 Variation in cost with respect to variation in buffer size

1500

2000

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7.2 Example 2



0.4 0.55 −2.6 1.65 D1 = , D0 = 1.5 1.4 1.2 −4.1 S0 =

6 7 −0.7 0.3 , S = 0.4 0.6 0.1 −0.7 α = (0.6, 0.4).

The fundamental rate of Markovian arrival process is calculated as λ = 0.6250. The coefficient of variation Cvar = 2λθ (−D0)−1 )e − 1 = 1.5323. The correlation 2 coefficient is Ccor = λθ (−D0)−1 D1(−D0)−1 e) − 1/Cvar = 0.7548. In this case a Markovian arrival process with positively correlated inter-arrival times is considered. The system behavior is studied by varying the different parameters. The values of the corresponding performance characteristics are tabulated in Tables 3 and 4. In Fig. 6 expected number of customers accumulated in the buffer is plotted against p and μ. Fig. 7a and b, respectively, shows graphically that the number of customers accumulated in the infinite waiting line decreases while the service rate of the initial server (μ) and p increases. This phenomenon is intuitively true. In ∗ ) is plotted Fig. 8 a probability that all servers in the first station are busy (Pbusy ∗ against the service rate of the initial server (μ). Evidently Pbusy goes on decreasing with the increase in μ. In Fig. 8b the expected number of customers accumulated in the intermediate buffer (EBF) is plotted against p and EBF monotonically increases with increase in p.

Table 3 Variation in system characteristics with respect to the increase in p

p 01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Eq1 0.0515 0.0466 0.0426 0.0394 0.0366 0.0342 0.0322 0.0304 0.0289 0.0275

Ec1 0.4734 0.4197 0.3763 0.3403 0.3098 0.2837 0.2609 0.2408 0.2229 0.2069

Es2 0.1751 0.1908 0.2049 0.2182 0.2311 0.2440 0.2571 0.2707 0.2849 0.3001

Ecs 0.1326 0.2238 0.2984 0.3606 0.4134 0.4589 0.4985 0.5333 0.5641 0.5917

Pe 0.7811 0.8344 0.8796 0.9191 0.9544 0.9866 1.0165 1.0448 1.0720 1.0986

Avf 1.6876 1.4925 1.3347 1.2038 1.0930 0.9976 0.9146 0.8414 0.7762 0.7176

Pe 0.0355 0.0406 0.0440 0.0465 0.0485 0.0502 0.0517 0.0531 0.0545 0.0560

Et 1 0.4855 0.4305 0.3860 0.3490 0.3178 0.2909 0.2675 0.2470 0.2286 0.2122

Ln1 0.9385 0.7377 0.5771 0.4460 0.3373 0.2462 0.1692 0.1038 0.0478 0.0000

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Table 4 Variation in system characteristics with increase in μ

μ 4 6 8 10 12 14 16 18 20 22

Eq1 0.0119 0.0055 0.0030 0.0018 0.0012 0.0008 0.0006 0.0004 0.0003 0.0003

Ec1 0.1485 0.0918 0.0625 0.0453 0.0344 0.0270 0.0217 0.0179 0.0150 0.0127

Es2 0.2091 0.1964 0.1906 0.1876 0.1859 0.1849 0.1842 0.1838 0.1835 0.1833

Ecs 0.5653 0.6317 0.6771 0.7101 0.7352 0.7549 0.7707 0.7838 0.7947 0.8040

Pe 0.9229 0.9199 0.9303 0.9431 0.9555 0.9667 0.9767 0.9855 0.9933 1.0001

Avf 0.8197 0.6905 0.5950 0.5222 0.4650 0.4189 0.3811 0.3495 0.3227 0.2997

Pe 0.0581 0.0627 0.0658 0.0680 0.0697 0.0710 0.0720 0.0729 0.0736 0.0742

Et 1 0.1523 0.0942 0.0641 0.0465 0.0353 0.0277 0.0223 0.0184 0.0154 0.0131

Ln1 0.1780 0.1395 0.1147 0.0974 0.0847 0.0749 0.0671 0.0608 0.0556 0.0512

0.96 0.95

EBF

0.94 0.93 0.92 0.91 0.5 8

0.4 6 0.3 p

4 0.2

2

μ

Fig. 6 Variation in number of customers accumulated in intermediate buffer with p

8 Conclusion We analyzed a hospital model two-stage tandem queue, where the service is provided in two different stations connected in series. First is the casualty clinic and second is the clinic of the specialist doctors. The waiting space at second station is limited, the capacity of this waiting space is optimally determined by designing an appropriate cost function. On numerical investigation, the cost function shows convex nature and increases with the increase in buffer size and reaches

T. S. Sinu Lal et al.

0.012

0.055

0.01

0.05

0.008

0.045 Eq1

Eq1

352

0.006

0.04

0.004

0.035

0.002

0.03

0

4

6

8

10

12

14

16

18

20

22

0.025 0.1

0.2

0.3

0.4

0.5

0.6

μ

p

(a)

(b)

0.7

0.8

0.9

1

Fig. 7 Variation in queue length with respect to the variations in p and μ. (a) μ versus Eq1. (b) p versus Eq1 0.5

0.926

0.45 0.9255

0.925

0.35

EBF

P*busy

0.4

0.3

0.9245 0.25 0.924

0.2

4

6

8

10

μ

(a)

12

14

0.9235 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

p

(b)

∗ . (b) Fig. 8 Variation in different system characteristics with respect to μ and p. (a) μ versus Pbusy p versus EBF

maximum and then becomes flat at an optimal point. Variations in different system characteristics with respect to various parameters are also investigated. Acknowledgments Sinu Lal T S thanks Kerala State Council for Science Technology and Environment (KSCSTE), Kerala, India for KSCSTE Research Fellowship 2015 (No 001/FSHPMAIN/2015/KSCSTE).

References 1. Balsamo, S., Person, V.D.N., Inverardi, P.: A review on queueing network models with finite capacity queues for software architectures performance prediction. Perform. Eval. 51(2–4), 269–288 (2003) 2. Baumann, H., Sandmann, W.: Multi-server tandem queue with Markovian arrival process, phase-type service times, and finite buffers. Eur. J. Oper. Res. 256(1), 187–195 (2017)

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3. Chakravarthy, S.R.: The batch Markovian arrival process: A review and future work. Adv. Probab. Theory Stoch. Process. 1, 21–49 (2001) 4. Chakravarthy, S.R., Krishnamoorthy, A., Joshua, V.C.: Analysis of a multi-server retrial queue with search of customers from the orbit. Perform. Eval. 63(8), 776–798 (2006) 5. Cinlar, E.: Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ (1975) 6. Ferng, H.W., Chang, J.F.: Connection-wise end-to-end performance analysis of queueing networks with MMPP inputs. Perform. Eval. 43(1), 39–62 (2001) 7. Gomez-Corral, A.: A tandem queue with blocking and Markovian arrival process. Queueing Systems 41, 343–370 (2002) 8. Gomez-Corral, A.: A matrix-geometric approximation for tandem queues with blocking and repeated attempts. Oper. Res. Lett. 30(6), 360–374 (2002) 9. Gomez-Corral, A., Martos, M.E.: Performance of two-stage tandem queues with blocking: The impact of several flows of signals. Perform. Eval. 63(9–10), 910–938 (2006) 10. Kim, C., Dudin, A.N., Dudin, S., Dudina, O.: Tandem queueing system eith impatient customers as a model of call center with interactive voice response. Perform. Eval. 70, 440–453 (2013) 11. Kim, C., Dudin, A., Dudina, O., Dudin, S.: Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution. Eur. J. Oper. Res. 235(1), 170–179 (2014) 12. Kim, C., Dudin, A., Dudin, S., Dudina, O.: Hysteresis control by the number of active servers in queueing system MMAP/PH/N with priority service. Perform. Eval. 101, 20–33 (2016) 13. Kim, C., Klimenok, V.I., Dudin, A.N.: Priority tandem queueing system with retrials and reservation of channels as a model of call center. Comput. Ind. Eng. 96, 61–71 (2016) 14. Krishnamoorthy, A., Deepak, T.G., Joshua, V.C.: Queues with postponed work. Top 12(2), 375–398 (2004) 15. Krishnamoorthy, A., Joshua, V.C., Babu, D.: A token based parallel processing queueing system with priority. In: International Conference on Distributed Computer and Communication Networks, pp. 231–239. Springer, Cham (2017, September) 16. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling, vol. 5. Siam (1999) 17. Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat. Stoch. Models 7(1), 1–46 (1991) 18. Mathew, A.P., Krishnamoorthy, A., Joshua, V.C.: A Retrial queueing system with orbital search of customers lost from an offer zone. In: Information Technologies and Mathematical Modelling. Queueing Theory and Applications, pp. 39–54. Springer, Cham (2018) 19. Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16(4), 764–779 (1979) 20. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Corporation (1994) 21. Perros, H.G.: A bibliography of papers on queueing networks with finite capacity queues. Perform. Eval. 10(3), 255–260 (1989)

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer Induced Interruption and Retrial of Customers Jomy Punalal and S. Babu

Abstract In this article, we consider a MAP/(PH,PH,PH)/1 model to which customers arrive, according to the Markovian arrival process. At the time of arrival, all customers viewed as ordinary. If the server is busy, the arriving customers enter an orbit of infinite capacity. Each customer in orbit tries, independently of each other, to access the server at a constant rate. Each customer in orbit, regardless of others, generates priority with inter occurrence time exponentially distributed with parameter γ . A priority generated customer is immediately taken for service if the server is free. Else such customer is placed in a waiting space A1 of capacity one which is reserved only for priority generated customers. We consider a customer induced interruption while service is going on. The interruption occurs according to a Poisson process. The interrupted customers will enter into a buffer B1 of finite capacity K and they will spend a random period for completion of interruption. The duration of the interruption of customers in B1 follows an exponential distribution. The service facility consists of one server and period of service times of ordinary, priority, and interruption completed customers follow phase-type distribution with appropriate representations. Various performance measures obtained and suitable profit function for getting optimal buffer size K is also derived. Keywords Retrial queues; Self-generation of priorities; Customer induced interruption; Markovian arrival process; Level dependant quasi-birth-death process; Matrix analytic method

Jomy Punalal () · S. Babu Department of Mathematics, University College, Thiruvananthapuram, Kerala, India © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_21

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1 Introduction There are a large number of probabilistic models on priority queues in literature (Gross and Harris [17] chapter 3, Jaiswal [9] chapter 7, Takagi [18] chapter 3). Studies on priority queues found many applications in health care systems (Brahimi [2], Taylor [20]). All these articles treat priority queues with some external priority rules. In many applications, this discipline may not be an accurate modeling approach. Self-generation of priorities of customers in queues has been introduced in literature by Gomez-Corral, Krishnamoorthy, and Viswanath [6]. The pioneer works on selfgeneration of priorities are Krishnamoorthy, S. Babu, and Viswanath [10, 11]. In classical queueing models, servers are always available to serve customers. In many practical queueing systems, servers get interrupted due to failure of the servers (see Avi-Itzhak [1], Gaver [5], Neuts [15], Krishnamoorthy [12], Takagi [18, 19], Wang [22]) or getting preempted due to the arrivals of high priority customers (see White [23], Jaiswal [8]). There is a survey on queues with interruption by Krishnamoorthy [13]. In this survey customer induced interruption is discussed in the final section. Varghese et al. [7] discuss customer induced interruption which is entirely different from service interruptions. When one customer is self-interrupted, the server is ready to offer services to the other waiting customers. Retrial queues are a special type of queueing system which provides re-service and blocking. Retrial queues are extensively investigated in Yang [24], Falin [3], and Falin and Templeton [4]. In this paper, we consider the MAP/(PH, PH, PH)/1 model with self-generation of priorities, customer induced interruption and retrial of customers. The stability of the system is established, and some system performance measures are derived. These measures are used to define an expected total profit function, and the effect of different system parameters in profit function is explained numerically and illustrated graphically.

2 Model Description Customers arrive to a retrial queueing system according to Markovian arrival process with representation (D0 , D1 ) of order n. An arriving customer enters service immediately, if the server is free and if the server is busy, customer enters an orbit of infinite capacity. Each customer in the orbit independently tries to access the server according to Poisson process with parameter σ . A retrial customer who finds the server busy returns to the orbit with probability δ and leaves the system with probability 1 − δ. A customer in the orbit can generate priority according to Poisson process with parameter γ . The priority generated customer is immediately taken for service if server is free. If the server is busy, priority generated customer is moved to a waiting space A1 of capacity one which is reserved only for priority generated customers. If the waiting space A1 is already occupied, the new priority generated customer will leave the system forever. We also consider a customer

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induced interruption when service is going on and the interruption occurs according to a Poisson process with parameter θ . The interrupted customers will enter into a buffer B1 of finite capacity K according to the availability of space and customer will lost forever if no space for the customer in buffer B1 . The service provided here is non-preemptive and the service times follow phase-type distribution with representations (α, T ), (β, S), (ν, V ) of orders m1 , m2 , m3 , respectively. Define T0 = −T e, S0 = −Se, and V0 = −V e where T0 , S0 , V0 represents service completions corresponding to the three service processes. When interruption occurs the customer currently in service will force to leave the service facility and the freed server is ready to offer services to other customers. The interrupted customer will spend a random period of time for completion of interruption and it follows exponential distribution with rate η and the interruption completed customers will move to a buffer B2 whose size is also K. We assume that the priority generated customers never undergo interruption and not more than one interruption is allowed for a customer during service. Also assume no customer is lost before entering to the orbit. Again we assume the sum of number of customers in buffers B1 and B2 should be less than or equal to K. When buffer B2 is full and a customer induced interruption happens, then the self-interrupted customer will lost from the system even though buffer B1 have free space. In particular, when buffer B2 is full, then B1 should be empty. A pictorial representation of the model is shown in Fig. 1.

3 Mathematical Formulation The model is studied as a Quasi Birth-Death (QBD) process and matrix geometric solution is obtained. For further analysis we use the following notations: N1 (t)= Number of customers in the orbit at time t. N2 (t)= ⎧ Number of busy servers at time t. ⎪ ⎪ ⎨1, server busy with ordinary customer at time t S(t) = 2, server busy with priority generated customer at time t ⎪ ⎪ ⎩3, server busy with customer fromB buffer at timet 2

N3 (t)= Number of priority generated customers waiting for service at time t. N4 (t)= Number of interruption completed customer in buffer B2 at time t. N5 (t)= Number of interrupted customers in buffer B1 at time t. M(t)= phase of service process at time t. A(t)= phase of arrival process at time t. Under the assumptions on arrival and service processes {χ(t) : t ≥ 0} where χ(t) = {N1 (t), N2 (t), S(t), N3 (t), N4 (t), N5 (t), M(t), A(t)} form a continuous time Markov chain on the state space  = L1 (i) ∪ L2 (i) ,where L1 (i) = {(i, 0, w, b2 , b1 , y) : i ≥ 0; w = 0, 1; b2, b1 = 0, 1, 2, · · · , K; b2 + b1 ≤ K; y = 1, 2, . . . , n}. L2 (i) = {(i, 1, s, w, b2 , b1 , x, y) : i ≥ 0. s = 1, 2, 3; w =

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If No waiting Space in A1 and server is busy, customer leaves system in search of emergency service

A1 Waiting Space for Priority

.

iγ

generated customers

is

1−d

d Arrival

Service Completion

M AP (D 0 , D 1 )n

Server

Ordinary service

(a, T )m1 , (b , S )m2 , (n , V )m3

Buffer of Interruption Completed Customers q

B2 b1 h B1 Buffer of Interrupted Customers

If B1 is full customer leaves system for ever Fig. 1 Pictorial representation of the model

0, 1; b2, b1 = 0, 1, 2, · · · , K; b2 + b1 ≤ K; x = 1, 2, . . . , ms , s = 1, 2, 3; y = 1, 2, . . . , n}. By partitioning the state space into levels with respect to the number of customers in the orbit, the generator of above Markov process⎤ is of the form: ⎡ A10 A0 . . . . . . ⎢ ⎥ ⎢A A A . . . . . ⎥ ⎢ 21 11 ⎥ 0 ⎢ ⎥ ⎢ . A A . . . . ⎥ 22 12 A0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. .. .. ⎢ ⎥ . . . Q= ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

. .

. .

.

.

⎥ ⎥

. A2N A1N A0 . . ⎥ ⎥ ⎥ . . A2N+1 A1N+1 A0 . ⎥ ⎥ .. .. ⎦ .. . . . . . .

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer. . .

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where A0 , A10 , A2i , A1i , for i = 1, 2, 3, · · · are square matrices of order 4(K + 1)(K + 2)mn and matrices defined as follows:

3.1 Matrix A10 Has the Following Transitions Entries of A10 are the transition rates within level 0. Let b = 0, 1; s = 1, 2, 3; w = 0, 1; b2 , b1 = 1, 2, 3, . . . K; b2 + b1 ≤ K; x = 1, 2, . . . ms ; y = 1, 2, . . . n J • (0, w, b2 , b1 , y) −−−−−−−→ (0, w, b2 , b1 , y) J • (0, b, s,/w, b2 , b1 , x, y) −−−−−−−→ (0, b, s, w, b2 , b1 , x, y) In ⊗ D0 − b1 ηIms ,n , if b1 = Kor b1 + b2 = Kor b1 + b2 < K J= In ⊗ D0 , otherwise b1 ηIms n • (0, b, s, w, b2 , b1 , x, y) −−−−−−−−−→ (0, b, s, w, b2 + 1, b1 − 1, x, y)

.

3.2 Matrix A0 Has the Following Transitions Entries of A0 are the transition rates from level i to (i + 1). Let b = 0, 1; s = 1, 2, 3; w = 0, 1; b2 , b1 = 1, 2, 3, . . . K; b2 + b1 ≤ K; x = 1, 2, . . . ms ; y = 1, 2, . . . n. In ⊗ D1 • (i, b, s, w, b2 , b1 , x, y) −−−−−−−→ (i + 1, b, s, w, b2 , b1 , x, y).

3.3 Matrix A2i Has the Following Transitions Entries of A2i are the transition rates from level i to (i − 1). Let s = 1, 2, 3; b2 , b1 = 1, 2, 3, . . . K; b2 + b1 ≤ K; x = 1, 2, . . . ms ; y = 1, 2, . . . n; b1 = 1, 2, 3, . . . K; b2 + b1 ≤ K iγ Ims ,n • (i, 0, s, 0, b2 , b1 , x, y) −−−−−−−→ (i − 1, 0, s, 1, b2 , b1 , x, y). A • (i, 0, s, 1, / b2 , b1 , x, y) −−−−−−−→ (i − 1, 0, s, 1, b2 , b1 , x, y); iγ Ims ,n , if b1 = K A= . (iγ + θ )Ims ,n , if b1 = K or b1 + b2 = K θ Ims ,n • (i, 0, s, 0, b2 , b1 , x, y) −−−−−−→ (i − 1, 0, s, 0, b2 , b1 + 1, x, y). θ Ims ,n • (i, 0, s, 0, b2 , b1 , x, y) −−−−−−−→ (i − 1, 0, s, 0, b2 , b1 , x, y); if b1 = K or b2 + b1 = K

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iσ Ims ,n • (i, 0, s, w, b2 , b1 , x, y) −−−−−−−→ (i − 1, 1, s, w, b2 , b1 , x, y) ; for w = 0, 1 B • (i, 1, s, 0, ⎧ b2 , b1 , x, y) −−→ (i − 1, 0, s, 0, b2 , b1 , x, y) ; ⎪ ⎪ ⎨T0 α ⊗ In , if s = 1 B = S0 β ⊗ In , if s = 2 and b2 = 0 ⎪ ⎪ ⎩V ν ⊗ I , if s = 3 and b = 0 0

n

2

C • (i, 1, s, 0, / b2 , b1 , x, y) −−→ (i − 1, 0, 3, 0, b2 , b1 , x, y) ; S0 β ⊗ In , if s = 2 and b2 = 0 C= V0 ν ⊗ In , if s = 3 and b2 = 0 D • (i, 1, s, 1, ⎧ b2 , b1 , x, y) −−→ (i − 1, 0, 2, 0, b2 , b1 , x, y) ; ⎪ ⎪ ⎨T0 α ⊗ In , if s = 1 D = S0 β ⊗ In , if s = 2 ⎪ ⎪ ⎩V ν ⊗ I , if s = 3 0

• • • •

• • • •

n

iγ (i, 0, 0, b2 , b1 ) −−−−→ (i − 1, 0, 1, b2, b1 ). iγ (i, 1, s, 0, b2 , b1 ) −−−−→ (i − 1, 1, s, 1, b2 , b1 ). iγ (i, 0, 1, b2 , b1 ) −−−−→ (i − 1, 0, 1, b2, b1 ). E (i, 1, s, 1, / b2 , b1 ) −−→ (i − 1, 1, s, 1, b2 , b1 ) ; iγ , if b1 = K E= . (iγ + θ ), if b1 = K or b1 + b2 = K θ (i, 1, s, 0, b2 , b1 ) −−→ (i − 1, 1, s, 0, b2 , b1 + 1). θ (i, 1, s, 0, b2 , b1 ) −−→ (i − 1, 1, s, 0, b2 , b1 ), if b1 = K or b2 + b1 = K . iσ (i, 0, w, b2 , b1 ) −−−−−−−→ (i − 1, 1, 1, w, b2 , b1 ) , for w = 0, 1 iσ (1 − δ) (i, 1, s, w, b2 , b1 ) −−−−−−−−−→ (i − 1, 1, s, w, b2 , b1 ) , for w = 0, 1

3.4 Matrix A1i Has the Following Transitions Entries of A1i are transitions within level i. Let b = 0, 1; w = 0, 1; b2, b1 = 1, 2, 3, . . . K; b2 + b1 ≤ K; x = 1, 2, . . . ms ; y = 1, 2, . . . n. Let † is a condition: b1 = K or b1 + b2 = K or b1 + b2 < K.

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer. . .

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F • (i, b, s, w, ⎧ b2 , b1 , x, y) −−→ (i, b, s, w, b2 , b1 , x, y) ; ⎪In ⊗ D0 − (iγ + iσ + θ + b1 η)Im ,n , if s = 0 and † ⎪ s ⎪ ⎪ ⎪ ⎪ ⎪ ⊗ D − (iγ + iσ + θ )I , if s = 0 and otherwise I n 0 ms ,n ⎪ ⎪ ⎪ ⎪ ⎪ T ⊕ D0 − (iγ + iσ + θ + b1 η)Ims ,n , if s = 0 and † ⎪ ⎪ ⎪ ⎨T ⊕ D − (iγ + iσ + θ )I 0 ms ,n , if s = 0 and otherwise F= ⎪ S ⊕ D0 − (iγ + iσ + θ + b1 η)Ims ,n , if s = 0 and † ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪S ⊕ D0 − (iγ + iσ + θ )Ims ,n , if s = 0 and otherwise ⎪ ⎪ ⎪ ⎪V ⊕ D0 − (iγ + iσ + θ + b1 η)Ims ,n , if s = 0 and † ⎪ ⎪ ⎪ ⎩V ⊕ D − (iγ + iσ + θ )I , if s = 0 and otherwise 0

ms ,n

4 System Stability Theorem 1 The system under discussion is stable. Proof Consider the Lyapunov test function defined by φ(s) = i where s is a state in level i. For a state s in level i , the mean drift ys is given by ys =



[φ(p) − φ(s)]qsp

p=s

=

   [φ(s " ) − φ(s)]qss " + [φ(s "" ) − φ(s)]qss "" + [φ(s """ ) − φ(s)]qss """ s"

s ""

s """

where s " , s "" , s """ vary over states belonging to levels i − 1, i, and i + 1, respectively. Then φ(s) = i, φ(s " ) = i − 1, φ(s "" ) = i, φ(s """ ) = i + 1 ys = −

 s"

qss" +



qss"""

s"""

⎧  ⎪ ⎪ −iγ − iσ + s""" qss""" , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−i(γ + σ (1 − δ)) − θ − [(T α ⊗ I )e] +  q n s 0 s""" ss""" , =  ⎪ ⎪ −i(γ + σ (1 − δ)) − θ − [(S0 β ⊗ In )e]s + s""" qss""" , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩−i(γ + σ (1 − δ)) − θ − [(V ν ⊗ I )e] +  q n s 0 s""" ss""" ,

when server is idle. when server is busy with OR. when server is busy with PG. when server is busy with IC.

where OR, PG, and IC denotes ordinary, priority generated, and interruption completed customers, respectively. Since s""" qss""" is bounded by some fixed constant for any s in level i ≥ 1 we can find a positive real number K such that  q < K for all s in level k ≥ 1 Thus for any  > 0 , we can find K ∗ ss""" s""" large enough that ys < − for any s belonging to level i ≥ K ∗ . Hence the theorem follows from Tweedie’s [21] result.



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4.1 Neuts–Rao Truncation Method When we apply this method our process χ transforms to χ¯ with infinitesimal generator ⎡

Q¯

A ⎢ 10 ⎢A ⎢ 21 ⎢ ⎢ . ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ . ⎢ ⎢ ⎢ . ⎢ ⎢ ⎢ . ⎢ ⎣ .

A0 . A11 A0 A22 A12 .. . . . . .

. . A0 .. .

. . . .. .

. . .

. . .

. . .

. A2N−1 A1N−1 A0 . . . . A2 A1 A0 . . . . A2 A1 A0 . . . . . . .. ..

. . . . . . .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

where A1 = A1N and A2 = A2N Let the steady state probability vector of the Markov process be x (x0 , x1 , x2 , . . . , xN−1 , xN , xN+1 , . . . ). We take i+1 xN+i = xN−1 RN

i = 0, 1, 2, . . .

=

(1)

where RN is the minimal solution of the matrix quadratic equation 2 RN A2N + RN A1N + A0 = 0

4.1.1 Choice of N To find the truncation level N, we use Neuts–Rao method (see [16]). As mentioned in [14], Elsner’s algorithm is used to determine the spectral radius η(N) of R(N). To minimize the effect of the approximation on the probabilities, N must be chosen such that |η(N) − η(N + 1)| < , where  is an arbitrarily small value. Again, x Q¯ = 0 leads xN−i = xN−i−1 RN−i

i = 1, 2, . . . N − 2

(2)

and x1 = x0 R1

i = 1, 2, . . . N − 2

(3)

where RN−i = −A0 (A1N−i + RN−i+1 A2N−i+1 )−1 and R1 = −A0 (A11 + R2 A22 )−1

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer. . .

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Finally from x0 A10 + x1 A21 = 0 we find x0 as the steady state distribution of finite state Markov chain with generator A10 + R1 A21 . Then from (1), (2) and (3), we get xi for i =1, 2, 3, . . . Now x is calculated by dividing each xi with the normalizing constant ∞ i=0 xi e.

5 Performance Measures Let ξ = (ξ0 , ξ1 , ξ2 , · · · ) be our steady state probability vector of the Markov process χ. For the evaluation of system performance measures we partition each ξi , i ≥ 0 as follows: Let ξi = (wi , xi , yi , zi ) where each vector corresponds to probability that the server is functioning i customers in the orbit.  • Probability that the server is idle. Pidle = ∞ i=0 wi e  • Probability that the server is busy with ordinary customer. Psbor = ∞ i=0 xi e •  Probability that the server is busy with priority generated customer. Psbpr = ∞ i=0 yi e • Probability  that the server is busy with interruption completed customers in B2 . Psbb2 = ∞ i=0 zi e • Probability that the server is idle with customers in the orbit. Pidleco = ∞ w e − w1 e i=0 i  • Expected number of customers in the orbit. Eor = ∞ i=1 iξi e •  Expected number of customers in the orbit when server is idle. Esidle = ∞ i=1 iwi e  • Successful retrial rate. Srr =  σ ∞ i=1 iwi e • Overall retrial rate. Orr = σ ∞ i=1 iξi e  • The fraction of successful rate of retrial. Fsrr =

Srr Orr

=

∞

i=1 ∞

i=1

iwi e iξi e

Let ξi = ζi (b, s, w, b2 , b1 ) where ζi (b, s, w, b2 , b1 ) is a row vector corresponding to N2 (t) = b, S(t) = s, N3 (t) = w, N4 (t) = b2 , N5 (t) = b1 with b = 0, 1; s = 1, 2, 3; w = 0, 1; b2 , b1 = 0, · · · , K; b2 + b1 ≤ K. • Probability that priority generated customers lost from the system. Pprl =

1  3  K K−b ∞   2

ζi (b, s, 1, b2 , b1 )e

i=1 b=0 s=1 b2 =0 b1 =0

• Probability that interrupted customers lost from the system. Pinl =

1  3  K ∞   i=1 b=0 s=1 b2 =0

ζi (b, s, 1, b2 , K − b2 )e

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• Expected number of ordinary customers in the orbit. Eorc =

∞  K K−b  2

iζi (0, 0, b2 , b1 )e +

i=0 b2 =0 b1 =0

+

∞  K K−b  2

∞  1  K K−b  2

iζi (1, 1, w, b2 , b1 )e

i=0 w=0 b2 =0 b1 =0

(i−1)ζi (1, 2, 0, b2 , b1 )e+

i=1 b2 =0 b1 =0

∞  K K−b  2

(i−2)ζi (1, 2, 1, b2 , b1 )e

i=2 b2 =0 b1 =0

+

1  K K−b ∞   2

iζi (1, 3, w, b2 , b1 )e.

i=0 w=0 b2 =0 b1 =0

• Expected number of priority generated customers in the orbit. Eprc =

∞  K K−b  2

ζi (0, 1, b2, b1 )e +

i=0 b2 =0 b1 =0

+

∞  K K−b  2

∞  K K−b  2

ζi (1, 1, 1, b2 , b1 )e

i=0 b2 =0 b1 =0

ζi (1, 2, 0, b2, b1 )e +

i=0 b2 =0 b1 =0

∞  K K−b  2

2ζi (1, 2, 1, b2 , b1 )e

i=0 b2 =0 b1 =0

+

K K−b ∞   2

iζi (1, 3, 1, b2 , b1 )e.

i=0 b2 =0 b1 =0

• Expected number of priority generated customers lost from the system. Eprl =

∞  1  3  K K−b  2

iζi (b, s, 1, b2 , b1 )e

i=1 b=0 s=1 b2 =0 b1 =0

• Expected number of interrupted customers lost from the system. Einl =

∞  1  3  K 

iζi (b, s, 1, b2 , K − b2 )e

i=1 b=0 s=1 b2 =0

• Expected number of interrupted customers in buffer B1 . Einb1 =

∞  1  3  1  K K−b  2 i=0 b=0 s=1 w=0 b2 =0 b1 =0

b1 ζi (b, s, w, b2 , b1 )e

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer. . .

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• Expected number of interruption completed customers in buffer B2 . Eincb2 =

∞  1  3  1  K K−b  2

b2 ζi (b, s, w, b2 , b1 )e

i=0 b=0 s=1 w=0 b2 =0 b1 =0

• Expected number of customers lost after retrials per unit time. Eclr = σ (1 − δ)

∞  3  1  K K−b  2

iζi (1, s, w, b2 , b1 )e

i=1 s=1 w=0 b2 =0 b1 =0

• Expected number of departure of ordinary customers after completing service. Edeor =

1  K K−b ∞   2

ζi (1, 1, w, b2 , b1 )e

i=0 w=0 b2 =0 b1 =0

• Expected number of departure of priority generated customers after completing service. Edepr =

1  K K−b ∞   2

ζi (1, 2, w, b2 , b1 )e

i=0 w=0 b2 =0 b1 =0

• Expected number of departure of interruption completed customers after completing service. Edeinc =

∞  1  K K−b  2

ζi (1, 3, w, b2 , b1 )e

i=0 w=0 b2 =0 b1 =0

6 Cost Analysis Based on the above system characteristics we propose an optimization problem and illustrating a numerical example. Define a revenue (profit) function as: ET P = r1 Edeor + r2 Edepr + r3 Edeinc − c1 Eorc − c2 Eprc − c3 Eincb2 − c4 Einb1 − c5 Einl − c6 Eprl − c7 Eclr − cf ixed , where r1 monetary units revenue obtained for each ordinary customer getting service and leaving the system without interruption, r2 monetary units revenue obtained for each priority generated customer getting service and leaving the system, r3 monetary units revenue obtained for each ordinary customer getting service and leaving the system with a customer induced interruption, c1 monetary units holding cost for each unit of time that an ordinary customer has to wait in the system, c2 monetary

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units holding cost for each unit of time that a priority generated customer has to wait in the system, c3 monetary units holding cost for each unit of time that an interruption completed customer has to wait in the buffer B2 , c4 monetary units holding cost of time that an interrupted customer has to wait in the buffer B1 , c5 monetary units cost for each customer lost due to no vacant space in buffer B1 at the time of interruption occurs, c6 monetary units cost for each priority generated customer lost due to no space in waiting space A1 at the time of priority generation occurs, c7 monetary units cost for each customer lost after retrial and cf ixed a miscellaneous fixed cost. Our goal is to find an optimum value for K (denoted by K; with all other parameters fixed) that maximizes the expected total profit, ET P.

7 Numerical Illustration Consider the three phase-type distribution with representations (α, T ), (β, S), (ν, V ) (For convenience we say (α, T ) as Type-I service, (β, S) as Type-II service and (ν, V ) as Type-III service) are defined by Type-I service; α = [0.3 0.7];

−15 3 T = ; 3 −15

12 ; T0 = 12

Type-II service; β = [0.4 0.6];

−8 4 S= ; 4 −8

4 S0 = ; 4

Type-III service; ν = [0.5 0.5];

−5.5 2.5 V = ; 2.5 −5.5

3 V0 = 3

In order to demonstrate the effect of correlation, we introduce four MAP arrival of customers. Denote the four MAP as: a MAPp , a MAPn , b MAPp , b MAPn and



−4.05 1.55 2.05 0.45 • a MAPp is defined by D0 = , D1 = 3.5 −5.5 1 1 = +0.00028752 Average arrival rate, λ = 2.3462, Correlation coefficient, c cor



−5.5 3.5 1 1 , D1 = • a MAPn is defined by D0 = 1 −3.5 1 1.5 Average arrival rate, λ = 2.3462, Correlation coefficient, ccor = −0.00028532

−5.15 2.10 2.60 0.45 , D1 = • b MAPp is defined by D0 = 4.05 −6.60 1.00 1.55 Average arrival rate, λ = 2.8822, Correlation ccor = +0.00040550

coefficient, −6.60 4.05 1.55 1.00 • b MAPn is defined by D0 = , D1 = 1.55 −4.60 1.00 2.05 Average arrival rate, λ = 2.8822, Correlation coefficient, ccor = −0.000411974

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer. . .

367

7.1 Optimum Buffer Size K We fix η = 10.6, θ = 5.7, γ = 25.0, σ = 0.5, δ = 0.8, m = 2 = n, r1 = 900, r2 = 900, r3 = 900, c1 = 2, c2 = 1, c3 = 50, c4 = 50, c5 = 10, c6 = 5, c7 = 1, cf = 10 and different service combinations shown in Table 1. (For example, System A means Type-I service given to ordinary customers, Type-II service given to priority generated customers, Type-III service given to interruption completed customers and System F means all customers are given Type-III service.) Then compute ET P for different K and selected arrival process a MAPp , a MAPn , b MAPp , b MAPn . Results are plotted in Figs. 2 and 3. From Figs. 2 and 3 we can see that K = 2 is the optimal buffer size for all the above considered cases.

7.2 Effect of θ in Einl for Different MAP and Buffer Sizes K We fix η = 0.6, γ = 25.0, σ = 0.5, δ = 0.8, m = 2 = n and compute Einl for different interruption rates θ and for buffer sizes K. Results are plotted graphically in Fig. 4. From Fig. 4 when interruption rate θ increases, Einl increases initially and then decreases. For the above set of fixed parameters here Einl is maximum when K = 2 and all other considered buffer sizes K = 1, 3, 4, 5, Einl is less than that at K = 2 for all arrival processes assumed in previous sections.

7.3 Effect of η in Einl for Different MAP and Buffer Sizes K We fix θ = 5.7, γ = 25.0, σ = 0.5, δ = 0.8, m = 2 = n and compute Einl for different rates η and for buffer sizes K. Results are plotted graphically in Fig. 4.

Table 1 Some selected phase-type service combinations

Service combination name System A System B System C System D System E System F

Ordinary customer Type I Type II Type II Type I Type II Type III

Priority generated customer Type II Type I Type III Type I Type II Type III

Interruption completed customer Type III Type III Type I Type I Type II Type III

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a

MAP p

50

50

40

40

30 20

MAP n

30 20

System A System B System C

10 0

a

60

ETP

ETP

60

1

2

3

4

5

6

System A System B System C

10 0

7

1

2

3

4

K MAP p

b

60

50

50

40

40

ETP

ETP

7

70

60

30

30

20

20 System A System B System C

10 0

6

MAP n

b

70

5

K

1

2

3

4

5

6

System A System B System C

10

7

K

0 1

2

3

4

5

6

7

K

Fig. 2 Graph of ET P vs. K for System A, B, C

From Fig. 4 when η increases, Einl increases initially and then decreases. For the above set of fixed parameters here Einl is maximum when K = 2 and all other considered buffer sizes K = 1, 3, 4, 6, 9, Einl is less than that at K = 2 for all arrival processes assumed in previous sections.

8 Conclusion A single-server queueing system with self-generation of priorities, customer induced interruption and retrial of customers is analyzed in this paper. Arrival of customers is according to Markovian arrival process and service times are different phase-type distributions. The interruption we discussed here is customer induced interruption. Performance measures required for an appropriate system designing were computed and numerically analyzed.

The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer. . . MAP p

a

60

60

50

50

40

40

ETP

ETP

a

30

30

10

10 System D System E System F

0 1

2

3

4

5

System D System E System F

0

6

1

K b

70

2

3

4

5

6

7

K

MAP p

b

70

60

60

50

50

40

40

ETP

ETP

MAP n

20

20

30

MAP n

30

20

20 System D System E System F

10 0

369

1

2

3

4

5

System D System E System F

10

6

0

7

1

2

3

4

K

5

6

7

K

Fig. 3 Graph of ET P vs. K for System D, E, F 0.09

0.08 0.07

K=1 K=2 K=3 K=4 K=6 K=9

0.08

0.06 0.07

E inl

E inl

0.05 0.04 K=1 K=2 K=3 K=4 K=5 K=6 K=7

0.03 0.02

0.05 0.04

0.01 0

0

5

q

10

0.06

15

0.03

0

2

4

6

8

10

h

12

14

16

18

20

Fig. 4 Graph of Einl vs. θ, η for different buffer size K

Acknowledgments The authors are thankful to Professor A. Krishnamoorthy for constructive suggestion and advice in the entire work of this paper. Support from the University Grants Commission (sanction no. FIP/12th Plan/KLKE029 TF-36) is gratefully acknowledged.

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References 1. Avi-Itzhak, B., Naor, P.: Some queuing problems with the service station subject to breakdown. Oper. Res. 11(3), 303–320 (1963) 2. Brahimi, M., Worthington, D.J.: Queueing models for out-patient appointment systems—a case study. J. Oper. Res. Soc. 42(9), 733–746 (1991) 3. Falin, G.: A survey of retrial queues. Queueing Systems 7(2), 127–167 (1990) 4. Falin, G., Templeton, J.G.C.: Retrial Queues, vol. 75. CRC Press, 1997 5. Gaver Jr., D.P.: A waiting line with interrupted service, including priorities. J. R. Stat. Soc. B (Methodol.), 73–90 (1962) 6. Gomez-Corral, A., Krishnamoorthy, A., Narayanan, V.C.: The impact of self-generation of priorities on multi-server queues with finite capacity. Stoch. Models 21(2–3), 427–447 (2005) 7. Jacob, V., Chakravarthy, S.R., Krishnamoorthy, A.: On a customer-induced interruption in a service system. Stoch. Anal. Appl. 30(6), 949–962 (2012) 8. Jaiswal, N.K.: Preemptive resume priority queue. Oper. Res. 9(5), 732–742 (1961) 9. Jaiswal, N.K.: Priority Queues. Elsevier (1968) 10. Krishnamoorthy, A., Babu, S., Narayanan, V.C.: MAP/(PH/PH)/c queue with self-generation of priorities and non-preemptive service. Stoch. Anal. Appl. 26(6), 1250–1266 (2008) 11. Krishnamoorthy, A., Babu, S., Narayanan, V.C.: The MAP/(PH/PH)/1 queue with selfgeneration of priorities and non-preemptive service. Eur. J. Oper. Res. 195(1), 174–185 (2009) 12. Krishnamoorthy, A., Pramod, P.K., Deepak, T.G.: On a queue with interruptions and repeat or resumption of service. Nonlinear Anal. Theory Methods Appl. 71(12), e1673–e1683 (2009) 13. Krishnamoorthy, A., Pramod, P.K., Chakravarthy, S.R.: Queues with interruptions: a survey. Top 22(1), 290–320 (2014) 14. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Corporation (1981) 15. Neuts, M.F., Lucantoni, D.M.: A Markovian queue with n servers subject to breakdowns and repairs. Manag. Sci. 25(9), 849–861 (1979) 16. Neuts, M.F., Rao, B.M.: Numerical investigation of a multiserver retrial model. Queueing Systems 7(2), 169–189 (1990) 17. Shortle, J.F., Thompson, J.M., Gross, D., Harris, C.M.: Fundamentals of Queueing Theory. Wiley (2018) 18. Takagi, H.: Queueing Analysis: Vacations and Priority System, vol. i. North-Holland, Amsterdam (1991) 19. Takagi, H.: Queueing Analysis: Vacations and Priority System, vol. iii. North-Holland, Amsterdam (1993) 20. Taylor, I.D.S., Templeton, J.G.C.: Waiting time in a multi-server cutoff-priority queue, and its application to an urban ambulance service. Oper. Res. 28(5), 1168–1188 (1980) 21. Tweedie, R.L.: Sufficient conditions for regularity, recurrence and ergodicity of Markov processes. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 78, pp. 125–136. Cambridge University Press, Cambridge (1975) 22. Wang, J.: An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47(10–11), 1713–1723 (2004) 23. White, H., Christie, L.S.: Queuing with preemptive priorities or with breakdown. Oper. Res. 6(1), 79–95 (1958) 24. Yang, T., Templeton, J.G.C.: A survey on retrial queues. Queueing Systems 2(3), 201–233 (1987)

Valuation of Reverse Mortgage D. Kannan and Lina Ma

Abstract This article provides an analytic valuation formula for reverse mortgage. We achieve this by utilizing the principle of balance between the expected gain and expected payment. The underlying model employs a jump-diffusion process to represent the dynamics of the house price, the Vasicek model to drive the instantaneous interest rate, and a bivariate distribution function to describe the longevity risk. We obtain, in particular, the formulas for the lump sum payment, joint annuity, increasing (decreasing) annuity, level annuity of reverse mortgage, and the valuation equation that the variable payment annuities satisfy. We then discuss the monotonicity of the lump sum, annuity, and annuity payment factors with respect to the parameters associated with the home price and the interest rate model. Finally, we analyze the sensitivity of the joint annuity with respect to the parameters associated with the home price, interest rate, and lifetime model. The numerical analysis supports our theoretical results. Keywords Reverse mortgage · Valuation · Joint annuity · Jump-diffusion · Vasicek model · Lifetime model

1 Introduction Reverse mortgage is an attractive financial lending product offered to any senior citizen who owns a house. It is categorized normally into two categories, namely collateral reverse mortgage and ownership conversion reserve mortgage (Ohgaki [7]). The collateral reverse mortgage is redeemable, while the ownership conversion reverse mortgage is not. Home equity conversion mortgage system is a typical D. Kannan () Department of Mathematics, University of Georgia, Athens, GA, USA e-mail: [email protected] L. Ma School of Finance, Capital University of Economics and Business, Beijing, China © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_22

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collateral reverse mortgage in the USA. In a collateral reverse mortgage, the elderly householder borrows annuity like periodical installment mortgage on his/her residential house. With the collateral reverse mortgage, the borrower is able to redeem the reverse mortgage by repaying the loan principal and accumulated interests through property sale at any time from the mortgage’s effective date to due date. Rente Viager is a typical ownership conversion reverse mortgage in France. In the ownership conversion reverse mortgage, the borrower enters into a contract with a lending institution to obtain an annuity until his/her death, and at death the pledged property ownership is transferred to the lender. Since the introduction of reverse mortgage, many scholars and practitioners have engaged in research, mainly on the basic principles, operation modes, feasibility, effectiveness, policies, laws, risks, and valuation. These aspects of reverse mortgage are well studied compared to the valuation problem. Hence, we focus our attention to the valuation problem. The valuation of reverse mortgage mainly includes three aspects: (1) determining a lump sum and annuity payments that the lender can pay before signing the reverse mortgage contract, (2) pricing the redemption right of the collateral reverse mortgage before signing the reverse mortgage contract, and (3) finding the value of reverse mortgage at any time t after signing the contract. The main idea behind the first aspect is to employ the principle of expected balance between gain and payment under the assumption of perfect competition market, which makes the discounted present value of payment of the lender be equal to a certain proportion of discounted present value of the mortgaged property (see, for example, Mitchell and Piggott [6]). The main valuation idea of the last two aspects is to apply the option pricing concept, which regards the mortgaged property as the underlying asset and the loan principal and accumulated interests as the strike price of underlying asset. When the contract expires, the lender or its successor determines whether to execute the option (i.e., redeem the pledged property) according to the difference between the price of pledged property and the loan principle and accumulated interests (see, for example, Tsay et al. [8]). The main risks involved with reverse mortgage include property-value risk, interest rate risk, and longevity risk. In general, the risk of house price is modeled in two ways. One is to assume directly that the dynamics of house price is driven by a forward stochastic differential equation, as in Tsay et al. [8]. The other is to fit the time series model based on the historical data of the house price, as discussed by Li et al. [3]. The literature on classical interest rate model is vast and we follow the Vasicek [9] model. While there are usually several ways to describe the longevity risk, we follow the Gompertz [2]. As pointed out earlier, this work focuses on the valuation problem of reverse mortgage. We provide analytic valuation formulas for the lump sum and three different annuity aspects of reverse mortgage. We also derive the valuation equation that the variable payment annuities satisfy. For our analysis, we appeal to the principle of balance between expected gain and expected payment. Taking into account of the influence of the parameters associated with the home price and interest rate models, we discuss the monotonicity of the lump sum, annuity, and annuity payment factors. Finally, we analyze the sensitivity of the joint annuity with

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respect to the parameters associated with the home price, interest rate, and lifetime model. Our numerical results show that the average return of home price exerts a dominating influence on the joint annuity, followed by the mean reversion level of interest rate. We observe here that both of them have more impact on the annuities of young applicants than those of old applicants. It is interesting to note that the initial age of male and that of female produce asymmetrical effect on the joint annuity. Remarkably, the dependence of joint lifetime significantly affects the joint annuity value. Organization of the article: Sect. 2 presents the models of risk factors. In Sect. 3, we first design the reverse mortgage predicated on the ownership conversion with fixed yearly payment until death, and then derive the valuation model for the lump sum and annuity payments under the principle of balance between expected gain and expected payment. Section 4 analyzes the monotonicity of the lump sum, annuity payments, and annuity payment factors with respect to the parameters involved in housing price and interest rate models. We provide in Sect. 5 some numerical results to examine how the housing price risk, interest rate risk, and longevity risk impact the lump sum, annuity payment, and the annuity payment factors. The main steps of the proofs of propositions are provided in the Appendix. This review article is based on the references Ma et al. [5] and Ma and Kannan [4].

2 Risk Factors We follow a stochastic model to determine the lump sum and annuity of the reverse mortgage without redemption right applied by a joint lives. All our random elements are defined on a complete filtered probability space (, F , P, {Ft }t ≥0), where (, F , P) is a complete probability space and {Ft }t ≥0 is a right continuous increasing family of sub σ -algebras of F with all the null events in F0 . The risk factors that the reverse mortgage without redemption right involves (1) we employ the jump-diffusion model to mimic the dynamics of home price, (2) the Vasicek model to drive the instantaneous interest rate, and (3) a bivariate distribution function to describe the dependent longevity risk of a joint-life (i.e. a couple).

2.1 House Price We assume that the house price h(t) (t ≥ 0) follows the exponential Lévy process ⎡ h(t) = h(0) exp ⎣

 0

h(0) = h0 .

t

$ μh (s)ds −

⎤ % N(t ) 1 2 σ + λh kh t + σh Wh (t) + Ji ⎦ , 2 h i=1

(2.1)

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Here: (1) the P-standard Brownian motion process {Wh (t), t ≥ 0} captures the unanticipated instantaneous change of house price. (2) As the Wiener process {Wh (t)} will not capture the abnormal shocks caused by sudden rise or drop in the house price, we model the rise/drop in the house price by independent Gaussian random jumps {Ji , i ≥ 0} with mean μJ and variance σJ2 , and we count the number of price jumps during the time interval (0, t] using a Poisson process {N(t), t ≥ 0} with intensity λh . (3) We assume that the processes {Wh (t), t ≥ 0}, {N(t), t ≥ 0}, and {Ji , i ≥ 0} are independent. (4) The drift coefficient μh (t) denotes the average rate of return. (5) The diffusion coefficient σh (> 0) represents the volatility of the house price. (6) The parameter kh is given by kh = exp (μJ + 12 σJ2 ) − 1.

2.2 Interest Rate We assume that the instantaneous short-rate dynamics follows the Vasicek model [9]. More precisely, the interest rate process {r(t), t ≥ 0} is governed by the following stochastic differential equation dr(t) = αr (μr − r(t))dt + σr dWr (t), r(0) = r0 ,

(2.2)

where {Wr (t), t ≥ 0} is a P-standard Brownian motion with Cov(dWr (t), dWh (t)) = ρhr dt. We assume that r0 , αr , μr , σr are positive constants. From Itô’s formula applied to eαr u r(u) we obtain r(t) = e−αr t r(0) + μr (1 − e−αr t ) + σr



t

e−αr (t −u)dWr (u), t ≥ 0.

(2.3)

0

The discount factor at time t, denoted by d(t), is defined as $  t % d(t) := exp − r(s)ds .

(2.4)

0

Some trivial computation yields $ E [d(t)] = exp +

σr2 4αr3

% σr2 1 − μr t + (μr − r0 )(1 − e−αr t ) 2 2αr αr

5 1 − (2 − e−αr t )2 .

(2.5)

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2.3 Joint Lives In our analysis, the initial time t = 0 represents the time at which the reverse mortgage is signed. Let x0 and y0 represent the age of the husband and wife at time 0, respectively. Let X and Y be the age-at-death of the husband and wife, respectively. By F (x, y) := P (X ≤ x, Y ≤ y) we denote the joint distribution function of random vector (X, Y ), with F1 (x) and F2 (y) denoting the respective marginal distributions. The bivariate distribution function can be specified by a copula function and two marginal distributions; that is, F (x, y) = C(F1 (x), F2 (y)), where C is a real-valued copula function that provides a link between the marginal distributions and the corresponding bivariate distribution. The copula function is given by (see Frees et al. [1]), C(u, v) =



(eαu − 1)(eαv − 1) 1 ln 1 + , α eα − 1

with the two marginal distributions following the Gompertz distribution:

m  x  − 1 F1 (x) = 1 − exp e σ1 1 − e σ1 , m 

y  − σ2 σ F2 (y) = 1 − exp e 2 1 − e 2 . The corresponding density functions for X and Y are given by m 

x  1 1 x−m − σ1 σ1 σ1 1 f1 (x) = e 1−e exp e , σ1 m 

y  2 1 y−m − σ2 σ2 σ2 2 1−e exp e f2 (y) = e . σ2 We consider a bivariate residual lifetime random vector (X − x0 , Y − y0 ), where X − x0 and Y − y0 represents the time-until-death of the husband and that of the wife, respectively. The joint distribution function for (X − x0 , Y − y0 ) is Fc (x, y) =

1 [F (x0 + x, y0 + y) − F (x0 + x, y0 ) − F (x0 , y0 + y) p0 +F (x0 , y0 )] ,

(2.6)

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where p0 = 1 − F1 (x0 ) − F2 (y0 ) + F (x0 , y0 ). Let T1 = min{X − x0 , Y − y0 },

T2 = max{X − x0 , Y − y0 },

(2.7)

then the density function fT (t) for T2 is fT (t) = Abbreviating

1 p0



dF (x0 + t, y0 + t) dF (x0 + t, y0 ) dF (x0 , y0 + t) − − . dt dt dt

dF (x0 +t,y0 +t ) dt

(2.8) as

dF dt ,

we have

dF f1 (x0 + t)eαF1 (x0 +t) (eαF2 (y0 +t) − 1) + f2 (y0 + t)eαF2 (y0 +t) (eαF1 (x0 +t) − 1) , = dt eα − 1 + (eαF1 (x0 +t) − 1)(eαF2 (y0 +t) − 1)

(eαF2 (y0 ) − 1)eαF1(x0 +t ) f1 (x0 + t) dF (x0 + t, y0 ) = α , dt e − 1 + (eαF1 (x0 +t ) − 1)(eαF2(y0 ) − 1) dF (x0 , y0 + t) (eαF1 (x0 ) − 1)eαF2(y0 +t ) f2 (y0 + t) = α . dt e − 1 + (eαF1 (x0 ) − 1)(eαF2(y0 +t ) − 1)

3 Valuation of Reverse Mortgage We start our valuation process by first introducing a reverse mortgage with the joint and γ annuities applied by the dependent joint lives. Then, the valuation models are built based on the principle of balance between expected gain and expected payment. Under the two-dimensional Gauss distribution and independence assumptions, we obtain the analytic valuation formulas for the lump sum, joint and γ annuities, increasing (decreasing) annuities, and level annuities of reverse mortgage without redemption right, and derive the valuation equation that the variable payment annuities satisfy.

3.1 Case of Joint and γ Annuities We shall now design a reverse mortgage with the joint and γ annuities. The joint life, i.e., a couple contract offers a yearly annuity payment until the last annuitant dies. The product that we design has the following basic features: (1) The lender starts the payments of annuity to the joint annuitants at the beginning of signing the

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contract, and the annuity payment is terminated upon the death of the last annuitant. While both annuitants are alive, the lender pays annuity amount A at the beginning of each year, and γ A while only one annuitant is alive. (2) When the last applicant dies, the lender will take over the annuitant’s pledged property, sell it in the market, and keep all of the proceeds from the sale of the property. The essence of reverse mortgage with the joint and γ annuities is to exchange the profit from selling the mortgaged house with the joint-life’s annuities until the last annuitants’ death. Upon the passing of the last annuitant, the lender will take over the homeowner’s mortgaged property and sell it. The cash acquired from the sale is used to repay loan (including annuities and accumulated interests) that the joint annuitants owe to the lender. Reverse mortgage possesses the non-recourse clauses, which is, that the lender may not reclaim the loan against the annuitants’ other assets or cash income. So, the lender will suffer a loss when the cash of selling the mortgaged property is less than those annuities and accumulated interests, otherwise the lender will make a profit. Next, we will provide a simple example to see how the reverse mortgage deals with the joint and γ annuities. Assume that the age-at-death of the husband and the wife be X = 65.7 and Y = 67.9 years, respectively. Let the initial age be, respectively, x0 = 65 and y0 = 64 years. Then the loan tenure is T2 = max{X − x0 , Y − y0 } = 3.9 years. This implies that the couple claims once cash payment A at the beginning of the first year of the contract. The wife as the last annuitant claims three times cash payments γ A at the beginning of the second, third, and fourth year of the contract, respectively. When the wife, the last survivor, dies at age of 67.9 years, the lender will take over the pledged house and sell it in the market. Most of the time, it is impossible to sell the pledged house as soon as the lender takes over it. Thus the time of selling out the pledged house usually lags behind that of taking over the pledged house for a time. In the following valuation models, we will take this delay time into consideration.

3.2 Valuation: Joint and γ Annuities In the valuation of the reverse mortgage we assume that the market is perfectly competitive and, price the reverse mortgage with the joint and γ annuity by the principle of balance between expected gain and expected payment. The terminology principle of balanced expected gain and payment means that the expected discounted present value of future sale of the pledged property is the same as the expected discounted present value of annuities that the lender pays during the whole loan period. At time T2 , the lender takes over the annuitants’ mortgaged property, and sells it at time T2 + t0 , where, recall that, t0 ≥ 0 is the delay time between the lender taking over the pledged property and the sale of that property. We assume that t0 is

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deterministic. Then the expectation of discounted present value of the sale price of the property (i.e., the lender’s expected gain) is E [h(T2 + t0 )d(T2 + t0 )] ,

(3.1)

where h(t) is the value of the mortgaged property at time t, and d(t) is the discount factor at time t given by Eq. (2.4). The expectation of discounted present value of the joint and γ annuities during the whole loan period (i.e., the lender’s expected payment) is ⎡ E⎣

T 1 

T 2 

Ad(k) +

⎤ γ Ad(k)⎦ ,

(3.2)

k=T1 +1

k=0

where x is the floor function (the largest integer not greater than x). Then, the principle of balance between expected gain and expected payment yields ⎡ E [h(T2 + t0 )d(T2 + t0 )] = E ⎣

T 1 

Ad(k) +

T 2 

⎤ γ Ad(k)⎦ .

(3.3)

k=T1 +1

k=0

In general, the explicit formula of annuity payment is difficult to obtain from Eq. (3.3). However, we can obtain the analytic annuity formula under the twodimensional Gaussian distribution and independence assumption. The following Proposition 1 presents the analytic formula for the expected discounted present value of the mortgaged property at any time t. Proposition 1 Assume that (a) the dynamics of home price follows the exponential Lévy process given by Eq. (2.1), and (b) the instantaneous short interest rate is governed by Eq. (2.2). Define Y (t) :=

 t  σr e−αr s 0

s

eαr u dWr (u) ds.

(3.4)

0

Assume also that (c) the joint distribution of (Wh (t), Y (t)) follows the twodimensional Gaussian distribution, and that (d) σh Wh (t) − Y (t) is independent  ) of N(t i=1 Ji . Then, the expectation of discounted present value of the mortgaged property at time t is given by E [h(t)d(t)] = G(t)D(t),

(3.5)

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where 

$ %5 1 1 −αr t 1 μ(s)ds − σh σr ρhr − G(t) = h0 exp t+ e , αr αr αr 0 % $ 2   1 σr − μr t + (μr − r0 ) 1 − e−αr t D(t) = exp 2 2αr αr

5 2 σr −αr t 2 . ) + 3 1 − (2 − e 4αr t

(3.6)

(3.7)

Proof A gist of the proof of Proposition 1 is given in the Appendix. The analytic valuation formula for the expected lump sum that the householder can borrow in average at time 0, and the analytic valuation formula for the joint annuity are given in Proposition 2. Proposition 2 Assume that (a) h(t)d(t), (t ≥ 0), is independent of T2 , (b) r(t) is independent of (T1 , T2 ), and (c) the pledged property is sold at time T2 +t0 . Then: - is (1) The expectation of the lump sum, denoted by G, -= G



+∞

G(x + t0 )D(x + t0 )fT (x)dx,

(3.8)

0

where G(x + t0 ), D(x + t0 ) and fT (x) are given by Eqs. (3.6), (3.7) and (2.8), respectively. (2) For the joint and γ annuity, the fixed amount A of annuity is given by A=

G , -1 F

(3.9)

where -1 = F

+∞ 

D(i) [1 − (1 − γ )Fc (i, +∞) − (1 − γ )Fc (+∞, i)

i=0

+(1 − 2γ )Fc (i, i)] , and Fc (x, y) is given by Eq. (2.6). Proof A concise proof is provided in the Appendix.

(3.10)

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3.3 Valuation: Variable Payment Annuities The phrase reverse mortgage with variable payment annuity tells us that the lender starts the payments of annuity to the joint lives at the beginning of signing the contract until the death of the last survivor, and that the annuity payment amount at year k, (k ≥ 1),  is Ak . Here, if the last annuitant passes away at k-th year, then a total amount ki=1 Ai of annuity payments has been paid. The increasing or decreasing annuity is a special case of the variable payment annuity with Ak ≡ A0 +d ·k (k = 1, 2, . . .). At the beginning of k-th period, the annuity payment is A0 + d · k, as long as at least one of the annuitants is alive. In the following, we call A0 the basic annuity, and d the annuity increment. The level annuity is a special case of the joint and γ annuity with γ = 1, and is also a special case of the variable payment annuity with Ak being the same constant for all k ≥ 1. Proposition 3 Assume that (a) h(t)d(t) (t ≥ 0) is independent of T2 , (b) r(t) is also independent of T2 , and (c) The pledged property is sold at time T2 + t0 . Then: (1) For the variable payment annuity, the annuity payments Ak (k = 1, 2, . . .) satisfy the following valuation equation 

+∞

G(x + t0 )D(x + t0 )fT (x)dx =

0

+∞ 

Ak+1 D(k)P (T2 ≥ k),

(3.11)

k=0

where P (T2 ≥ k) = 1 − Fc (k, k),

(3.12)

and D(k) is given by Eq. (3.7). (2) For the increasing (decreasing) annuity, A0 and d are determined by the simultaneous equations A0 =

-−d ·F -3 G , -2 F

d=

-2 - − A0 · F G , -3 F

(3.13)

kD(k)P (T2 ≥ k),

(3.14)

where -2 = F

+∞  k=0

D(k)P (T2 ≥ k),

-3 = F

+∞  k=0

- and P (T2 ≥ k) are, respectively, defined by Eqs. (3.7), (3.8), and and D(k), G (3.12).

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(3) For the level annuity, the fixed annuity amount A∗ is given by A∗ =

G , -2 F

(3.15)

-2 is given by Eq. (3.14). where F Proof We omit the proof of Proposition 3 as it parallels that of Proposition 2.

4 Effect of Parameters on the Annuity This and the following section show how the parameters associated with the house price, interest rate, and the delay duration in selling the pledged house would affect the various annuity payments.

4.1 Monotonicity Subject to the Parameters of House Price For our further analysis, we assume that the average rate of return μh (t) ≡ μh of the house price is deterministic. The next Proposition analyzes the monotonicity of the annuity payment, lump sum, and annuity payment factors w.r.t the parameters related with the house price model, including the average return rate μh , the volatility σh , the initial house price h0 , the correlation coefficient between the Brownian motion driving the house price and those driving the interest rate ρhr , and the delay time in selling the pledged house t0 . For the descriptions of the parameters μh , σh , ρhr , and h0 connected to the house price, we refer to Sect. 2. Proposition 4 Predicated on the parameters μh , σh , ρhr , and h0 of the house price, we have the following properties: -i (i = 1, 2, 3) are indepen4.1. Parameter μh : (a) The annuity payment factors F dent of μh ; (b) The lump sum G is an increasing function of μh ; and (c) The A, A0 , d and A∗ are all increasing functions of μh . -i (i = 1, 2, 3) are indepen4.2. Parameter σh : (a) The annuity payment factors F dent of the volatility σh of the house price. (b) If ρhr > 0, σr > 0, and αr = 0, - is a decreasing function of σh . (c) If ρhr < 0, σr > 0, then the lump sum G - is an increasing function of σh . (d) If and αr = 0, then the lump sum G ρhr > 0, σr > 0, and αr = 0, then A, A0 , d and A∗ all are decreasing functions of σh . (e) If ρhr < 0, σr > 0, and αr = 0, then A, A0 , d, and A∗ all are increasing functions of σh . -i , i = 1, 2, 3, are 4.3. Parameter ρhr : (a) The annuity payment factors F independent of ρhr . (b) If σh > 0, σr > 0, and αr = 0, then the lump sum G

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D. Kannan and L. Ma

is a decreasing function of ρhr . (c) If σh > 0, σr > 0, and αr = 0, then A, A0 , d, and A∗ are all decreasing functions of ρhr . -i , i = 1, 2, 3, are indepen4.4. Parameter h0 : (a) The annuity payment factors F - is an increasing function dent of the initial house price h0 . (b) The lump sum G ∗ of h0 . (c) The A, A0 , d, and A are all increasing functions of h0 . Proof The main steps of the proof are moved to the Appendix. Proposition 5 With respect to the delay time t0 , between acquiring the house and - and F -i (i = 1, 2, 3) have the selling the house, the factors A, A0 , d, A∗ , G, following properties: -i (i = 1, 2, 3) do not depend on t0 . (a) The annuity payment factors F (b) Now set $ % σh σr ρhr 2 2σr2 (r0 − μh ) $ := μr − r0 + + , αr αr2

(4.1)

αr2 σr2 σh σr ρhr √ z1 := − 2 μr − r0 − 2 + + $ , σr αr αr

(4.2)

αr2 σr2 σh σr ρhr √ − r − + − $ . μ r 0 σr2 αr2 αr

(4.3)

and z2 := −

(b-1) If any one of the following conditions is satisfied $ ≤ 0, $ ≥ 0, αr > 0, z1 ≥ 1, $ ≥ 0, αr > 0, z2 ≤ 0, - and the quantities A, A0 , d, and A∗ are all increasing then the lump sum G, functions of t0 . (b-2) If $ ≥ 0, αr > 0, z1 ≤ 0, z2 ≥ 1,

(4.4)

- and the quantities A, A0 , d, and A∗ are all holds, then the lump sum G decreasing functions of t0 . Proof A summary of proof is delegated to the Appendix.

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383

4.2 Monotonicity Subject to Parameters of Interest Rate The following Proposition 6 analyzes how the annuity payment, lump sum payment, and the annuity payment factors are affected by the parameters r0 , μr ,, σr involved in the interest rate mode. Proposition 6 With respect to the parameters of the interest rate model, the factors - F -i (i = 1, 2, 3) have the following properties: A, A0 , d, A∗ , G, - F -i , (i = 1, 2, 3), are decreasing 6.1. Initial Interest Rate r0 : If αr = 0, then G, functions of r0 . - F -i , (i = 1, 2, 3), are decreasing 6.2. Mean Reversion Level μr : If αr > 0, then G, - F -i , (i = 1, 2, 3), functions of μr . If the opposite case αr < 0 holds, then G, are increasing functions of μr . 6.3. Volatility σr : - is a decreasing function of σr in (a) If αr > 0, σh > 0 and ρhr ≥ 0, then G - is the interval σr ∈ (0, σh ρhr αr ]. If αr > 0, σh > 0, and ρhr ≤ 0, then G an increasing functions of σr . -i , (i = 1, 2, 3), are increasing functions (b) In case of αr = 0, σr > 0, then F of σr . (c) If αr > 0, σh > 0, and ρhr ≥ 0, then A, A0 , d, and A∗ are decreasing functions of σr in the interval σr ∈ (0, σh ρhr αr ]. Proof See Appendix.

5 Numerical Experiment In this section, we illustrate the impact of risks due to the house price, the interest rate, and the longevity on the annuity payment on the valuation of reverse mortgage. The following Table 1 provides, as the standard case, the parametric values involved in the models of house price, interest rate, and lifetime. The values of the parameters (m1 , m2 , σ1 , σ2 , α) come from the bivariate distribution function of the joint lifetimes (see Frees et al. [1]). For illustration purpose, we assume that the initial age of the male is 2 years greater than that of the female, that is, x0 = y0 + 2. Table 1 Parameters of the standard case

P ara V alue P ara V alue

μh 0.04 αr 0.5

σh 0.08 y0 x0 − 2

ρhr 0.3 m1 85.82

h0 $100 m2 89.40

t0 0 σ1 9.98

r0 0.04 σ2 8.12

μr 0.06 α −3.367

σr 0.01 γ 0.5

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5.1 Effect of House Price on Annuity Values The effect of the parameters of the house price and initial age on the joint annuity and γ annuity, keeping other parametric values fixed. Table 2 supports the following analysis. (a) The average return μh of house price: (1) When the initial age is fixed, the annuity increases significantly with the increase of the average return rate of house price μh ; this agrees with the theory established by Proposition 4. This is reasonable because the higher average return rate of house price implies the higher average gains that can be obtained by the lender when selling the mortgaged property in future. With the fair valuation principle, the lender is bound to pay enhanced annuities to the annuitants. (2) Compared to the applicant with higher initial age, the mean return has a stronger impact on the annuity of the annuitant with lower initial age. As the initial age increases, the annuity under different average return rates is stabilizing. While the average return rate of house price μh remains unchanged, the annuity increases with the increase of male initial age x0 . (b) The volatility σh in house price: (1) When the initial age is fixed, the annuity decreases with the increase of σh , (as supported by Proposition 4). This also is reasonable. After all, higher the volatility of house price, greater the market risk. In order to avoid the higher market risk, the lender will have to reduce the amount of annuity. (2) As the volatility of house price σh remains unchanged, the annuity increases with the increase of the initial age; that is, the older applicant will get better annuity. (c) The correlation coefficient ρhr between Brown motions: Here, the parameter ρhr denotes the correlation coefficient between the Brownian motion driving house price and that driving interest rate. (1) As the initial age is fixed, the annuity decreases with the increase of the correlation coefficient ρhr , as proved in Proposition 4. When the Brownian motion driving the house price and the Brownian motion driving the interest rate are completely negatively correlated (ρhr = −1), the annuity reaches the maximum. If they are completely positively correlated (ρhr = 1), the annuity reaches the minimum. In addition, the annuity values under the different correlation coefficients are very close, which implies that the influence of the correlation coefficient ρhr on the annuity is very weak. (2) Compared with the older applicants, the annuity of the younger applicant is more susceptible to the correlation coefficient. When the correlation coefficient ρhr is fixed, the annuity increases with the increase of the initial age, that is, the older applicant will receive a larger annuity. (d) The initial house price h0 : (1) As the initial age is fixed, the annuity increases obviously with the increase of initial house price. The higher initial house price implies that the lender reaps greater benefits while selling the mortgaged property in the future. With the fair valuation, the lender will pay better annuity to the borrower. As is clear from Table 2 that these annuity values get closer to each other in the case of lower initial age, and while the initial age increases these

x0 μh = 0.01 μh = 0.025 μh = 0.04 μh = 0.055 μh = 0.07 σh = 0.001 σh = 0.1 σh = 0.2 σh = 0.3 σh = 0.4 ρhr = −1 ρhr = −0.5 ρhr = 0 ρhr = 0.5 ρhr = 1 h0 = 100 h0 = 200 h0 = 300 h0 = 400 h0 = 500

50 1.072 1.833 3.193 5.658 10.191 3.247 3.179 3.112 3.046 2.982 3.438 3.342 3.248 3.156 3.068 3.193 6.385 9.578 12.771 15.963

55 1.429 2.281 3.708 6.124 10.271 3.762 3.694 3.626 3.559 3.494 3.955 3.858 3.763 3.671 3.582 3.708 7.415 11.123 14.830 18.538

60 1.913 2.865 4.356 6.722 10.52 4.411 4.342 4.274 4.207 4.141 4.603 4.506 4.412 4.319 4.229 4.356 8.712 13.068 17.424 21.780

Table 2 Effect of house price on annuity values

65 2.578 3.632 5.187 7.504 10.989 5.242 5.173 5.105 5.038 4.971 5.433 5.337 5.243 5.151 5.060 5.187 10.375 15.562 20.749 25.936

70 3.497 4.659 6.276 8.545 11.755 6.331 6.262 6.194 6.127 6.061 6.519 6.425 6.331 6.240 6.150 6.276 12.552 18.828 25.105 31.381

75 4.793 6.067 7.745 9.969 12.939 7.798 7.731 7.664 7.597 7.531 7.984 7.891 7.799 7.709 7.619 7.745 15.489 23.234 30.979 38.723

80 6.684 8.072 9.809 11.993 14.753 9.861 9.796 9.73 9.665 9.6 10.041 9.951 9.862 9.774 9.686 9.809 19.618 29.426 39.235 49.044

85 9.586 11.087 12.877 15.02 17.594 12.927 12.865 12.802 12.739 12.677 13.099 13.013 12.928 12.844 12.760 12.877 25.755 38.632 51.509 64.387

90 14.333 15.933 17.757 19.844 22.235 17.803 17.746 17.687 17.629 17.572 17.961 17.882 17.804 17.726 17.649 17.757 35.514 53.272 71.029 88.786

95 22.413 24.065 25.876 27.866 30.055 25.916 25.866 25.816 25.767 25.717 26.050 25.983 25.916 25.850 25.783 25.876 51.752 77.628 103.505 129.381

100 35.177 36.790 38.505 40.329 42.273 38.535 38.497 38.459 38.42 38.382 38.638 38.587 38.535 38.484 38.433 38.505 77.009 115.514 154.019 192.523

Valuation of Reverse Mortgage 385

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D. Kannan and L. Ma

annuity values gradually diverge. It means that the annuity for the older applicant is more affected by the initial house price than that of the younger applicant. (2) When the initial house price is fixed, the annuity increases with the increase of initial age; that is, the older applicant will be paid higher annuities every year as other factors, except for the initial age, are same.

5.2 Effect of Interest Rate on Annuity Values Table 3 portrays the following analysis. (a) The initial interest rate r0 : Keeping the initial age fixed, the annuity decreases slightly with the increase of the initial interest rate r0 , that is, the higher the initial interest rate the lower the annuity payment. The initial interest rate r0 has a less influence on the annuity of younger applicants than those of older applicants. In general, the initial interest rate r0 weakly affects the annuity payments. When the initial interest rate is fixed, the annuity increases with the increase of the initial age. (b) The average reversion level μr of interest rate: (1) Fixing the initial age, the annuity decreases with the increase of μr . The average reversion level μr impacts more the annuity of younger borrowers than that of the older. Generally, μr has a significant effect on the annuity. (2) With fixed average reversion level μr , the annuity increases with the increase of initial age. (c) The volatility σr of interest rate: The volatility σr of interest rate in Table 2 takes five values: 0.001, 0.01, 0.02, 0.03, 0.04. The corresponding annuity values with different σr almost coincide with each other under fixed initial male age. This indicates that the volatility of interest rate has weak effect on the annuity, while the volatility of interest rate is at a low level. It is known from the original data that: while the initial age kept fixed, the annuity decreases slightly, with the increase of volatility rate σr in case that σr ≤ σh ρhr αr = 0.012 (it is consistent with Proposition 6); and the annuity increases slightly with the increase of volatility rate σr in the case of σr ≥ σh ρhr αr = 0.012. Proposition 6 shows that the annuity amount decreases with increase of σr whenever σr ≤ σh ρhr αr . It implies that the valuation models can be used to determine the annuity payments as long as the volatility of interest rate σr can be controlled by the quantity σh ρhr αr (irrespective of the volatility rate). (d) The reversion speed αr of interest rate: As the initial age is fixed, the annuity decreases slowly with the increase of the reversion speed αr . While the reversion speed of interest rate is more than 0.75, the annuity is basically stable. The αr has a greater impact on the annuity of younger applicants than that of older applicants. While αr remains unchanged, the annuity increases with the increase of the initial age.

x0 r0 = 0.01 r0 = 0.04 r0 = 0.07 r0 = 0.1 r0 = 0.13 μr = 0.02 μr = 0.04 μr = 0.06 μr = 0.08 μr = 0.1 σr = 0.001 σr = 0.01 σr = 0.02 σr = 0.03 σr = 0.04 αr = 0.05 αr = 0.15 αr = 0.35 αr = 0.55 αr = 0.75

50 3.222 3.193 3.162 3.131 3.100 8.821 5.346 3.193 1.899 1.136 3.226 3.193 3.187 3.215 3.276 4.426 3.360 3.204 3.192 3.191

55 3.744 3.708 3.670 3.633 3.594 8.894 5.766 3.708 2.384 1.544 3.741 3.708 3.701 3.727 3.786 4.897 3.895 3.722 3.706 3.704

60 4.402 4.356 4.310 4.262 4.214 9.139 6.319 4.356 3.011 2.099 4.390 4.356 4.349 4.373 4.430 5.504 4.568 4.375 4.353 4.350

Table 3 Effect of interest rate on annuity values

65 5.246 5.187 5.128 5.067 5.006 9.602 7.056 5.187 3.831 2.852 5.222 5.187 5.179 5.201 5.256 6.299 5.430 5.213 5.184 5.177

70 6.354 6.276 6.198 6.118 6.037 10.355 8.052 6.276 4.917 3.881 6.311 6.276 6.267 6.287 6.339 7.358 6.557 6.310 6.271 6.260

75 7.851 7.745 7.638 7.529 7.419 11.515 9.428 7.745 6.393 5.309 7.779 7.745 7.734 7.753 7.801 8.804 8.073 7.791 7.737 7.720

80 9.959 9.809 9.657 9.503 9.349 13.285 11.396 9.809 8.477 7.360 9.843 9.809 9.797 9.814 9.858 10.848 10.191 9.872 9.797 9.771

85 13.101 12.877 12.652 12.426 12.199 16.045 14.354 12.877 11.588 10.462 12.910 12.877 12.865 12.879 12.918 13.892 13.318 12.963 12.860 12.820

90 18.104 17.757 17.410 17.064 16.717 20.550 19.083 17.757 16.557 15.470 17.788 17.757 17.744 17.754 17.786 18.719 18.244 17.869 17.733 17.671

95 26.418 25.876 25.337 24.801 24.268 28.178 26.987 25.876 24.839 23.870 25.903 25.876 25.863 25.867 25.888 26.728 26.369 26.009 25.845 25.757

100 39.304 38.505 37.713 36.931 36.156 40.236 39.351 38.505 37.695 36.919 38.526 38.505 38.492 38.491 38.502 39.186 38.942 38.641 38.470 38.364

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5.3 Effect of Joint Lifetime on Annuity Values In this subsection we discuss the impact on annuity value by joint lifetime. Table 4 shows that: (a) The modal value m1 of male lifetime: Since the parameters m1 and m2 in the respective Gompertz distributions have the same function, we consider only the parameter m1 . (1) When the initial age is fixed, the annuity decreases with the increase of m1 . The annuity for the older applicants is more sensitive to the change of m1 than that for the younger applicants. (2) When the modal value m1 is fixed, the annuity increases with the increase of the initial age, that is, older the applicant is, higher the annuity he will receive. (3) Smaller the parameter m1 becomes, greater the impact on annuity the parameter m1 will exert. (b) The dispersion coefficient σ1 of male lifetime: We shall treat only the parameter σ1 as σ2 shares the same function with σ1 . With fixed initial age, the annuity shows two different trends with the change of σ1 : (1) When the initial age is at a lower level (say, x0 = 50, 55, 60), the annuity increases first and then decreases with the increase of σ1 . (2) When the initial age is at a higher level (say, x0 = 70, 75, . . . , 100), the annuity decreases with the increase of σ1 . The annuity of an older applicant is more strongly affected by σ1 . As σ1 remains unchanged, the annuity increases with the increase of initial age x0 . (c) Parameter α, the dependence between male and female lifetime: As the initial age remains fixed, the change of annuity shows three different trends: (1) when the initial age is at a lower level (say x0 = 50, 55, 60), the annuity decreases with the increase of α; (2) when the initial age is at the middle level (say x0 = 65), the annuity decreases first and then increases with the increase of α; (3) as the initial age is at a higher level (say x0 = 70, 75, . . . , 100), the annuity increases with the increase of α. (4) In general, the impact of α on older applicants’ annuities is significantly stronger than that for young applicants’ annuities. As α is fixed, the annuity increases with the increase of the applicant age, that is, the older the applicants are, the greater annuity they will be paid. Keeping the initial age y0 (y0 = 50, 55, . . . , 100) of the female annuitant fixed, the annuity value increases with the increase of the male applicant’s initial age x0 , (50 ≤ x0 ≤ 100). The larger the initial age of female is, the stronger effect it will exert on the annuity amount. When the initial age x0 (x0 = 50, 55, . . . , 100) of male applicant is fixed, the annuity amount increases with the increase of the female’s initial age y0 , (50 ≤ y0 ≤ 100). The larger the initial age of male is, the stronger influence it will produce on the annuity value. We note that: the annuity value is approximately symmetrical, though not completely symmetrical, in the initial x0 and y0 . When the age difference between the male and female annuitants is the same, different influences are made on the annuity value in the case with the initial age y0 of female is greater than that of male and vice versa. Especially, when y0 − x0 = d (d > 0) the annuity value is greater

x0 m1 = 69 m1 = 79 m1 = 89 m1 = 99 m1 = 109 σ1 = 6 σ1 = 8 σ1 = 10 σ1 = 12 σ1 = 14 α = −5 α = −4 α = −3 α = −2 α = −1

50 3.69 3.401 3.081 2.658 2.205 3.138 3.176 3.193 3.191 3.175 3.223 3.205 3.185 3.160 3.134

55 4.337 3.976 3.567 3.05 2.515 3.66 3.699 3.708 3.691 3.658 3.736 3.720 3.700 3.677 3.652

60 5.134 4.701 4.176 3.534 2.891 4.342 4.37 4.356 4.312 4.251 4.378 4.365 4.350 4.334 4.316

Table 4 Effect of joint lifetime on annuity values

65 6.123 5.626 4.955 4.142 3.356 5.259 5.249 5.186 5.095 4.991 5.195 5.190 5.186 5.184 5.185

70 7.39 6.832 5.972 4.918 3.935 6.529 6.426 6.275 6.106 5.935 6.260 6.267 6.283 6.310 6.350

75 9.129 8.45 7.339 5.929 4.663 8.339 8.045 7.742 7.451 7.179 7.690 7.717 7.765 7.842 7.954

80 11.757 10.743 9.245 7.269 5.575 11.001 10.359 9.804 9.316 8.882 9.689 9.751 9.851 10.006 10.236

85 16.083 14.287 12.039 9.089 6.714 15.037 13.846 12.868 12.041 11.327 12.640 12.768 12.953 13.224 13.615

90 23.491 20.304 16.368 11.637 8.141 21.627 19.529 17.741 16.239 14.974 17.335 17.573 17.879 18.286 18.832

95 35.692 30.641 23.391 15.412 10.027 33.609 29.443 25.843 22.884 20.478 25.322 25.645 26.021 26.464 26.987

100 52.474 45.906 34.514 21.528 12.859 51.684 45.100 38.443 32.886 28.458 38.180 38.376 38.581 38.797 39.023

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than that when x0 − y0 = d. To validate this claim, we present the annuity values with the age difference of 5, 10, and 15 years in Table 5.

5.4 Effect of Other Parameters on Annuity Values (a) The delay time t0 in selling the pledged house: Table 6 shows that: When the initial age is fixed, the annuity slowly decreases with the increase of the delay time of selling the pledged house. The impact of delay time on the annuity is complex. While the other parameters in the valuation model change, the annuity may also increase with the increase of delay time. The delay time has a weak impact on the annuity. However, the effect is stronger on the older borrowers than on the younger borrowers. When the delay time of selling house remains unchanged, the annuity increases with the increase of the initial age. Facing the different delay time, the applicant with different initial age may get approximately the same annuity. (b) Parameter γ , the proportion coefficient of joint annuity: In order to evaluate the effect of the γ on the annuity, Table 6 presents the joint and γ annuity values. It shows that: As the initial age is fixed, the annuity payment decreases slowly with the increase of γ . The effect of γ on the annuity for older applicants is stronger than that of younger applicants. As γ remains unchanged, the annuity increases with the increase of initial age.

6 Appendix Proof of Proposition 1 It is easy to see that Y (t) follows the normal distribution with the mean 0 and variance σy2 (t) =

σr2 σr2 −αr t 2 1 − (2 − e . t + ) αr2 2αr3

(6.1)

Noting that Cov(dWh (t), dWr (t)) = ρhr dt, the covariance between Wh (t) and Y (t) is given by 1 Cov(Wh (t), Y (t)) = σr ρhr αr

$ % 1 1 −αr t − t+ e , αr αr

and hence the correlation coefficient ρ(t) between Wh (t) and Y (t) is ρ(t) =

σr ρhr √ αr σy (t) t

% $ 1 1 . t + e−αr t − αr αr

(6.2)

(x0 , y0 ) Annuity (x0 , y0 ) Annuity (x0 , y0 ) Annuity (x0 , y0 ) Annuity (x0 , y0 ) Annuity (x0 , y0 ) Annuity

(55,50) 3.490 (50,55) 3.590 (60,50) 3.653 (50,60) 3.846 (65,50) 3.811 (50,65) 4.086

(60,55) 4.084 (55,60) 4.206 (65,55) 4.293 (55,65) 4.537 (70,55) 4.492 (55,70) 4.852

(65,60) 4.837 (60,65) 4.995 (70,60) 5.103 (60,70) 5.430 (75,60) 5.342 (60,75) 5.837

Table 5 Effect of joint lifetime on annuity values

(70,65) 5.811 (65,70) 6.028 (75,65) 6.142 (65,75) 6.606 (80,65) 6.417 (65,80) 7.121

(75,70) 7.099 (70,75) 7.418 (80,70) 7.505 (70,80) 8.193 (85,70) 7.816 (70,85) 8.816

(80,75) 8.863 (75,80) 9.358 (85,75) 9.369 (75,85) 10.399 (90,75) 9.756 (75,90) 11.119

(85,80) 11.410 (80,85) 12.184 (90,80) 12.110 (80,90) 13.587 (95,80) 12.723 (80,95) 14.521

(90,85) 15.374 (85,90) 16.502 (95,85) 16.572 (85,95) 18.530 (100,85) 17.726 (85,100) 20.137

(95,90) 22.032 (90,95) 23.417 (100,90) 24.357 (90,100) 26.675

(100,95) 33.089 (95,100) 34.329

Valuation of Reverse Mortgage 391

x0 t0 = 0 t0 = 3 t0 = 6 t0 = 9 t0 = 12 γ = 1/2 γ = 2/3 γ = 3/4 γ = 4/5 γ =1

50 3.193 3.004 2.827 2.660 2.503 3.193 3.129 3.099 3.080 3.010

55 3.708 3.489 3.283 3.089 2.907 3.708 3.615 3.571 3.545 3.444

60 4.356 4.099 3.857 3.629 3.415 4.356 4.221 4.157 4.119 3.975

Table 6 Effect of other parameters on annuity values

65 5.187 4.881 4.593 4.322 4.067 5.187 4.990 4.897 4.843 4.638

70 6.276 5.907 5.558 5.230 4.922 6.276 5.987 5.851 5.773 5.481

75 7.745 7.290 6.860 6.455 6.074 0.745 7.314 7.116 7.002 6.582

80 9.809 9.236 8.692 8.179 7.696 9.809 9.157 8.862 8.694 8.082

85 12.877 12.133 11.420 10.746 10.112 12.877 11.868 11.421 11.168 10.260

90 17.757 16.752 15.772 14.843 13.967 17.757 16.162 15.468 15.079 13.701

95 25.876 24.475 23.057 21.702 20.422 25.876 23.349 22.262 21.657 19.534

100 38.505 36.573 34.487 32.466 30.553 38.505 34.755 33.141 32.242 29.088

392 D. Kannan and L. Ma

Valuation of Reverse Mortgage

393

Since the joint distribution of (Wh (t), Y (t)) follows the two- dimensional normal distribution with the correlation coefficient ρ(t) given by Eq. (6.2), we have with Eq. (6.1)  E {exp [σh Wh (t) − Y (t)]} =

+∞  +∞ −∞

−∞

exp(σh x − y)f (x, y)dxdy,

where f (x, y) =

1

A 2πσy (t) t (1 − ρ 2 (t))

 exp −

5 1 Sxy , 2(1 − ρ 2 (t))

and $ Sxy =

x √ t

%2

Under the transformation u =

x y + − 2ρ(t) √ t σy (t) x √

t

$

y σy (t)

%2 .

and v = y − ρ(t)σy (t)u, we obtain

E {exp [σh Wh (t) − Y (t)]}  +∞  +∞  √

 = exp σh t − ρ(t)σy (t) u − v g(u, v)dudv −∞



−∞

√ 1 2 1 2 = exp σh t − ρ(t)σy (t)σh t + σy (t) , 2 2

(6.3)

where /  4 1 1 2 v2 A g(u, v) = exp − u + 2 . 2 σy (t)(1 − ρ 2 (t)) 2πσy (t) 1 − ρ 2 (t) Noting {N(t), t ≥ 0} and {Ji , i ≥ 1} are independent, and Ji follows the normal distribution with mean μJ and variance σJ2 , we obtain N(t) E e i=1 Ji = exp(kh λh t). Setting 

t

m1 :=

$ μh (s)ds −

0

m2 := μr t +

% 1 2 σ + λh kh t, 2 h

  1 (μr − r0 ) e−αr t − 1 , αr

(6.4)

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one easily obtains ⎡ h(t) = h0 exp ⎣m1 + σh Wh (t) +

N(t )

⎤ Ji ⎦ ,

(6.5)

i=1



t

r(u)du = m2 + Y (t).

(6.6)

0

Since σh Wh (t) − Y (t) is independent of

N(t ) i=1

Ji , we have from Eqs. (6.3)–(6.6)

N(t) E [h(t)d(t)] = h0 em1 −m2 E eσh Wh (t )−Y (t ) E e i=1 Ji = G(t)D(t), where G(t) and D(t) is, respectively, defined by Eqs. (3.6) and (3.7). We obtain the Proposition 1. Proof of Proposition 2 The lump sum that the applicants can borrow at time t = 0 of signing the reverse mortgage contract is the random quantity h(T2 +t0 )d(T2 +t0 ). Since h(t)d(t) (t ≥ 0) is independent of T2 , - = E [h(T2 + t0 )d(T2 + t0 )] G  +∞ = E [h(x + t0 )d(x + t0 )] fT (x)dx 

0 +∞

=

G(x + t0 )D(x + t0 )fT (x)dx.

0

From the independence of r(t) and (T1 , T2 ), we have ⎡ E⎣

T 1 

Ad(k) +

⎤ γ Ad(k)⎦

k=T1 +1

k=0

=E

T 2 

+∞ i 



Ad(k)1{T1 =i} + E ⎣

+∞  i 

j +∞  +∞  

⎤ γ Ad(k)1{T1 =i,T2 =j } ⎦

i=0 j =i+1 k=i+1

i=0 k=0

=A

⎡

E[d(k)]P (T1 = i)

i=0 k=0

+γ A

j +∞  +∞   i=0 j =i+1 k=i+1

E[d(k)]P (T1  = i, T2  = j )

Valuation of Reverse Mortgage

=A

+∞ 

395

D(i)P (T1 ≥ i) + γ A

i=0

+∞ 

D(i)P (T1 < i, T2 ≥ i)

i=0

-1 =A·F where D(k) is characterized as in (3.7). Recalling that the probability density function for T2 is given by the Relation (2.8), we get Eq. (3.9). The proof is complete. Proof of Proposition 4 From the descriptions of the annuity payment factors -i (i = 1, 2, 3) (see Relations (3.10) and (3.14)), we note that these two annuity F payment factors are independent of μh . - (see Relation (3.8)), From the integrand in the definition of G ∂ [G(x + t0 )D(x + t0 )fT (x)] = (x + t0 )G(x + t0 )D(x + t0 )fT (x). ∂μh Since G(x + t0 ) > 0, D(x + t0 ) > 0, fT (x) ≥ 0, and x + t0 ≥ 0, the lump sum - is an increasing function of μh . Furthermore, from Eqs. (3.9), (3.13), and (3.15), G we note that A, A0 , d, and A∗ are increasing functions of μh . This obtains Part 4.1. -i (i = From Eqs. (3.10) and (3.14), defining the annuity payment factors F 1, 2, 3) we see that these annuity payment factors are independent of σh . Defining g1 (z) :=

 1  1 −z + 1 − e−αr z , αr αr

(6.7)

∂ [G(x + t0 )D(x + t0 )fT (x)] = σr ρhr G(x + t0 )D(x + t0 )fT (x)g1 (x + t0 ), ∂σh   where g1 (x +t0 ) = α1r −(x + t0 ) + α1r 1 − e−αr (x+t0) . When αr = 0 and z ≥ 0, we have g1 (z) ≤ 0. We thus proved Part 4.2. Since D(x + t0 ) and fT (x) are free from ρhr , ∂ [G(x + t0 )D(x + t0 )fT (x)] = σh σr G(x + t0 )D(x + t0 )fT (x)g1 (x + t0 ). ∂ρhr Noting that g1 (z) ≤ 0 when z ≥ 0, αr = 0, Part 4.3 follows. Since the Part 4.4 is obvious, we omit its proof. Proof of Proposition 5 Define g2 (z) :=

σr2 2 z + β1 z + β0 , (−∞ < z < +∞), 2αr2

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where σh σr ρhr σ2 + r2 , αr 2αr

β 0 = μh − μr −

β1 = μr − r0 +

σh σr ρhr σ2 − r2 . αr αr

Now ∂ [G(x + t0 )D(x + t0 )fT (x)] = G(x + t0 )D(x + t0 )fT (x)g2 (e−αr (x+t0) ), ∂t0 where g2 (e−αr (x+t0 ) ) =

σr2 −2αr (x+t0 ) e + β1 e−αr (x+t0) + β0 . 2αr2 α2

The minimum of g2 (z) is − 2σr2 $ ($ given by Eq. (4.1)). If the condition $ ≤ 0 r - is an increasing function of t0 . holds, we then have g2 (z) ≥ 0, and thus G Recall the definitions of z1 and z2 given above by the Relations (4.2) and (4.3), respectively. Now, if the condition $ ≥ 0 holds, then g2 (zi ) = 0, i = 1, 2. Moreover, it is obvious that 0 < exp(−αr (x + t0 )) ≤ 1 whenever αr > 0 and - is a decreasing function of t0 if the Condition x + t0 ≥ 0. Thus the lump sum G (4.4) holds. One similarly obtains the rest of the properties in Part (b-1). Proof of Proposition 6 First note that

∂ [G(x + t0 )D(x + t0 )fT (x)] 1 =− 1 − e−αr (x+t0 ) G(x + t0 )D(x + t0 )fT (x), ∂r0 αr +∞ -1 ∂F 1  =− (1 − e−αr k )D(k)[P (T1 ≥ k) + γ P (T1 < k, T2 ≥ k)], ∂r0 αr k=1

-2 ∂F 1 =− ∂r0 αr

+∞ 

(1 − e−αr k )D(k)P (T2 ≥ k),

k=1

+∞ -3 ∂F 1  =− k(1 − e−αr k )D(k)P (T2 ≥ k). ∂r0 αr k=1

1 −αr z ) αr (1 − e

≥ 0, ( αr = 0) and z ≥ 0, we obtain Part 6.1.   Defining g3 by g3 (z) := −z + α1r 1 − e−αr z . It is obvious that D(x)g3 (x), and Since

∂D(x) ∂μr

∂ [G(x + t0 )D(x + t0 )fT (x)] = G(x + t0 )D(x + t0 )fT (x)g3 (x + t0 ), ∂μr

=

Valuation of Reverse Mortgage

397

  where g3 (x + t0 ) = −(x + t0 ) + α1r 1 − e−αr (x+t0 ) . Noting g3 (z) ≤ 0 if αr > 0, z ≥ 0, and g3 (z) ≥ 0 if αr < 0, z ≥ 0, we obtain Part 6.2. For −∞ < z < +∞, define % % $ $ σr −2αr z 2σr σh ρhr σr σh ρhr −αr z e z + − + − g4 (z) = − 3 e αr 2αr αr3 αr2 αr2 $ % σh ρhr 3σr + − 3 , αr2 2αr % $ %

$ σr 1 σr 2 2σr g5 (y) = y+ y + σh ρhr − − σh ρhr . αr αr αr αr We have g5 (y) has two zero points y1 = 1 − σh ρσhrr αr and y2 = 1 if σh ρhr ≥ 0; and two zero points y1 = 1 and y2 = 1 − σh ρσhrr αr if σh ρhr ≤ 0. Now, $ $ % %

1 σr −2αr z 2σr σr dg4 (z) = e + σh ρhr − − σh ρhr e−αr z + = g5 (e−αr z ). dz αr αr αr αr Note that 0 < e−αr z ≤ 1 in case of z ∈ [0, +∞), αr > 0. In case of z ∈ [0, +∞), αr > 0, σh > 0 and ρhr ≥ 0, we have g5 (e−αr z ) ≤ 0 in the interval σr ∈ (0, σh ρhr αr ], then g4 (z) ≤ g4 (0) = 0 in the interval σr ∈ (0, σh ρhr αr ]. In case of z ∈ [0, +∞), αr > 0, σh > 0 and ρhr ≤ 0, we have g5 (e−αr z ) ≥ 0, then g4 (z) is an increasing function of z and g4 (z) ≥ g4 (0) = 0. Next we note ∂ [G(x + t0 )D(x + t0 )fT (x)] = G(x + t0 )D(x + t0 )fT (x)g4 (x + t0 ). ∂σr Defining

1 , g6 (z) := z + 1 − (2 − e−αr z )2 2αr we see that +∞ -1 ∂F σr  = 2 D(k)g6 (k)[P (T1 ≥ k) + γ P (T1 < k, T2 ≥ k)], ∂σr αr k=1

+∞ -2 ∂F σr  = 2 D(k)g6 (k)P (T2 ≥ k), ∂σr αr k=1

+∞ -3 ∂F σr  = 2 kD(k)g6 (k)P (T2 ≥ k). ∂σr αr k=1

Noting that g6 (z) ≥ 0 if z ∈ [0, +∞), we get Part 6.3. This completes the proof.

398

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References 1. Frees, E.W., Carrière, J., Valdez, E.: Annuity valuation with dependent mortality. J. Risk Insur. 63(2), 229–261 (1996) 2. Gompertz, B.: On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513C583 (1825) 3. Li, J.S.H., Hardy, M.R., Tan, K.S.: On pricing and hedging the no-negative equity guarantee in equity release mechanisms. J. Risk Insur. 77(2), 499–522 (2010) 4. Ma, L., Kannan, D.: Valuation of reverse mortgage with dependent joint life. Dyn. Syst. Appl. 27, 895 (2018) 5. Ma, L., Zhang, J., Kannan, D.: Fair pricing of reverse mortgage without redemption right. Dyn. Syst. Appl. 26, 473–498 (2017) 6. Mitchell, O.S., Piggott, J.: Unlocking housing equity in Japan. J. Jpn. Int. Econ. 18, 466–505 (2004) 7. Ohgaki, H.: Economic implication and possible structure for reverse mortgage in Japan. Rits University, pp. 1–14 (2003) 8. Tsay, J.T., Lin, C.C., Prather, L.J., Buttimer, R.J. Jr.: An approximation approach for valuing reverse mortgages. J. Hous. Econ. 25, 39–52 (2014) 9. Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)

Stationary Distribution of Discrete-Time Finite-Capacity Queue with Re-sequencing Rostislav Razumchik

and Lusine Meykhanadzhyan

Abstract The discrete-time re-sequencing model, consisting of one high and one low priority finite-capacity queue and a single server, which serves the low priority queue if and only if the high priority queue is empty, is being considered. Two types of customers, regular and re-sequencing, arrive at the system. The arrival and service processes are geometric, i.e. in each time slot at most one customer of each type may arrive at the system and at most one customer may be served. A regular customer upon arrival occupies one place in the high priority queue. An arriving resequencing customer moves one customer from the high priority queue (if it is not empty) to the low priority queue and itself leaves the system. A regular customer which sees the high priority queue full and a re-sequenced customer which sees the low priority queue full, are lost. Using the generating function method the recursive procedure for the computation of the joint stationary distribution of the number of customers in the high and in the low priority queues is derived. Keywords Queueing system · Discrete-time · Finite-capacity · Re-sequencing · Negative customers · Generating function

The reported study was funded by RFBR according to the research projects №20-07-00804 and

№19-07-00739.

R. Razumchik () Institute of Informatics Problems, FRC CSC RAS, Moscow, Russia Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia e-mail: [email protected]; [email protected] L. Meykhanadzhyan Financial University Under the Government of the Russian Federation, Moscow, Russia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_23

399

400

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1 Introduction Consideration is given to the discrete-time counterpart of continuous-time resequencing queue considered in [31]. The system consists of two finite-capacity queues (high and low priority) and a single server, which serves the low priority queue if and only if the high priority queue is empty. Two types of customers (regular and re-sequencing) arrive at the system. The arrival and service processes are geometric,1 i.e. in each time slot at most one customer of each type may arrive at the system and at most one customer may be served. A regular customer upon arrival occupies one place in the high priority queue and waits there for service. A re-sequencing customer upon arrival moves one customer from the high priority queue (if it is not empty) to the low priority queue and itself leaves the system. Whenever a re-sequencing arrival sees the high priority queue empty, it leaves the system having no effect on it. A regular customer which sees the high priority queue full and a re-sequenced customer which sees the low priority queue full, are lost. For this queueing system one is interested in the computation of the joint stationary distribution of the number of customers in the high and in the low priority queues. Nowadays discrete-time models keep receiving attention from the research community (see, for example, [4, 6, 10, 11, 20, 26–28, 35]). For some general overviews on discrete-time queueing models one can refer to [7, 25, 30, 34]. Our motivation behind the study of the particular queueing problem described in the previous paragraph is methodological. This can already be seen from the scope of the problem: the attention is restricted only to the computation of (quantities related only to) the stationary queues’ size distribution. Since the stationary probabilities satisfy the (finite) system of linear algebraic equations, numerous solution methods are available out there by which one can approach the problem and find the solution. Among those which are particularly suited for applied probability problems involving Markov chains, one can mention the following. Firstly, matrix analytic methods, which have been the topic of many papers published during the last decades (see, for example, [2, 3, 19, 21]). This algorithmic approach allows one to obtain numerical results for a great variety of models and the model considered here is not an exception (especially given that its transition probability matrix has the block tridiagonal form). Secondly, there exist general procedures for finding stationary distributions of finite irreducible discrete-time Markov chains, which involve the generalized inverses (see [16, 17]). Finally there is one traditional method which stands alone2 — the generating function method (GFM) (see, for example, [7, 8, 12] and many others). In the contrast to the algorithmic methods, the GFM may lead to explicit and closed-form expressions for the system’s performance characteristics and some useful probabilistic interpretations.

1 Also

sometimes called Bernoulli.

2 Another such method is the compensation

approach (see [1]). Yet we are unaware of any use cases of its application to problems with a finite state space.

Finite-Capacity Queue with Re-sequencing in Discrete Time

401

Fig. 1 The transition diagram for the discrete-time Markov chain representing the considered re-sequencing queue. The x-axis (y-axis) denotes the total number of customer in the high priority (low priority) queue when the server is busy. Empty system state is not shown

In this paper we approach the problem of the determination of the joint stationary distribution by the GFM. The main reason for that is that we seek to find the boundaries of tractability in using the GFM for the models with a finite state space. In this respect the considered discrete-time queueing problem gives rise to the new Markov chain on the finite two-dimensional grid (see Fig. 1), which to our best knowledge has not been analyzed before with the GFM. Its most peculiar feature is that the chain can jump not only to the adjoining states (horizontally, vertically, and diagonally) but can also skip one state in the west-north direction. From the system of balance equations for the joint stationary distribution, it can be seen that the system cannot be solved recursively. This is the starting point for this paper. It shall be seen in the next sections, that the double generating function of the joint stationary distribution allows one to deduce the new system of equations, which permits the recursive solution. The main idea behind the adopted method is the following. Due to the finite state space, in the expression (which a ratio of two polynomials) for the generating function all roots (and not only those which are of absolute value is less than 1) of the denominator are the roots of the numerator as well. By plugging the roots of the denominator into the numerator and after collecting the common terms, one obtains two polynomial functions with real coefficients in a single variable, say v, both of which are equal to zero for any value of v ∈ [0, 1]. Thus their coefficients must be equal to zero and this produces the system of equations, which can be used to determine the joint stationary distribution in a recursive manner. The rest of the paper is organized as follows. In Sect. 2, the model under the investigation is described in detail; we look at the case where the maximum queues’

402

R. Razumchik and L. Meykhanadzhyan

sizes are equal. The double probability generating function (PGF) for the joint stationary distribution is given in Sect. 3. Since it contains many unknowns, it is used only to derive some useful local balance relations. In Sect. 4 one applies the method described above to obtain the recursive procedure for the computation of the joint stationary distribution. Some conclusions are drawn in Sect. 5.

2 The Model Consider the discrete-time re-sequencing system consisting of two queues, each of capacity3 r < ∞: one high priority queue and one low priority queue. Two types of customers (regular and re-sequencing) arrive at the system independently of each other. Arriving regular customers are enqueued in the high priority queue and wait there for service. A re-sequencing customer4 moves one customer from the high priority queue (if it is not empty; otherwise the re-sequencing customer has no effect on the system) to the low priority queue and itself leaves the system. There is one server, which serves customers one by one and the service process is the same for both high and low priority customers. The service policy is nonpreemptive priority,5 i.e. upon a service completion a high priority customer is picked for service and, in case there are no high priority customers, a low priority customer is picked. Since the discrete-time setting is considered, one has to establish the precedence relations between arrivals, departures, and start of service. The following conventions are used:6 if the service of a customer has been finished in the slot, it leaves the system, and the server immediately takes the next customer from the high priority queue or, if the high priority queue is empty, from the low priority queue; then if a regular customer arrives in the same slot it is placed in the high priority queue if the server is busy and enters the server otherwise (its servicing begins immediately); then if a re-sequencing customer arrives in the same slot it moves a customer from the high priority queue to the low priority queue. If a re-sequencing customer finds the high priority queue empty, it immediately leaves the system without any effect on it. Finally, whenever a regular customer arrives and sees busy server and full high priority queue, it is lost; if a customer, which is being moved from the high priority queue to the low priority queue, sees that the latter is full, it is lost. 3 If 0 ≤ r ≤ 5, then some of the relations given below have to be left out. In order not to deal with such special cases, it is further assumed that r ≥ 6. 4 Also known in the literature as negative signal/customer, see [13–15]. 5 Since the waiting time characteristics of the regular customers are not studied in this paper (the waiting time distribution in the case of infinite-capacity queues had been studied in [29]), the service order in each queue and the re-sequencing order are irrelevant. For certainty one can consider all three orders to be head-of-queue. 6 In other words, EAS-IA setup (see, for example, [9] and [27, pp. 2–3]) with regular arrivals, start of service, and departures having precedence over re-sequencing customers.

Finite-Capacity Queue with Re-sequencing in Discrete Time

403

Regular and re-sequencing customers arrive at the system according to a geometric (Bernoulli) input flow, with the arrival probability in a slot equal to a and c, respectively. The service times of both high and low priority customers are assumed to constitute a set of i.i.d. random variables having geometric distribution with parameter b, which is also the probability of a service completion in a slot.

3 The Double Probability Generating Function In what follows the notations a = 1 − a, b = 1 − b, c = 1 − c are used for brevity. Introduce the discrete-time Markov chain (DTMC) by observing the system at epochs at the end of time slots (i.e., after all possible events in a slot have occurred). Let Ht be the total number of customers in the high priority queue plus server at time t and Lt —the total number of customers in the low priority queue at time t. Whenever the server is idle (i.e., Ht = 0), Lt is not defined. The stochastic process {(Ht , Lt ) : t = 0, 1, 2, . . . } is the irreducible aperiodic DTMC. It is convenient to define its state space as (0)∪{(k, m), 0 ≤ k ≤ r, 0 ≤ m ≤ r}, where the state (0) corresponds to the empty system; the state (k, m) corresponds to the state with the busy server, k customers in the high priority queue and m customers in the low priority queue. The transition diagram7 is given in Fig. 1. Define the joint stationary distribution π0 = lim P(Ht = 0), πkm = lim P(Ht = k + 1; Lt = m), 0 ≤ k, m ≤ r, t →∞

t →∞

which exists for all possible values of a, b, and c, with its associated double PGF (u, v) =

r  r 

uk v m πkm , 0 ≤ u, v ≤ 1.

k=0 m=0

From the system of balance equations8 one obtains the following expression for the double PGF (u, v): B(u, v) (u, v) = A(u, v),

(1)

where B(u, v) = u3 + u2 p(v) + uq(v) + r(v),

7 The state (0) is not included in the diagram. From the system’s description it can be seen that the only possible transition to (from) the state (0) is from (to) the state (0, 0). 8 The system consists of r 2 + 1 equations and due to the lack of space is not presented.

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* A(u, v) = (v − u ) 2

2

abc

* + (uv − u ) 2

abc + abc

+

abc

++

r 

v m π0m abcv abc abcv m=0 * + r r   abc abcv +(u − 1) u + v m πrm + v m π1m ur+1 (uv − u2 ) abc abc m=0 m=0 + * r  abc abc abc 2 abc + abc u + u+ vr π0r +(v − 1) uk πkr + (v − 1)u2 v r abc abc abc abc k=1 +(u − 1)(1 − v)ur+1 v r

abc abc

πrr + (1 − v)u2

ab abcv

π00 + (1 − v)uv r

abc abc

π1r ,

and the functions p(v) = p0 + vp1 , q(v) = q0 + vq1 , and r(v) = vr1 stand for p(v) = −

ab+ab+abc+abc abc

+

abc abc

v, q(v) =

abc abc

+

abc+abc abc

v, r(v) =

abc abc

v.

The given expression for (u, v) contains many unknowns but already in this form allows one to obtain some useful local balance relations. One set of such relations concerns the (marginal)  distribution {π·,m , 0 ≤ m ≤ r} of the low priority queue size, where π·,m = rk=0 πkm . By substituting u = 1 in (1) and using the method of equating the coefficients, one gets ab π0,m+1 + abπ1m + (1 − ab)π0m , 1 ≤ m ≤ r − 1, c   ab π01 + abπ10 . = b + ab π00 + c

π·,m = π·,0

Another set of useful relations is obtained if one puts v = 1 in (1) and then equates the coefficients of uk in the left-hand side and in the right-hand side of (1). This gives c πr,· = 0, c c πr,· = 0, c πk,· + p(1)πk+1,· + q(1)πk+2,· + r(1)πk+3,· = 0, 0 ≤ k ≤ r − 3,  πr−1,· + 1 + p(1) −  πr−2,· + p(1)πr−1,· + q(1) +

r where the notation πk,· = m=0 πkm is used. Thus πk,· = yk πr,· , 0 ≤ k ≤ r, for positive constants yk , which are uniquely determined from this system and which depend only on the values of a,b, and c. Since aπ0 = abπ00 and the normalization implies that rk=0 πk,· = 1 − π0 , we have πr,· = condition r (a − abπ00)/(a k=0 yk ). Let xm be such positive constants (depending only on

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the values of a, b, and c), that πrm = xm π00 , 0 ≤ m ≤ r. Then the latter relation between πr,· and π00 yields the closed-form expression for the probability π00 : π00 = a

r  m=0

xm

a r 

.

(2)

yk + ab

k=0

In (2) xm , 0 ≤ m ≤ r, are the only unknowns and in Sect. 4 it is shown how these quantities can be determined solely from the double PGF (u, v) given by (1). As a side result of manipulations with the PGF (u, v), one obtains such new9 relations, which allow the recursive computation of the whole joint distribution πkm .

4 Solution for the Joint Stationary Distribution Setting B(u, v) = 0 produces a cubic equation, which always has three roots, further denoted by u1 (v), u2 (v), and u3 (v). For the sake of brevity henceforth, when referred to the roots ui (v), the argument v is omitted. The roots ui may be real or complex numbers; in either case the following derivations remain true. Define the functions +k (v) =

u2 (u3 − u1 )(uk2 − uk1 ) − u3 (u2 − u1 )(uk3 − uk1 ) , k ≥ 1, (u2 − u1 )(u3 − u1 )(u3 − u2 )

(3)

with +k (v) ≡ 0 for k ≤ 0. The function +k (v) is the symmetric function of the roots ui and thus10 it can be expressed directly in terms of the coefficients p(v), q(v), and r(v) of the equation B(u, v) = 0, i.e. +k (v) is a polynomial function. In order to find its degree and its coefficients it is sufficient to notice that +1 (v) = −1 = λ10 and for k ≥ 2 the functions +k (v) satisfy the recurrence relation +k (v) = −p(v)+k−1 (v)− q(v)+k−2 (v) − r(v)+k−3 (v). Thus the degree of +k (v) is k − 1. Substitution of i +k (v) = k−1 i=0 v λki into the previous relation yields the recursive procedure λk0 = −q0 λk−2,0 −p0 λk−1,0 , λki = −r1 λk−3,i−1 −q0 λk−2,i −p0 λk−1,i −q1λk−2,i−1 −p1 λk−1,i−1 , 1 ≤ i ≤ k −3, λk,k−2 = −p0 λk−1,k−2 −q1λk−2,k−3 −p1 λk−1,k−3 , λk,k−1 = −p1 λk−1,k−2 , which can be used to compute the coefficients of the polynomial function +k (v) for any k ≥ 2. 9 “New” 10 See,

means that these relations cannot be seen directly from the system of balance equations. for example, [22, Chapter IX].

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Let us look closer at the double PGF (u, v) given by (1). It is the ratio of two polynomial functions and for each value of v (u, v) is the continuous function of u. Thus, since the left-hand side of (1) vanishes at points (u1 (v), v), (u2 (v), v), and (u3 (v), v), the right-hand side must vanish at these points too. This observation leads to the system of three equations: ⎧ A(u1 (v), v) = 0, ⎪ ⎨ A(u2 (v), v) = 0, ⎪ ⎩ A(u3 (v), v) = 0.

(4) (5) (6)

By expressing the term (1 − v)abπ00/(abcv) from (4) and substituting it firstly into  (5) and then into (6), one gets two new equations sharing the same term abc rm=0 v m+1 π1m /(abc), which does not depend on ui . Cancelation of this term yields the following equation:11 r 

v m π0m − "(v)

m=0

r 

v m πrm − (1 − v)v r

m=0

r 

-k (v)πkr

k=1

+(1 − v)v r (+r (v) − +r−1 (v)) p1 πrr = 0,

(7)

where the functions "(v) and -k (v) are defined by "(v) = +r (v) − +r+1 (v) + p1 v+r−1 (v) − p1 v+r (v), -k (v) = p1 +k (v) + q1 +k−1 (v) + r1 +k−2 (v). Since +k (v) is the polynomial function of degree k −1, then "(v) and -k (v) are the polynomial functions of degrees r and k − 1, respectively, i.e. "(v) = ri=0 v i ψi  i and -k (v) = k−1 i=0 v θki . The coefficients ψi and θki , 1 ≤ k ≤ r, can be computed directly from the coefficients λki : ψr = 0, ψ0 = λr0 − λr+1,0 , ψi = λri − λr+1,i + p1 λr−1,i−1 − p1 λr,i−1 , 1 ≤ i ≤ r − 1, θki = p1 λki + q1 λk−1,i + r1 λk−2,i , 0 ≤ i ≤ k − 3, θk,k−2 = p1 λk,k−2 + q1 λk−1,k−2 , θk,k−1 = p1 λk,k−1 . From the fact that the polynomial (of degree 2r) in the left-hand side of (7) is equal to zero for all values of v in [0, 1], it follows that all coefficients of v m , 0 ≤ m ≤ 2r,

11 Due

to the lack of space the details of these and some further derivations are omitted.

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407

must be equal to zero. Hence one obtains the following system of linear algebraic equations with constant coefficients: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(p1 λr,r−1 − θr,r−1 )πrr = 0,

(8)

(θr,r−1 − p1 λr,r−1 )πrr − θr−1,r−2 πr−1,r +(ψr−1 − θr,r−2 − p1 (λr−1,r−2 − λr,r−2 ))πrr = 0, (9) r−1 

ψi πr,m−i +

i=m−r

2r−1−m 

2r−m 

k=0

k=0

θr−k,m−r πr−k,r −

θr−k,m−r−1 πr−k,r

  + λr−1,m−r −λr,m−r −λr−1,m−r−1 +λr,m−r−1 p1 πrr = 0, r+2 ≤ m ≤ 2r−2,

r−1 

ψi πr,r+1−i +

r−2 

θr−k,1πr−k,r −

(10)

r−2 

θr−k,0 πr−k,r ⎪ ⎪ i=1 k=0 k=0 ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ p1 πrr + p1 π1r = 0, (11) + λ − λ − λ + λ r−1,1 r,1 r−1,0 r,0 ⎪ ⎪ ⎪ ⎪ ⎪ r−1 r−2 ⎪ ⎪   ⎪ ⎪ ⎪ ψi πr,r−i + θr−k,0 πr−k,r ⎪ ⎪ ⎪ ⎪ i=0 k=0 ⎪ ⎪ ⎪ ⎪ ⎪ +(λr−1,0 − λr,0 )p1 πrr − p1 π1r − π0r = 0, (12) ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪  ⎪ ⎪ ⎪ ψi πr,m−i − π0m = 0, 0 ≤ m ≤ r − 1. (13) ⎩ i=0

Analysis of this system shows, that Eqs. (8)–(11) can be used to express the probabilities πkr , 1 ≤ k ≤ r − 1, in terms of the probabilities πrm , 2 ≤ m ≤ r. Indeed, by summing (8) and (9) one gets the relation between πr−1,r and πrr ; next, summation of (8), (9), and (10) for m = 2r − 2 yields the relation between πr−2,r , πr−1,r , πr,r−1 , and πrr , and so on. After some tedious but simple algebra one finds the general expressions12 πkr =

r−1 

βkm πrm + αk πrr , 2 ≤ k ≤ r − 1,

(14)

m=k+1

π1r = −

r 

r−1 

m=2 j =r+1−m

12 Here

 θk0 ψj πrm + πkr + (λr−1,0 − λr0 )πrr , p1 p1

and henceforth the agreement

r

k=2

k−1 m=k

≡ 0 is used.

(15)

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in which the constants αk and βkm are computed recursively by ψk −θr,k−1 −p1 (λr−1,k−1 −λr,k−1 )  ψm −θm,k−1 αm + , 2 ≤ k ≤ r −1, θk,k−1 θk,k−1 r−1

αk =

m=k+1

βk,k+1 =

ψr−1 , 3 ≤ k ≤ r − 2, θk,k−1 r−1 

βkm =

ψj

j =k+r−m

θk,k−1

−

m−1  j =k+1

θj,k−1 βj m , k +2 ≤ m ≤ r −1, 3 ≤ k ≤ r −3. θk,k−1

To summarize, Eq. (7) obtained from the system (4)–(6) allowed one to obtain explicit balance relations between the boundary probabilities πkr and πrm (and between π0m and πrm , see (13)). Now it will be shown that the system (4)–(6) can also be used to obtain the expressions for the probabilities πrm in terms of π00  only. By expressing the term abc rm=0 v m π1m /(abc) from (4) and substituting it firstly into (5) and then into (6), one gets two new equations sharing the same term abc rm=0 v m π0m /(abc), which does not depend on ui . Elimination of this term leads to the equation

!0 (v)

r 

v m πrm + (1 − v)v r

m=0

+ abcv

r 

!k (v)πkr + abc!r+1 (v)(1 − v)v r πrr

k=1 r+2

(1 − v)π0r − ab(1 − v)vπ00 − abc(1 − v)v r+1 π1r = 0,

(16)

where the functions !k (v) are defined by   !0 (v) = −ab(cv−1)vr(v)+r−1 (v)−ab v 2 −cv 3 +cr(v) +r+1 (v)   −ab cv 3 +cp(v)v 2 −cr(v)−cq(v)v+cq(v)v 2 +cr(v)v +r (v)

!k (v) =

+abc (v + p(v)) v+r+2 (v) + abcv+r+3 (v),

(17)

abc(p1 !∗k (v)

(18)

+ q1 !∗k−1 (v)

+ r1 !∗k−2 (v)),

1 ≤ k ≤ r,

!r+1 (v) = −(p(v)+q(v)+r1 +v)v+r (v)+(1−v)(r(v)+r−1 (v)−v+r+1 (v)), (19) and !∗k (v) = (vq(v)+r(v))+k (v)+vr(v)+k−1 (v)−v 2 +k+1 (v). Recall that +k (v) is the polynomial function. Thus the functions !k (v) are polynomials as well. Since the degree of +k (v) is k −1, the degrees of !0 (v) and !r+1 (v) are both equal to r + 2, and the degree of !k (v) for 1 ≤ k ≤ r is equal to k+2. Note that the lowest  degree i of monomials in each !k (v) is equal to 1. By substituting !0 (v) = r+2 i=1 v φ0i ,

Finite-Capacity Queue with Re-sequencing in Discrete Time

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 r+2 i i !k (v) = k+2 i=1 v φki , and !r+1 (v) = i=1 v φr+1,i in the left-hand side and in the right-hand side of (17)–(19) and using the method of equating the coefficients, one obtains the recursive procedure for the computation of all coefficients13 φki : φ0i = abλr+2,i−2 − abλr+1,i−2 + abcp0 λr+2,i−1 + abcλr+3,i−1 +abλr,i−1 − abcλr+1,i−1 − abcr1 λr−1,i−3 +abcλr+1,i−3 − abc(1 + p1 + q1 )λr,i−3 +abr1 λr−1,i−2 − (abcp0 + abcr1 + abcq0 − abcq1 )λr,i−2 , 1 ≤ i ≤ r + 2, φki = abc(−p1 λk+1,i−2 + p1 (r1 + q0 )λk,i−1 + (p1 − 1)q1 λk,i−2 +(p1 r1 + q12 − r1 )λk−1,i−2 + (q1 r1 + q0 q1 )λk−1,i−1 + 2q1 r1 λk−2,i−2 +r1 (r1 + q0 )λk−2,i−1 + r12 λk−3,i−2 ), 1 ≤ i ≤ k + 2, 1 ≤ k ≤ r, φr+1,i = λr+1,i−2 − (p0 + q0 + r1 )λr,i−1 − (p1 + q1 + 1)λr,i−2 + r1 λr−1,i−1 −r1 λr−1,i−2 − λr+1,i−1 , 1 ≤ i ≤ r + 2.

The polynomial (of degree 2r + 3) in the left-hand side of (16) is equal to zero for all values of v in [0, 1]. Hence the coefficients of v m must be equal to zero. Consideration of the coefficients of v 1 , v 2 , . . . , v r+1 yields the following relations: ⎧ φ01 πr0 − abπ00 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ φ02 πr0 + φ01 πr1 + abπ00 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪  ⎪ ⎪ ⎪ φ0,m+1−i πri = 0, 2 ≤ m ≤ r − 1, ⎨

(20) (21) (22)

i=0

⎪ ⎪ ⎪ r−1 r−1 ⎪   ⎪ ⎪ ⎪ φ π + φk1 πkr + (φ11 − abc)π1r ⎪ 0,r+1−i ri ⎪ ⎪ ⎪ i=0 k=2 ⎪ ⎪ ⎪   ⎩ + φ01 + abcφr+1,1 + φr1 πrr = 0.

(23)

By substituting πrm /π00 = xm in (20)–(23) one gets the recursive procedure for the computation of xm . Indeed, from (20) it follows that x0 = ab/φ01 . Relation (21) gives x1 = −(ab + φ02 x0 )/φ01 and relations (22) yield xm =  14 − m−1 φ i=0 0,m+1−i xi /φ01 , 2 ≤ m ≤ r − 1. Finally, the value xr is computed from (23), since the last two terms in the left-hand side of (23) can be expressed through πrm (see (14) and (15)).

that, by definition, λki ≡ 0 for i < 0 and i ≥ k. expression is too cumbersome to be given here and thus is omitted.

13 Note 14 Its

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The values xm , 0 ≤ m ≤ r, obtained from the system (20)–(23), are used to compute the probability π00 by (2). Once it is done, the whole joint stationary distribution πkm can be determined from (11)–(13) and the system of balance equations. The respective procedure in pseudocode is given below. Procedure for the recursive computation of the joint stationary distribution πkm 15 procedure STEADYSTATEDISTRIBUTION(πri ,πir ,π0i ,0 ≤ i ≤ r) for m = 0 → r − 1 do πr−1,m = −(1/c + p0 − p1 )πrm for k = r − 2 → 1 do πk0 = −(p0 πk+1,0 + q0 πk+2,0 ) for m = 1 → r − 1 do πr−2,m = −(p0 πr−1,m + p1 πr−1,m−1 + q0 πrm + (p1 + q1 )πr,m−1 ) for k = r − 3 → 1 do πk,r−1 = − cc p(1)πkr +πk−1,r +q(1)πk+1,r +q1πk+1,r−1  +r1 (πk+2,r +πk+2,r−1 ) for m = 1 → r − 2 do π1m = − abc /c+π2,m−1 ) ab (p0 + q1 )π0m +r1 (π  0,m+1 +p1 π0,m−1 +q1 π1,m−1 for k = 2 → r −  3 do πkm = − ab ab p0 πk−1,m +p1πk−1,m−1 +r1 πk+1,m−1 +πk−2,m +q1 πk,m−1 values of πri , πir , π0i , 0 ≤ i ≤ r, computed from (12)–(15) and (20)–(23), are the input for the procedure. 15 The

5 Conclusion The technique used in the paper to obtain the recursive procedure for the joint stationary distribution is not new,16 but is rarely used. It is suitable for exact arithmetic implementation but sometimes may suffer from the numerical instability (in the considered model such case is when the re-sequencing arrival probability c is much greater than the regular arrival probability a). On the one hand, it must be admitted that the technique is not well suited for the computation of the whole joint stationary distribution πkm . But on the other hand, as can be seen from Sect. 3, the whole distribution πkm is not needed if one is only interested in the computation of

16 It had been used before for the analysis of some other types of queueing systems (see, for example, [5, 18, 33]).

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the system’s main performance characteristics like loss probabilities, moments of queues’ sizes, etc. The severe limitation of the technique is the memoryless assumption of arrival and service processes and the extension to a more general case is the open question. Yet sometimes the technique proves to be useful because it yields recursive solutions to problems, which seemed not to have such. As an example,17 one can mention the heterogeneous Markov ordered entry queue with two finite-capacity queues (see [23, Chap. 3] and [33]). Finally it is worth mentioning that the adopted technique permits one noteworthy modification. All the relations ((8)–(13) and (20)–(23)), which eventually allow the computation of xm , depend on the values of λki , being the coefficients of the polynomials +k (v) given by (3). The larger the value of k, the higher the degree of +k (v) is. By assuming that the degree of +k (v) is min(k − 1, n) with n < r, one reduces the degrees of the polynomials "(v), -k (v), and !k (v), which are needed to compute xm . Consequently, this leads to the simplification of calculations18 but surprisingly not always at the expense of accuracy loss.19

References 1. Adan, I.J.B.F., Wessels, J., Zijm, W.H.M.: A compensation approach for two-dimensional Markov processes. Adv. Appl. Probab. 25(4), 783–817 (1993). https://doi.org/10.2307/ 1427792 2. Akar, N., O˘guz, N.C., Sohraby, K.: A novel computational method for solving finite QBD processes. Commun. Stat. Stoch. Models 16(2), 273–311 (2000). https://doi.org/10.1080/ 15326340008807588 3. Alfa, A.S.: Discrete time queues and matrix-analytic methods. TOP 10, 147–185 (2002). https://doi.org/10.1007/BF02579008 4. Atencia, I.: A discrete-time queueing system with changes in the vacation times. Int. J. Appl. Math. Comput. Sci. 26(2), 379–390 (2016). https://doi.org/10.1515/amcs-2016-0027 5. Avrachenkov, K.E., Vilchevsky, N.O., Shevlyakov, G.L.: Priority queueing with finite buffer size and randomized push-out mechanism. In: Proceedings of the 2003 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, San Diego, pp. 324–335 (2003). https://doi.org/10.1145/781027.781079 6. Barbhuiya, F.P., Gupta, U.C.: Discrete-time queue with batch renewal input and random serving capacity rule: GI X /GeoY /1. Queueing Syst. Theory Appl. 91(3), 347–365 (2019). https://doi.org/10.1007/s11134-019-09600-7

17 Another example worth mentioning here is the computation of the joint stationary distribution in the two M/M/1/r queues running in parallel with coupled arrivals. Although in this problem the technique does not help, it leads to some insights into the interdependence between the equilibrium probabilities (see [24]). 18 In the sense, that the number of terms in Eqs. (8)–(13) and (20)–(23) will be smaller. 19 Even though the whole joint stationary distribution π km cannot be computed accurately under this assumption, some performance characteristics (like loss probabilities, mean waiting times) can be. The example of one such study is [32].

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7. Bruneel, H., Kim, B.G.: Discrete-Time Models for Communication Systems Including ATM. Kluwer Academic Publishers, Dordrecht (1993). https://doi.org/10.1007/978-1-4615-3130-2 8. Chaudhry, M.L.: Exact and approximate numerical solutions of steady-state single-server bulkarrival discrete-time queues: GeomX /G/1. Int. J. Math. Stat. Sci. 62, 133–185 (1993) 9. Chaudhry, M.L., Gupta, U.C.: Queue-length and waiting-time distributions of discrete-time GI X /Geom/1 queueing systems with early and late arrivals. Queueing Systems 25, 307–324 (1997). https://doi.org/10.1023/A:1019144116136 10. Claeys, D., De Vuyst, S.: Discrete-time modified number- and time-limited vacation queues. Queueing Systems 91(3), 297–318 (2019). https://doi.org/10.1007/s11134-018-9596-8 11. De Clercq, S., Laevens, K., Steyaert, B., Bruneel, H.: A multi-class discrete-time queueing system under the FCFS service discipline. Ann. Oper. Res. 202(1), 59–73 (2013). https://doi. org/10.1007/s10479-011-1051-8 12. Dester, P.S., Fricker, C., Tibi, D.: Stationary analysis of the shortest queue problem. Queueing Systems 87(3–4), 211–243 (2017). https://doi.org/10.1007/s11134-017-9556-8 13. Do, T.V.: An initiative for a classified bibliography on G-networks. Perform. Eval. 68(4), 385– 394 (2011). https://doi.org/10.1016/j.peva.2010.10.001 14. Gelenbe, E.: G-networks: a unifying model for neural and queueing networks. Ann. Oper. Res. 48(5), 433–461 (1994). https://doi.org/10.1007/bf02033314 15. Gelenbe, E., Glynn, P., Sigman, K.: Queues with negative arrivals. J. Appl. Prob. 28(1), 245– 250 (1991). https://doi.org/10.2307/3214756 16. Hunter, J.J.: A survey of generalized inverses and their use in stochastic modelling. Adv. Probab. Stoch. Process. 1, 79–90 (2000) 17. Hunter, J.J.: Generalized inverses of Markovian kernels in terms of properties of the Markov chain. Linear Algebra Appl. 447, 38–55 (2014). https://doi.org/10.1016/j.laa.2013.08.037 18. Ilyashenko, A., Zayats, O., Muliukha, V., Laboshin, L.: Further investigations of the priority queuing system with preemptive priority and randomized push-out mechanism. In: Balandin, S., Andreev, S., Koucheryavy, Y. (eds.) Internet of Things, Smart Spaces, and Next Generation Networks and Systems, vol. 8638, pp. 433–443. Springer, Heidelberg (2014). https://doi.org/ 10.1007/978-3-319-10353-2_38 19. Kapodistria, S., Palmowski, Z.: Matrix geometric approach for random walks: stability condition and equilibrium distribution. Stoch. Models 33(4), 572–597 (2017). https://doi.org/ 10.1080/15326349.2017.1359096 20. Krishnamoorthy, A., Pramod, P.K., Chakravarthy, S.R.: Queues with interruptions: a survey. TOP 22(1), 290–320 (2012). https://doi.org/10.1007/s11750-012-0256-6 21. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia (2000). https://doi.org/10.1137/1.9780898719734 22. Littlewood, D.E.: The Skeleton Key of Mathematics: A Simple Account of Complex Algebraic Theories. Courier Corporation, North Chelmsford (2002) 23. Medhi, J.: Stochastic Models in Queueing Theory. Academic, Amsterdam (2003) 24. Meykhanadzhyan, L., Matyushenko, S., Pyatkina, D., Razumchik R.: Revisiting joint stationary distribution in two finite capacity queues operating in parallel. Inf. Appl. 11(3), 106–112 (2017). https://doi.org/10.14357/19922264170312 25. Miyazawa, T., Takagi, H.: Advances in discrete-time queues. Queueing Systems 18, 1–3 (1994) 26. Morozov, E., Fiems, D., Bruneel, H.: Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions. Perform. Eval. 68(12), 1261–1275 (2011). https://doi.org/10.1016/j.peva.2011.07.002 27. Nobel, R.: Retrial queueing models in discrete time: a short survey of some late arrival models. Ann. Oper. Res. 247(1), 37–63 (2015). https://doi.org/10.1007/s10479-015-1904-7 28. Ozawa, T., Kobayashi, M.: Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process. Queueing Systems 90(3–4), 351–403 (2018) https://doi.org/10.1007/s11134-018-9586-x

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29. Pechinkin, A., Razumchik, R.: Waiting characteristics of queueing system Geo/Geo/1 with negative claims and a bunker for superseded claims in discrete time. In: International Congress on Ultra Modern Telecommunications and Control Systems, Moscow, pp. 1051–1055 (2010). https://doi.org/10.1109/ICUMT.2010.5676508 30. Pechinkin, A.V., Razumchik, R.V.: Queueing Systems in Discrete Time. Fizmatlit, Moscow (2018, in Russian). ISBN:978-5-9221-1791-3 31. Razumchik, R.: Analysis of finite capacity queue with negative customers and bunker for ousted customers using Chebyshev and Gegenbauer polynomials. Asia-Pac. J. Oper. Res. 31(04), 1450029 (2014). https://doi.org/10.1142/S0217595914500298 32. Razumchik, R.: Algebraic method for approximating joint stationary distribution in finite capacity queue with negative customers and two queues. Inf. Appl. 9(4), 68–77 (2015). https:// doi.org/10.14357/1992264150407 33. Razumchik, R., Zaryadov, I.: Stationary blocking probability in multi-server finite queuing system with ordered entry and Poisson arrivals. In: Vishnevsky V., Kozyrev D. (eds.) Distributed Computer and Communication Networks. DCCN 2015. Communications in Computer and Information Science, vol. 601, pp. 344–357. Springer, Cham (2016). https://doi.org/10.1007/ 978-3-319-30843-2_36 34. Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation. Discrete-Time Systems, vol. 3. North-Holland, New York (1993) 35. Ushakumari, P.V., Krishnamoorthy, A.: The queueing system BD /GD /∞. Optim. J. Math. Program. Oper. Res. 34(2), 185–193 (1995). https://doi.org/10.1080/02331939508844104

The Polaron Measure Chiranjib Mukherjee and S. R. S. Varadhan

Abstract {x(t) − x(s)} are the increments of the three dimensional Brown e−|t−s| ian motion over the intervals [s, t]. F (T , ω) = −T ≤s i | C = c − 1)P(C = c − 1).

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Thus, we have P(N − C > i | C = c − 1) . E[N − C | C = c − 1]

qc−1,i =

It should be noted that under the condition C = c − 1, N − C represents the number of customers in the system that are not receiving the service. Thus, qc−1,i (i = 0, 1, 2, . . . ) is the probability that a waiting customer find i other customers waiting in front of him under the condition that c − 1 servers are active (See Burke [4]). Let QRes denote the discrete random variable following this distribution. Thus our decomposition result is summarized as follows: d

(c)

Q(c) = QON−−I DLE + QRes . Remark 8 Tian et al. [29, 31] obtain a similar result for a multiserver model with Poisson arrival and vacation, i.e., αi = (c − i)α and β(z) = z. However, the random variable with the distribution qc−1,i here is not given a clear physical meaning in [29, 31].

6 Performance Measures We derive the power consumption and the mean queue length for our model and the corresponding one without setup time.

6.1 Power Consumption The cost per unit time for states SETUP, ON, and IDLE of a server are denoted by Cset up , Crun , and Cidle , respectively. The power consumption of our system is given by * POn−−off = Cset up 1 −

c−1 

+ πi,i − Πc (1) + Crun cρ,

i=0

where cρ = λ/μ is the mean number of running servers. For comparison, we also plot the curves for the conventional M/M/c queue under the same setting. It should be noted that in the conventional M/M/c system, an idle server is not turned off. As a result, the cost for power consumption is given by POn−−idle = Crun cρ + Cidle (c − cρ).

Batch Arrival Multiserver Queue with State-Dependent Setup

435

6.2 Mean Queue Length The mean number of waiting customers for our model is given by E[QOn−−off ] =

c 

Π " (1).

i=0

Let E[QOn−−idle ] denote the mean queue length MX /M/c queue without setup time which could be obtained from the analysis in [8].

7 Numerical Experiments In this section, we consider the case where αi = α, i.e., staggered setup policy. Furthermore, we consider fixed batch size, i.e., β(z) = zk for k = 1, 2, . . . . In this case ρ = kλ/μ. It means that a batch consists of k customers. In all the figures, the curves for On–Idle policy are indicated by “On–Idle” and other curves are of the On–Off model. Furthermore, we fix μ = 1, c = 10 in all the numerical examples.

7.1 Power Consumption Against ρ We set the costs as follows: Cset up = 1, Crun = 1, and Cidle = 0.6. In this section we investigate the power consumption against the traffic intensity. Figures 2, 3, and 4 show the power consumption against the traffic intensity for α = 0.1, 1, and 10, respectively. We observe from these three figures that the On–Off policy always outperform the On–Idle policy. However, from the performance point of view, the waiting time in the former is expected to be longer than the latter. Thus, we will investigate the impact of setup time on the total cost of the system next section. An important observation is that keeping the traffic intensity the same, power consumption decreases with the batch size k. This suggests that it is more efficient to design the system where customers arrive in group with a large batch size. Furthermore, we also observe from these figures that the power consumption decreases with α.

7.2 Power Consumption Against α In this section, we set the costs as follows: Cset up = 5, Crun = 1, and Cidle = 0.6. It should be noted that in this setting the power consumption of a setup server is five times bigger than that of a running server. Figure 5 shows the power consumption

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T. Phung-Duc 10

9.5

System Power Consumption

9

8.5

8

7.5

7 ON-IDLE k=1 k=3 k=5 k=7 k=9

6.5

6 0.5

0.6

0.7

0.8

0.9

1

Traffic Intensity

Fig. 2 Power consumption against ρ (α = 0.1)

10

9.5

System Power Consumption

9

8.5

8

7.5

7 ON-IDLE k=1 k=3 k=5 k=7 k=9

6.5

6 0.5

0.6

0.7

0.8 Traffic Intensity

Fig. 3 Power consumption against ρ (α = 1)

0.9

1

Batch Arrival Multiserver Queue with State-Dependent Setup

437

10

9.5

System Power Consumption

9

8.5

8

7.5

7 ON-IDLE k=1 k=3 k=5 k=7 k=9

6.5

6 0.5

0.6

0.7

0.8

0.9

1

Traffic Intensity

Fig. 4 Power consumption against ρ (α = 10)

k = 1, ON-IDLE k=1 k = 3, ON-IDLE k=3 k = 5, ON-IDLE k=5 k = 7, ON-IDLE k=7 k = 9, ON-IDLE k=9

14

Total Power Cost

12

10

8

6

0.01

0.1

1 Setup rate (D)

Fig. 5 Power consumption against α (ρ = 0.5)

10

100

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against α for ρ = 0.5. We observe that the power consumption increases with the setup rate α. Furthermore, there exists some threshold αT such that the On–Off model is more power-saving than the On–Idle model if α > αT . However, if α < αT , it is more power-saving to keep the server idle even when there is no waiting job.

7.3 Queue Length Figure 6 shows the queue length against the setup rate α. It should be noted that for the model without setup, the queue length does not depend on α. We observe that the queue length of the model with setup time decreases with the setup rate and tends to the queue length of the On–Idle model. Furthermore, the queue length increases with the batch size k. 1000 k = 1, ON-IDLE k=1 k = 3, ON-IDLE k=3 k = 5, ON-IDLE k=5 k = 7, ON-IDLE k=7 k = 9, ON-IDLE k=9

Queue Length

100

10

1

0.1

0.01 0.01

0.1

Fig. 6 Queue length against α (ρ = 0.5)

1 Setup rate (D)

10

100

Batch Arrival Multiserver Queue with State-Dependent Setup

439

8 Concluding Remarks In this paper, we have considered the MX /M/c queueing system with state-dependent setup rates. A server is turned off immediately after serving a job and there is no waiting customer. If there are some waiting customers, OFF servers are turned on according to a policy such that the setup rate depends on the number of active servers. This policy covers two important cases: the staggered setup policy where the servers are setup one by one and the vacation model where the server goes for vacation once it has no job to process and returns to normal mode after the vacation time. Using a generating function approach, we have obtained the generating functions of the queue length. We also have obtained recursive formulae for computing the factorial moments of the number of waiting jobs. Numerical experiments have shown some insights into the performance of the system. Furthermore, it is also important to consider the case where a fixed number of servers are always kept ON in order to reduce the delay of customers. It is also interesting to find the relation between the decomposition formula in this paper with that of Fuhrmann and Cooper [10]. We have obtained generating functions for the joint queue lengths. A possible future work may be to obtain the tail asymptotic for the joint queue lengths.

References 1. Adan, I.J., Van Leeuwaarden, J.S.H., Winands, E.M.: On the application of Rouche’s theorem in queueing theory. Oper. Res. Lett. 34, 355–360 (2006) 2. Artalejo, J.R., Economou, A., Lopez-Herrero, M.J.: Analysis of a multiserver queue with setup times. Queue. Syst. 51, 53–76 (2005) 3. Barroso, L.A., Holzle, U.: The case for energy-proportional computing. Computer 40, 33–37 (2007) 4. Burke, P.J.: Delays in single-server queues with batch input. Oper. Res. 23, 830–833 (1975) 5. Chen, Y., Das, A., Qin, W., Sivasubramaniam, A., Wang, Q., Gautam, N.: Managing server energy and operational costs in hosting centers. ACM SIGMETRICS Perform. Eval. Rev. 33, 303–314 (2005) 6. Choudhury, G.: On a batch arrival Poisson queue with a random setup time and vacation period. Comput. Oper. Res. 25, 1013–1026 (1998) 7. Choudhury, G.: An M X /G/1 queueing system with a setup period and a vacation period. Queue. Syst. 36, 23–38 (2000) 8. Cromie, M.V., Chaudhry, M.L., Grassmann, W.K.: Further results for the queueing system M X /M/c. J. Oper. Res. Soc. 30, 755–763 (1979) 9. Dean, J., Ghemawat, S.: MapReduce: simplified data processing on large clusters. Commun. ACM 51(1), 107–113 (2008) 10. Fuhrmann, S.W., Cooper, R.B.: Stochastic decompositions in the M/G/1 queue with generalized vacations. Oper. Res. 33(5), 1117–1129 (1985) 11. Gandhi, A., Harchol-Balter, M.: How data center size impacts the effectiveness of dynamic power management. In: Proceedings of 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1164–1169 (2011)

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12. Gandhi, A., Harchol-Balter, M.: M/G/k with staggered setup. Oper. Res. Lett. 41, 317–320 (2013) 13. Gandhi, A., Harchol-Balter, M., Adan, I.: Decomposition results for an M/M/k with staggered setup. ACM SIGMETRICS Perform. Eval. Rev. 38, 48–50 (2010) 14. Gandhi, A., Harchol-Balter, M., Adan, I.: Server farms with setup costs. Perform. Eval. 67, 1123–1138 (2010) 15. Gandhi, A., Gupta, V., Harchol-Balter, M., Kozuch, M.A.: Optimality analysis of energyperformance trade-off for server farm management. Perform. Eval. 67, 1155–1171 (2010) 16. Gandhi, A., Harchol-Balter, M., Kozuch, M.A.: The case for sleep states in servers. In: Proceedings of the 4th Workshop on Power-Aware Computing and Systems (2011). Article no. 2 17. Greenberg, A., Hamilton, J., Maltz, D.A., Patel, P.: The cost of a cloud: research problems in data center networks. ACM SIGCOMM Comput. Commun. Rev. 39, 68–73 (2008) 18. Mazzucco, M., Dyachuk, D.: Balancing electricity bill and performance in server farms with setup costs. Future Gener. Comput. Syst. 28, 415–426 (2012) 19. Meisner, D., Gold, B.T., Wenisch, T.F.: PowerNap: eliminating server idle power. ACM Sigplan Not. 44, 205–216 (2009) 20. Mitrani, I.: Service center trade-offs between customer impatience and power consumption. Perform. Eval. 68, 1222–1231 (2011) 21. Mitrani, I.: Trading power consumption against performance by reserving blocks of servers. In: Computer Performance Engineering. Springer, Berlin (2013), pp. 1–15 22. Mitrani, I.: Managing performance and power consumption in a server farm. Ann. Oper. Res. 202, 121–134 (2013) 23. Phung-Duc, T.: Impatient customers in power-saving data centers. In: Proceedings of 21th International Conference on Analytical and Stochastic Modeling Techniques and Applications (ASMTA 2014), Lecture Notes in Computer Science LNCS 8499. Springer, Cham (2014), pp. 185–199 24. Phung-Duc, T.: Server farms with batch arrival and staggered setup. In: Proceedings of the Fifth Symposium on Information and Communication Technology (SoICT). ACM, New York (2014), pp. 240–247 25. Phung-Duc, T.: Multiserver queues with finite capacity and setup time. In: International Conference on Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science LNCS 9081. Springer, Cham (2015), pp. 173–187 26. Phung-Duc, T.: Exact solutions for M/M/c/setup queues. Telecommun. Syst. 64(2), 309–324 (2017) 27. Schwartz, C., Pries, R., Tran-Gia, P.: A queuing analysis of an energy-saving mechanism in data centers. In: Proceedings of International Conference on Information Networking (ICOIN) (2012), pp. 70–75 28. Takagi, H.: Priority queues with setup times. Oper. Res. 38, 667–677 (1990) 29. Tian, N., Li, Q.L., Gao, J.: Conditional stochastic decompositions in the M/M/c queue with server vacations. Stoch. Models 15, 367–377 (1999) 30. Wolfgang, B.: Analysis of M/G/1-queues with setup times and vacations under six different service disciplines. Queue. Syst. 39, 265–301 (2001) 31. Zhang, Z.G., Tian, N.: Analysis of queueing systems with synchronous single vacation for some servers. Queue. Syst. 45, 161–175 (2003)

Weak Convergence of Probability Measures of Trotter–Kato Approximate Solutions of Stochastic Evolution Equations T. E. Govindan

Abstract The paper considers semilinear stochastic evolution equations in real Hilbert spaces. The goal here is to establish the weak convergence of probability measures induced by mild solutions of Trotter–Kato approximating equations. Keywords Stochastic evolution equations in infinite dimensions · Existence and uniqueness of a mild solution · Trotter–Kato approximations · Weak convergence of probability measures 2000 Mathematics Subject Classification 60H10

1 Introduction Stochastic evolution equations (SEEs) in infinite dimensions have been investigated by several authors, see Ichikawa [8], Da Prato and Zabczyk [2], and Govindan [7] and the references cited therein for details. SEEs are well known to model real world problems arising from many areas of science, engineering, and finance. The aim of this paper is to study weak convergence of probability measures induced by Trotter–Kato approximate mild solutions of SEEs in a real separable Hilbert space X of the form, see, for instance, Govindan [6]: dx(t) = [Ax(t) + f (t, x(t))]dt + g(t, x(t))dw(t),

t > 0,

x(0) = x0 ,

(1.1) (1.2)

where A is the infinitesimal generator of a strongly continuous semigroup {S(t) : t ≥ 0} of bounded linear operators on X; f : R + × X → X (R + = [0, ∞)),

T. E. Govindan () Department of Mathematics, ESFM-IPN, Mexico City, Mexico © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_26

441

442

T. E. Govindan

g : R + ×X → L(Y, X) and {w(t), t ≥ 0} is a Y -valued Wiener process. In Eq. (1.2), x0 is an X-valued random variable F0 -measurable and satisfies E|x0 |p < ∞, p ≥ 2. Kunze and van Neerven [10] and Govindan [6] studied Trotter–Kato approximations of Eq. (1.1). Such approximations have been considered earlier for other classes of stochastic equations, see Kannan and Bharucha-Reid [9] and Govindan [4, 5]. Motivated by Kannan and Barucha-Reid [9] and Govindan [4], the approximation result given in Theorem 3.2 in Govindan [6], see Sect. 3 below, can be used to derive another approximation result, that is, the weak convergence of a probability measure Pn induced by a mild solution of Trotter–Kato approximating equation to the probability measure P induced by the mild solution of Eq. (1.1). So, the objective of this paper is to consider weak convergence of induced probability measures. The rest of the paper is organized as follows: In Sect. 2, we give the preliminaries and essentially work in the framework of Ichikawa [8] and Govindan [6]. The main results are presented in Sect. 3. An example is given in Sect. 4.

2 Preliminaries Let X, Y be a pair of real separable Hilbert spaces and L(Y, X) the space of bounded linear operators mapping Y into X. For convenience, we shall use the notations | · | and (·, ·) for norms and scalar products for both the Hilbert spaces. We write L(X) for L(X, X). Let (, F , P ) be a complete probability space. A map x :  → X is a random variable if it is strongly measurable. Let x :  → X be a square integrable random variable, that is, x ∈ L2 (, F , P ; X). The covariance operator of the random element x is Cov[x] = E[(x − Ex) ◦ (x − Ex)], where E denotes the expectation and g ◦ h ∈ L(X) for any g, h ∈ X is defined by (g ◦ h)k = g(h, k), k ∈ X. Then Cov[x] is a self-adjoint nonnegative trace class (or nuclear) operator and trCov[x] = E|x − Ex|2, where tr denotes the trace. The joint covariance of any pair {x, y} ⊂ L2 (, F , P ; X), is defined as Cov[x, y] E[(x − Ex) ◦ (y − Ey)]. Let I be a subinterval of [0, ∞). A stochastic process {x} with values in X is a family of random variables {x(t), t ∈ I }, taking values in X. Let Ft , t ∈ I , be a family of increasing sub σ -algebras of the sigma algebra F . A stochastic process {x(t), t ≥ 0} is adapted to Ft if x(t) is Ft measurable for all t ∈ I. A stochastic process {w(t), t ≥ 0} in a real separable Hilbert space Y is a Wiener process if (a) w(t) ∈ L2 (, F , P ; Y ) and Ew(t) = 0 for all t ≥ 0, (b) Cov[w(t) − w(s)] = (t − s)W, W ∈ L+ 1 (Y ) is a nonnegative nuclear operator, (c) w(t) has continuous sample paths, and (d) w(t) has independent increments. The operator W is called the incremental covariance  (operator) of the Wiener process ∞ w(t). Then w has the representation w(t) = n=1 βn (t)en , where {en }(n = 1, 2, 3, . . .) is an orthonormal set of eigenvectors of W, βn (t), n = 1, 2, 3, . . .

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

443

are mutually independent real-valued  Wiener processes with incremental covariance λn > 0, W en = λn en and trW = ∞ n=1 λn . In the sequel, we will use the notation A ∈ G(M, α) for an operator A which is the infinitesimal generator of a C0 -semigroup {S(t) : t ≥ 0} of bounded linear operators on X satisfying ||S(t)|| ≤ M exp(αt), t ≥ 0 for some positive constants M ≥ 1 and α, where || · || denotes the operator norm. Now we make the system (1.1)–(1.2) more precise: Let A : D(A) ⊆ X → X (D(A) is the domain of A) be the infinitesimal generator of a strongly continuous semigroup {S(t) : t ≥ 0} in X. Let the functions f and g with f : R + × X → X, and g : R + × X → L(Y, X) be Borel measurable maps. Next, we introduce the notion of a mild solution for the system (1.1)–(1.2). Definition 2.1 A stochastic process x : [0, T ] → X defined on the probability space (, F , P ) is called a mild solution of Eq. (1.1) if (i) x is jointly measurable and Ft -adapted and its restriction to the interval [0, T ] T satisfies 0 |x(t)|2 dt < ∞, P − a.s., and (ii) x(t) satisfies the integral equation 

t

x(t) = S(t)x0 +

S(t − s)f (s, x(s))ds

0



t

+

S(t − s)g(s, x(s))dw(s),

t ∈ [0, T ],

P − a.s..

0

Note that the second integral in the last equality is the Itô stochastic integral. For the definition and properties of this integral, see Ichikawa [8]. See also Da Prato and Zabczyk [2] and Govindan [7].

3 Weak Convergence of Probability Measures In this section, we shall establish weak convergence of induced probability measures associated with the Trotter–Kato approximations of Eq. (1.1). Let us state the following basic assumptions used in the rest of the paper. Hypothesis (H1) The nonlinear functions f (t, x) and g(t, x) satisfy the following Lipschitz and linear growth conditions for all t ≥ 0: For p ≥ 2, |f (t, x) − f (t, y)| ≤ L1 |x − y|,

L1 > 0,

x, y ∈ X,

|g(t, x) − g(t, y)| ≤ L2 |x − y|,

L2 > 0,

x, y ∈ X,

|f (t, x)|p ≤ L3 (1 + |x|p ),

L3 > 0,

x ∈ X,

|g(t, x)| ≤ L4 (1 + |x| ),

L4 > 0,

x ∈ X.

p

p

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Note that the constants Li , i = 1, 2, 3, 4 do not depend on t. Theorem 3.1 (Govindan [6]) Let A ∈ G(M, α) and the assumption (H1) hold. Then, Eq. (1.1) has a unique mild solution x ∈ C([0, T ], Lp (, X)), p ≥ 2. Moreover, for any p ≥ 1, we have sup0≤t ≤T E|x(t)|2p ≤ kp,T (1 + E|x0 |2p ), where kp,T is a positive constant. Consider the Trotter–Kato approximations of Eq. (1.1): dxn (t) = [An xn (t) + f (t, xn (t))]dt + g(t, xn (t))dw(t),

t > 0,

(3.1)

xn (0) = x0 , where An , n = 1, 2, 3, . . . , is the infinitesimal generator of a strongly continuous semigroup {Sn (t) : t ≥ 0} of bounded linear operators on X. For each n ≥ 1, as before, one can define a mild solution xn ∈ C([0, T ], Lp (, X)), p ≥ 2 of Eq. (3.1) so that xn (t) satisfies the stochastic integral equation  t xn (t) = Sn (t)x0 + Sn (t − s)f (s, xn (s))ds 0



t

+

Sn (t − s)g(s, xn (s))dw(s),

t ∈ [0, T ],

P − a.s..

0

The following hypothesis is needed to consider the next result. Hypothesis (H2) (i) Let An ∈ G(M, α) for each n = 1, 2, 3, . . . , (ii) As n → ∞, An x → Ax for every x ∈ D, where D is a dense subset of X, (iii) There exists a γ with Reγ > α for which (γ I − A)D is dense in X, then the closure A of A is in G(M, α). A consequence of the Trotter–Kato theorem is the following. Proposition 3.1 (Pazy [11], Theorem 4.5, p. 88) Let the hypothesis (H2) hold. If Sn (t) and S(t) are the C0 -semigroups generated by An and A, respectively, then lim Sn (t)x = S(t)x,

x ∈ X,

n→∞

(3.2)

for all t ≥ 0, and the limit in (3.2) is uniform in t for t in bounded intervals. Theorem 3.2 (Govindan [6]) Suppose that the hypotheses (H1) and (H2) are satisfied. Then, there exists a unique mild solution xn in C([0, T ], Lp (, X)) for each n = 1, 2, 3, . . . , of Eq. (3.1), and for each T > 0, sup E|xn (t) − x(t)|2 → 0

0≤t ≤T

where x(t) is a mild solution of Eq. (1.1).

as

n → ∞,

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

445

As mentioned in the introduction, the Trotter–Kato approximation result given in Theorem 3.2 can be used to derive another interesting approximation result. For this, first observe that the solutions x and xn , n = 1, 2, 3, . . . , are elements of C([0, T ], Lp (, X)), p ≥ 2. Let P and Pn be the probability measures on C([0, T ], Lp (, X)) induced by x and xn , respectively. We shall show in what follows that Pn converges weakly to P as n → ∞. Towards this, note that Theorem 3.2 implies that every finite dimensional (joint) distribution of Pn converges weakly to the corresponding one of P . Our claim will be proved once we establish the tightness of the family {Pn , n = 1, 2, 3, . . .}. In order to prove the weak convergence result, we need to make assumptions different from Hypothesis (H2). First observe that a closed operator A generates a strongly continuous analytic semigroup on a Banach space X if and only if A is densely defined and sectorial, that is, there exist M ≥ 1 and ω ∈ R, the real line, such that {λ ∈ C : Reλ > ω} is contained in the resolvent set ρ(A) and sup ||(λ − ω)R(λ, A)|| ≤ M,

(3.3)

Reλ>ω

the constants M and ω are called the sectoriality constants of A; in this context, we say A is sectorial of type (M, ω). We now make the following further assumptions, see Kunze and van Neerven [10]: Hypothesis (H3) (i) The operators A and An , for each n = 1, 2, 3, . . . , are densely defined, closed, and uniformly sectorial on X in the sense, there exist numbers M ≥ 1 and ω ∈ R such that A and each An is sectorial of type (M, ω). (ii) The operators An converge to A in the strong resolvent sense: lim R(λ, An )x = R(λ, A)x

n→∞

for all Reλ > ω and x ∈ X. Under the Hypothesis (H3)(i), the operators A and An generate strongly continuous analytic semigroups S(t) and Sn (t), respectively, satisfying the uniform bounds ||S(t)||, ||AS(t)||,

||Sn (t)|| ≤ Meωt ,

||An Sn (t)|| ≤

M" t

eωt ,

t ≥ 0, t > 0.

The following Trotter–Kato type approximation theorem is well known. For the proof of part (i), see Arendt et al. [1, Theorem 3.6.1] and for part (ii), see Kunze and van Neerven [10].

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Proposition 3.2 (Kunze and van Neerven [10]) Let the Hypothesis (H3) hold. (i) For all t ∈ [0, ∞) and x ∈ X, we have lim Sn (t)x → S(t)x,

n→∞

and the convergence is uniform on compact subsets of [0, ∞) × X. (ii) For all t ∈ (0, ∞) and x ∈ X, we have lim An Sn (t)x → AS(t)x,

n→∞

and the convergence is uniform on compact subsets of (0, ∞) × X. Theorem 3.3 Let the Hypotheses (H1) and (H3) hold. Then, there exists a unique mild solution xn in C([0, T ], Lp (, X)) for each n = 1, 2, 3, . . . , of Eq. (3.1), and for each T > 0, sup E|xn (t) − x(t)|2 → 0

0≤t ≤T

as

n → ∞,

where x(t) is a mild solution of Eq. (1.1). Moreover, Pn converges weakly to P as n → ∞. Proof The existence and uniqueness of a mild solution xn of Eq. (3.1) in C([0, T ], Lp (, X)) for each n = 1, 2, 3, . . . , of Eq. (3.1), follows from Theorem 3.1. Next, the proof of sup E|xn (t) − x(t)|2 → 0

0≤t ≤T

as

n → ∞,

is exactly identical as in Theorem 3.2 wherein Proposition 3.1 was employed to show that sup E|Sn (t)x − S(t)x|2 → 0

0≤t ≤T

as

n → ∞,

(3.4)

for all t ≥ 0, x ∈ X, and the limit in (3.4) is uniform in t for t in bounded intervals. However, to prove our theorem, we shall employ Proposition 3.2 (i) to show (3.4) instead of Proposition 3.1. It remains to show that Pn converges weakly to P as n → ∞. The proof is divided into three steps. Step 1

We claim that for each 0 < T < ∞, we have sup sup E|xn (t)|p < ∞. n 0≤t ≤T

(3.5)

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

447

To show this, consider for each n = 1, 2, 3, . . . , 

t

xn (t) = Sn (t)x0 +

Sn (t − s)f (s, xn (s))ds

0



t

+

Sn (t − s)g(s, xn (s))dw(s),

t ∈ [0, T ],

P − a.s..

0

Now using Lemma 1.9 from Ichikawa [8], it follows that E|xn (t)| ≤ 3 p

p−1

 t p    p  Sn (t − s)f (s, xn (s))ds  E|Sn (t)x0 | + E  0

 t p 5   + E  Sn (t − s)g(s, xn (s))dw(s) 0  ≤ 3p−1 M p exp(pωT )E|x0|p 

t

+ T p−1 M p exp(pωT )E

|f (s, xn (s))|p ds

0



t

+ c(p, T )M exp(pωT )E p

5 |g(s, xn (s))| ds , p

0

where c(p, T ) > 0 is a constant. Hence Hypothesis (H1) yields  E|xn (t)|p ≤ 3p−1 M p exp(pωT )E|x0 |p  + M p exp(pωT )[T p−1 L3 + c(p, T )L4 ]  p−1 ≤3 M p exp(pωT )E|x0 |p

t

5 (1 + E|xn (s)|p )ds

0

+ M p exp(pωT )[T p−1 L3 + c(p, T )L4 ]T 5  t p p−1 p + M exp(pωT )[T L3 + c(p, T )L4 ] E|xn (s)| ds , 0

for each n = 1, 2, 3, . . .. An appeal to Bellman–Gronwall’s lemma proves the claim (3.5). Step 2 For an arbitrarily fixed 0 < T < ∞, we claim that, for each n = 1, 2, 3, . . . and 0 ≤ s < t ≤ T , there exists a constant C > 0 such that E|xn (t) − xn (s)|4 ≤ C(t − s)2 .

(3.6)

448

T. E. Govindan

First, by Theorem 2.4 from Pazy [11], we have 

t

|(Sn (t) − Sn (s))x0 |2 ≤ T

|Sn (u)An x0 |2 du

s

≤ T (M exp(ωT ))2 E|x0 |2 (t − s). So, |(Sn (t) − Sn (s))x0 |4 ≤ C1 (t − s)2 ,

(3.7)

for some constant C1 > 0. Next, consider 



t

s

Sn (t − u)f (u, xn (u))du −

0

Sn (s − u)f (u, xn (u))du

0



s

=



t

[Sn (t − u) − Sn (s − u)]f (u, xn (u))du +

0

Sn (t − u)f (u, xn (u))du

s

= I1 + I2 ,

say.

Now, by Hypothesis (H1), we get 

t

E|I2 |4 ≤ (t − s)3 E

M 4 e4ω(t −u)|f (u, xn (u))|4 du

s



t

≤ (t − s)3 M 4 exp(4ωT )E

  L3 1 + |xn (u)|4 du

s 4

≤ T M exp(4ωT )L3 (T + sup E|xn (t)|4 )(t − s)2 , 0≤t ≤T

and   E|I1 |4 = E 



s 0

t −u s−u

 t   3  ≤T E  0

 t   3  ≤T E  0

4  Sn (v)An f (u, xn (u))dvdu

t −u s−u

4  Sn (v)An f (u, xn (u))dv  du

4  Sn (v + s − u)An f (u, xn (u))dv  du * +

t −s 0

≤ T 5 M " e4ωT L3 T + sup E|xn (t)|4 (t − s)2 . 4

0≤t ≤T

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

449

Hence, from Step 1,  t    E Sn (t − u)f (u, xn (u))du − 0

s 0

4  Sn (s − u)f (u, xn (u))du ≤ C2 (t − s)2 , (3.8)

where C2 > 0 is a constant. Lastly, consider the stochastic integral term: 

t

 Sn (t − u)g(u, xn (u))dw(u) −

0

s

Sn (s − u)g(u, xn (u))dw(u)

0



s

=

[Sn (t − u) − Sn (s − u)]g(u, xn (u))dw(u)

0



t

+

Sn (t − u)g(u, xn (u))dw(u)

s

= I3 + I4 ,

say.

By applying Lemma 7.2 from Da Prato and Zabczyk [2, p. 182] and exploiting Hypothesis (H1), we get $

t

E|I4 |4 ≤ KE

%2 |Sn (t − u)g(u, xn (u))|2 du

s

$

t

≤ KM 4 exp (4aT )E

%2 |g(u, xn (u))|2 du

s

≤ 2KM exp (4aT )T L4 (T + sup E|xn (t)|4 )(t − s)2 , 4

2

0≤t ≤T

where K > 0 is a constant, and   E|I3 |4 = E 

s 0



t −u

s−u

$ t    ≤ KE  0

$ t    ≤ KE  0

t −u s−u t −s 0

" 4 4ωT

≤ T KM e 2

4  Sn (v)An g(u, xn (u))dvdw(u) 2 %2  Sn (v)An g(u, xn (u))dv  du 2 %2  Sn (v + s − u)An g(u, xn (u))dv  du

  L4 T + sup E|xn (t)|4 (t − s)2 , 0≤t ≤T

450

T. E. Govindan

Thus, using (3.5), we obtain  t  E  Sn (t − u)g(u, xn (u))dw(u) 0



s

− 0

4  Sn (s − u)g(u, xn (u))dw(u) ≤ C3 (t − s)2 ,

(3.9)

where C3 > 0 is a constant. Combining all the estimates (3.7)–(3.9), the claim (3.6) follows. Step 3 In Step 2, it was shown that xn (t) converges to x(t) uniformly on compact intervals of [0, ∞) as n → ∞. Then, the family {Pn } of probability measures is tight on C([0, T ], Lp (, X)). This together with Theorem 3.2 implies that Pn → P weakly on C([0, T ], Lp (, X)). The proof is complete. As an application of Theorem 3.3, we consider a classical limit theorem on the dependence of the stochastic evolution equation (1.1) on a parameter. For this, we shall follow Gikhman and Skorokhod [3, pp. 50–54]. Consider the family of stochastic evolution equations dxn (t) = [An xn (t) + fn (t, xn (t))]dt + gn (t, xn (t))dw(t),

t > 0,

(3.10)

xn (0) = x0 , where An , n = 1, 2, 3, . . . is the infinitesimal generator of a strongly continuous semigroup {Sn (t) : t ≥ 0} of bounded linear operators on X. For each n = 1, 2, 3, . . . , one can define a mild solution xn ∈ C([0, T ]; Lp (, X)), p ≥ 2 as before that satisfies the stochastic integral equation 

t

xn (t) = Sn (t)x0 +

Sn (t − s)fn (s, xn (s))ds

0



t

+

Sn (t − s)gn (s, xn (s))dw(s),

t ∈ [0, T ],

P − a.s..

0

Hypothesis (H4) For each n = 1, 2, 3, . . . , the nonlinear functions fn (t, x) and gn (t, x) satisfy the following Lipschitz and linear growth conditions for all t ≥ 0: For p ≥ 2, |fn (t, x) − fn (t, y)| ≤ L"1 |x − y|,

L"1 > 0,

x, y ∈ X,

|gn (t, x) − gn (t, y)| ≤ L"2 |x − y|,

L"2 > 0,

x, y ∈ X,

|fn (t, x)| ≤ p

|gn (t, x)|p ≤

L"3 (1 + |x|p ), L"4 (1 + |x|p ),

L"3 L"4

> 0,

x ∈ X,

> 0,

x ∈ X.

Note that the constants L"i , i = 1, 2, 3, 4 do not depend on t.

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

451

We now make the following further assumption. See Gikhman and Skorokhod [3, p. 52]. Hypothesis (H5) For each N > 0, sup |fn (t, x) − f (t, x)| → 0

|x|≤N

and

sup |gn (t, x) − g(t, x)| → 0

|x|≤N

as n → ∞ for each t ∈ [0, T ]. Theorem 3.4 Suppose that the hypotheses (H1), (H3), (H4), and (H5) hold. Then, there exists a unique mild solution xn in C([0, T ], Lp (, X)) for each n = 1, 2, 3, . . . , of Eq. (3.10), and for each T > 0, sup E|xn (t) − x(t)|2 → 0

0≤t ≤T

as

n → ∞,

where x(t) be the mild solutions of Eq. (1.1). Proof The existence and uniqueness of a mild solution xn of Eq. (3.10) in C([0, T ], Lp (, X)) for each n = 1, 2, 3, . . . , of Eq. (3.10), follows from Theorem 3.1. Next, the proof of sup E|xn (t) − x(t)|2 → 0

0≤t ≤T

as

n → ∞,

is exactly identical as in Theorem 4.1 from Govindan [6] wherein Proposition 3.1 was employed to show that sup E|Sn (t)x − S(t)x|2 → 0

0≤t ≤T

as

n → ∞,

(3.11)

for all t ≥ 0, x ∈ X, and the limit in (3.11) is uniform in t for t in bounded intervals. We shall instead employ Proposition 3.2 (i). This completes the proof. Let P and Pn∗ be the probability measures on C([0, T ], Lp (, X)) induced by the mild solution x of Eq. (1.1) and mild solution xn of Eq. (3.10), respectively. We shall show, in what follows, that Pn∗ converges weakly to P as n → ∞. Theorem 3.5 Let all the Hypotheses of Theorem 3.4 hold. Then Pn∗ converges weakly to P as n → ∞. Proof The proof follows as in Theorem 3.3 and is divided into three steps. Step 1

We claim that for each 0 < T < ∞, we have sup sup E|xn (t)|p < ∞. n 0≤t ≤T

(3.12)

452

T. E. Govindan

Consider for each n = 1, 2, 3, . . ., 

t

xn (t) = Sn (t)x0 +

Sn (t − s)fn (s, xn (s))ds

0



t

+

Sn (t − s)gn (s, xn (s))dw(s),

t ∈ [0, T ],

P − a.s..

0

As before, from Lemma 1.9 from Ichikawa [8] and Hypothesis (H4), for each n = 1, 2, 3, . . ., it follows that  E|xn (t)|p ≤ 3p−1 M p exp(pωT )E|x0 |p 7 6 + M p exp(pωT ) T p−1 L"3 + c(p, T )L"4 T 5  7 t 6 p−1 " p " p + M exp(pωT ) T L3 + c(p, T )L4 E|xn (s)| ds , 0

where c(p, T ) > 0 is a constant. An appeal to Bellman–Gronwall’s lemma proves (3.12). Step 2 For an arbitrarily fixed 0 < T < ∞, we claim that, for each n = 1, 2, 3, . . . and 0 ≤ s < t ≤ T , there exists a constant C " > 0 such that E|xn (t) − xn (s)|4 ≤ C " (t − s)2 .

(3.13)

As in Step 2 of Theorem 3.3, we have |(Sn (t) − Sn (s))x0 |4 ≤ C1" (t − s)2 ,

(3.14)

for some constant C1" > 0. Next, consider 

t



s

Sn (t − u)fn (u, xn (u))du −

0

 =

Sn (s − u)fn (u, xn (u))du

0 s



t

[Sn (t − u) − Sn (s − u)]fn (u, xn (u))du +

0

Sn (t − u)fn (u, xn (u))du

s

= J1 + J2 ,

say.

By Hypothesis (H4), we get * 4

E|J2 | ≤ T M

4

exp(4ωT )L"3

+ T + sup E|xn (t)| 0≤t ≤T

4

(t − s)2 ,

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

453

and   E|J1 |4 = E 

s



0

t −u s−u

" 4 4ωT

≤ T 5M e

4  Sn (v)An fn (u, xn (u))dvdu *

+

L"3 T + sup E|xn (t)|4 (t − s)2 . 0≤t ≤T

Hence, from Step 1,  t    E Sn (t − u)fn (u, xn (u))du − 0

s 0

4  Sn (s − u)fn (u, xn (u))du ≤ C2" (t − s)2 , (3.15)

where C2" > 0 is a constant. Lastly, consider 

t



s

Sn (t − u)gn (u, xn (u))dw(u) −

0

Sn (s − u)gn (u, xn (u))dw(u)

0



s

=

[Sn (t − u) − Sn (s − u)]gn (u, xn (u))dw(u)

0



+

t

Sn (t − u)gn (u, xn (u))dw(u)

s

= J3 + J4 ,

say.

By Lemma 7.2 from Da Prato and Zabczyk [2, p. 182] and exploiting Hypothesis (H4), we get E|J4 |4 ≤ 2K " M 4 exp (4aT )T 2 L"4 (T + sup E|xn (t)|4 )(t − s)2 , 0≤t ≤T

where K " > 0 is a constant, and   E|J3 | = E 

s

4

0



t −u s−u

4  Sn (v)An gn (u, xn (u))dvdw(u)

" 4 4ωT

≤ T 2 KM e

L"4 (T + sup E|xn (t)|4 )(t − s)2 , 0≤t ≤T

454

T. E. Govindan

Thus, using (3.12), we obtain  t  E  Sn (t − u)gn (u, xn (u))dw(u) 0



−

s

0

4  Sn (s − u)gn (u, xn (u))dw(u) ≤ C3" (t − s)2 ,

(3.16)

where C3" > 0 is a constant. Combining all the estimates (3.14)–(3.16), (3.13) follows. Step 3 In this step, we use similar arguments as in Theorem 3.3. The proof is complete.

4 An Example In this section, we discuss an example from Kunze and van Neerven [10]. Consider the stochastic partial differential equation of the form: ∂z (t, x) = Az(t, x) + f (z(t, x)) ∂t +

K 

gk (z(t, x))

k=1

z(t, x) = 0,

x ∈ ∂O,

z(0, x) = ϕ(x),

wk (t), ∂t

x ∈ O,

t ≥ 0,

(4.1)

t ≥ 0,

x ∈ O,

where O is a bounded open domain in R d and wk (t) are independent real-valued standard Wiener processes. Here, A is the second-order divergence form differential operator defined by Az(x) =

%  $ d d d   ∂ ∂z ∂z (x) + bj (x) (x), aij (x) ∂xi ∂xj ∂xj i=1

j =1

j =1

whose coefficients a = (aij ) and b = (bj ) satisfy suitable boundedness and uniform ellipticity conditions. The functions f and gk are Lipschitz continuous. (s)| Let ||f ||Lip = supt =s |f (t|t)−f denote the Lipschitz seminorm of a function f . −s| Hypothesis (H6) Let a, an ∈ L∞ (O; R d×d ) and b, bn ∈ L∞ (O; R d ). Let f, fn , gk , gk,n : R → R be Lipschitz continuous. Assume that there exist finite constants κ, C such that: (i) a, an are symmetric and ax · x, an x · x ≥ κ|x|2 for all x ∈ R d ,

Weak Convergence of Probability Measures of Trotter–Kato Approximate. . .

(ii) (iii) (iv) (v)

455

||a||∞ , ||an ||∞ , ||b||∞ , ||bn ||∞ ≤ C, ||f ||Lip, ||fn ||Lip, ||gk ||Lip, ||gk,n ||Lip ≤ C, limn→∞ an = a, limn→∞ bn = b a.e. on O, and limn→∞ fn = f , limn→∞ gk,n = gk pointwise on O.

Similarly, An can be defined. Let Sn (·) and S(·) be strongly continuous analytic semigroups generated by An and A, respectively. In order to reformulate the above equation in the abstract setting on the Banach space Lr (O), 1 < r < ∞, we use a variational approach. Consider the sesquilinear form  a[u, v] := (a∇u) · ∇v + (b · ∇u)vdx O

on the domain D(a) := H01 (O). The sectorial operator A on Lr (O) associated with a generates a strongly continuous analytic semigroup {S(t) : t ≥ 0} extrapolates to a consistent family of strongly continuous analytic semigroups {S (r) (t) : t ≥ 0} on Lr (O). Let us denote (r) their generates by A(r). The forms an and the associated semigroups Sn (t) with (r) generators An are defined likewise. Lemma 4.1 (Kunze and van Neerven [10]) Let the Hypothesis (H6)(i), (ii), and (r) (iv) hold. Then, the operators A(r) and An satisfy Hypothesis (H3). Lemma 4.2 (Kunze and van Neerven [10]) Suppose that Hypothesis (H6) (iii) and (v) hold. Then (i) the maps f, fn : Lr (O) → Lr (O) defined by [f (z)](x) := f (z(x)),

[fn (z)](x) := fn (z(x)),

satisfy Hypotheses (H1) and (H4), and (ii) the maps g, gn : Lr (O) → L(R K , Lr (O)) defined by [g(z)h](x) :=

K  k=1

gk (z(x))(ek , h),

[gn (z)h](x) :=

K 

gk,n (z(x))(ek , h),

k=1

K where {ek }K k=1 is the standard unit basis of R , satisfy Hypotheses (H1) and (H4).

Hence, Eq. (4.1) can be expressed in the abstract setting as Eq. (1.1) with A, f, and g as defined above.

456

T. E. Govindan

Acknowledgement This research is supported by SIP from IPN, Mexico.

References 1. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser-Verlag, Basel (2001) 2. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) 3. Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972) 4. Govindan, T.E.: Autonomous semilinear stochastic Volterra integrodifferential equations in Hilbert spaces. Dyn. Syst. Appl. 3, 51–74 (1994) 5. Govindan, T.E.: Trotter-Kato approximations of semilinear stochastic evolution equations. Bol. Soc. Matemat. Mexicana 12, 109–120 (2006) 6. Govindan, T.E.: On Trotter-Kato approximations of semilinear stochastic evolution equations in infinite dimensions. Stat. Probab. Lett. 96, 299–306 (2015) 7. Govindan, T.E.: Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications. Probability Theory and Stochastic Modelling Series, vol. 79. Springer, Switzerland (2016) 8. Ichikawa, A.: Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl. 90, 12–44 (1982) 9. Kannan, D., Bharucha-Reid, A.T.: On a stochastic integrodifferential evolution equation of Volterra type. J. Integr. Eqns. 10, 351–379 (1985) 10. Kunze, M., van Neerven, J.M.A.M.: Approximating the coefficients in semilinear stochastic partial differential equations. J. Evol. Eqns. 11, 577–604 (2011) 11. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983

Stochastic Multiphase Models and Their Application for Analysis of End-to-End Delays in Wireless Multihop Networks Vladimir Vishnevsky and Andrey Larionov

Abstract This paper presents a study of applying open queueing networks with MAP /P H /1/N nodes for estimation of performance characteristics of wireless networks with linear topology using either relay, or DCF channels. Basic properties of such queueing networks are outlined along with Markovian arrival processes (MAPs) and phase-type (PH) distributions fitting methods. Due to exponential growth of the system state space in MAP /P H /1/N → · · · → •/P H /1/N queueing networks, the exact calculation of its characteristics is practically impossible for an arbitrary large number of nodes, and we propose an algorithm which finds approximated results by iterative estimations of node parameters using departure processes approximations with MAPs of smaller order. We use this approach to get numerical results, which are further compared with the data obtained by MonteCarlo method. The comparison shows that the results obtained by both methods are very close to each other, while the iterative approach requires significantly less time. The paper provides results of fitting transmission delays using PH distributions and end-to-end delays estimations for wireless networks with simple relay and IEEE 802.11 DCF channels. All numerical results are validated using a simulation model. Keywords DCF · Wireless relay networks · Markovian arrival processes · PH distributions · Queueing networks · Multihop wireless networks

1 Introduction Wireless networks are often used as permanent or temporary backbone networks, for example, for transmitting data from sensors, organizing communication along roads or pipelines, or for connecting base stations in cellular networks. Most often, radio relay or IEEE 802.11 channels are used to build such networks. In both cases, it is possible to build a network capable of transmitting sufficiently large amounts V. Vishnevsky · A. Larionov () V.A. Trapeznikov Institute of Control Sciences of RAS, Moscow, Russia © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_27

457

458

V. Vishnevsky and A. Larionov

Fig. 1 An example of a wireless multihop network with linear topology and its model

of information. However, errors that occur when transmitting signals in the air, interference from neighboring stations, as well as features of the channel access protocols can significantly affect the performance of the wireless networks. For its evaluation, one can use various methods—from the construction of test-beds with wireless equipment to analytical and simulation modeling. In cases where it is necessary to estimate network parameters such as end-to-end delays or station utilization, queueing models can be used. The use of queuing theory allows to abstract from the complex features of data transmission protocols and applications, describing the transmission time, intervals, and packet sizes with random functions. A typical example of a wireless network with linear topology and its queueing model is shown on Fig. 1. One of the promising ways to model telecommunication networks is open queuing networks MAP /P H /1/N → · · · → •/P H /1/N. In these queueing networks applications traffic is modeled with Markovian arrival processes (MAP) [13], and channel transmission delays with phase-type (PH) distributions. The use of Markovian arrival processes allows to take into account the correlation present in the traffic of real network applications [8, 12], while PH distributions provide enough means to model rather complex transmission delays. Various questions of applying these queueing systems for wireless networks performance evaluation were studied in previous work [20–22]. The key problem in MAP /P H /1/N → · · · → •/P H /1/N queueing network analysis is the exponential growth of the state space as the number of stations increases. Because of this, numerical methods have to be used for the queueing system properties estimation in case of large networks. The most widely used approach here is Monte-Carlo method, in which the results are estimated from the repeated sampling of the target characteristics values. Another possible approach, described in this paper, is departure processes replacement with approximated MAPs of smaller order. These approximated MAPs are used as arrival processes for the next node input, and have the same first moments and lags values as the original departure MAP of higher order. Below we use both approaches for tandem queueing network evaluation and compare the calculated end-to-end delays values. It will be shown, that iterative approximation approach provides good precision and requires significantly less time for computation.

Multiphase Models for Analysis of End-to-End Delays in Wireless Networks

459

Two types of wireless networks channels are considered in this paper: radio relay and channels operating under IEEE 802.11 DCF (Distributed Coordination Function). Radio relay channels use frequency duplexing, directed antennas and it is assumed that no collisions take place during transmissions; channel access is very similar to the mechanism used in wired networks. On the contrary, IEEE 802.11 DCF channel access is based on CSMA/CA scheme and assumes a competition with possible collisions. In case of relay channels, the transmission delay is defined by the payload transmission time plus constant intervals, preambles, and headers transmissions. In case of DCF channels, transmission delay is also defined with random channel listening intervals, acknowledgements, and possible packet retransmissions. For each type of the channel we find a PH distribution that models transmission delays. To do this, we run simulation models of the channels, collect transmission delays samples and use them to find the fitting PH distributions. These PH distributions are used in MAP /P H /1/N → · · · → •/P H /1/N queueing networks. For each queueing network we find end-to-end delays using Monte-Carlo method and iterative algorithm with departures approximation, and compare the results with end-to-end delays estimated from a multihop wireless network simulation. The paper is organized as follows. In Sect. 2 we outline the related work, in particular—approaches to estimation of transmission delays in wireless channels, and MAP and PH fitting methods. In Sect. 3 we briefly describe the relay and DCF channels and provide parameters values used in numerical experiment. Then, in Sect. 4 we outline MAP /P H /1/N systems, MAP/PH fitting methods and describe an iterative algorithm for estimation of tandem properties using departure processes approximations. Section 5 provides numerical experiments results and Sect. 6 concludes the paper.

2 Related Work Since this work aims at the performance analysis of multihop wireless networks using open queuing networks and methods for their approximate estimation, here we consider studies related to estimating the packets transmission delays in wireless channels, as well as methods of MAP flows and PH distributions fitting. A large number of papers are devoted to the performance analysis of IEEE 802.11 networks. In many cases the proposed methods are based on a Markov chain proposed by Bianchi in paper [2]. This chain is originally used by the author for the network throughput estimation in saturated mode, when there is always another packet to transmit. Based on this model, many methods have been developed for estimating the time of data transmission in DCF channels. For instance, in [1, 6, 15] transmission delays are analyzed when network operates in saturated mode, paper [19] takes into account queueing delay, and papers [7, 18] estimate delays when network is unsaturated. A significant number of papers are also devoted to the methods of fitting MAP flows [3–5, 9–11, 14, 16] and PH distributions [3, 17]. Some authors [3, 10, 16]

460

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focus on moments matching methods, others use different approaches based on EM (Expectation Maximization) algorithm [9, 14, 17]. In this paper, we will use the moments matching method to find a smaller order MAP flow when reducing the state space of the departure flow. For fitting PH distributions of the packets transmission delay, we will use G-FIT [17] method. Initial data for fitting PH distributions of packet transmission delay will be obtained analytically for a relay network and using a simulation model for a network with DCF channels. Note that instead of simulation modeling, in further studies one can use the estimates obtained from the works cited above.

3 Channel Access in Wireless Networks The basic channel access mechanism used in IEEE 802.11 networks from the earliest versions of the standard is Distributed Coordination Function (DCF). This mechanism is based on the CSMA/CA access scheme and involves listening to the channel for a random time before transmitting data. The success of the transmission is confirmed by the receiver, in the absence of confirmation, retransmission is performed. Transmission errors (collisions) may occur due to simultaneous transmissions of multiple stations. You can read more about DCF in the IEEE 802.11 standard, or in one of the many papers (e.g., [2]). In our model example, we consider a simplified version of DCF (see Fig. 2), not using the RTS/CTS mechanism, not limiting the number of retransmissions, and also ignoring the post-backoff, since its accounting would lead to dependencies of successive service times and the need to use Markovian service process (MSP) instead of PH distributions.

Fig. 2 Basic channel access under control of DCF

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Fig. 3 Basic channel access in wireless relay channel Table 1 Parameters values used in the experiment

Parameter PHY header MAC DATA header IP header MAC ACK frame Slot duration (σ ) IFS DIFS SIFS CWmin CWmax Bitrate IP header Payload avg. size Payload min. size Payload max. size Traffic rates for relay Traffic rates for DCF

Value 128 272 160 112 (+PHY header) 50 128 128 28 16 1024 1000 160 6344 2000 10,688 120,250,500 20,406,080,120,250

Unit bit bit bit bit μs μs μs μs – – kbps bit bit bit bit kbps kbps

Used in DCF, Relay DCF, Relay DCF, Relay DCF DCF Relay DCF DCF DCF DCF DCF, Relay DCF, Relay DCF, Relay DCF, Relay DCF, Relay Relay DCF

In radio relay channels (see Fig. 3) access is deterministic. In many cases it does not involve sensing the channel for a random time interval, interference with other stations is omitted; there are also no acknowledgements. In our model example we assume an ideal channel, i.e. if transmissions are not colliding, they are transmitted successfully (bit error rate, BER, is assumed zero). In a more general case, it is certainly desirable to use more realistic channel models. In our numerical experiment we are going to use channel access parameters values shown in Table 1. These parameters are basically the same as the values from [2], which was required for the model validation. However, higher-speed modern IEEE 802.11 versions can be modeled if these parameters are updated. We assumed that IFS (interframe space) for a relay channel is equal to DIFS (DCF interframe space). We also omit preambles since in the ideal channel they are simply an additional constant that can be counted in another interval (IFS for relay channel or DIFS for DCF). We also show the application traffic parameters in the bottom of the table. Note that uniformly distributed payload sizes are used in both models,

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while in relay networks modeling we will also use exponentially distributed and constant payloads with the same mean values.

4 Open Queueing Tandem Networks with MAP /P H/1/N Nodes In this section we will briefly describe Markovian arrival processes and phasetype distributions, MAP fitting using moments and lags matching and present an algorithm of approximate estimation of tandem queueing network properties with departure process approximation with a MAP of a given size.

4.1 Markovian Arrival Processes (MAP) and Phase-Type (PH) Distributions A Markovian arrival process is defined by an irreducible continuous-time Markov chain νt , t ≥ 0 with a finite state space {0, . . . , W }. The process νt , t ≥ 0 is in state ν during exponentially distributed time with parameter λν , ν ∈ 0, W . After the time expires the chain jumps from state ν to state ν˜ with probability p0 (ν, ν) ˜ if the transmission is unobserved and p1 (ν, ν˜ ) otherwise. An observed transmission generates a message. It is also assumed that the process cannot stay in the same state ν˜ = ν without message generation. Matrices D0 , D1 are used to define the MAP: (D0 )ν,ν " =

/ −λν ,

if ν = ν "

λν p0 (ν, ν " ), otherwise

(D1 )ν,ν " = λν p1 (ν, ν " ). The matrix D = D0 + D1 defines an infinitesimal generator of the random process νt , t ≥ 0. Its stationary probability vector θ is obtained from the system θ D = 0,

θ 1 = 1,

where 0 is a row vector of zeros and 1 is a column vector of ones. The steady-state probability vector π of a discrete-time Markov chain embedded at arrival instants with a generator P = (−D0 )−1 D1 can be obtained as the solution of the following linear system: πP = π ,

π 1 = 1.

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The average arrival intensity of a MAP is λ = 1/π(−D0 )−1 1. The k-th moment and lag-k correlation can be expressed as mk = k! π (−D0 )−k 1, lk =

k ≥ 1,

λ2 π(−D0 )−1 P k (−D0 )−1 1 − 1 , λ2 π (−D0 )−2 1 − 1

(1) k ≥ 1.

(2)

A phase-type (PH) distribution is defined as a hitting time of the absorbing state in a continuous-time Markov chain with a single absorbing state. Formally, a random variable X is said to have PH distribution X ∼ P H (S, τ ) if τ ∈ RV is a probability distribution and S ∈ RV ×V is a subinfinitesimal matrix defining initial states probabilities and transition rates between non-absorbing states, respectively. The background Markov chain has the following generator matrix: $ % S −S1 0 1 The k-th moment E[Xk ], X ∼ P H (S, τ ) can be found via the expression mk = k! τ (−S)−k 1,

k ≥ 1.

(3)

Markovian arrival processes and MAP /P H /1/N queues satisfy the following properties [20, 21]: 1. The result of sifting a MAP with constant probability is also a MAP; 2. The composition of a finite number of MAPs is a MAP; 3. The departure process of MAP /P H /1/N system is also a MAP. Note that MAP /P H /1/N queue can lose packets due to the queue overflow and the flow of lost packets is also a MAP. Taking into account these properties it can be shown that a departure process form the first server is a MAP and consequently the arrival processes to all succeeding servers are also MAPs as well as the departure processes. Thus an iterative procedure can be built to compute parameters of a queueing network , [21]. However, the order of departure MAP at i-th phase, i = 1, K, is O(W1 ij =1 Vj (Nj + 2)). So it is impossible to use this procedure without approximations in practical cases for arbitrary network sizes.

4.2 MAP and PH Fitting For PH fitting we will make use of expectation-maximization method (EM) implemented in G-FIT algorithm [17], in which the PH distribution is found in the form of hyper-Erlang distribution. To fit MAP, we will use the generalized method of moments.

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In moments matching method elements of matrices D0 , D1 are considered unknown. Assuming we know the first Km moments and the first Kl lags, the Eqs. (1) and (2) are used to build the equations system for MAP. Then the MAP fitting may be described as a solution of the optimization problem constrained by the values of the moments and lag-k autocorrelation coefficient values. Let mKm be the vector of the first Km moments of MAP, lKl be the vector of the first Kl lags given in (1) and (2) correspondingly; μ and ν be the vectors of moments and lags of a random process to fit correspondingly. Using this notation the problem of MAP fitting can be formulated via solution of a nonlinear algebraic system 

mKm (D0 , D1 ) = μ, lKl (D0 , D1 ) = ν.

(4)

System (4) should be solved for D0 and D1 such that D = D0 + D1 is an infinitesimal generator and D0 is a subgenerator. By these restrictions, the system may have no solution for some pairs (μ, ν) and the order N, thus a MAP with such lags and moments does not exist. It should be noticed that there are no known closed form margins for the moments and lags values for MAPs and PH distributions of an arbitrary order making the problem much harder. We suggest that approximate solution of the system can be brought to an optimization problem as follows. Define a loss function L (·) = (| · |)2 and a loss functional Q(D0 , D1 ) = L (mKm (D0 , D1 ) − μ) + L (lKl (D0 , D1 ) − ν).

(5)

Then a proper MAP is found as a solution of (D0 , D1 ) = arg min Q(D0 , D1 ). D0 ,D1

The problem described is generally nonconvex which leads to local optima solutions and requires additional effort to randomize the initial vectors and look for the best solution.

4.3 Computing End-to-End Delays with Departure Approximation We assume that the wireless network contains K channels, so the queueing network consists of K queues. Let us denote the arrival MAP at each phase as Yi (Y1 is known), departure flow as Yi" , and approximated departure MAP as Yi"" . We can now describe an iterative algorithm: Besides this approximation procedure, we will also use Monte-Carlo method by implementing the tandem queue in a discrete-event simulation system to compare results.

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Result: End-to-end delay T i := 1; T := 0; while i ≤ K do if i > 1 then "" ; Yi := Yi−1 end find matrices D0" , D1" of the departure MAP Yi" (see [20, 21]); compute response time T i ; compute moments and mKm and lags l Kl of Yi" ; find Yi"" := MAP (D0"" , D1"" ) by solving (4) with loss (5); T := T + T i ; i := i + 1; end Algorithm 1: Iterative procedure for MAP /P H /1/N tandem properties estimation with departure process approximation

5 Numerical Results In the numerical experiment we estimate end-to-end delays by performing the following four steps: 1. Fit PH distributions for the given channel using samples collected from the channel simulation; 2. Estimate end-to-end delays using Monte-Carlo method with PH distributions from the first step; 3. Estimate approximate end-to-end delays again using iterative Algorithm 1; 4. Compute end-to-end delays using wireless network simulation. Simulation model used in steps 1 and 4 takes into account actual channel access protocols and models interactions between nodes. Description of the model is beyond the scope of this work. It should be noted that it was implemented in Python language, allows to investigate the process of data transmission in detail, and is significantly simpler than the models implemented in NS-3 or OMNeT++. Source code and documentation are available at GitHub.1 In all experiments we use the same Markovian arrival process A0 , which was fitted using moments matching method from a sample of values of random variable |γ |, γ ∼ N (a0 , σa ). To model various arrival rates, we scale this MAP by multiplying its matrices D0 , D1 by the corresponding constants. For DCF channels we assume only uniformly distributed payload sizes and data rates from 20 kbps up to 250 kbps. For radio relay channels we consider different

1 Simulation

model source code: https://github.com/larioandr/pycsmaca.

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payload size distributions (uniform, exponential and constant) and data rates from 120 kbps and up to 500 kbps.

5.1 Channel Delay Fitting with PH Distributions We simulate channels shown on Fig. 4 to collect samples for PH distributions fitting. Several stations may compete for the channel access in DCF network, and all stations have applications generating data with the same rate. Any two or more simultaneous transmissions cause collision and retransmission after a longer random backoff. In radio relay channel there are no competing stations. Assuming that in the multihop wireless network with DCF channels stations are placed far enough from each other to neglect interference between two-hop neighbors. Then each station in the network competes with either one neighbor (for the boundary stations), or with two neighbors (for intermediate stations), see Fig. 1. In case of a one-hop network there are no competing stations, as well as in the network with radio relay channels. For PH fitting we use G-FIT algorithm [17]. Figure 5 illustrates mean values and standard deviations of transmission delays in DCF channels depending on arrival traffic rate for 1, 2, and 3 competing stations, and Fig. 6 shows the density functions of the transmission delay in DCF channel, obtained from the fitted PH distribution and the collected samples from the channel simulation. Regarding radio relay channel, Fig. 7 shows the probability density functions for the original distributions of transmission delays along with the fitted PH distributions. Analytic expression for the channel delay used here is τ = (ξ + H )/B + IFS, where ξ —random variable describing payload size, H —total length of MAC, PHY, and IP headers, B—channel bitrate, and IFS is the interframe space; all constants values are given in Table 1.

Fig. 4 Network topologies used in channel delay fitting

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Fig. 5 Average transmission delay in DCF channels

Fig. 6 Probability density functions of transmission delay in DCF channels with 1, 2, and 3 colliding stations under various user data bitrates

Fig. 7 Probability density functions of transmission delay in relay channel under various payload size distributions

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5.2 Estimation of End-to-End Delays in Wireless Relay Networks For each type of payload size distributions we found end-to-end delays in networks with radio relay channels using PH distributions found and described above. In approximate estimation using iterative algorithm we approximated departure processes with Poisson process (it can be described as a MAP of order 1) and with a more generic MAP of order 3. The results are shown in Fig. 8. Payload transmission takes more than 90% of service time in the relay channel, so the system utilization can be roughly approximated as the relation of the payload bitrate to the channel bitrate. If utilization is under 0.25, it can be seen that the results obtained using both Monte-Carlo and iterative departure approximation methods are fairly close to each other in all cases except the exponential payload size distribution. The cause of error here is the service time dependency existing in the wireless network (payload size is fixed for each packet), but ignored in the queueing network, where service times at different nodes are independent. Note that in case of other payload size distributions the results obtained using Monte-Carlo method are very close to the results obtained with iterative departure approximation method with MAPs of order 3. However, if the departure processes are fitted with Poisson processes, then the error grows with the growth of the utilization coefficient.

Fig. 8 End-to-end delays in wireless networks with relay channels containing from 1 to 10 nodes

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5.3 Estimation of End-to-End Delays in Networks with DCF Access End-to-end delays were estimated for a wireless network with DCF channels in the same way as for the network with radio relay channel, but using only uniform payload size distribution. In contrast to the queueing model for a radio relay network, here we used different PH distributions for different nodes: boundary servers used PH distributions fitted for the channel with two competing stations, while intermediate servers used PH distributions fitted for the channel with three competing stations. In a basic case of a network of size 1, we used PH distribution obtained from the channel without competing stations. End-to-end delays estimation results are shown in Fig. 9. It can be seen that both iterative departure approximation and Monte-Carlo methods provide fairly accurate results until payload bitrate reaches 120 kbps. The possible reason for the increase in error is the drop in service time as the distance from the source in the real network increases with the load (see Fig. 10). Ways to take this effect into account in the queueing model are expected to be considered in the future work.

Fig. 9 End-to-end delays in wireless networks with DCF channels containing from 1 to 10 nodes

Fig. 10 Node response time in a network of size 10 with DCF channels

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6 Conclusion The paper presented the results of applying the open queuing networks to calculate end-to-end delays in wireless networks with radio relay and IEEE 802.11 DCF channels. Methods for fitting realistic PH distributions of packet transmission times were described, and also in a numerical experiment it was shown that under low utilization open queuing networks MAP /P H /1/N → · · · → •/P H /1/N can be used to adequately evaluate end-to-end delays. At the same time, the dependence of the service times at neighboring stations leads to an error with increasing utilization. The key conclusion of this work is that the method of iterative properties estimation with approximation of the departure flow can be used to calculate the parameters of a queuing network. The approximation by the Poisson arrival process leads to errors earlier than the approximation by the “real” MAP flow (the third order was considered in the work), but under a small load it also allows to get an adequate result. At the same time, the approximation method has a much smaller time complexity: for example, it took about 10 min to obtain estimates for the radio relay network using the Monte-Carlo method on a computer with an i7 processor, and using the iterative approximation algorithm, with both Poisson and MAP-3 flows, only about 1 min 44 s. In the case of a network with DCF channels, the times were 5 min 16 s versus 1 min, respectively. The calculation algorithm plays a significant role here: when evaluating using the Monte-Carlo method, it was necessary to model a network of each size independently, while the approximation method uses an iterative algorithm that allows one to obtain results for networks of a size not exceeding a specified in one pass. In addition, in the Monte-Carlo method, it is necessary to generate a sufficiently large number of events (packet arrivals and service terminations), while in iterative departure approximation method all properties are computed analytically. All numerical experiments, the results of which are presented in this work, some additional data, including the matrices of the fitted PH distributions and measurement details are available in the repository2 on GitHub. All simulation and analytical models and numerical experiment itself are written in Python language. Acknowledgement This work was partly financially supported by the Russian Foundation for Basic Research, grant No. 18-57-00002.

References 1. Banchs, A., Serrano, P., Azcorra, A.: End-to-end delay analysis and admission control in 802.11 DCF WLANs. Comput. Commun. 29(7), 842–854 (2006) 2. Bianchi, G.: Performance analysis of the IEEE 802.11 distributed coordination function. IEEE J. Sel. Areas Commun. 18(3), 535–547 (2000) 2 Experiment

code: https://github.com/larioandr/2019-icaap-queues-model.

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3. Bobbio, A., Horvath, A., Telek, M.: Matching three moments with minimal acyclic phase type distributions. Stoch. Model. 21, 303–326 (2005) 4. Bodrog, L., Heindl, A., Horvath, G., Telek, M.: A Markovian canonical form of second-order matrix-exponential processes. Eur. J. Oper. Res. 190, 459–477 (2008) 5. Casale, G., Zhang, E.Z., Smirni, E.: Trace data characterization and fitting for Markov modeling. Perform. Eval. 67, 61–79 (2010) 6. Chatzimisios, P., Vitsas, V., Boucouvalas, A.: Throughput and delay analysis of IEEE 802.11 protocol. In: Proceedings 3rd IEEE International Workshop on System-on-Chip for Real-Time Applications, pp. 168–174. IEEE, Piscataway (2002) 7. Dong, L.F., Shu, Y.T., Chen, H.M., Ma, M.D.: Packet delay analysis on IEEE 802.11 DCF under finite load traffic in multi-hop ad hoc networks. Sci. China Ser. F Inf. Sci. 51(4), 408– 416 (2008) 8. Heyman, D., Lucantoni, D.: Modelling multiple ip traffic streams with rate limits. IEEE ACM Trans. Netw. 11, 948–958 (2003) 9. Horvath, G., Okamura, H.: A fast EM algorithm for fitting marked markovian arrival processes with a new special structure. In: Computer Performance Engineering, pp. 119–133. Springer, Berlin (2013) 10. Horvath, G., Buchholz, P., Telek, M.: A map fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation. In: Second International Conference on the Quantitative Evaluation of Systems, pp. 124–133 (2005) 11. Horvath, G., Reinecke, P., Telek, M., Wolter, K.: Heuristic representation optimization for efficient generation of PH-distributed random variates. Ann. Oper. Res. 239, 643–665 (2016) 12. Klemm, A., Lindermann, C., Lohmann, M.: Modelling IP traffic using the batch Markovian arrival process. Perf. Eval. 54, 149–173 (2008) 13. Neuts, M.: A versatile markovian point process. J. Appl. Probab. 16, 764–779 (1979) 14. Okamura, H., Dohi, T.: Faster maximum likelihood estimation algorithms for markovian arrival processes. In: IEEE Sixth International Conference on the Quantitative Evaluation of Systems (QEST’09) (2009) 15. Sakurai, T., Vu, H.: MAC access delay of IEEE 802.11 DCF. IEEE Trans. Wirel. Commun. 6(5), 1702–1710 (2007) 16. Telek, M., Horvath, G.: A minimal representation of Markov arrival processes and a moments matching method. Perf. Eval. 64, 1153–1168 (2007) 17. Thummler, A., Buchholz, P., Telek, M.: A novel approach for fitting probability distributions to real trace data with the EM algorithm. In: International Conference on Dependable Systems and Networks (2005) 18. Tickoo, O., Sikdar, B.: Modeling queueing and channel access delay in unsaturated IEEE 802.11 random access MAC based wireless networks. IEEE/ACM Trans. Netw. 16(4), 878– 891 (2008) 19. Vardakas, J., Papapanagiotou, I., Logothetis, M., Kotsopoulos, S.: On the end-to-end delay analysis of the IEEE 802.11 distributed coordination function. In: Second International Conference on Internet Monitoring and Protection (ICIMP 2007), pp. 16–16. IEEE, Piscataway (2007) 20. Vishnevski, V., Larionov, A., Ivanov, R.: An open queueing network with a correlated input arrival process for broadband wireless network performance evaluation. In: International Conference on Information Technologies and Mathematical Modelling, pp. 354–365 (2016) 21. Vishnevsky, V., Dudin, A., Kozyrev, D., Larionov, A.: Methods of performance evaluation of broadband wireless networks along the long transport routes. In: Communications in Computer and Information Science, vol. 601, pp. 72–85. Springer, Berlin (2016) 22. Vishnevsky, V., Larionov, A., Semenova, O., Ivanov, R.: State reduction in analysis of a tandem queueing system with correlated arrivals. In: Communications in Computer and Information Science, vol. 800, pp. 215–230. Springer, Berlin (2017)

Variance Laplacian: Quadratic Forms in Statistics Garimella Rama Murthy

Abstract In this research paper, it is proved that the variance of a discrete random variable, Z can be expressed as a quadratic form associated with a Laplacian matrix i.e. Variance [Z] = XT GX G is Laplacian matrix whose elements are expressed in terms of probabilities. We formally state and prove the properties of Variance Laplacian matrix, G. Some implications of the properties of such matrix to statistics are discussed. It is reasoned that several interesting quadratic forms can be naturally associated with statistical measures such as the covariance of two random variables. It is hoped that VARIANCE LAPLACIAN MATRIX G will be of significant interest in statistical applications. The results are generalized to continuous random variables also. It is reasoned that cross-fertilization of results from the theory of quadratic forms and probability theory/statistics will lead to new research directions. Keywords Variance · Laplacian matrix · Eigenvalues · Eigenvectors · Quadratic form

1 Introduction Structured matrices such as Toeplitz matrix naturally arise in various application areas of mathematics, science, and engineering. Specifically, in probability theory as well as statistics, the autocorrelation matrix of an Auto-Regressive (AR) random process is a Toeplitz matrix. Auto-Regressive stochastic processes find many applications in stochastic modeling. Motivated by practical considerations, detailed

G. Rama Murthy () Mahindra Ecole Centrale, Hyderabad, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_28

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research efforts went into understanding the properties of Toeplitz matrices (such as connections to orthogonal polynomials). For instance, considerable research effort went into efficiently inverting a Toeplitz matrix (such as Levinson–Durbin algorithm). In the research area of Graph theory, a structured matrix called Laplacian naturally arises [1]. It is defined utilizing the adjacency matrix of a graph (which essentially summarizes the adjacency information associated with the vertices of graph). Thus, Graph Laplacian was subjected to detailed study and several new properties of it are discovered. Some of these properties have graph-theoretic significance. Effectively, researchers are interested in discovering the connections between concepts in probability/statistics and structured matrices. Discrete random variables find many applications in Statistics. Thus, a curious natural question is to see whether structured matrices are naturally associated with scalar measures of discrete random variables, such as the moments.

2 Review of Related Literature In the field of mathematics, research related to quadratic forms have long history dating back to the time of Fermat, Bhaskara, others. Several interesting results such as the Rayleigh’s theorem were discovered and proved. Quadratic forms have connections to such diverse areas such as topology, differential geometry, etc. To the best of our knowledge, the author discovered for the first time that the variance of a discrete random variable can be expressed as the quadratic form associated with a Laplacian matrix (of probabilities) [3, 4]. This discovery motivated the author to express other statistical/probabilistic measures as quadratic forms. This line of research enables cross-fertilization of ideas between probability theory/Statistics and the theory of quadratic forms.

3 Variance of a Discrete Random Variable: Laplacian Quadratic Form Consider a discrete random variable, Z with probability mass function {p1 , p2 . . . , pN }. The variance of Z is given by   Variance (Z) = Var(Z) = E Z 2 − (E(Z))2

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475

Let the values assumed by the random variable Z be given by {T1 , T2 . . . , TN }. Let the associated vector of values assumed by Z be denoted by T . Hence, we have that Var(Z) =

N 

* Ti2 pi

−

i=1

=

N 

N 

+2 Ti pi

i=1

Ti2 pi

i=1

−

N  N 

Ti Tj pi pj

i=1 j =1

t = T [D − P˜ ]T

where D is a diagonal matrix whose diagonal elements are {p1 , p2 , . . . , pN } and P˜ij = pi pj for all 1 ≤ i, j ≤ N. T Let G = D − P-. Hence, we have that Var(Z) = T GT Thus, we have shown that variance of discrete random variable Z constitutes a quadratic form associated with the matrix G. We now state the following well known definition: Definition 1 ([1]) A square matrix is called a Laplacian matrix if and only if all diagonal elements of it are all positive, all non-diagonal elements are non-positive, and all the row sums are all zero. Now, we prove that the square matrix G is a Laplacian matrix. Lemma 1 The square matrix G is a Laplacian matrix. Proof From the definition of G, we readily have that Gii = pi − pi2 = pi (1 − pi ) Also, we have that Gij = −pi pj for i = j . Further N  j =1

Gij = Gii +

N 

Gij = pi (1 − pi ) −

j =1j =i

N=1 

pi pj

j =1j =i

= pi (1 − pi ) − pi (1 − pi ) = 0 Hence, the square matrix G is a Laplacian matrix.

Q.E.D.

Note In the case of specific discrete random variables (such as Bernoulli, Poisson, Binomial, etc.), the associated Laplacian matrix can easily be determined. Also, if the number of values assumed by the random variables is atmost 5, the eigenvalues of Laplacian matrix (roots of the associated characteristic polynomial) can be determined by algebraic formulas (Galois Theory).

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Example 1 Specifically when the dimension of G is 2 (i.e. the random variable, Z is Bernoulli random variable), we determine its eigenvalues and eigenvectors explicitly. Let Probability {z = 0} = q. Then we have that

q(1 − q) −q(1 − q) G= −q(1 − q) q(1 − q)    The eigenvalues are 0, 2 q − q 2 ) .

/

The orthonormal basis of eigenvectors are 1 2,



√1 2 √1 2

  ,

√1 2 −1 √ 2

4 . When q =

spectral radius is 12 .

Note Suppose we consider a discrete random variable Z which assumes the values {+1, −1}. In such case, it is easy to show that Variance (Z) = 4q(1 − q) Example 2 We now consider random variable whose probability   discrete uniform 1 1 1 mass function is given by N , N , . . . , N . The Variance Laplacian associated with it is given by ⎡ N−1

⎤

N2 −1 N2

−1 N2 N−1 N2

··· ···

−1 N2 −1 N2

−1 N2

−1 N2

... ···

N−1 N2

⎢ ⎢ G=⎢ ⎢ .. ⎣ .

.. .

⎥ ⎥ .. ⎥ ⎥ . ⎦

Since, the sum of absolute values of elements in every row is same, the spectral radius Sp(G) can be determined (using well known result in linear algebra). Sp(G) =

2(N − 1) , N2

Trace (G) =

N −1 , N

Determinant (G) = 0

Since G is a right circulant matrix, from linear algebra, its eigenvalues as well as eigenvectors can be explicitly determined. Note The matrix, −G constitutes a generator matrix of a finite state space Continuous Time Markov Chain (CTMC). Thus a discrete random variable can be associated with a CTMC. In general, since, G is a symmetric matrix, it is completely specified by the eigenvalues and eigenvectors. Now, we briefly summarize few properties of G matrix that readily follow. 6 7t Let e be a column vector of 1" s (ONES) i.e. e = 1 1 . . . .

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• From Lemma 1, we have that Ge ≡ 0. Hence ‘0’ is an eigenvalue of G and the corresponding eigenvector is e. T • Since Variance [Z] is non-negative, we have that the quadratic form T GT ≥ 0 for all vectors of T . Hence the Laplacian matrix G is a positive semi-definite matrix. Thus, all eigenvalues of G are real and non-negative. We now derive an important property of G in the following lemma. Lemma 2 The spectral radius, μmax i.e. largest eigenvalue of G is less than or equal to 1/2. Proof From linear algebra (particularly matrix norms), it is well known that the spectral radius of any square matrix A i.e. Sp(A) is bounded in the following manner: Minimum absolute row sum (A) ≤ Sp(A) ≤ Maximum absolute row sum (A) But, in the case of Laplacian matrix G, we have that N    Gij  = 2pi (1 − pi ) for all i j =1

Hence, using the above fact from linear algebra, we have that Max Min {2pi (1 − pi )} . {2pi (1 − pi )} ≤ Sp(G) ≤ i i Max {2pi (1 − pi )}. i Let f (pi ) = {2pi (1 − pi )} = 2pi − 2pi2 . We now calculate the stationary Points of f (pi ). Let f " (pi ) = 2 − 4pi = 0. Hence pi = 12 is the unique critical point in feasible region. "" Also we have   that f (pi ) = −4. Thus the critical point is maximum of f (pi ). Thus, f 12 = 12 . Hence we readily have that spectral radius of G i.e. Using the fact that, pi" s are probabilities, we now bound

Sp(G) ≤

1 2

Q.E.D.

Note The function f (pi ) constitutes the well known logistic map whose properties were investigated by several researchers. Goals • Goal 1: In view of the above discovery related to the variance of a discrete random variable (i.e. Laplacian quadratic form), we would like to discover other quadratic forms which naturally arise in probability/statistics.

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• Goal 2: Once the interesting quadratic forms are identified, the results from the theory of quadratic forms (for instance Rayleigh’s Theorem) are applied to statistical/probabilistic quadratic forms. On the other hand, results related to statistical/probabilistic quadratic forms are invoked to derive new results in the theory of quadratic forms (such as inequalities between quadratic forms). • We now derive a specific inequality associated with quadratic forms based on statistical/probabilistic quadratic forms. Consider a vector K whose components are all positive real numbers. It readily follows that by means of the following normalization procedure, it can be converted into a probability vector p (i.e. vector whose components are probabilities and sum to one i.e. probability mass function of a random variable, say Z). Let the vector of values assumed by the random variable Z be T . K p = N

i=1 Ki

=

K α

But, we know that the variance of discrete random variable Z is non-negative.  T  Hence T diag(p) − ppT T ≥ 0, where diag(p) is a diagonal matrix whose components are all the components of vector p. It readily follows that (on using the above normalization equation), we have the following inequality    T   T T α T (diag(K))T ≥ T KK T for all T , K, α We now state the following Theorem, useful in bounding the variance of Z. Rayleigh’s Theorem The local/global optimum values of a quadratic form associated with a matrix B evaluated on the unit Euclidean hypersphere (constraint set) are the eigenvalues of B and they are attained at the corresponding eigenvectors of B. Using Rayleigh’s theorem, we arrive at the following result. Lemma 3 Variance (Z) ≤

2 1 2 L − norm(T ) . 2

Proof Formally, if the vector of values assumed by the random variable i.e. T lies on the unit Euclidean hypersphere, then we have that T

μmin ≤ T GT ≤ μmax ≤

1 , 2

if L2 − norm(T ) = 1

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Suppose L2 − norm(T ) = 1. Then, we readily have that

T L2 −norm(T )

is a vector

whose − norm is equal to one and the Rayleigh’s Theorem can be applied to the quadratic form based on it. Thus, it follows that L2

  2 2 μmin L2 − norm(T ) ≤ Variance(Z) ≤ μmax L2 − norm(T ) Hence, by applying the earlier upper bound on spectral radius, we have

Variance(Z) ≤

2 1 2 L − norm(T ) 2

Q.E.D.

Corollary The non-zero lower bound on Variance (Z) is given by (using μmin ). 2  μmin L2 − norm(T ) ≤ Variance(Z) Property (iv) Now, we consider sum of eigenvalues of G i.e. Trace(G). It readily follows that Trace(G) =

N  i=1

N  N N     2 2 pi − pi = 1 − pi (1 − pi ) = pi = μi i=1

i=1

i=1

Since Trace(G) is the sum of eigenvalues, we have the following obvious bounds: Nμmin ≤ Trace(G) ≤ Nμmax The following Lemma provides an interesting upper bound on Trace(G). Lemma 4 Let G be N × N matrix. Then Trace(G) has the following upper bound. % $ 1 Trace(G) ≤ 1 − N variable Z. Proof Let {p1 , p2 , . . . , pN } be the probability mass function of random 2 We now apply the Lagrange-multipliers method to bound N i=1 pi . The objective function for the optimization problem is given by J (p1 , p2 , . . . , pN ) =  N 2 i=1 pi with the constraint that the probabilities sum to one. Hence the Lagrangian is given by L (p1 , p2 , . . . , pN ) =

N  i=1

* pi2

+α

N  i=1

+ pi − 1

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Now, we compute the critical point and the components of the Hessian matrix: δ2 L δL = 2pi + α, 2 = 2 for all i, δpi δpi

δ2 L = 0 for all i = j δpi δpj

Hence, there is a single critical point and the Hessian matrix is positive definite at the critical point. Thus, we conclude that the objective function has a unique minimum and occurs at δL =0 δpi

i.e.

pi =

−α 2

Using the constraint that probabilities sum to one, we have α = −2 N . Thus, the global minimum occurs at pi = N1 for all i " . Equivalently, we have the following upper bound on Trace(G). % $ 1 Trace(G) ≤ 1 − N

Q.E.D.

Corollary We the second smallest eigenvalue, μ2 of G. It is clear that  now bound  1 Trace(G) ≤ 1 − N . Further (N − 1)μ2 ≤ Trace(G). Hence μ2 ≤ N1 . Thus we have

$ 1 1 and μi ∈ , μ2 ∈ 0, N N

1 2

for i ≥ 3.

Q.E.D.

Note The upper bound on Trace(G) is attained for uniform probability mass function i.e. pi = N1 for all i. , where Note The finite condition number of Laplacian matrix G is defined as μμmax min μmin is the smallest non-zero eigenvalue of G and μmax is the spectral radius of G. Using the content of Lemma 2 and above corollary, the following lower bound on finite condition number of G follows: as μmax ≥ 2Npmin (1 − pmin ) μmin where pmin is the minimum of all the probabilities in the PMF of random variable Z.

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3.1 Connections to Statistical Mechanics Note The expression for Trace(G) has familiar relationship to Tsallis Entropy concept from statistical mechanics. We have the following Definition: Definition Tsallis entropy of a probability mass function {p1 , p2 , . . . , pN } is defined as + * N  k q 1− pi Sq (p) = q −1 i=1

where “k” is Boltzmann constant and q is real number. We, thus readily have that Trace(G) = kS2 (p) where p specifies the probability mass function [Property (iv)]. Note It readily follows that Trace(G) is the DC/constant contribution to the variance Laplacian based quadratic form evaluated on the unit hypercube (i.e. set of all vectors whose components are +1 or −1). We readily have that T

T GT = Trace(G) + terms dependent on T Trace(G) is exactly equal to the scaled Tsallis entropy, kS2 (p) associated with the probability mass function of the discrete random variable. N q In the following lemma, we derive interesting results related to i=1 pi . Specifically, the set of inequalities can have interesting consequences for Tsallis entropy. Lemma 5 Consider probability mass function {p1 , p2 , . . . , pN }. The following inequalities hold true: N  2m+1 m+1 p ≤ N for all integer m. But i=1 i i=1 pi N  i=1

* pi2m+1

≥

N 

+2 pim+1

for all m.

i=1

Hence ⎛ +2 ⎞ *N  k ⎝ S2m+1 (p) ≤ pim+1 ⎠ . 1− 2m i=1

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Proof Since pi" s are probabilities, we readily have that pi2m+1 ≤ pim+1 for any integer m.  2m+1 m+1 Thus, N ≤ N for all integer m. i=1 pi i=1 pi   m Now, consider a random variable Z which assume the values p1m , p2m , . . . , pN i.e. values assumed are higher integer powers of the probabilities in the associated PMF. We know that the variance of Z is non-negative.   Variance(Z) = Var(Z) = E Z 2 − (E(Z))2 ≥ 0   Thus it readily follows that E Z 2 ≥ (E(Z))2 and hence N 

pi2m+1

≥

*N 

i=1

+2 pim+1

for all m.

i=1

$  2 % N m+1 k Hence S2m+1 (p) ≤ 2m 1 − or equivalently i=1 pi   m S2m+1 (p) ≤ Sm+1 (p) − (Sm+1 (p))2 . 2k Corollary Suppose the random variable Z assumes probability values qi" s different from pi" s. Then, using the fact that Variance of Z is non-negative, we have the following inequality N 

qi2 pi ≥

i=1

*N 

+2 qi pi

i=1

It should be noted that both sides of inequality are convex combinations of real numbers Q.E.D.   Note Suppose the values assumed by the random variable are p1m , p1m , · · · , p1m , N 1 2 then using the idea in the above proof, we have that N 

1

i=1

pi2m−1

* ≥

N 

1

i=1

pim−1

+2

In the above inequalities, the probabilities can be rational numbers less than one. Hence the above inequalities hold true between rational numbers.

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 2   Now, we compute the Trace G (in the same spirit of Trace G ) and briefly study its properties. It readily follows that, treating G a vector, we have that N 2   2  μ2i Trace G = L2 -norm(G) = i=1

  2 i.e. treating the set of eigenvalues leading to eigenvalue vector, Trace G is the square of L2 -norm of such vector (of eigenvalues, the smallest of which is zero). Also, from the theory of matrix norms, the L2 -norm of a matrix is related to the spectral radius. We have N N N      2 2 2 Trace G = pi (1 − pi ) + pi2 pj2 i=1i=j j =1

i=1

=

N  i=1

⎡

= 2⎣

pi2

N 

pj2 +

j =1 N N  

i=1i=j j =1

N N  

pi2 pj2

i=1i=j j =1

⎤

pi2 pj2 ⎦ =

N 

μ2i (with μmin = 0)

i=1

  2 Hence, Trace G is divisible by 2. Using the definition of Tsallis entropy Sq (p), it can be readily seen that

 2 1 2 3 Trace G = 2 2 (S2 (p))2 − (S2 (p)) + (S4 (p)) k k k Now, we derive interesting property related to the eigenvectors of G. Lemma 6 The right eigenvectors g " s (whose transpose are the left eigenvectors) of the variance Laplacian G that are different from the all-ones vector (i.e. e which lies in the right null space of G) are such that they lie in the null space of matrix of all ones, S i.e. Sij = 1 f orall i, j Proof Since G is a symmetric matrix, the set of eigenvectors forms an orthonormal basis. Also, the eigenvector corresponding to the ZERO eigenvalue of G is the column vector of all ONES. Hence, we readily have the following fact: g Ti e = 0 for all i. Thus, the components of all other eigenvectors sum to zero. Also, it readily follows that g T Sg = 0. Since S is a rank one matrix with only non-zero eigenvalue being “N” (with e being the associated eigenvector), all the vectors g " s lie in the null space of S (in fact they form the basis of the null space of S).

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  Hence L1 − norm g i is divisible by 2 for all eigenvectors g " s. Also, let g be an eigenvector of G, other than all-ones vector i.e. e. We have that *N  i=1

+2 gi

=

N 

gi2 + 2 (pairwise product of distinct components of g) = 0.

i=1

Sg = −1, where S˜ is a matrix all of whose diagonal Hence if follows that g T elements are zero and all the non-diagonal elements are 1. Since L2 − norm of g is one, it readily follows that pairwise product of distinct components of g = − 12 . Q.E.D. Similar result can be derived based on the Lp − norm of g. Details are avoided for brevity. We now propose an interesting orthonormal basis which satisfies all the properties required of the set of eigenvectors of an arbitrary Laplacian matrix. Definition Hadamard basis (orthonormal) is the normalized set of rows/columns of a symmetric Hadamard matrix, Hm . For instance, it is well known that H2 =

1 1 . Hence the Hadamard basis is given by 1 −1 /

√1 2 √1 2

  ,

√1 2 −1 √ 2

4

Note Two +1, −1 vectors are orthogonal if and only if the number of +1’s is equal to the number of −1’s. Such vectors exist if and only if the dimension of vectors is an even number. Further the sum of elements in such vectors is zero (as required by the eigenvectors of an arbitrary Laplacian matrix which is not necessarily a variance Laplacian matrix). Note In view of Rayleigh’s Theorem, if the orthonormal basis of eigenvectors of a Variance Laplacian G is the Hadamard basis, then the global maximum value of associated quadratic form evaluated on the unit hypercube is attained at the eigenvector corresponding to its spectral radius.

3.2 Spectral Representation of Symmetric Laplacian Matrix G We now arrive at the spectral representation of variance Laplacian matrix G i.e.  T T " " G = P DP = N i=2 μi f i f i where μi s are eigenvalues with μ1 = 0 and f i are normalized eigenvectors of G. It should be noted that the column vector of all ones

Variance Laplacian: Quadratic Forms in Statistics

485

 T i.e. e = 1 1 . . . is an eigenvector corresponding to the zero eigenvalue and √1 e N is the associated normalized eigenvector. We know that G is completely specified by the probability mass function of the associated discrete random variable i.e. {p1 , p2 , . . . , pN } . Hence we have that N 

  μi fij2 = pj 1 − pj for 1 ≤ j ≤ N (i.e. diagonal elements of G.)

i=2

Also, we have that N

i=2 μi fil fim

= −pl pm for l = m and 1 ≤ l ≤ N, 1 ≤ m ≤ N (i.e. off diagonal elements of G).

The orthogonal matrix P is of the following form: ⎡

√1 N √1 N √1 N

⎢ ⎢ ⎢ ⎢ P =⎢ ⎢ . ⎢ . ⎣ .

√1 N

T

f21 f31 · · · fN1

⎤

⎥ f22 f32 · · · fN2 ⎥ ⎥ f23 f33 · · · fN3 ⎥ ⎥ .. .. .. .. ⎥ ⎥ . . . . ⎦ f2N f3N · · · fNN

T

Since, we have that P P = P P = I , the L2 -norm of rows, column vectors of P is one.  T The residue matrices i.e. E i = f i f i are such that N i=1 E i = I . Hence

N 

E i = Q with Qii =

i=2

N −1 1 for all i and Qij = − for i = j N N

Also, we have readily that, N  i=2

fij2 =

N  N −1 1 and fil fim = − for l = m and 1 ≤ l ≤ N, 1 ≤ m ≤ N N N i=2

Note In the spirit of properties of Laplacian G, we can derive new results related to Graph Laplacian. Thus new results in spectral graph theory can be readily derived. Abstract Vector Space of Random Variables Consider a collection of discrete random variables. All of them assume same values. Specifically consider two random variables X, Y . From research literature [2], E(XY ) (i.e. expected value of their product) can be regarded as an inner product between the random variables

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X, Y (regarded as abstract vectors). Suppose T be the set of values assumed by the T random variables X, Y . It readily follows that E(XY ) = T P˜ T , where P˜ can be considered as a symmetric matrix. Using Dirac notion E(XY ) = T , P-T . It readily follows that the inner product E(XY ) is zero i.e. the associated random variables are orthogonal if T lies in the null space of the symmetric matrix P˜ . Thus, the null space of the matrix P˜ determines the space of orthogonal random variables.

3.3 Connections to Stochastic Processes Let us first consider a discrete time, discrete state space stochastic process i.e. a countable collection of discrete random variables. In view of the above results, the variance values of random variables constitute a sequence of quadratic forms. Thus, the sequence of scalar variance values constitute an infinite sequence of real/complex numbers. We consider the following special cases: • Consider the case where the random process is a strict sense stationary random process. Hence, the sequence of variance values (i.e. the associated quadratic forms) form a constant sequence (DC sequence). • Consider the case where the random process constitutes a homogeneous Discrete Time Markov Chain (DTMC). Since such a process exhibits an equilibrium behavior, the sequence of variance values of the discrete random variables (i.e. associated quadratic forms) converges to an equilibrium variance value (based on the equilibrium probability mass function).

4 Other Interesting Quadratic Forms in Probability/Statistics In this section, we investigate several other quadratic forms which are naturally associated with measures such as covariance/Correlation of two random variables which assume same values.  N • In general, quadratic form is of the form β = N i=1 j =1 Ti Tj Bij , where Bij has statistical or probabilistic significance e.g. B could be Toeplitz autocorrelation matrix of an Auto-Regressive process. In fact B could be the state transition matrix of a Discrete Time Markov chain (DTMC). Further B could be −Q, where Q is the generator matrix of a CTMC. • Variance Laplacian related investigation naturally leads to studying the following more general quadratic form associated with two jointly distributed random variables X, Y that are “symmetric” in the sense that their “marginal probability mass functions” are exactly same and the values assumed by them are same. Let

Variance Laplacian: Quadratic Forms in Statistics

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the common marginal probability mass function of the two random variables be {p1 , p2 , . . . , pN }. In the spirit of Laplacian G, we are motivated to introduce, a more general Laplacian matrix, H i.e. H = D − P-, where D = diag (p1 , p2 , . . . , pN ) i.e. a diagonal matrix and P˜ij = Probability {X = i, Y = j } i.e. matrix of joint probabilities. With such definition H need not be symmetric but still is Laplacian. Suppose, P˜ is a symmetric matrix (a stronger condition which ensures that the random variables X, Y are “symmetric”), H will be a symmetric, Laplacian matrix. Let the common vector of values assumed by the random variables, X, Y be T . Hence, the quadratic form associated with H is given by T T H T . Explicitly, we have the following novel measure associated with jointly distributed random variables X, Y . T

θ = T HT =

N 

Ti2 pi −

i=1

N  N 

Ti Tj Prob{X = i, Y = j }

i=1 j =1

    = E X2 − E(XY ) = E Y 2 − E(XY ) Note If X, Y are independent and identically distributed random variables, then the above measure is the common variance of them. Also, if X, Y s are same, then θ is zero. • We now introduce the concept of “symmetrization” of Jointly Distributed Random variables based on the following well known result associated with quadratic forms: T -T = 1 T T (P˜ + Pˇ T )T i.e. symmetric quadratic form. T P 2

We now introduce a new definition. Definition Two jointly distributed random variables with Joint PMF matrix P˜ (not necessarily symmetric) are “symmetrized” when they are associated with the  symmetric joint PMF matrix 1 P˜ + Pˇ T . 2

T

Lemma 7 Laplacian quadratic form T H T is always positive semi-definite. Proof It readily follows that E(XY ) is non-positive, then “θ ” is non-negative. Thus, the more interesting case is when E(XY ) is non-negative. In this case, we invoke a well known result in the abstract vector space of random variables. From [2], the following definition is well known.

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Definition The second moment of the random variables X, Y i.e. E(XY ) is defined as their inner product. Further, the ratio E(XY ) B     E X2 E Y 2 is the cosine of their angle, β i.e. say cos(β). Hence, it is well known that | cos(β)| ≤ 1. Thus, in the case of random variables X, Y whose joint probability mass function matrix, P˜ is symmetric, we have that     |E(XY )| ≤ E X2 . Thus, if E(XY ) ≥ 0, E(XY ) ≤ E X2 T

Thus, the Laplacian quadratic form T H T is always positive semi-definite. Q.E.D. Corollary In this case, the covariance of random variables considered above can be bounded in the following manner: Cxy = E(XY ) − (E(X))2     Since Variance is non-negative, we have that E X2 ≥ (E(X))2 or − E X2 ≤ Q.E.D. −(E(X))2 . Hence, Cxy ≥ −θ . We now briefly consider familiar scalar measures routinely utilized in probabilistic/statistical investigations and provide them with quadratic form interpretation. • Covariance: By definition, covariance of two random variables X, Y is given by Cxy = E(XY ) − E(X)E(Y ) suppose the random variables X, Y assume the same vector of values T . Then we have the following quadratic form interpretation of covariance of X, Y . Cxy =

N  N 

Ti Tj Prob{X = i, Y = j } −

i=1 j =1 T T = T P-T − T J˜T ,

N  N 

Ti Tj pi pj

i=1 j =1

where J˜ij = pi pj

T = T (P- − J-)T

Thus, we have a quadratic form that is not Laplacian. • From the above  discussion, it readily follows that given a random variable, X (E(X))2 , E X2 are arbitrary quadratic forms.

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Correlation Matrix of Finitely Many Random Variables Let us consider finitely many real valued discrete random variables, all of which assume the same set of finitely many⎡values. The correlation matrix of such random variables is ⎤ R11 · · · R1N ⎢ .. ⎥, where R = E X X . From the above given by c RN = ⎣ ... ... ij i j . ⎦ RN1 · · · RNN discussion, it is clear that the elements of RN are quadratic forms in the set of values assumed by the random variables T (diagonal elements are Laplacian quadratic forms where as other elements are not necessarily Laplacian). It is well known that RN is non-negative definite. Using the above discussion, the T correlation matrix RN can be written as RN = T oP oT , where “o” is suitably defined product like Kronecker or Schur product. It should be noted that P is the associated block symmetric matrix of probabilities.

5 Conclusions In this research paper, it is proved that the variance of a discrete random variable constitutes the quadratic form associated with a Laplacian matrix (whose elements are expressed in terms of probabilities). Various interesting properties of the associated Laplacian matrix are proved. Also, other quadratic forms which naturally arise in statistics are identified. It is shown that cross-fertilization of results between the theory of quadratic forms and statistics/probability theory leads to new research directions.

References 1. Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1994) 2. Papoulis, A., Pillai, S.U.: Probability, Random Variables and Stochastic Processes. TataMcGraw Hill, New Delhi (2002) 3. Rama Murthy, G.: Time optimal spectrum sensing. IIIT Technical Report No. 63, December (2015) 4. Rama Murthy, G., Singh, R.P., Chilamkurti, N.: Wide band time optimal spectrum sensing. Int. J. Internet Technol. Secur. Trans. 10(4), (2020)

On the Feynman–Kac Formula B. Rajeev

Abstract In this article given y : [0, η) → H , a continuous map into a Hilbert space H , we study the equation t

y(t) ˆ = e0

c(s,y)ds ˆ

y(t),

where c(s, ·) is a given “potential” on C([0, η), H ). Applying the transformation y → yˆ to the solutions of the SPDE and SDE underlying a diffusion, we study the Feynman–Kac formula. Keywords S " valued process · Diffusion processes · Hermite–Sobolev space · Path transformations · Quasi-linear SPDE · Feynman–Kac formula · Translation invariance Subject Classification 2010 60G51, 60H10, 60H15

1 Introduction One of the well-known formulas at the boundary of probability and analysis is t the Feynman–Kac formula u(t, x) = Ex (f (Xt )e− 0 V (Xs )ds ) which represents the solution u(t, x) of the evolution equation for the operator L − V , where L is the infinitesimal generator of a diffusion (Xt , Px ), x ∈ Rd , V (x) ≥ 0 the potential, and f the initial value [5]. We refer to [6, 8, 9] for basic material on this topic. It is also known that this formula defines a sub-Markovian semi-group whose underlying process (Xˆ t ) is obtained from (Xt ) by the operation known as “killing” according B. Rajeev () Department of Theoretical Statistics and Mathematics, Indian Statistical Institute, Bangalore, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_29

491

492

B. Rajeev t

to the multiplicative functional Mt := e− 0 V (Xs )ds [14]. It may be of interest to have an answer to the following natural question: is it possible to have a “pathwise” ˆ The special case when (Xt ) satisfies an Itô stochastic construction of the process X. differential equation (SDE) is of interest. However, it turns out that it is the SPDE satisfied by the distribution valued process (Yt ) := (δXt ) [10, 11] rather than the SDE for (Xt ) that is more relevant for our purposes. To motivate our “pathwise” construction we proceed as follows. Let H be a separable real Hilbert space and consider C([0, η), H ), the space of continuous functions on [0, η), 0 < η ≤ ∞, with values in H . Let u(t) ∈ C([0, η), H ) be the solution of the following evolution equation in H , viz. ∂t u(t) = Lu(t) t

with L : H → H , a linear operator. Consider u(t) ¯ := u(t)e 0 c(s,u) ds , where c(s, ·) : C([0, η), H ) → R is a given function (the potential) for each 0 ≤ s < η. Then integrating by parts, it is easy to see that u¯ solves ∂t u(t) ¯ = Lu(t) ¯ + c(t, u)u(t). ¯ We would have a good and proper evolution equation for u(t) ¯ if we were able to write c(t, u) = c(t, ˆ u) ¯ for some c(t, ˆ ·) : C([0, η), H ) → R. If the map t ˆ u) := u → S(u) := u¯ ≡ u(t)e 0 c(s,u) ds were invertible, then we may define c(t, c(t, R(u)), where R(u) = S −1 (u) so that c(t, ˆ u) ¯ = c(t, u). It is easy to see that the inverse R is a path transformation R : C([0, η), H ) → C([0, η), H ) induced by the potential c : [0, η) × C([0, η), H ) → R as follows: For a given y ∈ C([0, η), H ), R(y) ∈ C([0, η), H ) is the solution yˆ of the equation y(t) ˆ = y(t)e−

t 0

c(s,y) ˆ ds

.

In Sect. 2, we prove existence and uniqueness to the above equation in Theorem 2.2, using a fixed point argument. Thus the map R is well defined and injective. Since −c satisfies the conditions of Theorem 2.2 whenever c does, the map R is also onto. From a modeling point of view, R(y) maybe viewed as a perturbation, induced by the potential c(t, y), of the trajectory of a particle represented by y(·). We deal with real Hilbert spaces as we consider applications only to the theory of diffusions. However, complex Hilbert spaces and complex valued potentials (with the corresponding interpretation of “amplitude” and “phase”) may also be of interest. Given a diffusion (Xt , 0 ≤ t < η, Px , x ∈ Rd ), we try to realize the Feynman– Kac formula by applying the above transformation to the paths of the diffusion. We remark here that we could choose H = Rd but this does not lead to the Feynman– Kac formula (see Remark 3.2). However if we look at the process (Yt ) := (δXt ) up to time η, then this is a semi-martingale in a Hilbert space Sp —the Hermite– Sobolev space—and is indeed the unique solution of a quasi- linear stochastic partial

On the Feynman–Kac Formula

493

differential equation (SPDE) [10, 11]; one may then look at the process (Yˆt ) := t − 0 V (Xs )ds δXt ) and using the rules of stochastic calculus write an SPDE for Yˆ . (e Note that we can write V (Xs ) = V , δXs  =: c(s, δX ), 0 ≤ s < η, if V belongs to a suitable class of test functions. In Sect. 3, we show that when (Yt ) satisfies a quasi-linear SPDE in Sp then (Yˆt ) is the solution of a new SPDE with a potential term, viz. c(t, Yˆ ) and whose coefficients are defined on the path space C([0, η), Sp ) using the coefficients of the original equation and the transformation discussed above. This transformation works at both levels, viz. the SPDE and the PDE underlying the diffusion, although the “Kac functional” (we use the terminology from [1]) induced by the potential function V (x) is necessarily different in the two cases (see the discussion on diffusions in Sect. 5). In Sect. 4, we allow c(·, ·) to depend also on x ∈ Rd and we show that the above transformation may also be applied directly to the solutions of a class of nonlinear PDEs. We conclude in Sect. 5 with a discussion on two classes of examples in both of which the functional c(t, x, y) depends on x albeit in different ways. The second example that we discuss in Sect. 5 concerns diffusion processes and shows also the connections that can arise between the transformations of the solutions to the SPDE and the solutions to the associated PDE. In Sects. 3, 4, and 5, we work in the framework of [11] to which we refer for results relating to SPDEs, the related notations and references. See also Example 7 of [11] where we had briefly indicated the results in Sect. 2.

2 A Transformation on Path Space Let H be a separable real Hilbert space with norm denoted by . We consider for 0 ≤ T < ∞, the space C([0, T ], H ) of continuous functions y : [0, T ] → H with the sigma field Bt , 0 ≤ t ≤ T generated by the coordinate maps up to time t. Let η > 0. We denote by C([0, η), H ) the set of continuous maps y : [0, η) → H . We denote the norm on C([0, s], H ) by ys := sup y(u). Fix T > 0. Let c : [0, T ] × C([0, T ], Sp ) → R satisfy

u≤s

C1. For 0 ≤ t ≤ T , |c(t, y1 ) − c(t, y2 )| ≤ βy1 − y2 t , where β = β(T ) depends only on T . We note that as a consequence of this condition we have the following: for 0 ≤ s ≤ T , and y1 , y2 ∈ C([0, T ], H ), y1 (u) = y2 (u), 0 ≤ u ≤ s implies c(s, y1 ) = c(s, y2 ). C2. For α > 0 and T > 0 there exists a constant M(α, T ) such that |c(t, y)| ≤ M(α, T ) for 0 ≤ t ≤ T and y ∈ B(0, α) ≡ B(0, α, T ) := {y ∈ C([0, T ], H ), yT ≤ α}.

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We note that if c(t, y) satisfies the above conditions, then so does −c(t, y). Let α(t) ≡ α(t, y) := e

t − c(s,y)ds 0

for y ∈ C([0, T ], H ). Given a y ∈ C([0, η), H ) for some η > 0, and 0 < T < η we consider the following equation for yˆ in C([0, T ], H ), viz. y(t) ˆ = y(t)α(t, y) ˆ = y(t)e

t − c(s,y)ds ˆ

(2.1)

0

for 0 ≤ t ≤ T . We first derive an a priori estimate for the distance between two solutions of (2.1) corresponding to inputs y1 and y2 . Lemma 2.1 Let η > 0. Let y1 , y2 ∈ C([0, η), H ) and suppose yˆ1 , yˆ2 are the corresponding solutions of (2.1). Then for every 0 < T < η, we have the following estimate, viz. yˆ1 − yˆ2 T ≤ My1 − y2 T eMy2 T βe , δ

where δ > βT yˆ1 − yˆ2 T and M := e

T 0

(2.2)

|c(s,y1 )|ds .

Proof Let 0 < T < η and δ, M as above. Then 

T

|c(s, yˆ1 ) − c(s, yˆ2 )| ds ≤ βT yˆ1 − yˆ2 T < δ

0

Consequently, using the elementary estimate |1 − ex | ≤ eδ |x|, |x| < δ we have for any 0 ≤ t ≤ T , |1 − e

T 0

(c(s,yˆ1 )−c(s,yˆ2 ))



T

ds| ≤ eδ β

|yˆ1 − yˆ2 |s ds.

0

Then we have yˆ1 − yˆ2 T = (y1 − y2 )e− +y2 e−

· 0

· 0

c(s,yˆ1 ) ds

c(s,yˆ1 ) ds

≤ y1 − y2 T e × sup |1 − e

T 0

t

(1 − e

·

0 (c(s,yˆ1 )−c(s,yˆ2 ))

|c(s, yˆ1 )|ds + y2 T e

0 (c(s,yˆ1 )−c(s,yˆ2 ))

ds

t ≤T



T

≤ My1 − y2 T + My2 T e β 0 δ

0

)T

|c(s,yˆ1 )|ds

|

δ

≤ My1 − y2 T eT My2 T βe

T

ds

yˆ1 − yˆ2 s ds

On the Feynman–Kac Formula

495



where the last step follows from Gronwal’s inequality.

Let y1 ∈ C([0, t1 ], H ), y2 ∈ C([0, t2 ], H ), where 0 ≤ t1 < t1 + t2 < T . In the proof of the following theorem we need a construction of “concatenation” y1 2 y2 ∈ C([0, T ], H ) of the paths y1 and y2 : y1 2 y2 (s) := y1 (s)|[0,t1 ] (s) + (y2 (s − t1 ) − y2 (0) + y1 (t1 ))|(t1,t1 +t2 ] (s) +(y2(t2 ) − y2 (0) + y1 (t1 ))|(t1 +t2 ,T ] (s), where |A is the indicator of the set A. The following theorem is our main result. Theorem 2.2 Let η > 0 and let c(t, y) satisfy C1 and C2 above for every T , 0 ≤ T < η. Then for a given y ∈ C([0, η), H ) there exists a unique yˆ ∈ C([0, η), H ) satisfying Eq. (2.1) for every T , 0 ≤ T < η. Proof It suffices to show existence and uniqueness of Eq. (2.1) on [0, T ] for every T < η. Using uniqueness, we can then patch up the solutions on different intervals to get the required solution. So let 0 < T < η. Uniqueness of the solution on [0, T ] is immediate from (2.2). To show existence on [0, T ] suppose we have the decomposition (0, T ] = m−1 > (Tn , Tn+1 ]. Fix n, 0 ≤ n ≤ m − 1. Define y(0) ˆ = y(0). Suppose y(t), ˆ t ∈ n=0

[0, Tn ] has been defined. We use an inductive procedure to extend yˆ to the interval (Tn , Tn+1 ] as follows: We first solve the following equation on [0, Tn+1 − Tn ], viz. yˆn (t) = y(t + Tn )α(Tn , y)α ˆ n (t, yˆn ) − y(Tn )α(Tn , y) ˆ = yn (t)αn (t, yˆn ) + an , where for y ∈ C([0, Tn+1 − Tn ], H ) and t ∈ [0, Tn+1 − Tn ], αn (t, y) := e

t − c(s+Tn ,y2y)ds ˆ 0

, yn (t) := y(Tn + t)α(Tn , y), ˆ

ˆ and an := −y(Tn )α(Tn , y). We extend yˆ to the interval (Tn , Tn+1 ] as follows: y(t) ˆ := yˆn (t − Tn ) + y(T ˆ n ), t ∈ (Tn , Tn+1 ]. Then provided yˆ satisfies Eq. (2.1) in [0, Tn ], we have ˆ n) y(t) ˆ := yˆn (t − Tn ) + y(T ˆ n (t − Tn , yˆn ) − y(Tn )α(Tn , y) ˆ + y(T ˆ n) = y(t)α(Tn , y)α = y(t)α(Tn , y)α ˆ n (t − Tn , yˆn ) = y(t)α(t, y), ˆ

(2.3)

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where in the third equality we have used the assumption that yˆ satisfies Eq. (2.1) in [0, Tn ]. As for the fourth equality, we use the fact that yˆ on the interval (Tn , Tn+1 ] is the concatenation of yˆ ∈ C([0, Tn ], H ) and yˆn ∈ C([0, Tn+1 − Tn ], H ), i.e. y(t) ˆ = yˆ 2 yˆn (t), t ∈ (Tn , Tn+1 ]. Thus it suffices to solve (2.3) on [0, Tn+1 − Tn ] for a suitable partition 0 = T0 < T1 < · · · < Tm = T of [0, T ]. So let α > sup y(s), and c(., .) satisfy C1 and C2 on [0, T ] for some M(α) := s≤T

M(α, T ) and β. Let  > 0 be such that eM(3α)T < α2 . By uniform continuity of y on [0, T ] we can divide [0, T ] into a finite number (say m) of subintervals [Tn , Tn+1 ] with Tm = T such that y(t1 ) − y(t2 ) ≤ 

∀t1 , t2 ∈ [Tn , Tn+1 ], n = 0, · · · m − 1.

Next we choose δ > 0 such that |ex − 1| < eδ |x| for |x| < δ. By refining the partition if necessary we may assume without loss of generality that αM(3α)e(M(α)T )+δ (Tn+1 − Tn )
0, R : C([0, t], H ) → C([0, t], H ) is a homeomorphism. In particular, for every t > 0, the map R : (C([0, t], H ), Bt ) → (C([0, t], H ), Bt ) is a measurable isomorphism. Proof To see that R is 1–1, suppose that R(y1 ) = R(y2 ). Then since this implies yˆ1 = yˆ2 , we also have y1 = y2 . That R is onto follows from the observation that t c(s, y) ˆ ds 0 if yˆ ∈ C([0, η), H ) is given and if we define y(t) := y(t)e ˆ , then clearly R(y) = y. ˆ · Note that for a given y ∈ C([0, η), H ), R −1 (y) = y(·)e 0 c(s,y) ds . Since R −1 has the same form as R it suffices to show that R is continuous. But this is clear from (2.2). The last statement follows from the continuity of R and the fact that the Borel sigma field on C([0, η), H ) is the same as Bt .



3 Application to Stochastic PDEs In this section we discuss the applications to SPDEs of the results in the previous section. We work in the framework of the Hermite–Sobolev spaces, Sp , p ∈ R, and we refer to [4, 7, 11] for the results and notations that we use. Let S, S " denote, respectively, the Schwarz space of rapidly decreasing smooth functions and its dual. We refer to [2, 3] for results on stochastic calculus in Hilbert spaces. We work on a probability space (, F , P ) on which is given an r-dimensional Brownian motion (Bt ). Let (FtB )t ≥0 be the filtration of (Bt ). We now consider solutions of the SPDE dYt = L(Yt ) dt + A(Yt ) · dBt ; Y0 = Y,

(3.4)

where L, Ai , i = 1, · · · r are quasi-linear partial differential operators of the form d d  1  t 2 L(y) := (σ σ )ij (y)∂ij y − bi (y)∂i y 2 i,j =1

Ai (y) := −

d 

j =1

σj i (y)∂j y,

j =1

where σij , bi : Sp → R and Y :  → Sp is independent of (Bt ). In [11] we have proved existence and uniqueness of solutions to (3.4) and shown that for a given Y :  → Sp , a unique solution (Yt , η) exists under a Lipschitz condition on the coefficients σij and bi . Here 0 < η ≤ ∞ is the lifetime of the process and if σij , bi are uniformly bounded on Sp , then η = ∞ almost surely (see [11, Proposition 5.2]). Let η > 0 be a fixed positive number, which later will also be allowed to be random. Let c(·, ·) : [0, η) × C([0, η), Sp ) → R satisfy C1, C2 on bounded

On the Feynman–Kac Formula

499

intervals [0, T ], T < η. Given y ∈ C([0, η), Sp ) let yˆ be the solution of (2.1) given by Theorem 2.2 with H = Sp . Let σij , bi : Sp → R and L, Ai as above. Then, the transformation y → yˆ induced by the map c(·, ·) and Eq. (2.1) induces a corresponding transformation of maps σij (·), bi (·) → σˆ ij (·, ·), bˆi (·, ·) as follows: σˆ ij , bˆi : [0, η) × C([0, η), Sp ) → ˆ bˆi (s, y) := bi (y(s)). ˆ Define c(s, ˆ y) := R are given by σˆ ij (s, y) := σij (y(s)), ˆ ˆ c(s, y), ˆ 0 ≤ s < η, y ∈ C([0, η), Sp ). Let L(t, y) and Ai (t, y) be maps from [0, η) × C([0, η), Sp ) to Sp−1 defined as follows: d d   ˆ y) := 1 L(s, bˆi (s, y)∂i ys + c(s, (σˆ σˆ t )ij (s, y)∂ij2 ys − ˆ y)ys 2 i,j =1

Aˆ i (s, y) := −

d 

j =1

σˆ j i (s, y)∂j ys

j =1

Let (Yt , η) be a strong solution (see [11]) of Eq. (3.4) with initial value Y and η now a random variable. Then for each ω ∈ , the trajectory Y· (ω) ∈ C([0, η(ω), Sp ). Define for 0 ≤ t < η(ω) t

Yˆt (ω) := Yt (ω)e 0

c(s,Y (ω))ds

.

ˆ Aˆ i be as above. We take Yˆt (ω) := δ, t ≥ η, where δ is a “coffin Let σˆ ij , bˆi , c, ˆ L, state.” By the continuity of c(·, ·) and> the definition of a strong solution [11], (Yˆt ) is a continuous FtB -adapted, Sˆp := Sp {δ} valued process. Theorem 3.1 Let (Yt , η) be a strong solution of (3.4) and let c(·, ·) satisfy C1 and C2. Then (Yˆt )0≤t d4 sufficiently large, the S−p valued process Yt := δXtx . Then (Yt ) satisfies the SPDE (3.4) with operators L, Ai , i = 1, · · · r given as in Sect. 3 with coefficients σij , bi : S−p → R given as σij (φ) := σ¯ ij , φ, etc., for φ ∈ S. On the other hand, let now V ∈ Sp and define c(., .) : [0, ∞) × C([0, ∞), S−p ) → R as c(t, y) := < V , y(t) >, y ∈ C([0, ∞), S−p ). t

c(s,Y )ds

Let Yˆt := Yt e 0 , where c(s, Y ) := V , Ys  = V (Xsx ). If V is bounded above by K, then we have EYˆt p ≤ eKt EYt p < ∞. ˆ ·), Aˆ i (t, ·) defined as in We note that (Yˆt ) satisfies the SPDE (3.5) with L(t, ˆ Sect. 3 with the coefficients σˆ ij , bi , cˆ all defined through the corresponding σij , bi , c defined above. We define PtV : S → S " as follows. For f ∈ Sp , we define t

PtV f (x)

:= E(e 0

V (Xsx )ds

f (Xtx ))

∗

and let the kernel PtV (x) ∈ S−p be defined as ∗ PtV (x)

t

:= E Yˆt = E(e 0

V (Xsx )ds

δXtx ).

Then the following calculation show that (PtV )∗ (x) satisfies (5.9). Let f ∈ S. ˆ y) we have Firstly we note that from the definition of L(t, t

ˆ Yˆ ) = e 0 f, L(t, t

= e0

V (Xsx )ds

V (Xsx )ds

t

f, L(δXtx ) + e 0 (L¯ + V )f (Xtx ),

V (Xsx )ds

V (Xtx )f, δXtx 

On the Feynman–Kac Formula

505

Hence from the equation satisfied by Yˆt we get ∗ f, PtV (x) = PtV f (x) = f, E Yˆt  = f (x) +

t = f (x) +

s

E[e 0

V (Xux )du



t

ˆ Yˆ )ds f, L(s,

0

(L¯ + V )f (Xsx )]ds

0

t = f (x) +

PsV ((L¯ + V )f )(x)ds

0

t = f (x) +

∗ f, (L¯ ∗ + V )PsV (x) ds.

0

It follows by uniqueness of solutions of (5.9) that with the coefficients σ¯ ij , b¯i , V (x) as above and p sufficiently large, we have the following special case of Corollary 4.2: For each x ∈ Rd , ∗

PtV (x) = Pt ∗ (x)et V (x) = et V (x) EδXtx , where the equality holds in S−p . Acknowledgments The author would like to acknowledge the financial support from the SERB (Science and Engineering Research Board, India) through the MATRIX project No. MTR/2017/000750.

References 1. Chung, K.L., Varadhan, S.R.S. (1980) Kac functionals and Schrodinger equations. In: Bhatia, R., Bhat, A.G., Parthasarathy, K.R. (eds.) Collected Papers of S.R.S. Varadhan, vol. 2, pp. 304–315. Hindustan Book Agency, New Delhi 2. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) 3. Gawarecki, L., Mandrekar, V.: Stochastic Differential Equations with Applications to Stochastic Partial Differential Equations. Springer, Berlin (2011) 4. Itô, K.: Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. CBMS 47. SIAM, Philadelphia (1984) 5. Kac, M.: On the distribution of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949) 6. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2010) 7. Kallianpur, G., Xiong, J.: Stochastic Differential Equations in Infinite Dimensional Spaces. Lecture Notes, Monograph Series, vol. 26. Institute of Mathematical Statistics, Hayward (1995)

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8. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1998) 9. Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications (Universitext). Springer, Berlin (2010) 10. Rajeev, B.: Translation invariant diffusions in the space of tempered distributions. Indian J. Pure Appl. Math. 44(2), 231–258 (2013) 11. Rajeev, B.: Translation invariant diffusions and stochastic partial differential equations in S " (2019). http://arxiv.org/abs/1901.00277 12. Rajeev, B., Thangavelu, S.: Probabilistic representations of solutions to the forward equation. Potential Anal. 28, 139–162 (2008) 13. Rajeev, B., Vasudeva Murthy, A.S.: Existence and uniqueness for 2nd order quasi linear parabolic PDE’s. Preprint 14. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1999) 15. Stroock, D.W.: Partial Differential Equations for Probabilists. Cambridge University Press, Cambridge (2008) 16. Yosida, K.: Functional Analysis. Springer, Berlin (1979)

Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities Ekaterina Lisovskaya , Svetlana Moiseeva and Ekaterina Pankratova

, Michele Pagano

,

Abstract In the paper, we consider a queuing system with n types of customers. We assume that each customer arrives at the queue according to a renewal process and takes a random resource amount, independent of their service time. We write Kolmogorov integro-differential equation, which, in general, cannot be analytically solved. Hence, we look for the solution under the condition of infinitely growing a service time, and we obtain multi-dimensional asymptotic approximations. We show that the n-dimensional probability distribution of the total resource amounts is asymptotically Gaussian, and we look at its accuracy via Kolmogorov distance. Keywords Renewal arrival process · Different types of servers · Queueing system

1 Introduction The globalization of modern managed systems sets new tasks at the hardware, structural, and organizational level. Such systems include both global computer and complex socio-economic relations. In addition to the fact that they are highly heterogeneous, they can also comprise a large number of various objects by

E. Lisovskaya () Tomsk State University, Tomsk, Russian Federation e-mail: [email protected] S. Moiseeva Tomsk State University, Tomsk, Russian Federation M. Pagano Department of Information Engineering, University of Pisa, Pisa, Italy e-mail: [email protected] E. Pankratova V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russian Federation © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 V. C. Joshua et al. (eds.), Applied Probability and Stochastic Processes, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-15-5951-8_30

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highly connected cooperations. For example, the actively developing conceptions of Internet of Things (IoT), Internet of Everything (IoE), and Internet of Nano Things (IoNT) involve the interaction of both objects and subjects of the social environment [3, 9, 15]. In this regard, an integrated approach is needed to solve multi-dimensional problems of managing complex technical and social objects in a dynamically changing environment. Cellular networks are transformed from a planned set of large base-stations to an irregular deployment of heterogeneous infrastructure elements. In paper [2], authors developed a tractable, flexible, and accurate model for a heterogeneous cellular network consisting of K level of randomly located base-station, where each level may differ in terms of average transmit power and supported data rate. It should be noted that the number of publications has been devoted to modeling of wireless communication systems by the resource queueing system [1, 4, 5]. However, the main results were obtained assuming that requests to resources is deterministic. Thus, considering new models of heterogeneous resource queues is currently relevant [7, 8, 12]. Important task of modeling connection networks is cost criterion, which defines the quality of the system operation. A tandem queueing systems with heterogeneous customers is analyzed in the paper [16]. The authors computated the stationary distribution of the system states under the fixed set of the thresholds—the most difficult part of solving the problem of minimizing the cost. Similarly, in our article, the problem of finding a stationary probability distribution of the total volumes of occupied resources in a heterogeneous queue is solved. The considered heterogeneous resource queue can be applied when analyzing the performance indicators of radio resource separation schemes of next-generation telecommunication [6, 14].

2 Problem Statement 2.1 Mathematical Model Consider the queueing system (see Fig. 1) with unlimited number and n different servers types, also assume that each customer carries a random capacity (or needed some resource). Customers arrive in the system according to a renewal arrival process, given by distribution function A(z) of random variable between time point of customers arriving, which has a finite mean and variance (a and σ 2 ). Each arriving customer randomly selects its type according to the set of n  pi = 1. Further, the customer goes probabilities pi (i = 1, . . . , n), and besides i=1

to the conforming server, staying there for a random time with distribution function

Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities

509

Fig. 1 Queueing system with n servers types

Fig. 2 Dynamic screening of the arrival process

Bi (x), and also taking random resources amount vi > 0 with distribution function Gi (y). Queueing system with such service discipline was considered by the authors in [13]. However, it does not take into account that each customer requires a random amount of resources. Denote by {V1 (t), . . . , Vn (t)} the each type’s customers total capacity in the system at time t. This process is non-Markovian, therefore, we use the dynamic screening method for its investigation. Let the system be empty at moment t0 , and let us fix any time moment T in the future as shown in Fig. 2. The set of dynamic probabilities S1 (t), . . . , Sn (t) represents that a customer arriving at time t have the i-type and it will be served at the moment T , i.e. Si (t) = pi (1 − Bi (T − t)), for t0 ≤ t ≤ T . Denote by {W1 (t), . . . , Wn (t)} the each type total customers capacities screened before the moment t. It is easy to prove the property for the probability distribution stochastic processes [10]: P {V1 (T ) < w1 , . . . , Vn (T ) < wn } = P {W1 (T ) < w1 , . . . , Wn (T ) < wn }, wi ≥ 0.

The above n-dimensional process is non-Markovian, then we will add the residual time from t to the next arrival z(t).

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2.2 Kolmogorov Integro-Differential Equation For the probability distribution of (n + 1)-dimensional Markovian process {z(t), W1 (t), . . . , Wn (t)}: P (z, w1 , . . . , wn , t) = P {z(t) < z, W1 (t) < w1 , . . . , Wn (t) < wn } , z, w1 , . . . , wn > 0, we can write the following Kolmogorov integro-differential equation: ∂P (z, w1 , . . . , wn , t) ∂P (z, w1 , . . . , wn , t) = ∂t ∂z  ∂P (0, w1 , . . . , wn , t) (A(z) − 1) + A (z) Si (t) ∂z i=1 ⎡w i ∂P (0, w1 , . . . , wi − yi , . . . , wn , t) ×⎣ ∂z n

+

0

×dGi (yi ) −

∂P (0, w1 , . . . , wn , t) . ∂z

(1)

We define the initial conditions in the form  R(z), w1 = . . . = wn = 0, P (z, w1 , . . . ,n , t0 ) = 0, otherwise, 1 z (1 − A(u)) du is the stationary probability distribution of the a0 values of the random process z(t). To solve (1), we introduce the partial characteristic function: where R(z) =

∞ h(z, v1 , . . . , vn , t) =

∞ e

0

j v1 w1

ej vn wn P (z, dw1 , . . . , dwn , t),

... 0

√ where j = −1 is the imaginary unit. Then, we obtain the following equation: ∂h(z, v1 , . . . , vn , t) ∂h(z, v1 , . . . , vn , t) ∂h(0, v1 , . . . , vn , t) = + ∂t ∂z ∂z   n  ∗ Si (t)(Gi (vi ) − 1) , × A(z) − 1 + A(z) i=1

(2)

Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities

511

where G∗i (vi )

∞ =

ej vi y dGi (y), 0

with the initial condition h(z, v1 , . . . , vn , t0 ) = R(z).

(3)

3 Asymptotic Analysis In general, Eq. (2) cannot be solved analytically, but it is possible to find approximate solutions under suitable asymptotic conditions; in this paper we consider the case that the service times of the different types of customers growth proportionally to each other. We state and prove the following theorem. Theorem 1 The asymptotic characteristic function of the stationary probability distribution of the process {V1 (t), . . . , Vn (t)} has the form / h(v1 , . . . , vn ) ≈ exp λ

n  i=1

+κ

(i)

j vi a1 pi bi + λ

n n   i=1 m=1

n  (j vi )2 i=1

2

(i)

a2 pi bi

4 j vi j vm (i) (m) a1 a1 pi pm Kim , 2

(4)

where λ = (a)−1 , κ = λ3 (σ 2 − a 2 ), (a and σ 2 being the mean and the variance of the interval time, respectively), and a1(i)

∞ =

ydGi (y),

a2(i)

∞ =

0

∞ bi = (1 − Bi (x))dx; 0

y 2 dGi (y), 0

Kim

∞ = (1 − Bi (x))(1 − Bm (x))dx. 0

Proof At first, we prove a secondary statement.

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Lemma 1 The first-order asymptotic characteristic function of the process {z(t), W1 (t), . . . , Wn (t)} is given by ⎧ ⎫ t n ⎨  ⎬ (i) h(z, v1 , . . . , vn , t) ≈ R(z) exp λ j vi a1 Si (θ )dθ . ⎩ ⎭ i=1

t0

Proof Let bi = bqi for some real values qi > 0 and b → ∞. Put ε=

1 , vi = εyi , tε = τ, t0 ε = τ0 , T ε = T˜ , Si (t) = S˜i (τ ), bqi h(z, v1 , . . . , vn , t) = f1 (z, y1 , . . . , yn , τ, ε).

Then, from the expressions (2) and (3), we get ε

∂f1 (z, y1 , . . . , yn , τ, ε) ∂f1 (z, y1 , . . . , yn , τ, ε) ∂f1 (0, y1 , . . . , yn , τ, ε) = + ∂τ ∂z ∂z   n  ∗ S˜i (τ )(Gi (εyi ) − 1) , × A(z) − 1 + A (z) i=1

(5) with the initial condition f1 (z, y1 , . . . , yn , τ0 , ε) = R(z). Let ε → 0; then Eq. (5) becomes: ∂f1 (z, y1 , . . . , yn , τ ) ∂f1 (0, y1 , . . . , yn , τ ) + (A(z) − 1) = 0. ∂z ∂z and hence f1 (z, y1 , . . . , yn , τ ) can be expressed as f1 (z, y1 , . . . , yn , τ ) = R(z)Φ1 (y1 , . . . , yn , τ ),

(6)

where Φ1 (y1 , . . . , yn , τ ) is some scalar function, satisfying the condition Φ1 (y1 , . . . , yn , τ0 ) = 1. Now let z → ∞ in (5): ∂f1 (0, y1 , . . . , yn , τ, ε)  ˜ ∂f1 (∞, y1 , . . . , yn , τ, ε) = Si (τ )(G∗i (εyi ) − 1). ∂τ ∂z n

ε

i=1

Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities

513

Then, we substitute here the expression (6), take advantage of the Taylor expansion   ej εs = 1 + j εs + O ε2 ,

(7)

divide by ε and perform the limit as ε → 0. Since R " (0) = λ, we get the following differential equation:  ∂Φ1 (y1 , . . . , yn , τ ) (i) = Φ1 (y1 , . . . , yn , τ )λ S˜i (τ )jyi a1 . ∂τ n

(8)

i=1

Taking into account the initial condition, the solution of (8) is ⎧ ⎫ τ n ⎨  ⎬ Φ1 (y1 , . . . , yn , τ ) = exp λ jyi a1(i) S˜i (θ )dθ . ⎩ ⎭ i=1

τ0

By substituting Φ1 (y1 , . . . , yn , τ ) from (6), and then we can write h(z, v1 , . . . , vn , t) = f1 (z, y1 , . . . , yn , τ, ε) ≈ f1 (z, y1 , . . . , yn , τ ) = R(z)Φ1 (y1 , . . . , yn , τ ) ⎧ ⎫ τ n ⎨  ⎬ = R(z) exp λ jyi a1(i) S˜i (θ )dθ ⎩ ⎭ i=1

τ0

⎫ ⎧ t n ⎬ ⎨  (i) j vi a1 Si (θ )dθ . = R(z) exp λ ⎭ ⎩ i=1

t0

Let h2 (z, v1 , . . . , vn , t) be a solution of the following equation: ⎧ ⎫ t n ⎨  ⎬ (i) h(z, v1 , . . . , vn , t) = h2 (z, v1 , . . . , vn , t) exp λ j vi a1 Si (θ )dθ . ⎩ ⎭ i=1

t0

(9)

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Substituting this expression into (2) and (3), we get the following equivalent problem:  ∂h2 (z, v1 , . . . , vn , t) (i) j vi a1 Si (t) + λh2 (z, v1 , . . . , vn , t) ∂t n

i=1

=

∂h2 (z, v1 , . . . , vn , t) ∂h2 (0, v1 , . . . , vn , t) + ∂z ∂z   n   ∗  × A(z) − 1 + A(z) Si (t) Gi (vi ) − 1 ,

(10)

i=1

with the initial condition h2 (z, v1 , . . . , vn , t0 ) = R(z).

(11)

By performing the following changes of variable ε2 =

1 , vi = εyi , tε = τ, t0 ε = τ0 , T ε = T˜ , Si (t) = S˜i (τ ), bqi

(12)

h2 (z, v1 , . . . , vn , t) = f2 (z, y1 , . . . , yn , τ, ε). In (10) and (11), we get the following problem:  ∂f2 (z, y1 , . . . , yn , τ, ε) (i) + f2 (z, y1 , . . . , yn , τ, ε)λ j εyi a1 S˜i (τ ) ∂τ n

ε2

i=1

=

∂f2 (z, y1 , . . . , yn , τ, ε) ∂f2 (0, y1 , . . . , yn , τ, ε) + ∂z ∂z   n  × A(z) − 1 + A(z) S˜i (τ )(G∗i (εyi ) − 1) ,

(13)

i=1

with the initial condition f2 (z, y1 , . . . , yn , τ0 , ε) = R(z). As a generalization of the approach used in the previous subsection, the asymptotic solution of this problem f2 (z, y1 , . . . , yn , τ ) = lim f2 (z, y1 , . . . , yn , τ, ε). ε→0

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Letting ε → 0 in (13), we get the following equation: ∂f2 (z, y1 , . . . , yn , τ ) ∂f2 (0, y1 , . . . , yn , τ ) + (A(z) − 1) = 0. ∂z ∂z Hence, we can express f2 (z, y1 , . . . , yn , τ ) as f2 (z, y1 , . . . , yn , τ ) = R(z)Φ2 (y1 , . . . , yn , τ ),

(14)

where Φ2 (y1 , . . . , yn , τ ) is some scalar function that satisfies the condition Φ2 (y1 , . . . , yn , τ0 ) = 1. The solution f2 (z, y1 , . . . , yn , τ ) can be represented in the expansion form f2 (z, y1 , . . . , yn , τ ) = Φ2 (y1 , . . . , yn , τ )   n    (i) ˜ j εyi a1 Si (τ ) + O ε2 , × R(z) + f (z)

(15)

i=1

where f (z) is a suitable function, and besides f (∞) = const, let be f (∞) = 0. By substituting the previous expression and the Taylor–Maclaurin expansion (7) in (13), taking into account that R " (z) = λ(1 − A(z)), it is easy to verify that κ f (z) = 2

z

z (1 − A(u)) du + λ

0

(R(u) − A(u)) du. 0

Letting z → ∞ in (13), by the definition of the function f2 (z, y1 , . . . , yn , τ, ε), we obtain lim

z→∞

∂f2 (z, y1 , . . . , yn , τ, ε) = 0, ∂z

and, taking into account the expansion ej εs = 1 + j εs +

  (j εs)2 + O ε3 , 2

we can write  ∂f2 (∞, y1 , . . . , yn , τ, ε) + f2 (∞, y1 , . . . , yn , τ, ε)λ S˜i (τ )j εyi a1(i) ∂τ n

ε2

i=1

% $ n   ∂f2 (0, y1 , . . . , yn , τ, ε)  ˜ (j εyi )2 (i) (i) a2 + O ε 3 . = Si (τ ) j εyi a1 + ∂z 2 i=1

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By substituting here the expansion (15) and taking the limit as z → ∞, we get  ∂Φ2 (y1 , . . . , yn , τ ) (i) j εyi a1 S˜i (τ ) + Φ2 (y1 , . . . , yn , τ )λ ∂τ n

ε2

i=1

= Φ2 (y1 , . . . , yn , τ )λ

n  i=1

% $ (j εyi )2 (i) a2 S˜i (τ ) j εyi a1(i) + 2

+ Φ2 (y1 , . . . , yn , τ )f " (0)

n 

S˜i (τ )j εyi a1(i)

i=1

×

n  m=1

% $   (j εym )2 (m) a2 + O ε3 . S˜m (τ ) j εym a1(m) + 2

After simple remakes, and taking into account that κ = 2f " (0), we get the following differential equation for Φ2 (y1 , . . . , yn , τ ):  n  (jyi )2 (i) ∂Φ2 (y1 , . . . , yn , τ ) = Φ2 (y1 , . . . , yn , τ ) λ a2 S˜i (τ ) ∂τ 2 i=1  n n   jyi jym (i) (m) a1 a1 S˜i (τ )S˜m (τ ) , +κ 2 i=1 m=1

whose solution (with the given initial condition) can be expressed as ⎧ τ n ⎨  (jyi )2 (i) a2 Φ2 (y1 , . . . , yn , τ ) = exp λ S˜i (θ )dθ ⎩ 2 i=1

τ0

⎫ τ n  n ⎬  jyi jym (i) (m) a1 a1 +κ S˜i (θ )S˜m (θ )dθ . ⎭ 2 i=1 m=1

τ0

Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities

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Substituting this expression into (14) and performing the inverse substitutions of (12) and (9), we get the following expression for the asymptotic characteristic function of the process {z(t), W1 (t), . . . , Wn (t)}: ⎧ t n ⎨  (i) h(z, v1 , . . . , vn , t) ≈ R(z) exp λ j vi a1 Si (θ )dθ ⎩ i=1

+λ

n  (j vi )2 i=1

2

t0

a2(i)

t Si (θ )dθ t0

⎫ t n  n ⎬  j vi j vm (i) (m) Si (θ )Sm (θ )dθ , +κ a1 a1 ⎭ 2 i=1 m=1

t0

For z → ∞, t = T and t0 → −∞ we get the characteristic function of the process {V1 (t), . . . , Vn (t)} in the steady state regime / h(v1 , . . . , vn ) ≈ exp λ

n 

j vi a1(i) pi bi

i=1

+λ

n  i=1

4 n  n  (j vi )2 (i) j vi j vm (i) (m) a2 pi bi + κ a1 a1 pi pm Kim . 2 2 i=1 m=1

The structure of this characteristic function implies that the n-dimensional process {V1 (t), . . . , Vn (t)} is asymptotically Gaussian with mean

a = λ a1(1)p1 b1 a1(2)p2 b2 . . . a1(n) pn bn and covariance matrix

K = λK(1) + κK(2) , where

K(1)

⎡ (1) 0 a p1 b1 ⎢ 2 (2) 0 a2 p2 b2 ⎢ =⎢ ⎣ ... ... 0 0

⎤ ... 0 ⎥ ... 0 ⎥ ⎥, ... ... ⎦ (n) . . . a2 pn bn

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⎡

K(2)

(1) (1)

(1) (2)

a a p1 p1 K11 a1 a1 p1 p2 K12 ⎢ 1(2) 1(1) (2) (2) ⎢ a1 a1 p2 p1 K21 a1 a1 p2 p2 K22 =⎢ ⎣ ... ... (n) (1) (n) (2) a1 a1 pn p1 Kn1 a1 a1 pn p2 Kn2

⎤ (1) (n) . . . a1 a1 p1 pn K1n ⎥ (2) (n) . . . a1 a1 p2 pn K2n ⎥ ⎥. ⎦ ... ... (n) (n) . . . a1 a1 pn pn Knn

4 Simulation Results The result (4) was obtained under the asymptotic condition for an unlimited increase of the service time (bi → ∞). We conducted several simulation experiments [11], changing all the systems parameters (i.e., the laws that characterize the incoming flow, the service time, and the customers resource, as well as the probabilities pi ), in order to investigate their practical applicability. Since the different values of the source data show similar results, for example, we present only one of them. Thus, we assume that the arrival renewal process is characterized by a uniform distribution of the interval time in the [0.5, 1.5], corresponding to a fundamental rate of arrivals λ = 1 customers per time unit. The remaining distribution laws and their parameters are presented in Table 1, according to the customers type. We compared the asymptotic distributions with the empiric ones by Kolmogorov distance: Δ = sup |Fem (x) − Fas (x)| , x

where Fem (x) is the distribution function built on the basis of simulation results, and Fas (x) is the Gaussian approximation given by (4). Table 2 shows the results for the marginal distributions of the total resource amount for each customers types (Δ1 and Δ2 , respectively) and for two-dimensional distributions (Δ). As expected, the asymptotic results become more precise when the service time parameter b increases. This conclusion is also confirmed by Figs. 3 and 4, which compare the asymptotic approximations with the empirical histograms for the total resource amount of each type of customers for two different values of b.

Table 1 Types of customers and their distribution laws

Type First Second

Distribution laws Probability p1 = 0.7 p2 = 0.3

Service time Gamma (0.5b, 0.5) Gamma (1.5b, 1.5)

Resources Exponential (2) Exponential (1)

Heterogeneous System GI/GI(n) /∞ with Random Customers Capacities Table 2 Kolmogorov distance

10 0.136 0.041 0.136

b Δ1 Δ2 Δ

0.2

50 0.035 0.020 0.035

100 0.024 0.014 0.024

0.06

theoretical simulation

0.18

20 0.072 0.027 0.072

519

200 0.017 0.010 0.017

theoretical simulation

0.05

0.16 0.14

0.04 f(v1)

f(v1)

0.12 0.1

0.03

0.08 0.02

0.06 0.04

0.01

0.02 0

0 0

2

4

6

8 v1

10

12

14

90

80

70 v1

60

50

40

16

100

(b)

(a)

Fig. 3 Distributions of the total resource amount for the first type of customers. (a) b = 20. (b) b = 200 0.14

0.04

theoretical simulation

theoretical simulation

0.035

0.12

0.03

0.1 0.08

f(v2)

f(v2)

0.025 0.02

0.06 0.015 0.04

0.01

0.02

0.005

0

0 0

5

10 v2

(a)

15

20

20

30

40

50

60 v2

70

80

90

100

(b)

Fig. 4 Distributions of the total resource amount for the second type of customers. (a) b = 20. (b) b = 200

5 Conclusion In this work we considered a queue with n customers types under the assumption that arrival points correspond to a renewal process and each customer occupies a random resource amount. At first we constructed the system of Kolmogorov differential equations, which in the general case cannot be solved analytically. Hence, we obtained the approximations of probability distributions in case of infinitely growing service time by asymptotic analysis method, and we noticed that the n-dimensional probability distribution of the total resource amount is asymptotically Gaussian.

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Finally, by discrete-event simulation we tested the approximation reliability, by considering the Kolmogorov distance as accuracy measure. Acknowledgement This publication has been prepared with the support of the University of Pisa PRA 2018-2019 Research Project “CONCEPT—COmmunication and Networking for vehicular CybEr-Physical sysTems”.

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