Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms: k-out-of-n Reliability Models and Applications (Infosys Science Foundation Series) 9819994446, 9789819994441

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Infosys Science Foundation Series in Mathematical Sciences

Vladimir M. Vishnevsky · Dharmaraja Selvamuthu · Vladimir Rykov · Dmitry V. Kozyrev · Nika Ivanova · Achyutha Krishnamoorthy

Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms k-out-of-n Reliability Models and Applications

Infosys Science Foundation Series

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

The Infosys Science Foundation Series in Mathematical Sciences, a Scopusindexed book series, focuses on high-quality content in the domain of pure and applied mathematics, biomathematics, financial mathematics, operations research and theoretical computer science. With this series, Springer hopes to provide readers with monographs, textbooks, edited volume, handbooks and professional books of the highest academic quality on current topics in relevant disciplines. Literature in this series will appeal to a wide audience of researchers, students, educators and professionals across the world.

Vladimir M. Vishnevsky · Dharmaraja Selvamuthu · Vladimir Rykov · Dmitry V. Kozyrev · Nika Ivanova · Achyutha Krishnamoorthy

Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms k-out-of-n Reliability Models and Applications

Vladimir M. Vishnevsky Moscow, Russia Vladimir Rykov Moscow, Russia Nika Ivanova Moscow, Russia

Dharmaraja Selvamuthu Department of Mathematics Indian Institute of Technology Delhi New Delhi, India Dmitry V. Kozyrev Moscow, Russia Achyutha Krishnamoorthy Centre for Research in Mathematics CMS College Kottayam Kottayam, India

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 978-981-99-9444-1 ISBN 978-981-99-9445-8 (eBook) https://doi.org/10.1007/978-981-99-9445-8 Mathematics Subject Classification: 60Kxx, 60K25, 60K37, 68M15, 68T07 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 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 Paper in this product is recyclable.

Foreword

Reliability of components and systems plays an important role in our day-to-day activities. Ever since the seminal book by Barlow and Proschan, the field has grown significantly with applications spanning different industries and businesses. In terms of reliability, most of the technical systems and biological objects are redundant systems. Redundancy of individual elements of a complex system is one of the common ways to enhance the system’s reliability. A typical form of redundancy is a k-out-of-n configuration, that is, a model describing a class of systems consisting of n units in parallel redundancy. The k-out-of-n models have wide practical applications and are effectively used to study the reliability of complex systems. Although a large number of publications, both purely theoretical and applied studies, are devoted to these issues, there is practically no systematic presentation of studies of k-out-of-n models in the form of a monograph. Therefore, this monograph focusing mainly on different k-out-of-n models, methods for their study, and their applications to assess the reliability of tethered high-altitude unmanned telecommunication platforms, is very timely and useful for students and researchers interested in this topic. The authors of the book are well-known experts in the development of stochastic models of modern broadband wireless telecommunication systems and networks, and, in particular, in assessing the reliability of such networks based on unmanned aerial vehicles (UAVs). The content of this monograph is based on the original results of the authors, reflected in their recent publications. The book gives a description of new mathematical methods and approaches (based on decomposable semi-regenerative processes, simulation and machine learning methods, and inventory models) to the study of the complex k-out-of-n systems. This enables one to carry out the needed numerical calculations of the reliability indicators. A distinctive feature of this book is that the theoretical studies presented in this book are aimed at solving urgent practical problems within a promising direction in the development of unmanned tethered high-altitude telecommunication platforms. The mathematical models, methods, and algorithms presented in the book, not only will make a significant contribution to the development of reliability theory and theoretical foundations of unmanned tethered v

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Foreword

and autonomous UAV-based aerial communications networks but also be effectively used in the study of the mathematical models of other real systems. The material of the book can be useful for students, graduate and postgraduate students, and researchers in the field of reliability theory, as well as specialists in the design and implementation of tethered high-altitude unmanned telecommunication platforms. This will be a good addition to the studies on k-out-of-n systems. September 2022

Srinivas R. Chakravarthy Professor Emeritus of Industrial Engineering and Mathematics, Departments of Industrial and Manufacturing Engineering and Mathematics Kettering University Flint, MI, USA

Acknowledgements

This book has been funded by the Russian Science Foundation, Russia, and the Department of Science and Technology, India within scientific project No. 22-4902023 “Development and study of methods for reliability enhancement of tethered high-altitude unmanned telecommunication platforms”.

vii

Introduction

At present, tethered high-altitude unmanned telecommunication platforms, the longterm operation of which is ensured by transmitting electricity from the ground to the board via a thin cable, have found wide application and have been widely developed [146, 149, 156, 165]. Tethered high-altitude platforms occupy an intermediate position between satellite systems and terrestrial systems, the equipment of which (cellular base stations, radio relay, radar equipment, etc.) is located in high-rise structures. Tethered high-altitude platforms are highly economical compared to expensive satellite systems, and surpass the terrestrial telecommunications systems in terms of the vastness of the telecommunications and video coverage area. Given the breadth of practical application of tethered unmanned high-altitude platforms, both in civil and defense industries, research centers in the leading countries of the world are intensively working on the design and implementation of a new generation of such platforms. Tethered platforms of a new generation should be able to lift and hold a payload of significant weight (up to 15 kg) at heights of 100–150 m, for which it is necessary to ensure the transmission of high-power energy (up to 10–15 kW) from the ground to the unmanned aerial platform. Such platforms, that is especially important for long-term operation without lowering to the ground, must have high-reliability characteristics, both of the main components and of the high-altitude platform as a whole [154]. It should be noted that the reliability study of autonomous unmanned aerial vehicles (UAVs) is not so important compared to the tethered unmanned platforms, since autonomous UAVs, after completing a flight task, which usually takes no more than 1–2 h, return to the base, where preventive maintenance and repair works are carried out. For tethered high-altitude unmanned platforms, reliability indicators are of crucial importance due to the fact that the main area of its application is solving problems related to long-term operation (tens of hours) without lowering the unmanned flight module to the ground. For the ground-to-air power transmission system, this is achieved by selecting highly reliable electronic components and reserving “bottlenecks” in both groundbased (AC–AC 380V/1000V) and airborne (AC–DC 1000V/48V) voltage converters ix

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Introduction

[157]. In order to increase strength of a thin cable-rope, in a turbulent atmosphere, a Kevlar thread is used along with copper wires and optical fiber. Various measures are envisaged in the event of emergency situations. For example, when power feeding from the ground from an industrial source of 380V/50 Hz is interrupted (even for a short time), the system of accident-free landing of the copter is implemented by switching to a backup on-board battery, which ensures the safety of the unmanned module itself and the payload. When operating in mountainous conditions (or near high-rise buildings), when GPS/GLONASS signals are weakened or completely disappear, a switching to a backup local navigation system with ground beacons [2, 160, 161] is provided. A description of the hardware and software of the tethered high-altitude telecommunication platform “Albatros”, which was developed under the guidance and participation of a number of authors of the current book, is given in [156, 159]. As it can be seen from the above, the task of ensuring the high reliability of tethered unmanned high-altitude platforms is a complex task that includes both questions of the mathematical reliability theory and technical (engineering) issues of improving reliability (taking into account wind loads on the flight module, taking into account the weight of the cable-rope, the problem of voltage stabilization in the ground-toboard power energy transmission, redundancy of the navigation system, etc.). In this book, we will focus only on the part of the problem that is modeled using k-out-of-n reliability models. The high reliability of the unmanned flight module is achieved by selecting propulsion systems with a long time between failures, redundancy of individual elements of the control system, and, most importantly, the use of a multi-rotor architecture (for example, in a quadcopter, the failure of one engine leads to a complete breakdown, whereas with an eight-rotor design, after failure of two engines, the copter can continue to work). The failure of one or more engines leads to an increase in the load on the others, which leads to the possibility of a faster termination of their operation. In addition, the failure of the entire system depends on the location of the failed engines, for example, the failure of nearby engines is more likely to lead to system failure than the failures of distant engines. To study the reliability of this type of complex system in the world literature, k-out-of-n mathematical models are effectively used, which have wide practical applications [140]. Taking into account the features of the functioning of a high-altitude unmanned platform significantly complicates the study of the k-out-of-n models, considered in this book, compared to the known ones, and is an urgent task. A significant number of articles are devoted to the study of k-out-of-n systems, an overview of which is given in Sect. 1.2. However, a systematic presentation of the results in the field of the theory of k-out-of-n systems obtained in recent years and their applications for the reliability assessment of high-altitude unmanned platforms is currently absent in the world literature. The sufficient completeness of the mathematical results and the practical needs of the developers of robotic systems based on tethered high-altitude platforms made it expedient to write the proposed book, which closes this gap.

Introduction

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The content of this book is based on the original results of the authors, references to which are given in the Bibliography section. Materials of lecture courses given to students and postgraduate students of the Moscow Institute of Physics and Technology, the RUDN University, and the Indian Institute of Technology Delhi were also used. The monograph includes an introduction, five chapters, and a bibliography. Chapter 1 is devoted to tethered high-altitude telecommunications platforms and the study of their reliability using k-out-of-n models. In Sect. 1.1, a description of tethered high-altitude telecommunication platforms is presented. In Sect. 1.2, the description of the mathematical model of k-out-of-n systems, various problems of studying their reliability, as well as the application of these models to the object of study are considered. In Sect. 1.3, general notations and assumptions are given, which, as necessary, will be supplemented and refined in the relevant chapters. Chapter 2 examines the behavior of the system before the first failure and its reliability function. Sections 2.1 and 2.2 of this chapter consider the reliability function of the k-out-of-n model under conditions where recovery of intermediate failures is either allowed or not. Section 2.1 examines the reliability function of a system without intermediate recovery and consists of the following parts. In the first Subsect. 2.1.2, after some preliminary information, a k-out-of-n model is considered, the failure of which depends only on the number of failed components. In Sect. 2.1.3, the reliability of a more complex system is investigated, the failure of which depends not only on the number of failed components but also on their location. Subsections 2.1.2 and 2.1.3 are based on a probabilistic analysis of the states of a system whose components are identical, and their lifetimes have an arbitrary distribution. Subsection 2.1.4 extends the previous models to the case where component failure not only leads to the failure of the entire system but also to an increase in the load on the remaining in operation state components, which leads to a decrease in their residual lifetime. This situation is modeled using the component failure hazard function. An algorithm is proposed for calculating the reliability function of such a system, the components of which are also identical, and their lifetimes have an arbitrary distribution. Subsection 2.1.5 considers the reliability function of a k-out-of-n model under conditions that failures of its components lead to an increase in the load on the remaining ones and, consequently, to a change in their residual lifetimes. Order statistics of the system’s components’ lifetime are used to model this situation. Section 2.2 is devoted to the study of the k-out-of-n model before its first failure, taking into account the possibility of component-wise recovery after failure. It is assumed that the lifetime is exponentially distributed, the repair time has an arbitrary distribution, and the repair itself is carried out by one repair team unit. To study the reliability of such a system, the Markovization method is used, which is based on the introduction of additional variables. This method allows you to build a multidimensional Markov process and study it. Subsection 2.2.1 considers a two-dimensional Markov process, for which the system of Kolmogorov partial differential equations and the corresponding initial and boundary conditions are written in Subsect. 2.2.2. The presented system is proposed to be solved using the method of characteristics.

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The algorithm for solving and calculating the reliability function is also presented in this section. Section 2.3 is devoted to solving the relevant problem of studying an analytical model for the reliability analysis of the flight module of a tethered multi-rotor highaltitude platform in the form of a homogeneous hot standby system consisting of n elements operating in a random environment. The considered k-out-of-n model takes into account the layout of the failed components. A general Markov reliability model of the considered system operating in a random Markov environment is proposed. Relations for calculating stationary and non-stationary reliability measures for such a system are given. A numerical study and comparison of the reliability characteristics of a system operating in a stable and random environment with two states is carried out. The results of the numerical study, presented in the form of tables and graphs, showed both common features and differences in the operation of systems in a random and a stable environment. Chapter 3 is devoted to the study of repairable k-out-of-n systems and the calculation of their stationary and non-stationary reliability characteristics. Section 3.1 provides additional description and notation, and clarifies two possible scenarios for recovering a system after its failure. Section 3.2 is devoted to the calculation of nonstationary probabilities of system states for partial and full repairs, respectively, in Subsects. 3.2.1 and 3.2.2. To calculate such probabilities, the Markovization method is also used to construct a two-dimensional Markov process and the characteristic method is to calculate the probabilities from the constructed system of Kolmogorov partial differential equations. Algorithms for calculating non-stationary characteristics are also presented in the corresponding subsections. Section 3.3 considers stationary probabilities similarly for the case of partial and full system repair in Subsects. 3.3.1 and 3.3.2, respectively. An explicit form of the stationary probabilities of the system states is presented in terms of the Laplace transform of the component repair time. The corresponding expressions are obtained by solving the ordinary system of Kolmogorov differential equations using the method of constant variation. Chapter 4 is devoted to the development of mathematical methods of preventive maintenance (PM) organization for the k-out-of-n model using different optimization criteria. Because the detailed information about lifetimes of the system’s components are usually not available, and only one or two of their moments are known, in the chapter, we focus on the problems of the decision-making about PM sensitivity to the information about system’s component lifetime distributions. In Sect. 4.1, we consider PM organization with respect to maximization of the availability criteria. We deal with the model when the system failures depend not only on the number of its failed components but also on their position at the system. The problem set, and the notations, are specified in Subsect. 4.1.1. The properties of the process that describe the model behavior and the general algorithm for the PM comparison are proposed in Subsect. 4.1.2. Two final subsections are devoted to the numerical study of models when the system failure depends not only on the number of its failed components (Subsect. 4.1.3) but also on their position at the system (Subsect. 4.1.4).

Introduction

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Section 4.2 continues investigations of the previous one for a cost-type criterion. In Subsect. 4.2.1, the problem set and the notations are specified. The main result and the general procedure for the PM comparison are proposed in Subsect. 4.2.2 and the numerical study of the PM choice for some examples are represented in the last two subsections. In the previous chapters, the study of k-out-of-n models was carried out using the method of introducing additional variables. Other methods are also possible and were considered by the authors of this monograph (Chap. 5). Section 5.1 is devoted to the study of k-out-of-n repairable systems using the theory of decomposable semiregenerative processes. Stationary and non-stationary reliability characteristics are calculated for the case of two-system recovery after its full failure. Section 5.2 considers the application of a combination of simulation and machine learning techniques to the reliability analysis of k-out-of-n systems, the models of which are presented in Subsects. 2.1.2 and 2.1.3. Arguing the expediency of applying the new research method, some numerical examples are considered and an analysis of the sensitivity of systems to its initial parameters is carried out. A step-by-step description of the application of machine learning techniques is given, as well as the results of training and testing the corresponding models. Section 5.3 considers some aspects of applying machine learning methods to the problem of calculating the stationary characteristics of a k-out-of-n model in the case of partial repair and arbitrary initial distributions of component and system life and repair times. As the first step of such investigation in Sect. 5.3.2 some numerical and sensitivity analyses are shown. After some numerical analysis, in Sect. 5.3.3 the full description of neural network construction as well as data collection for training and testing is presented. The training results with some tests are shown in Sect. 5.3.4. Section 5.3.5 presents the comparative analysis of the results of analytics, simulation, and neural network prediction. In Sect. 5.4, two new directions of research are introduced. The first one is referred to as reliability-inventory, which is introduced to improve the system reliability by incorporating a stock of spare units for replacing failed components of the system. This is introduced in Sect. 5.4.1 In Sect. 5.4.2, the notion of interdependent component systems is introduced. This is done by assuming that the component’s lifetime passes through various stages before each one fails. It is assumed that during each component’s aging process, it transits among different states of a Markov chain and finally gets absorbed into a state which is referred to as the failed state of that component. For different components, the driving Markov chains are distinct. The sojourn time in each state is exponentially distributed with a parameter which depends on the state in which the component is in and the state to be visited next, as determined by the Markov chain rule. Thus, we get a semi-Markov process associated with each component of the system. If the driving Markov chains are independent then the component lifetimes are independent; otherwise, they turn out to be dependent, which we refer to as interdependence among the components of the system.

Contents

1 On Tethered High-Altitude Unmanned Telecommunication Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 System Description and Object of Study . . . . . . . . . . . . . . . . . . . . . . . 1.2 Review of k-Out-of-n Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Notations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Reliability Function of a Complex k-Out-of-n Model . . . . . . . . . . . . . . . 2.1 Reliability Function of a Non-repairable k-Out-of-n Model . . . . . . . 2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Reliability Function of a System, Failure of Which Depends Only on Number of its Failed Components . . . . . . 2.1.3 Reliability Function of a Complex Heterogeneous System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Reliability Function of a Load-Sharing k-Out-of-n Model Due to its Components Failure . . . . . . . . . . . . . . . . . . . 2.1.5 Reliability Function of a k-Out-of-n Model Under Decreasing Components’ Residual Lifetimes . . . . . . . . . . . . . 2.2 On Reliability Function of a k-Out-of-n Model with Component-Wise Repairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Reliability Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 9 9 10 10 12 14 21 33 34 35 39 41 41 42 47

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3 Reliability Characteristics for Repairable k-Out-of-n Model . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Time-Dependent Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Partial Repair Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Full Repair Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stationary Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Partial Repair Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Full Repair Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 59 65 68 69 74

4 Preventive Maintenance for k-Out-of-n Model . . . . . . . . . . . . . . . . . . . . 4.1 Preventive Maintenance of the k-Out-of-n Model with Respect to Availability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Problem Set and Notations . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Process J and the General Procedure of the PM Quality Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 The PM of the k-Out-to-n Model When Its Failure Depends Only on Number of Its Failed Components . . . . . . 4.1.4 The PM of the k-Out-of-n Model, Which Failure Depends on the Location of Its Failed Components . . . . . . . 4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Problem Setup, Assumptions, and Notations . . . . . . . . . 4.2.2 The Problem Solution and the General Procedure for Comparing the Quality of PM Strategies . . . . . . . . . . . . . 4.2.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 New Approaches to Reliability Investigation of k-Out-of-n Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Application of Decomposable Semi-regenerative Processes to the Study of Repairable k-Out-of-n Models . . . . . . . . . . . . . . . . . . 5.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 State of Problem Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Partial Repair Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Full Repair Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Application of Simulation and ML Methods to the Study of Non-repairable k-Out-of-n Models . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Numerical Example and Sensitivity Analysis . . . . . . . . . . . . . 5.2.3 Methods and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Training and Testing Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80 82 85 89 92 93 95 98 105 105 106 109 111 121 131 131 132 134 135

Contents

5.3 Application of Simulation and ML Methods to the Study of Repairable k-Out-of-n Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Numerical and Sensitivity Analysis Using Simulation . . . . . 5.3.3 Design and Configuration of a Neural Network . . . . . . . . . . . 5.3.4 Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Comparative Analysis of the Results of Analytics, Simulation, and Neural Network Prediction . . . . . . . . . . . . . . 5.4 Some New Directions of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Reliability-Inventory and Its Extensions . . . . . . . . . . . . . . . . . 5.4.2 Analysis of the Reliability of a k-Out-of-n System with Interdependent Components . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Acronyms

AC c.d.f. CTMC DC DM DSRP EMRM Erl ESMM ESRP Exp Γ GW GLONASS GPS HAP i.i.d. LnN LT LTE LST MAE MAP MC MDP m.g.f. ML MRM MSE MTTF p.d.f.

Alternating Current cumulative distribution function Continuous-Time Markov Chain Direct Current Decision-Maker Decomposable Semi-Regenerative Process Embedded Markov Renewal Matrix Erlang Distribution Embedded Semi-Markov Matrix Embedded Semi-Regenerative Process Exponential Distribution Gamma Distribution Gnedenko–Weibull Distribution Global Navigation Satellite System Global Positioning System High-Altitude Platform independent identically distributed Lognormal Distribution Laplace Transform Long-Term Evolution Laplace–Stiltjes Transform Mean Absolute Error Markovian Arrival Process Markov Chain Markov Decision Process moment generating function Machine Learning Markov Renewal Matrix Mean Square Error Mean Time to Failure probability density function xix

xx

PH PM RMSE RP r.v. SMM SRP s.s.p.’s t.d.s.s.p.’s THAP TRM U UAV

Acronyms

Phase-Type Distribution Preventive Maintenance Root Mean Square Error Regenerative Process random variable Semi-Markov Matrix Semi-Regenerative Process steady-state probabilities time-dependent system state probabilities Tethered High-Altitude Platform Transition Rate Matrix Uniform Distribution Unmanned Aerial Vehicle

Chapter 1

On Tethered High-Altitude Unmanned Telecommunication Platforms

This chapter is devoted to the tethered high-altitude telecommunications platforms and their reliability study using .k-out-of-.n models. Section 1.1 gives a detailed description of the structure and main components of a tethered unmanned highaltitude telecommunication platform and a description of its aerial part—the flight module, as an object of study. Section 1.2 is dedicated to the study of mathematical models of .k-out-of-.n systems, including their relation to the object of study. In Sect. 1.3, general notations and assumptions are given, which, as and when necessary, will be supplemented and refined in the relevant chapters.

1.1 System Description and Object of Study The system under consideration is a telecommunications high-altitude platform (HAP) on the base of a tethered UAV (see Fig. 1.1). Tethered UAVs have been witnessed in various applications in both civil and military sectors. They are intended to perform long-term flights and persistent missions such as reconnaissance tasks, surveillance, electronic warfare activity, firefighting, long-range broadband wireless communications, deployment of modern telecommunication infrastructure in emergency situations, and infrastructure monitoring and guarding the critical facilities and state borders [88]. The platform consists of the following main components: 1. Unmanned multi-rotor flight module (see Fig. 1.2). 2. Ground module, which includes an AC voltage converter, a system for diagnostics of the parameters of the high-altitude platform and an intelligent wrench. 3. Ground-to-board energy transmission system. 4. Control and stabilization system.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 V. M. Vishnevsky et al., Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9445-8_1

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1 On Tethered High-Altitude Unmanned Telecommunication Platforms

Fig. 1.1 The structure of the tethered platform system

Fig. 1.2 Unmanned multi-rotor flight module

5. On-board payload equipment including 4G/LTE base station; radar and radio relay equipment; equipment for video surveillance and environmental monitoring, etc. 6. Cable-rope, including copper wires for transmission of high-voltage, high-frequency signals and optical fiber for data transmission [149, 153]. The choice of the mathematical model, its parameters, and a set of initial data depends on the use case. We consider the following application scenario—using a tethered high-altitude telecommunications platform to provide wireless broadband communications for a long operating time in remote rural areas without terrestrial

1.2 Review of k-Out-of-n Models

3

access network infrastructure. The UAV-carried base station of cellular communication Long-Term Evolution (LTE) should provide services to mobile users. Long-term operation of the whole system is provided by a continuous power supply of propeller engines of the flight module and the on-board telecommunications equipment from the ground-based energy sources. For tethered HAPs, it is advisable to use a multirotor scheme (here we consider an eight-rotor UAV), where the platform’s angular position is controlled by changing the thrust of the propellers and changing their rotation frequency. Thus, to sum up, the object of our study is an unmanned multi-rotor flight module that may malfunction due to the failure of the propeller engines. The following is an overview of .k-out-of-.n systems that can be considered as a HAP model.

1.2 Review of . k-Out-of-.n Models The redundancy is one of the ways for the system reliability enhancement. A typical form of redundancy is a .k-out-of-.n configuration. A .k-out-of-.n .(1 ≤ k ≤ n) model is a system that consists of .n components. It may be described in two ways, depending on the definition of the parameter .k, as follows. The parameter .k may represent the number of components in the system that must work in order for the entire system to work, referred to as a .k-out-of-.n : G model. Or .k may represent the number of components in the system that must fail before the entire system fails, referred to as a .k-out-of-.n : F model [140]. Of course, these descriptions are closely connected and each of them is dual to another. Based on these definitions, a .k-out-of-.n : G model is equivalent to a .(n − k + 1)-out-of-.n : F model. Similarly, .k-out-of-.n : F model is equivalent to a .(n − k + 1)-out-of-.n : G model. Remark also that for .k = 1 the model looks like a series system, and for .k = n becomes a parallel system. Since for our aims, it is more convenient to consider the subset of failed components of the system, we use the second type of system description omitting the symbol “. F”. Reliability study of .k-out-of-.n models is interesting both from theoretical and practical points of view. From a theoretical point of view, it gives the wide possibility for new mathematical methods and applications. From a practical point of view, there are many investigations devoted to the reliability-centric analysis of .k-out-of.n-type systems. The application of such models can be seen in many real-world phenomena, including telecommunication, transmission, transportation, manufacturing, and service applications. One of the applications of such kind of models is the one described in the previous subsection unmanned multi-rotor flight module of the HAP. Due to wide applications of .k-out-of-.n models in many spheres of human activity including engineering, telecommunication, medicine, biology, etc., many papers have been devoted to their study. There is an extensive literature on the study of such systems (see, for example, Trivedi [145], Chakravarthy et al. [9], and the bibliography therein). Further, the study of such systems received significant development, studies

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1 On Tethered High-Altitude Unmanned Telecommunication Platforms

of redundant .k-out-of-.n models with several types of failures appeared [33, 172]. To study heterogeneous systems in the 1980s, Ushakov [147, 148] proposed the method of universal generating functions, which has recently become very popular and has found various applications (see, for example, [75]). Theoretical studies of the reliability of .k-out-of-.n systems have been considered in a number of publications [6, 80, 87, 171, 173]. Classical .k-out-of-.n systems find wide practical applications in various fields such as data transmission, redundant networks, production control, transport systems, etc. [59, 127, 133, 144]. In particular, Chakravarthy et al. [9], Krishnamoorthy and others [64–66] studied the reliability of systems of the .k-out-of-.n type, in which the service device (repair team), in addition to restoring failed components of the main system, also provided service for external requests. Reference [65] considers a variant of the model from [64], which also considers the . N -strategy for servicing failed components, but the priority in servicing the main requirements is somewhat limited by the assumption that the service of external requirements is not interrupted at the time when the number of failed components reaches . N . The [66] article continues the study of the previous work. We study a .k-out-of-.n system with a single server that also provides the service of external requirements. It is supposed to use the N-strategy to serve the main requirements only after reaching N failed components. Once the basic requirement service has started, it continues until all failed components have been restored. In recent papers, Yuge et al. [170] considered the reliability of a .k-out-of-.n system with joint failures using a multivariate exponential distribution. Zhang and others [172] analyzed the availability and reliability of a .k-out-of-.(M + N ) warm standby system with two types of failures. Kuo and Zuo [72] introduced a wide class of models of reliability systems, such as parallel, serial, redundant, multi-state systems subject to preventive maintenance (recoverable), etc. Trivedi [145] introduced the concept of restart delay (reboot delay) and considered its impact on the reliability or availability of the model of recoverable systems. Wang et al. [162] considered a one-line machine repair model with server walks. Ke et al. [47] investigated the problem of repair with several unreliable repair devices. Wang et al. [163] addressed the problem of warm standby with latency, failover, and failover failovers. [33] considers a system consisting of .m components of type .a and .n components of type .b, respectively. The uptimes of components of types .a and .b are independent identically distributed random variables with distribution functions . Fa (t) and . Fb (t), respectively. The components of types .a and .b are stochastically independent. The system is operable if and only if at least .k components of type .a and .r components of type .b are operable. Simple formulas are obtained for the reliability function of the considered system. In a series of papers by Dudin, Krishnamoorthy, et al. [18, 62–66], the .k-out-of-.n non-reliable queuing system with different types of service and repair strategies were considered. Some optimal opportunistic maintenance strategies were introduced and investigated in [90, 91]. As for the development of the multi-state system—one can find it in [29]. Recent developments on optimal PM policies can be found in [23, 25, 26, 37, 86]. For a review on the recent investigations on maintenance optimization, see [46].

1.2 Review of k-Out-of-n Models

5

Usually, the detailed initial information about lifetimes of the system units is not available, and only one or two of their moments are known. Therefore, it is fundamentally important to study the sensitivity of the system reliability indicators with respect to the shape of their unit lifetime distributions. Research in this direction can be found in a series of our papers, references to which one can find in Chap. 9 of [108], as well as in [132]. For an overview of the recent research methods for queuing and reliability systems, one can find it in [102]. In some of our works (see, for example [42, 43, 119, 121, 132]), analytical and numerical methods were used to study the sensitivity analysis of the reliability characteristics of the .k-out-of-.n-type models to the shape of their components’ lifetime and repair distribution functions. Most research assumes that any system PM and repair lead to its full renovation. This means that, after each completed repair, the system becomes a “new” one. As a result of this assumption, the mathematical formulation of the problem can be conducted within the framework of regenerative or semi-regenerative processes. A probabilistic study of a real-world .k-out-of-.n model often helps to develop an optimal strategy to maintain high level of system reliability. Except for redundancy, another method to improve the reliability of systems is the organization of preventive maintenance (PM). The PM optimization problem is a part of the general theory of stochastic systems control. The latter originated and developed within the framework of the decision-making theory, in particular, the Markov Decision Processes (MDP). One of the first research studies in this direction belongs to Kolmogorov and was associated with product acceptance control. Famous scientists, such as Bellman, Blackwell, Derman, Dynkin, Ross, Shiryaev, and others, took part in the construction and development of this theory. An overview of the first steps in these investigations can be found in [142]. The current state of research in this area, jointly with their applications, can be found in [5]. Without going into much detail, the main result of the MDP theory is as follows. The optimal strategy for MDP, with respect to additive optimization functional, belongs to the class of non-randomized Markov strategies. This result opens up some perspectives for constructing optimal strategies and specific numerical methods. The basic ideas have been developed by Howard [41], Wolf, and Danzig [166], and can be found in some monographs, for example [92]. Regular applications of the theory of controllable stochastic processes to queuing and reliability models began with [99, 110], (see also [100]). There, for the first time, a definition of the concept of a controllable queuing system was proposed, and an overview of the earlier studies on this topic was given. Later, several special monographs appeared, analyzing this topic [38, 51, 138] and others. As a part of the general theory of the stochastic system control, the specificity of applications of the PM optimization problem leaves an imprint on their formulations and ways for solutions. One can find an overview of various approaches and results of studying PM of complex stochastic systems in Gertsbakh’s monograph [30, 31]. There are different types of PM scenarios based on the possibility of observing system states, available information about the system, and other factors. Some of these scenarios are (a) periodic replacement of units; (b) age replacement of units;

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1 On Tethered High-Altitude Unmanned Telecommunication Platforms

and (c) PM based on system state observation. Other scenarios are also possible, and different criteria are used for the best PM choice. Despite the vastness of existing research on systems, there are only a few review articles on this topic. In the book by Kuo and Zuo 2003 [72], investigations on reliability of both non-repairable and repairable .k-out-of-.n system, as well as their weighted and consecutive models are presented. The book [96] 2019 covers some new .k-out-of-.n models and their investigation, including reliability analysis, preventive maintenance, optimal construction, and warranty analysis. A 2010 Eryilmaz’s review [24] is focused on reliability studies of consecutive .k-out-of-.n system, including investigations dated between 1999 and 2010. A short review by Krishnan [61] published in 2020 covers works on reliability of .k-out-of-.n system since 1981–2012. Nevertheless, despite numerous studies of .k-out-of-.n models, new problem statements put forward by the study of tethered unmanned platforms lead to the need to develop new models that form the content of the presented monograph.

1.3 Notations and Assumptions To study the .k-out-of-.n model, we introduce the following notations: • .P{·}, E[·]—symbols of probability and expectation, symbols.Pi {·}, Ei [·] are used for conditional probability and expectation, given initial state of the process is .i; • .μ = E[X ] is the expectation of some random variable (r.v.) . X ; • .σ is a standard deviation of some r.v.; σ • .v = is the coefficient of variation of a r.v.; μ • .j = ( j1 , j2 , . . . jn ) is the system state, where. ji = 0 if the component works,. ji = 1 if it fails; • . E = is the set of states, consisting of disjoint subsets . E 0 and . E¯ 0 —the subsets of “DOWN” and “UP” states, correspondingly; • . Ai (i = 1, 2, . . . ) is the series of components’ lifetimes, which is supposed to be independent identically distributed (i.i.d.) r.v.’s; • . A(t) = P{Ai ≤ t} is their common cumulative distribution function (c.d.f.); ∫∞ • .a = (1 − A(t))dt < ∞ is mean lifetime of a component; 0

• .α is intensity of the components’ failure according to a Poisson flow (. A(t) = 1 − e−αt ∼ E x p(α)); • .λ j = (n − j)α is the system failure intensity in its . jth state in case of Poisson flow of the lifetime; • . Bi (i = 1, 2, . . . ) is the series of components’ partial repair time, which is supposed to be i.i.d. r.v.; • . B(t) = P{Bi ≤ t} is their common c.d.f.; b(x) • .β(x) = is conditional partial repair density of components, given 1 − B(x) elapsed repair time .x;

1.3 Notations and Assumptions

7

˜ is Laplace transform (LT) of the probability density function (p.d.f.) .b(x); • .b(s) ∫∞ • .b = (1 − B(t))dt < ∞ is mean partial repair time of a failed component; 0

• . Fi (i = 1, 2, . . . ) is the series of full system repair time in the case of full system repair scenario, which is supposed to be i.i.d. r.v.; • . F(t) = P{Fi ≤ t} is their c.d.f.; ∫∞ • . f = (1 − F(t))dt < ∞ is the mean full repair time of a failed system; 0

f (x) is the conditional full repair density of components, given 1 − F(x) elapsed repair time .x; • .T is the time to the system failure; • . R(t) = P{T > t} is the system reliability function; ∫∞ • .E[T ] = μT = R(t)dt is the mean time to the first system failure (MTTF).

• .φ(x) =

0

Some additional special notations will be introduced in appropriate sections.

Chapter 2

Reliability Function of a Complex k-Out-of-n Model

.

.

This chapter considers the reliability function of a .k-out-of-.n model under conditions where the recovery of intermediate failures is either allowed or not. Various scenarios of the dependence of the system failure on the failure of its components are considered. Analytical expressions and algorithms are presented for calculating the reliability function and the mean system lifetime before the first failure. The results are accompanied by numerical examples. Section 2.1 considers the reliability function of a .k-out-of-.n model up to its first failure without intermediate recovery of the failing components. Section 2.2 deals with the model where the component-wise repair after their failures is allowed. A reliability analysis of a .k-out-of-.n model in a random environment with account for the increase in the functional load and the location of the failed components is considered in Sect. 2.3. The material of this chapter is based on papers [84, 119, 122, 132, 151].

2.1 Reliability Function of a Non-repairable . k-Out-of-.n Model Section 2.1 is aimed to the reliability investigation of some non-repairable .k-out-ofn models. Section 2.1.2 presents reliability measures of a .k-out-of-.n model, which failure depends only on the number of its failed components. Some complication of this model is considered in Sect. 2.1.3. Within this model, the failure of the system depends not only on the number of failed elements but also on their location relative to each other. Moreover, it is assumed that the components’ lifetime distributions are different. Section 2.1.4 considers a.k-out-of-.n model, same as in Sect. 2.1.2. However, within this model, a component’s failure leads to a redistribution of the load on the remaining operation ones, which leads to a decrease in the residual lifetime of the surviving components.

.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 V. M. Vishnevsky et al., Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9445-8_2

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2 Reliability Function of a Complex .k-Out-of-.n Model

2.1.1 Preliminaries Consider a complex.k-out-of-.n model, whose failure depends not only on the number of failed components but also on their position in it. For reliability of such a model, we use vector notation for the system components states .j = ( j1 , j2 , . . . jn ), where . ji = 1, if .ith component is in “DOWN” state, and . ji = 0, if .ith component is in . “UP” state. In this case, . j = 1≤i≤n ji is the number of working/failed components. Further, . E = {j = ( j1 , j2 , . . . jn ) : ( ji ∈ (0, 1))} denotes the system set of states and . E 0 and . E¯ 0 denotes subsets of its “DOWN” and“UP” states accordingly. Note that the description of these sets depends on concrete applications and should be specified additionally. Define a random process.J = {J(t) : t ≥ 0} with the set of space. E by the relation J(t) = j, if at time t system is in the state j ∈ E

.

and denote by .T lifetime of the system, .

T = inf{t : J(t) ∈ E 0 } = sup{t : J(t) ∈ E¯ 0 }.

In this section, we are interested in the calculation of the system reliability function .

R(t) = P{T > t | J(t) ∈ E¯ 0 }.

2.1.2 Reliability Function of a System, Failure of Which Depends Only on Number of its Failed Components Denoted by . Dni (t), an event that exactly .i from .n components of the system are in “DOWN” state up to time .t. For the homogeneous (with identical components) system, its probability is i .P{Dn (t)}

. . n = (1 − A(t))n−i A(t)i , i

(2.1)

and appropriate the system reliability function is

.

R(t) = P{T > t} =

k−1 . . . n (1 − A(t))n−i A(t)i , i i=0

(2.2)

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

11

and mean time to first system failure (MTTF) is . E[T ] = μT =

.

∞

R(t)dt =

0

2.1.2.1

k−1 . . . . n i=0

i

∞

(1 − A(t))n−i A(t)i dt.

(2.3)

0

An Example

Consider a homogeneous .4-out-of-.8 model, whose identical components have exponentially distributed lifetimes, . A(t) ∼ E x p(α), with the same mean value .μ = α −1 . For this example, it holds .

R(t) = 56e−5αt − 140e−6αt + 120e−7αt − 35e−8αt ,

E[T ] =

.

140 120 35 533 56 − + − = = 0.634524α −1 . 5α 6α 7α 8α 840α

(2.4) (2.5)

Consider also the case of non-exponential lifetime distributions . A(t) and coefficient of variation .v. Suppose that mean lifetime .a = 1 and its coefficient of variation takes the following values, .v = 0.1, 0.5, 1, 5, 10. The considered distributions as well as their parameters for the fixed.a and.v are presented in Table 2.1. Here,.α means the shape parameter, .β is the scale parameter, and .μ and .σ represents the mean and standard deviation of a lognormal distribution derived from the corresponding normal distribution. Figure 2.1 shows the reliability function . R(t) with distributions above, and Table 2.2 represents MTTF of the model. Black, red, and blue colors correspond to .., GW, and LnN distributions. Curve behavior of reliability function corresponds to the values of MTTF. The lower .v, the higher the reliability of the model . R(t). Conversely, the lowest reliability falls on the highest .v. Moreover, as .v < 1, the reliability

Table 2.1 Lifetime distributions and their parameters Gamma Gnedenko–Weibull Distributions .. (.α, β) GW (.α, β) = 0.1

.α

= 100, .β = 0.01

.α

.v

= 0.5

.α

= 4, .β = 0.25

.α

.v

=1

.α

= 1, .β = 1

.α

.v

=5

.α

= 0.04, .β = 25

.α

.v

= 10

.α

= 0.01, .β = 100

.α

.v

.β

= 12.1, .β = 1.04321

Lognormal LnN (.μ, σ ) .μ

= −0.00498, = 0.09975 = 2.1, .β = 1.12906 .μ = −0.11157, .σ = 0.47238 = 1, .β = 1 .μ = −0.03466, .σ = 0.83255 = 0.31, .β = 0.12462 .μ = −1.62905, .σ = 1.80502 = 0.2332, .μ = −2.30756, = 0.02675 .σ = 2.14828 .σ

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2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.1 Reliability function of a homogeneous .4-out-of-.8 model Table 2.2 MTTF of the .4-out-of-.8 model .v = 0.1 .v = 0.5

.v

=1

.v

=5

.v

= 10

..

.0.9819

.0.8631

.0.6345

.0.0035

.0.0001

GW LnN

.0.9947

.0.8806

.0.6345

.0.0617

.0.0157

.0.9808

.0.8496

.0.6641

.0.2007

.0.1114

function . R(t) is asymptotically insensitive to the shape of lifetime distribution, but sensitive to its coefficient of variation. As .t ≈ 2, the model is fully unreliable.

2.1.3 Reliability Function of a Complex Heterogeneous System For investigation of the reliability function of complex heterogeneous system, where lifetimes distributions . Ai (t) are different and the system’s failure depends on the position of the failed components in the system, denoted by .

pj (t) =

.

(1 − Ai (t))1− ji Ai (t) ji

1≤i≤n

the probability of its state .j = ( j1 , . . . , jn ) at the time .t. At that, the probabilities of UP and DOWN states of the system to the time .t equal correspondingly to

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

P(U P) =

.

.

13

.

pj (t), P(D O W N ) =

j∈ E¯ 0

pj (t),

j∈E 0

where . E 0 is the set of DOWN states, and . E¯ 0 is the set of UP states of the system. The appropriate reliability function of the system equals to .

R(t) =

. .

(1 − Ai (t))1− ji Ai (t) ji .

(2.6)

j∈E 0 1≤i≤n

2.1.3.1

An Example

Consider the .4-out-of-.8 model, whose components’ lifetimes have the same distribution with different mean values. Suppose that the system fails if any .4 components fail or any .3 adjacent components fail. According to the notation introduced above, this is a .(3, 4)-out-of-.8 model. Let . Ai (t) ∼ E x p(αi ), where .αi , means the failure intensity of an .ith component, .i = 0, 8. We number the system states as .

.

j=

(1 − ji )2n−i

1≤i≤n

and obtain the expression for the reliability function using formula (2.6):

.

R(t) = e

−

8 .

αi t

i=1

+ (1 − e−α8 t )e

... + (1 − e−α8 t )(1 − eα7 t )e

−

−

6 . i=1

7 . i=1

αi t

αi t

+ (1 − e−α7 t )e

. . 6 . − αi +α8 t i=1

+ (1 − e−α8 t )(1 − e−α6 t )e

... + (1 − e−α1 t )(1 − e−α2 t )(1 − e−α4 t )e

−

8 . i=3

αi t

+ ...

. . 5 . − αi +α7 t i=1

.

+ ... (2.7)

Substituting .αi = α in this expression we obtain a homogeneous case, and formulas for . R(t) and .E[T ] take the following form: .

R(t) = 48e−5αt − 116e−6αt + 96e−7αt − 27e−8αt . E[T ] =

.

509 = 0.605952α −1 , 840α

If the system’s failure doesn’t depend on the adjacency of the failed components and .αi = α, these reliability measures turn to expressions (2.4)–(2.5) as expected. In order to consider these results graphically (Fig. 2.2), let .αi = 1 + 0.2 · i, i = 1, 8, which in practice can be due either to different initial elapsed lifetimes of the

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2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.2 Reliability function for the heterogeneous models (. Ai (t) ∼ E x p(αi )) Table 2.3 MTTF for the heterogeneous .4-out-of-.8 model Case 1

Case 2

.E[T ]

.0.38204

.0.36217

rotors, or to the fact that they were produced by different manufacturers. Here Case 1 corresponds to the reliability function (2.7) for the heterogeneous .(3, 4)-out-of-.8 model, and Case 2 shows the reliability function for the heterogeneous .4-out-of-.8 model which doesn’t take into account the adjacency of the failed components. Case 3 curve is given just for the reference and corresponds to the homogeneous .4-out-of-.8 model. This figure shows that despite different scenarios of the system’s failure, the reliability function curves are very close to each other. MTTF values also confirm this fact (Table 2.3). The curve for Case 1 is expectedly steeper than the one for Case 2 because Case 1 additionally takes into account system failures due to failure of adjacent components.

2.1.4 Reliability Function of a Load-Sharing . k-Out-of-.n Model Due to its Components Failure 2.1.4.1

Preliminaries and Approach

In the real practice, component failure can be the cause of both system failure and increase or redistribution of the load on the components remaining in operation. However, the failure of any component can lead to an increase in the load on the remaining ones and, consequently, to a decrease in their residual lifetime. In this Sect., we study the main characteristics of reliability .k-out-of-.n models in conditions

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

15

where the failure of one of its components leads to a redistribution of the load on the remaining components and, consequently, to a proportional decreasing in their residual lifetime. One of the ways to model the reliability of the system components and the whole system, as well as their changes in connection with the redistribution of the load because of the failure of one of the components, is to use the hazard rate function. Suppose that the residual lifetime of the remaining in-operation components is changed as follows. After the .ith failure (failure of the .ith component), all surviving components “age” for some time .wi , which corresponds to a jump in the components’ hazard rate function with the value of its shift to the time .wi , takes place. In terms of the system components’ hazard rate function, this means that on the semi-interval .[Si , Si+1 ) between the .ith and .(i + 1)th failures the components’ hazard rate function has the form α (u) = αi−1 (wi + u).

(2.8)

. i

Denoted by . Ai (i = 1, 2, . . . , n) lifetimes of the system components that are supposed to be i.i.d. r.v.’s and by . A(t) = P{Ai ≤ t} their common c.d.f. It is well known that the reliability function of a component .r (t) = 1 − A(t) connected with its hazard rate function .α(t) by the following expression: α(t) =

.

a(t) a(t) = , r (t) 1 − A(t)

where .a(t) is a probability density function (p.d.f.) of r.v.’s . Ai . In terms of hazard rate function, the c.d.f. . A(t) of the component lifetime has a form

.

. . t . . . t . A(t) = 1 − exp − α(u)du and r (t) = exp − α(u)du . 0

0

For the system reliability study, denoted by . j the system state in which . j components are in the failure state, and by . E = {0, 1, 2, . . . , k} the system set of states. It is supposed that, in the initial time, all components are in UP states, that means that the initial system state is . j = 0.

2.1.4.2

Main Result

To calculate the system reliability function after each component failure, denoted by • . I0 = {1, 2, . . . , n}, the initial set of components; • . S1 = min{Ai : i ∈ I0 }, the time of the first failure, .i 1 = arg min{Ai : i ∈ I0 }, the number of the first failed component, . I1 = I0 \ {i 1 };

16

2 Reliability Function of a Complex .k-Out-of-.n Model

• analogously by induction, . Sl , the time moment of .lth failure, .il , the number of failed component in this time, and. Il = Il−1 \ {il }, the set of remaining in operation (surviving) components after this step, S = min{Ai : i ∈ Il−1 }, il = arg min{Ai : i ∈ Il−1 },

Il = Il−1 \ {il }.

. l

It is well known that the reliability function of any component with a hazard rate function .α0 (u) = α(u) equals . . t . .r 1 (t) = exp − α0 (u)du . 0

For the .k-out-of-.n model, the system hazard rate function in the initial time with such components is .λ0 (u) = nα0 (u). Further, it is supposed that .λi (u) = (n − i)αi (u), i = 0, k − 1, where .i is the number of failures (number of failed components). Therefore, the system reliability function up to the first failure . S1 is as follows: . . t . . R1 (t) = P{min [Ai : i = 1, n] > t} = exp − λ0 (u)du . (2.9) 0

According to the rule (2.8), the hazard rate functions of every survived component are shifted by the constant .w1 . Thus, they are equal to .α1 (t) = α0 (w1 + t) as .t > S1 . Thus, for any .i ∈ I1 , the conditional (with respect to the time of the first failure . S1 ) reliability function of the survived components equals to .r 2 (t

P{Ai > t} P{Ai > t, Ai > S1 } = = | S1 ) = P{ Ai > t | S1 } = (2.10) P{ Ai > S1 } P{Ai > S1 } . . . . . t . . . t . exp − 0t α0 (u)du . . . = exp − α0 (u)du = exp − α1 (u)du . = S1 S1 exp − 0S1 α0 (u)du

These expressions follow due to the independence of component lifetimes for any component .i ∈ I1 as .t ≥ S1 . Moreover, the last equality follows from the fact that in the failure time . S1 , the hazard rate functions of all surviving components are changed from .α0 (u) to .α1 (u). Analogously to the reliability function of each component, the system reliability function after the first component’s failure should be calculated. Similar to the previous argumentation, all survived after the first failure components, are independent, and have the same hazard rate function .α1 (u). Thus, the conditional (with respect to the first failure time) system reliability function equals to .

R2 (t | S1 ) = P{S2 > t | S1 } = P{min[ Aq : q ∈ I1 ] > t | S1 } = (2.11) . . . t . . . t λ1 (u)du . = exp − (n − 1)α1 (u)du = exp − S1

S1

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

17

Analogously, at the moment of time . Sl of the .lth component’s failure, two events occur: – the failed component .il leaves the set of working ones, . Il = Il−1 \ {il }; – the hazard rate function .αl−1 (u) of components .i ∈ Il−1 is replaced by .αl (u) = αl−1 (wl + u). The conditional (with respect to the .lth component’s failure time . Sl ) reliability function of any survived components .i ∈ Il equals to r

. l+1

. . t . (t | Sl ) = P{Ai > t | Sl } = exp − αl (u)du .

(2.12)

Sl

The conditional reliability function of a system after .lth failure (with respect to its failure time . Sl ) equals to .

. . t . Rl+1 (t | Sl ) = P{Sl+1 > t | Sl } = exp − λl (u)du .

(2.13)

Sl

The above results deal with some component conditional failure time (with respect to the previous one). Based on these results, the next theorem considers the appropriate unconditional reliability function. Theorem 2.1 Reliability functions up to the.lth failure.(l = 1, . . . , k − 1) are recursively determined by the following relations: . .

t

Rl+1 (t) = −

Rl+1 (t | x)d Rl (x) + Rl (t),

(2.14)

0

where, for the recursion beginning, the function . R1 (x) is determined from (2.9) and, for its continuation, the function . Rl (t | x) is determined from (2.13) . . t . . . t . . R1 (t) = exp − λ0 (u)du , Rl+1 (t | x) = exp − λl (u)du . 0

x

Proof To prove the theorem, we use the total probability formula. According to it, the conditional probability . Rl+1 (t | Sl ) is defined only over the domain .{Sl ≤ t}. Therefore, to calculate the unconditional one, we should integrate it over the domain .{Sl ≤ t} given by the first term in (2.14). The second term represents the probability .P{Sl > t} of the complement set .{Sl > t}. . . Based on the previous results, the following general procedure for the system reliability function calculation can be proposed. Algorithm 2.1 Beginning. Determine: Integers .n, k, real .wi ; distribution . A(t) of components lifetime or/and their hazard rate function .α(t), as well as the procedure of their changing after system component failure.

18

2 Reliability Function of a Complex .k-Out-of-.n Model

Work. Step 1. Put .C0 = 0, α0 (t) = α(t), λ0 (t) = nα0 (t) and, according to (2.9), calculate the reliability function up to the first failure time . S1 = min{Ai : i ∈ I0 } and .i 1 = arg min{Ai : i ∈ I0 }, . . t . . R1 (t) = exp − λ0 (u)du . 0

Step 2. For .l from .1 to .k − 1, calculate I = Il−1 \ {il }, Cl = Cl−1 + wl , αl (u) = αl−1 (wl + u) = α(Cl + u),

. l

λ (u) = (n − l)αl (u) = (n − l)α(Cl + u).

. l

Step 3. Taking into account expression (2.13), the conditional system reliability function of the next failure time . Sl+1 = min{Ai : i ∈ Il } for .t > Sl equals to .

. . t . Rl+1 (t | Sl ) = exp − λl (u)du , Sl

thus calculate its unconditional form . . t . . t . Rl+1 (t) = − exp − λl (u)du d Rl (Sl ) + P{Sl > t}. 0

Sl

Step 4. For .l = k − 1, the needed system characteristics will be found • System reliability function .

R(t) = Rk (t),

• System lifetime expectation .

∞

E[Sk ] =

R(t)dt,

.

0

• Variance

.

∞

Var[Sk ] = −

.

(t − E[Sk ])2 d R(t).

0

.

Stop. . 2.1.4.3

Numerical Experiments

The proposed approach for calculating the reliability function is applicable for various continuous distribution functions of the lifetime of system elements, as well as arbitrary parameters .k and .n.

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

19

Consider as an example a .3-out-of-.6 model. First, consider an exponential distribution components’ lifetime, . A(t) ∼ E x p(α). In this case, the hazard rate function is a constant value, .α(t) = α. Thus, the reliability function takes the result, which coincide with those obtained by the corresponding four-state Markov case with no component-wise repair and absorbing state .

R(t) = e−6αt − 6e−6αt (1 − eαt ) − 15e−6αt (1 − eαt )2 .

(2.15)

As known, the exponential distribution has no memory. Thus, considering such an example does not show the effect of load redistribution on survived components . after failure. Then, suppose .. . a . GW θ, , .(1 + 1/θ ) σ = a −1 · .v = a

.

that

.

A(t) ∼

Gnedenko–Weibull

.(1 + 2 · θ −1 ) − 1, α(t) = θ .(1 + θ −1 )2

.

.(1 + θ −1 ) a

.θ

.

GW =

t θ−1 ,

where .θ is the shape parameter of GW distribution calculated according to the preset value of the coefficient of variation .v, .σ is the standard deviation. It is well known that the behavior of the hazard rate function for GW distribution depends on its shape parameter. If .θ = 1, GW distribution turns to the exponential one with meantime.a and the coefficient of variation.v = 1. So, the reliability function has the form (2.15). The coefficient of variation .0 < v < 1 leads to the value of the parameter .θ > 1. In this case, the hazard rate function is monotonically increasing. The coefficient of variation .v > 1 leads to the value of parameter .θ < 1, then the hazard rate function becomes decreasing. Since such a situation does not correspond to the practical problem presented in this monograph, we will not consider this case. Figure 2.3 demonstrates the behavior of hazard rate function.α(t) with different shape parameter .θ if .a = 1. Suppose that the mean time .a = 1. The coefficient of variation .v is used according to the hazard rate function is monotonic and increases. Thus, the shape parameter of GW distribution is .θ = 1, 2, 3 and it leads to the coefficient of variation .v = 1, 0.5227, 0.3634. Since the .3-out-of-.6 model fails due to the failure of any three components, we have two constants, that define the aging time of survived components. So, in the first experiment, consider the reliability of the system under the condition about ‘aging’ time .wi = 0.1, i = 1, 2. That is, the failure of system’s components leads to the aging of surviving ones for the equal time. Figure 2.4 presents the reliability function . R(t). According to this graph, the higher reliability coincides with the lower value .v. Thus, the sensitivity of the reliability characteristics of the.3-out-of-.6 model to the value of the coefficient of variation of components’ lifetime is observed.

20

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.3 Hazard rate function .α(t) of GW distribution

Fig. 2.4 Reliability function . R(t) with . A(t) ∼ GW

To study the reliability of a system, such a characteristic as a quantile is often used. This measure shows how long the system will be reliable with a fixed probability. The quantiles .qγ = R −1 (γ ) of the reliability function are presented in Fig. 2.5. Here black bottoms correspond to .γ = 0.999, gray bottoms are for .γ = 0.99. All the values for quantiles .γ = 0.999; 0.99; 0.9 are presented in Table 2.4. The quantile values also emphasize the significant influence of the coefficient of variation on the highly reliable performance of the .3-out-of-.6 model.

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

21

Fig. 2.5 Reliability function . R(t) with GW distribution and quantiles Table 2.4 Quantiles of reliability function (GW distribution) .v = 1 .v = 0.5227 .q0.999

.0.04

.0.17

.v

= 0.3634

.0.24

.q0.99

.0.09

.0.28

.0.37

.q0.9

.0.22

.0.46

.0.57

2.1.5 Reliability Function of a . k-Out-of-.n Model Under Decreasing Components’ Residual Lifetimes 2.1.5.1

Preliminaries and Approach

This Sect. is devoted to the problem of system failure, associated with a change in the residual components’ lifetime, depending on the increase in load after the failure of any component. Suppose that the failure of .ith component for .i < k leads to increasing in load to others and therefore to decreasing their residual lifetimes. It can be modeled by multiplying the residual lifetime of the surviving components by some weighting factor .wi < 1, (i = 1, k − 1). To model such a situation, order statistics are used. Evident, that if a .k-out-of-.n model’s failure depends only on a number of its failed components, it coincides with the .kth order statistics from .n i.i.d. r.v.’s . Ai (i = 1, n) with given c.d.f. . A(t). For simplicity further, we will denote order statistics . A(1) ≤ · · · ≤ A(k) ≤ · · · ≤ A(n) of i.i.d. r.v.’s . Ai (i = 1, n) by . X i that means . X i = A(i) and . X 1 ≤ · · · ≤ X k ≤ · · · ≤ X n . Distributions of order statistics are studied well (see, for example, [15]), where it was shown that the joint p.d.f. . f n (x1 , . . . xn ) of all order statistics . X 1 ≤ X 2 ≤ · · · ≤ X n from .n i.i.d. r.v.’s . A1 , A2 , . . . , An with given p.d.f. .a(x) has the form f (x1 , x2 , . . . , xn ) = n!a(x1 )a(x2 ) · · · a(xn ) (x1 ≤ x2 ≤ · · · ≤ xn ).

. n

(2.16)

22

2 Reliability Function of a Complex .k-Out-of-.n Model

By integration of this p.d.f. with respect to last .n − k variables one can simply find the joint p.d.f. . f k (x1 , . . . xk ) of the first .k order statistics . X 1 ≤ X 2 ≤ · · · ≤ X k from the .n i.i.d. r.v.’s . Ai (i = 1, n) in the domain .x1 ≤ x2 ≤ ... ≤ xk in the form f (x1 , x2 , ..., xk ) =

. k

n! a(x1 )a(x2 ) . . . a(xk )(1 − A(xk ))n−k . (n − k)!

(2.17)

However, if the failure of one of the system’s components leads to changing in residual times of all survived components, then their distributions are also changed.

2.1.5.2

Transformation of Order Statistics

Following the proposed model of the influence of components’ failures on the residual lifetime of survivors, they are reduced by multiplying by some constant .wi depending on the number of failed components. Denoted by .Yi (i = 1, k), the time of .ith component failure under the conditions of increasing the load on survived components. To simplify these values’ representation in terms of order statistics . X 1 ≤ X 2 ≤ · · · ≤ X n , we introduce the following notations: .

W1 = (1 − w1 ), W2 = w1 (1 − w2 ), . . . , Wk−1 = w1 · · · wk−2 (1 − wk−1 ), Wk = w1 · · · wk−1 .

(2.18)

In these notations, the following theorem holds. Theorem 2.2 The time to the considered system failure .Yk is a linear function of order statistics of the following form: Y = W1 X 1 + W2 X 2 + · · · + Wk−1 X k−1 + Wk X k .

. k

(2.19)

Proof To calculate the time of the system failure, we slightly expand the problem statement and calculate the successive time moments .Yi (i = 1, k) of failures of the system’s components under conditions of increasing load on the surviving compo( j) nents. To do that, we utilize a recursive procedure and denote by . X i the expected time moment of the .ith failure after the failure of the . jth component .(i > j). Thus, to start the induction, we have . X i(0) = X i . After the first failure of a component at time .Y1 = X 1(0) = X 1 , all residual lifetimes of surviving components that equal. X i − X 1 for.i > 1 decrease by a factor of.w1 , and therefore the expected failure times . X i(1) for .i > 1 take the form .

X i(1) = X 1(0) + w1 (X i(0) − X 1(0) ) = (1 − w1 )X 1 + w1 X i , i = 2, k. (2.20)

Therefore, .Y2 = X 2(1) = (1 − w1 )X 1 + c1 X 2 . ( j−1) ( j−1) , the residual lifetimes. X i − Similarly, after the. jth failure at time.Y j = X j ( j−1)

Xj

of all surviving components for all.i > j decrease by a factor.w j , 0 < w j < 1

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

23

(. j = 1, k) and the expected failure times of components take the following form: .

( j)

Xi

( j) Xi

( j−1)

, ∀i ≤ j,

( j−1) Xj

+ w j (X i

= Xi =

( j−1)

( j−1)

− Xj

( j−1)

) = (1 − w j )X j

( j−1)

+ w j Xi

, ∀i > j.

Thus, the expected failure times of the system components .Y j ( j = 1, k) under ( j) ( j−1) ( j = 1, k). Expressing . X i conditions of load redistribution equal to .Y j = X j in terms of the original order statistics, we obtain the following expression for .i > j: .X

( j) = (1 − w1 )X 1 + w1 (1 − w2 )X 2 + w1 w2 (1 − w3 )X 3 + · · · i

· · · +w1 · · · w j−1 (1 − w j )X j + w1 · · · w j X i = =

j .

w1 · · · wl−1 (1 − wl )X l + w1 · · · w j X i .

(2.21)

l=1

Supposing that the last expression true for given . j and for all .i > j: .X

( j+1) = (1 − w1 )X 1 + w1 (1 − w2 )X 2 + w1 w2 (1 − w3 )X 3 + · · · i

· · · +w1 · ... · w j−1 (1 − w j )X j + w1 · ... · w j (1 − w j+1 )X j+1 + + w1 · ... · w j w j+1 X i = =

=

j . l=1 j+1 .

w1 · · · wl−1 (1 − wl )X l + w1 · · · w j (1 − w j+1 )X j+1 + w1 · · · w j+1 X i =

w1 · · · wl−1 (1 − wl )X l + w1 · · · w j+1 X i .

l=1

Hence, by the principle of mathematical induction, the equality (2.21) holds for any. j. In terms of the original order statistics. X i (i = 1, k), we obtain for all. j = 1, k: .Y j

( j−1)

= Xj

= (1 − w1 )X 1 + w1 (1 − w2 )X 2 + · · · + w1 · · · w j−2 (1 − w j−1 )X j−1 + w1 · · · w j−1 X j ,

which, using the notation introduced earlier, leads to (2.19) for . j = k, which com. pletes the proof. .

2.1.5.3

Distribution of the System Failure Time

Turns now to the calculation of the c.d.f. . FYk (y) of the system time failure .Yk under the condition of redistributing the load on the components. We will do that by taking into account the expression (2.19) for the time of the system failure in terms of order statistics . X i and using the formula (2.17) for the joint distribution of the first .k-order statistics.

24

2 Reliability Function of a Complex .k-Out-of-.n Model

To simplify the representation of this c.d.f. we introduce the following notations: .z 0

= y,

z i = z i (y; x1 , ..., xi ) =

y − W1 x1 + W2 x2 − ... − Wi xi (i = 1, k − 1). (2.22) Wi+1

With these notations the following theorem holds. Theorem 2.3 The distribution of the system failure time for . y > 0 is .

FYk (y) = P{Yk < y} = .z0 .z1 .zk−1 n! = a(x1 )d x1 a(x2 )d x2 . . . a(xk )(1 − F(xk ))n−k d xk . (2.23) (n − k)! 0

x1

xk−1

Proof According to Theorem 2.2 (see formula (2.19)) the time .Yk of the system failure is the linear function of the first .k order statistics Y = W1 X 1 + W2 X 2 + · · · + Wk−1 X k−1 + Wk X k .

. k

Therefore, for c.d.f. . FYk (y) of r.v. .Yk in terms of p.d.f. . f k (x1 , . . . , xk ) of the first .k order statistics, we obtain .

FYk (y) = P{Yk < y} = = P{W1 X 1 + W2 X 2 + · · · + Wk−1 X k−1 + Wk X k < y} = . . = ··· f k (x1 , x2 , ..., xk )d x1 ...d xk ,

(2.24)

D(x1 ,...,xk ;y)

where integration domain is . D(x 1 , ..., x k ;

y) = {0 ≤ x1 ≤ · · · ≤ xk , W1 x1 + W2 x2 + W3 x3 + · · · + Wk−1 xk−1 + Wk xk ≤ y}.

Let us represent this multidimensional integral as an iterated one. Taking into account that .x1 ≤ x2 ≤ · · · ≤ xk the integration domain can be transformed by the following way. For the last variable .xk from the inequality .

W1 x1 + W2 x2 + W3 x3 + · · · + Wk−1 xk−1 + Wk xk ≤ y,

(2.25)

it follows that x ≤

. k

y − W1 x1 − W2 x2 − W3 x3 − · · · − Wk−1 xk−1 = z k−1 (y; x1 . . . xk−1 ). w1 · · · wk−1

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

25

Further, taking into account that .xk−1 ≤ xk , from the last inequality, it follows that x

. k−1

≤ xk ≤

y − W1 x1 − W2 x2 − W3 x3 − · · · − Wk−1 xk−1 . w1 · · · wk−1

From which with the simple algebra, one can find x

. k−1

≤

y − W1 x1 − W2 x2 − · · · − Wk−2 xk−2 = z k−2 (y; x1 . . . , xk−2 ). w1 · · · wk−2

Following by the same way, we obtain for variable .x2 the inequality .

y ≥ (1 − w1 )x1 + w1 (1 − w2 )x2 + w1 w2 x3 ≥ ≥ (1 − w1 )x1 + w1 (1 − w2 )x2 + w1 w2 x2 = (1 − w1 )x1 + w1 x2 ,

from which it follows that x ≤

. 2

y − (1 − w1 )x1 , w1

and at last .

y ≥ (1 − w1 )x1 + w1 x1 = x1 .

It means that .0 ≤ x1 ≤ y. This argumentation shows that the integration domain D(x1 , . . . , xk ; y) in terms of notations (2.22) can be represented as

.

.

D(x1 , . . . , xk ; y) = {xi−1 ≤ xi ≤ z i (y; x1 , . . . xi−1 ) (i = 1, k)}.

Thus, using the formula (2.17) for p.d.f. . f k (x1 , . . . , xk ) for the first .k order statistics and the above form of the integration domain, we can rewrite integral (2.24) for . y ≥ 0 as n! . FYk (y) = (n − k)! that ends the proof. .

.y

.z1 a(x1 )d x1

0

.zk−1 a(x2 )d x2 · · ·

x1

a(xk )(1 − A(xk ))n−k d xk ,

xk−1

.

As a consequence of the theorem, the main system reliability characteristics can be calculated. Remark 2.1 Based on the system failure time distribution, any other system reliability characteristics can be calculated, such as: – its reliability function . R(y) = .1 − FY (y); ∞ – mean system lifetime .E[T ] = 0 R(t)dt; – variation .var[T ].

26

2 Reliability Function of a Complex .k-Out-of-.n Model

2.1.5.4

A Special Case: Exponential Distribution

In a special case, when the system components . Ai (i = 1, n) lifetimes have exponential (. E x p) distribution with parameter .α the integral (2.23) can be calculated analytically, but the calculations are enough cumbersome. We show it for the given value of .k = 2. But for exponential distribution of the system components’ lifetime, we propose another approach for the system lifetime distribution. It is based on the lack of memory property of any exponentially distributed r.v. Denoted by .Ti , the time interval between .i − 1st and .ith components failures, .i = 1, k − 1 (. T0 = 0). Then due to the lack of memory property of exponential distribution the time to the .kth failure .Yk is the sum Y = T1 + T2 + · · · + Tk ,

. k

of .k independent exponentially distributed r.v.’s .Ti with parameters λ = nα, λi = w1 w2 · · · wi−1 (n − i + 1)α = w¯ i (n − i + 1)α, i = 2, k,

. 1

where for simplicity additional notations are used: . w¯ i =

.

1, w1 · · · wi−1 ,

i = 1, i = 2, k.

The moment generating function (m.g.f.) of the system’s lifetime in this case has the following form: . . . . . . φ (s) = E e−sYk = E e−sTi =

. k

1≤i≤k

1≤i≤k

w¯ i λi . s + w¯ i λi

To apply the above theorem and the proposed approach, let us consider the simplest example of a .k-out-of-.n model with .k = 2. In this case, suppose .w1 = w. Thus, according to (2.17), the joint distribution of r.v.’s . X (1) , X (2) is f (x1 , x2 ) =

. 2

n! n! a(x1 )a(x2 )(1 − A(x2 ))n−2 = α 2 e−αx1 e−(n−1)αx2 . (n − 2)! (n − 2)!

Calculate c.d.f. . FY2 (y) of the r.v. .Y2 = (1 − w)X (1) + w X (2) ,

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

.

27

. . y − (1 − w)X (1) = FY2 (y) = P{(1 − w)X (1) + w X (2) < y} = P X (2) < w .y = n(n − 1)α 2

y−(1−w)x1 w

e−αx1 d x1

.

e−(n−1)αx2 d x2 =

x1

0

(n−1)α nw n−1 e−nαy − e− w y , = 1+ nw − (n − 1) nw − (n − 1)

and therefore its p.d.f. for . y ≥ 0 is f (y) =

. Y2

. n(n − 1)α . − (n−1)α y e w − e−nαy . nw − (n − 1)

Note that this result holds for .w .= (n − 1)/n and in this case the distribution is mixture of exponential distributions. The point .w = (n − 1)/n is a singular point for which c.d.f. of the r.v. .Y2 is Erlang distribution, .

FY2 (y) = 1 − e−nαy − nαye−nαy , y > 0,

with p.d.f. .

pY2 (y) = n 2 λ2 ye−nλy , y > 0.

Remark 2.2 The singularity in the system lifetime c.d.f. calculation arises because for some special values of the coefficient .wi (here for .w = (n − 1)/n) the moment generating function of the system lifetime has multiple roots that leads to changing of the shape of distribution. With the help of another approach, one can find m.g.f. of the system lifetime in the form n(n − 1)α 2 . .φ2 (s) = s 2 + (2n − 1)αs + n(n − 1)α 2 By expanding this expression into simple fractions, we find φ (s) =

. 2

n(n − 1)α n(n − 1)α − , s + nα s + (n − 1)α

then, by calculating the inverse function, we get . . f (y) = n(n − 1)α e−(n−1)α y − e−nαy ,

. 2

which is the same as the result above for .w = 1. The analytical calculations of the reliability characteristics are not always possible. Nevertheless, their numerical analysis in the wide domain of initial data is possible. Therefore, in the next section, a procedure for the numerical calculation of

28

2 Reliability Function of a Complex .k-Out-of-.n Model

different reliability characteristics of the considered model will be proposed. Further, in Sect. 2.1.5.6, this procedure will be used for the numerical analysis of the model with some examples.

2.1.5.5

The General Procedure of the System Reliability Characteristics Calculation and Numerical Experiments

Based on the results of the previous section, the general procedure of the problem solution can be implemented with the help of the following algorithm. Algorithm 2.2 Beginning. Determine: Integers .n, k, real .wi (i = 1, k), distribution . A(t) of the system components’ lifetime and its p.d.f. Work. Step 1. Taking into account that the system’s failure moment .Yk according to formula (2.19) equals Y = W1 X 1 + W2 X 2 + · · · + Wk−1 X k−1 + Wk X k ,

. k

calculate the following: ⎧ ⎪ ⎨1 − wi , . Wi = w1 · · · wi−1 (1 − wi ) ⎪ ⎩ w1 · · · wk−1

i = 1, i = 2, k − 1, i = k.

Step 2. Taking into account that, according to formula (2.17), the joint distribution density of first .k order statistics . X 1 ≤ X 2 ≤ · · · ≤ X k holds f

. X 1 X 2 ...X k

(x1 , x2 , ..., xk ) =

n! a(x1 )a(x2 ) . . . a(xk )(1 − A(xk ))n−k , (n − k)!

with which following to (2.23) calculate the reliability function

. R(y)

= 1 − FYk (y) = 1 −

n! (n − k)!

.y

.z 1 a(x1 )d x1

0

z k−1 .

a(x2 )d x2 · · · x1

a(xk )(1 − A(xk ))n−k d xk ,

x k−1

where the limits of integration are determined by the relation (2.22) z = y, z i = z i (y; x1 , ..., xi ) =

. 0

y − W1 x1 + W2 x2 − ... − Wi xi (i = 1, k − 1). w1 w2 ...wi

Find the values of the constants .wi (singular points at which the denominator of the c.d.f. . FYk (y) turns into .0) for which the c.d.f. changes its appearance. Step 3. From the system reliability function . R(y), calculate

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

29

– mean time to the system failure .∞ μT = E[Yk ] =

R(y)dy;

.

0

– its variance 2 .σT

.∞ d = Var[Yk ] = (y − μT )2 f (y)dy, where f (y) = FY (y), dy k 0

and coefficient of variation v=

.

σ . μ .

Stop. .

Remark 2.3 The algorithm can also be used for other different problem solutions, for example, to analyze the sensitivity of the system reliability function and its characteristics to the shape of the system components’ lifetime distribution. Further, the algorithm will be applied to some examples.

2.1.5.6

Numerical Experiments

According to Algorithm 2.2, calculate the reliability function of a .2-out-of-.6 model. Since such a system fails due to the failure of two components, we have only one constant that defines the decreasing residual lifetime of surviving components. Therefore, hereafter, suppose .w1 = w. Consider .GW.distribution as the . distribution of the sysa tem’s components’ lifetime, . A(t) ∼ GW θ, , with the corresponding .(1 + θ −1 ) c.d.f. . . .θ . t.(1 + θ −1 ) . A(t) = 1 − exp − , t > 0, a where • .a is a fixed mean components’ lifetime, • .θ is the shape parameter of .GW distribution calculated based on the preset value of the coefficient.of variation, .(1 + 2 · θ −1 ) σ • .v = = a −1 · − 1 is the coefficient of variation, a .(1 + θ −1 )2 • .σ is the standard deviation.

30

2 Reliability Function of a Complex .k-Out-of-.n Model

Also consider Erlang (. Erl) distribution, . A(t) ∼ Erl (l, θ ) with p.d.f. a(y) =

.

θ l l−1 −θ y y e , y > 0. .(l)

In this case, the distribution’s parameters can be represented via corresponding mean a and coefficient of variation .v as follows:

.

l = v−2 , θ = (av2 )−1 .

.

For numerical experiments, we consider the reliability function and its characteristics of .2-out-of-.6 model for given distributions with fixed mean .a and different values .v. Thus, we can analyze the influence of the coefficient of variation of the repair time on the reliability characteristics of the system. In other words, investigate its sensitivity. Suppose that the mean lifetime of the component .a = 1. If .θ = 1, .GW and . Erl distributions transform into exponential one with the mean time .a and the coefficient of variation .v = 1. In this case, its reliability function is 5e−6t − 6w · e− w . . R(t) = 5 − 6w 5t

(2.26)

From formula (2.26), it is clear that .w = 56 leads to changing of the shape of distribution. Since calculating the coefficient .θ for .GW through the value .v is quite difficult, we define the parameter .θ so that .v ≈ 0.5. Moreover, if .θ of .GW takes non-integer values, it is not always possible to obtain a closed-form reliability function . R(t) according to Algorithm 2.2 (the integrand takes a complex form). Thereby, define .θ = 2, then, the coefficient of variation .v = 0.5227. For . Erl distribution, suppose that .v = 0.5, that leads to .θ = 4. Suppose .w = 0.1; 0.5; 1. Figure 2.6 presents the reliability function of the .2-outof-.6 model with different distributions as well as .w and .v. Here, solid line means .v = 1 and reliability function (2.26), dotted one is for . GW with .v = 0.5227 and dash-dotted is for . Erl with .v = 0.5. The legend of the figure denotes the color of line of different .w. The figure shows that the higher reliability coincides with the lower value of .v. The case .w = 1 means no load from failed components to surviving ones, thus this case corresponds to the highest reliability for different .v compared to the values .w < 1. Moreover, dependence of the reliability function curve on the shape of lifetime distribution is observed. On a small interval . y, the system’s reliability as . A ∼ Erl is higher than as . A ∼ GW for each .w, despite close values .v. This may indicate the sensitivity of the reliability function not only to the shape of the lifetime distribution but also to the corresponding value of the coefficient of variation. According to the algorithm, calculate other reliability characteristics of the .2out-of-.6 model (Tables 2.5 and 2.6). These characteristics correspond to the system

2.1 Reliability Function of a Non-repairable k-Out-of-n Model

31

Fig. 2.6 Reliability function of a .2-out-of-.6 model Table 2.5 .E[Y2 ] of a .2-out-of-.6 model .w = 0.1 = 0.5 (. A ∼ Erl) .v = 0.5227 (. A ∼ GW ) .v = 1 (. A ∼ E x p) .v

.0.4925

.w

= 0.5

.0.5670

.w

=1

.0.6668

.0.4316

.0.5251

.0.6420

.0.1867

.0.2667

.0.3667

Table 2.6 .vsys of a .2-out-of-.6 model .w .vcomp .vcomp .vcomp

= 0.5 (. A ∼ Erl) = 0.5227 (. A ∼ GW ) = 1 (. A ∼ E x p)

= 0.1

.0.3802

.w

= 0.5

.0.3049

.w

=1

.0.2986

.0.4813

.0.3854

.0.3641

.0.8993

.0.7289

.0.7100

reliability behavior shown in Fig. 2.6. The lower value of .v leads to the higher value of the system lifetime expectation .E[Y2 ], and the lower value of .w leads to the lower value of .E[Y2 ]. Moreover, as .v ≈ 0.5, the relative error between the considered distributions is .14.11% for .c = 0.1, .7.98% for .c = 0.5 and .3.86% for .c = 1. To distinguish coefficients of variation of the components and the whole system, denote them as.vcomp and.vsys , respectively. Thus, Table 2.6 shows the following. With a decrease in .c, the coefficient of variation of the system .vsys grows and tends to the value of the coefficient of variation of each system element .vcomp . The increasing .vcomp leads to increasing .vsys for all distributions and .w. Thus, the coefficient of variation of the system .vsys confirms that as .c tends to .0 and .vcomp tends to .1, variability with respect to the average lifetime of the system .E[Y2 ] grows.

32

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.7 Reliability function with .v = 1 and quantiles (. A ∼ E x p)

Fig. 2.8 Reliability function with .v = 0.5227 and quantiles (. A ∼ GW )

To study the reliability of a system, a characteristic such as a quantile is often used. This measure shows how long the system will be reliable with a fixed probability. The quantiles .qγ = R −1 (γ ) of the reliability function are presented in Figs. 2.7, 2.8, 2.9. In all cases, red bottoms correspond to .γ = 0.99, whereas black bottoms correspond to .γ = 0.9. All the values for quantiles .γ = 0.999; 0.99; 0.9 are presented in Table 2.7 for different distributions. The values of the table show that for the presented quantiles .qγ , the shape of system components’ lifetime plays a critical role. So, for example, as .w = 0.1 and . A ∼ Erl, a given reliability level .0.9 will last about .8 times longer than for .w = 0.1 and . A ∼ E x p. At that for .q0.999 , the difference for similar case

2.2 On Reliability Function of a k-Out-of-n Model with Component-Wise Repairs

33

Fig. 2.9 Reliability function with .v = 0.5 and quantiles (. A ∼ Erl) Table 2.7 Quantiles of reliability function .qγ .w .q0.999

.q0.99

.q0.9

∼ Exp . A ∼ GW . A ∼ Erl .A ∼ Exp . A ∼ GW . A ∼ Erl .A ∼ Exp . A ∼ GW . A ∼ Erl .A

= 0.1

.0.0026

.w

= 0.5

.0.0059

.w

=1

.0.0083

.0.0419

.0.0804

.0.1027

.0.1019

.0.1641

.0.1945

.0.0088

.0.0192

.0.0271

.0.0804

.0.1458

.0.1859

.0.1576

.0.2345

.0.2784

.0.0344

.0.0691

.0.0972

.0.1813

.0.2784

.0.3517

.0.271

.0.3574

.0.424

is almost .40 times. As the coefficient .w increases, this difference decreases for all values of the quantiles and lifetime distributions of the components. As .w = 1 this difference is reduced by about two times.

2.2 On Reliability Function of a . k-Out-of-.n Model with Component-Wise Repairs In this section, the reliability analysis is presented in case of repair of failed components before the full system failure. During the system’s life cycle, its components are repaired with the help of a single repair facility. It is supposed that the components’ lifetimes have an exponential distribution and their repair times have a general distribution. In Sect. 2.2.1 for the considered system a two-dimensional Markov process

34

2 Reliability Function of a Complex .k-Out-of-.n Model

with the help of the markovization method is proposed. The system of Kolmogorov forward partial differential equations as well as an algorithm of the reliability function calculation are given in Sect. 2.2.2. A numerical investigation is also performed in Sect. 2.2.3.

2.2.1 Preliminaries Consider a component-wise repairable homogeneous .k-out-of-.n model up to its first failure (till its first entering to the system failure state .k). The failed components of the system are repaired by a single repair facility. Suppose that • the components fail according to a Poisson flow with intensity .α; • the random repair times of components are i.i.d. and their common c.d.f. . B(t) is absolutely continuous with p.d.f. .b(t) = B . (t). To study the considered .k-out-of-.n model, we use the so-called markovization method based on the introduction of supplementary variables [12] (see about it applications in reliability models also in [129]). To construct the appropriate Markov process, we introduce as a supplementary variable . X (t)—the elapsed repair time of the component under repair, and consider a two-dimensional stochastic process .

Z = {Z (t) = (J (t), X (t)), t ≥ 0},

where the value . J (t) represents the number of failed components at time .t, .

J (t) = j, if at time t the system is in state j ∈ E,

and the supplementary variable . X (t) means elapsed time—the time spent by the repair facility for repairing the component being repaired. The state space of the process . Z is + .E = {0, ( j, R ) : j = 1, k}. Due to the supplementary variable, the process . Z is a Markov one. Denote its microstate p.d.f.’s with respect to the supplementary variable in domain .0 ≤ x ≤ t < ∞ by .π j (t; x)d x = P{J (t) = j, x < X (t) ≤ x + d x} ( j = 1, k). and appropriate macro-state probabilities for .t ≥ 0 by .t π j (t) = P{J (t) = j} =

π j (t; x)d x.

.

0

2.2 On Reliability Function of a k-Out-of-n Model with Component-Wise Repairs

35

Fig. 2.10 Transition graph of the process . Z with absorption

Taking into account the probabilistic origin of these functions, they should be supposed as non-negative .π j (t; x) ≥ 0, π j (t) ≥ 0. Denote also by .T the system lifetime and by . R(t) the system reliability function, .

T = inf{t : J (t) = k},

R(t) = P{T > t}.

For the study of reliability function of the considered system, we should use the process . Z with absorption state .k. Its transition graph is presented in Fig. 2.10. Here, we denote .λi = (n − i)α, (i = 0, k − 1) the system failure intensity when .i components of .n fail.

2.2.2 Reliability Function The system of Kolmogorov forward partial differential equations for process . Z with absorption state .k in the scope .0 < x < t < ∞ has the following form: . t d π0 (t) = −λ0 π0 (t) + π1 (t, x)β(x)d x, . dt 0 . . ∂ ∂ + π1 (t; x) = −(λ1 + β(x))π1 (t; x), ∂t ∂x . . ∂ ∂ + π j (t; x) = −(λ j + β(x))π j (t; x) + λ j−1 π j−1 (t; x), j = 2, k − 1, ∂t ∂x .t d πk (t) = λk−1 πk−1 (t, x)d x (2.27) dt 0

jointly with initial π (0) = 1,

. 0

(2.28)

36

2 Reliability Function of a Complex .k-Out-of-.n Model

and boundary conditions .

t

π (t, 0) = λ0 π0 (t) + π2 (t; x)β(x)d x, 0 . t π j (t, 0) = π j+1 (t; x)β(x)d x ( j = 2, k − 2),

. 1

0

πk−1 (t, 0) = 0.

(2.29)

This system of differential equations was constructed by the usual method of comparison of the corresponding system state probabilities at close time epochs .t and .t + .t and by passing to the limit ..t → 0. Remark 2.4 The last boundary condition appears due to the fact that because the last state .k is an absorbing one, the process never enters the state .k − 1 with the elapsed repair time equal to zero. The reliability function is connected with the probability of absorbing state by the expression, . R(t) = P{T > t} = 1 − P{T ≤ t} = 1 − πk (t), (2.30) which is its closed-form representation. To solve the hyperbolic system (2.27), the method of characteristics (see [89, 97]) can be used. This method consists in the construction of a family of curves with respect to a parameter .u in the definition domain .0 ≤ x ≤ t < ∞ of system (2.27), that satisfy the system of ordinary differential equations: .

dt dx = 1, = 1, du du

(2.31)

which are called characteristics of Eq. (2.27). (In (2.31), 1’s are the coefficients before both partial derivatives in (2.27), . ∂t∂ and . ∂∂x . In general case, each of the equations (2.31) should be equal to its own coefficient). In the considered case, it is convenient to choose the variable .x as the parameter .u. Then, substituting the coefficients before each partial differential equation of system (2.27) for their representation from (2.31) with respect to parameter .x, one can get the system of ordinary differential equations, which determine the functions .π j (. j = 1, k − 1) along the characteristics: .

d π1 (x) = −(λ1 + β(x))π1 (x), dx d π j (x) = −(λ j + β(x))π j (x) + λ j−1 π j−1 (x), ( j = 2, k − 1). (2.32) dx

The initial and boundary conditions in the form (2.29) should be used for continuation of the solution of the given equations from the characteristics to all the definition domains of the partial differential equations’ system.

2.2 On Reliability Function of a k-Out-of-n Model with Component-Wise Repairs

37

Based on this idea in the paper [132] and below, an algorithm for the system of equations (2.27) and the reliability function calculation has been proposed. The working of the algorithm has been verified for the models .2-out-of-.n and .3-out-of-.n systems that represent special cases of HAP flight module (see Chap. 1). Algorithm 2.3 Step 1. To solve the second of equations (2.27), rewrite the equations for characteristics (2.31) and the first of equations (2.32) in symmetric form as dt = d x = −

.

dπ1 . (λ1 + β(x))π1

(2.33)

⇒ ⇒

(2.34) (2.35)

The first integrals of system (2.33) are .dt = d x dπ1 = −(λ1 + β(x))π1 d x

.

t − x = C, π1 = D1 e−λ1 x (1 − B(x)).

The general solution of a partial differential equation can be represented as a smooth function of its first integrals [97]. Thus, following this rule, the general solution of equation (2.33) along the characteristics .C = t − x is .

D1 = h 1 (C)

⇒

π1 (t, x) = h 1 (C)e−λ1 x (1 − B(x)),

(2.36)

where .h 1 is an arbitrary smooth function, which should be found from the boundary conditions. Step 2.1. The equations for characteristics (2.31) and for the other functions .π j of (2.32) (for . j = 2, k − 1) in symmetric form are dt = d x = −

.

dπ j , ( j = 2, k − 1). (λ j + β(x))π j + λ j−1 π j−1

(2.37)

One of the first integrals of these equations for . j = 2 is the same as before dt = d x

.

⇒

t − x = C.

(2.38)

The other first integral of equations (2.37) for . j = 2 along the characteristic .t − x = C, using the expression for .π1 (t, x) from (2.36), should be found from the following equation: dπ2 = −(λ2 + β(x))π2 + λ1 h 1 (C)e−λ1 x (1 − B(x)). . (2.39) dx For its solution, one should use the constant variation method. General solution of the homogeneous part of this equation .

d πˇ 2 = −(λ2 + β(x))πˇ 2 dx

38

2 Reliability Function of a Complex .k-Out-of-.n Model

has the form

πˇ = D2 e−λ2 x (1 − B(x)),

. 2

and changing the constant . D2 by function . D2 (x) the constant variation method for unknown function . D2 (x) gives the following equation: .

D2. (x) = λ1 h 1 (C)e−(λ1 −λ2 )x ,

which particular solution is .

D2 (x) = −

λ1 h 1 (C)e−(λ1 −λ2 )x . λ 1 − λ2

Thus, the general solution of the heterogeneous equation (2.39) has the form .

. λ1 −(λ1 −λ2 )x h 1 (C)e .π2 = D2 − e−λ2 x (1 − B(x)) = λ 1 − λ2 . . λ1 −λ1 x −λ2 x − h 1 (C)e (1 − B(x)). = D2 e λ 1 − λ2

(2.40)

Putting . D2 = h 2 (C), where .h 2 is an arbitrary smooth function, one can obtain the general solution along the characteristic of the corresponding partial differential equation in the form . π (t, x) = h 2 (t − x)e−λ2 x − h 1 (t − x)

. 2

. λ1 e−λ1 x (1 − B(x)). λ 1 − λ2

(2.41)

Step 2.2. (for cases when .k > 3). The same approach should be used for the other equations. One of the first integrals for any of equations of (2.37) for .k = 3, k − 1 is the same as before .dt = d x ⇒ t − x = C. The construction of general solution for equations .

dπ j = −(λ j + β(x))π j + λ j−1 π j−1 ( j = 3, k − 1) dx

gives the following result: . .π j (t, x)

=

h j (t − x)e−λ j x +

j−1 . (−1)i + i=1 .

⎞ λ j−1 · · · λ j−i h i e−λi x ⎠ (1 − B(x)), (λi − λi+1 ) · · · (λi − λ j )

(2.42) .

2.2 On Reliability Function of a k-Out-of-n Model with Component-Wise Repairs

39

where .h 1 , . . . h j are arbitrary smooth functions. Using the substitution, hˆ = (−1)i

. i

λ j−1 · · · λ j−i h i , i = 1, j − 1, hˆ j = h j , (λi − λi+1 ) · · · (λi − λ j )

(2.43)

one can represent the general solution of the considered system of partial differential equations (Kolmogorov forward equations) (2.27) in the form π j (t, x) =

. j .

.

. hˆ i (t − x)e

−λi x

(1 − B(x)) ( j = 1, k − 1).

(2.44)

i=1

Note, that the functions .hˆ 1 , ..., hˆ k−1 are connected by the equation 0=

k−1 .

.

hˆ j (t),

(2.45)

j=1

which can be obtained from the last boundary condition in (2.29) by substitution of x = 0, j = k − 1.

.

Step 3. Find LT .h˜ j (s) of all functions .h j (t) ( j = 1, k − 1) in terms of LT .π˜ 0 (s) using the representation of the functions .π j in the form (2.44) and the boundary conditions (2.29). Step 4. Find LT .π˜ 0 (s) from the first of equations (2.27). Step 5. Find LT .

1 1 ˜ R(s) = − π˜ k (s) = (1 − λk−1 π˜ k−1 (s)) s s

of the reliability function . R(t) using its representation (2.30). Further, using analytical or numerical methods, by applying the inverse Laplace transform, find the reliability function . R(t). . Stop. .

2.2.3 An Example As before, as an example, consider a .3-out-of-.6 model. According to [132] the reliability function . R(t) in terms of LT takes the form: .

C3 (s) · s 2 + C2 (s) · s + C1 (s) ˜ , R(s) = .

(2.46)

40

2 Reliability Function of a Complex .k-Out-of-.n Model

where ˜ + 5α) − 5b(s ˜ + 4α), = 1 + 5b(s . . ˜ + 5α) − 25b(s ˜ + 4α) , C2 (s) = 3α 5 + 23b(s . . ˜ + 5α) − 125b(s ˜ + 4α) , C1 (s) = 2α 2 37 + 98b(s . . ˜ + 5α) − 5b(s ˜ + 4α) + s(1 − 5b(s ˜ + 5α) − 5b(s ˜ + 4α) , . = 6α(5α + s)(4α + s) 1 + 4b(s

.C 3 (s)

from where the expectation of the system lifetime is ˜ ˜ 37 + 98b(5α) − 125b(4α) ˜ . E[T ] = R(0) = ˜ ˜ 60α(1 + 4b(5α) − 5b(4α))

.

Remark 2.5 It should be noted that, by substituting the exponential distribution of repair time, this result coincides with those obtained with a simple birth and death process with absorbing state .k, .

˜ R(s) =

74α 2 b2 + (1 + bs)2 + 5αb(2 + 3bs) . 120α 3 b2 + 74α 2 b2 s + s(1 + bs)2 + 5αbs(2 + 3bs)

Present the reliability function graphically. Suppose that the repair time distribution is Erlang (. Erl(l, θ )) with corresponding mean time .b, coefficient of variation .v ˜ and LT .b(s), . .l √ θ −1 ˜ .b = l · θ , v = l/l, b(s) = . s+θ

Fig. 2.11 . R(t) of .3-out-of-.6 model, . B ∼ Erl

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment Table 2.8 Mean system lifetime .E[T ], . B ∼ Erl .v = 0.5 .v = 0.7

.v

.E[T ]

.0.7083

.0.6522

.0.6725

41

=1

The reliability function for such an example for .v = 1, 0.7, 0.5 is presented in Fig. 2.11 with mean time to system failure (Table 2.8). The figure shows that during time .t the behavior of reliability function is asymptotically insensitive to different .v, however, according to the table, the expectation of the system lifetime .E[T ] is slightly growing with the increasing .v.

2.3 On Reliability Analysis of a . k-Out-of-.n Model in a Random Environment The preliminary’s subsection gives a brief overview of classical and modern works, devoted to reliability study of such systems. Section 2.3.2 presents a general stochastic model of a homogeneous redundant system operating in a random (Markov) environment. To perform its reliability study, a two-dimensional stochastic process is introduced, its transition graph is presented and the corresponding system of Kolmogorov equations for the system state probabilities together with its solution are given. Relations for calculating stationary and non-stationary reliability measures for such a system are given. Section 2.3.3 demonstrates the results of numerical analysis for a special case of the flight module of the tethered HAP based on a hexacopter, consisting of six homogeneous rotors, operating in a random environment that takes two states.

2.3.1 Preliminaries The possibility of long-term operation of tethered unmanned high-altitude platforms, which is one of the main advantages over autonomous UAVs, puts forward a number of new requirements towards reliability of both individual nodes and the high-altitude platform as a whole. The multi-rotor architecture of such platforms allows the platform with .n rotary-wing engines to remain operational even after .k − 1 engines fail. However, the failure of a part of the engines causes an increase in the load on the remaining engines, which leads to a decrease in their reliability. Moreover, the performance of the system depends on the location of the failed engines. Thus, the worst case for this system is the failure of nearby rotors. Consequently, when developing models for assessing reliability and searching for the optimal architecture of such multi-rotor systems, it is necessary to take into account the factor of “failure

42

2 Reliability Function of a Complex .k-Out-of-.n Model

dependence”. Thus, the considered systems can be modeled using heterogeneous k-out-of-.n models, whose component failures depend on the configuration of the failed components. Due to the wide area of practical application, numerous works have been devoted to the study of .k-out-of-.n systems. In [151], an overview of classical and modern works, devoted to the considered class of systems was proposed [9, 33, 145, 170, 172], several variants of such a model were considered, including those that take into account the dependence of system failures on the configuration of failed components, and an algorithm was developed that allows calculating the reliability function of such a system, the mean and variance of its uptime. Earlier, in [57], the results of calculating the reliability characteristics for a system of the (2, 3)-out-of-6 type were obtained taking into account the increase in the functional load on the remaining elements, as well as taking into account the location of the failing elements. The results obtained were compared with similar results of classical models for analyzing the reliability of redundant systems, and it was concluded that, despite the fact that the latter give expectedly higher estimates of reliability characteristics, they do not take into account the heterogeneity of the failure structure of multi-rotor high-altitude modules. High-altitude modules of tethered unmanned HAPs, like most technical systems, operate in a changing environment, which can be of both regular and random nature, and the frequency of these changes can be either commensurate with or be significantly higher or less than the system failure rate. The influence of these factors on the reliability of the system is of considerable interest in the context of the rapidly developing technical capabilities of the modern world. There are a number of works devoted to the study of the behavior of queuing systems operating in a random environment. A fairly detailed survey of modern works on this topic can be found, for example, in [48]. In [124], the influence of environmental variability on the stationary reliability characteristics of technical systems, as well as on the distribution of their time to failure, was investigated. External factors affecting the uptime of tethered unmanned HAPs are, in particular, weather conditions (rain, snow, hail, wind load). The current subsection contains the results of [84] and is devoted to the study of a Markov reliability model of a .k-out-of-.n-type system operating in a random Markov environment, taking into account the increase in the functional load and the location of its failed elements.

.

2.3.2 General Model We consider a model of a .k-out-of-.n-type system consisting of .n homogeneous elements (rotors), assuming that the system operates in a random environment that accepts .m states. Such a hot standby system is efficient as long as at least .(n − k + 1) of .n elements are operating, i.e., the system is considered inoperative if .k elements fail, and.k0 if the.k failed elements are not located next to each other (.2 ≤ k0 ≤ k − 1). It is assumed that the elements of the system operate and fail independently of each other.

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

43

The states of the system within the .(k0 , k)-out-of-.n model can be described by vectors .z = {z c , z 1 , ..., z n }, the first component of which describes the state of the external environment and takes .m values (.z c = 1, m), and the components . z i (i = 1, n) indicate the states of the system elements and take two values: .0— if an element is operable, and .1—otherwise. Denote by . E = {(z c , z 1 , ..., z n ); z c = 1, m, z i = {0, 1}, (i = 1, n)} the phase space of possible states of such a system, n .|E| = m × 2 , and by . E 0 , E 1 —subsets of its operable and failure states, respectively.

2.3.2.1

System Model

We consider a.(k0 , k)-out-of-.n model of a homogeneous system under the assumption that all elements of the system have the same distributions of uptime, and failures of some elements lead to an increase in the functional load on the rest of the operable elements. We will consider the operation of the system before its first failure, taking into account the location of the failed elements, without limiting the number of repair devices. The system is inoperative when .k0 adjacent elements fail or when .k of any elements fail. Since the system under consideration is homogeneous, it is allowed to enlarge its states by combining states with the same number of failed elements. In this case, the state space of the system has the form . E = {(z c , z i ); z c = 1, m, (i = 0, k)} with the total number of states . N = (k + 1) × m, where: z = 1, m—the state of the external environment in which the system operates; z = 0—the initial state of the system when all elements are operational; ... . z k0 = k 0 , (k 0 = 2, k − 1)—the state of the system, in which .k 0 elements not standing next to each other are faulty; ... . z k−1 = k − 1—the state of the system in which .(k˘1) elements that are not nearby are faulty; . z k = k—the state of the system, in which either .k 0 elements standing nearby or any .k elements are faulty. . c

. 0

Suppose that the durations of uptime for all elements and their recovery time have exponential distributions. We will denote through .α0 , β0 the parameters of the uptime distribution of the elements and distribution of the time of their repair, and through .αi (i = 1, k), the parameters of the uptime distribution of the remaining elements after the failure of .i elements, respectively, in a stable external environment (it can be seen that .α0 < α1 < · · · < αk0 < · · · < αk ). We also denote by .αc 0 , βc 0 , αc i (i = 1, k) the corresponding parameters when the system operates in a random environment in a state .c (c = 1, m).

44

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.12 Transition graph of the process . X (t) when the system operates in the .cth state of the random environment

It is assumed that when the state of the external environment changes, the system components instantly change the failure rates. Suppose further that changes in the external environment are described by a homogeneous Markov process with a finite number .m of states. On the state space . E, we introduce a two-dimensional random process . X (t) = (Z 0 (t), Z (t)), the first component of which takes .m possible values and describes the state of the environment, and the second—the number of failed system elements at time .t and takes .k + 1 possible values .0, k. Figure 2.12 shows the transition graph of the process . X (t) when the system is operating in the.cth state of the external environment, where:.wc i = (n − i)αc i , (i = 0, k)—the intensity of transitions of the process . X (t) due to failures of not adjacent elements. Starting from the state .k0 − 1, it becomes possible for the process . X (t) to go directly to the failure state .k due to the failure of the neighboring-elements of the system. The intensities of the corresponding transitions are equal .2αc j for the state . j, . j = k 0 − 1, k − 2. According to the assumptions made, the system can only go to “neighboring” states. In what follows, matrix notation is used, where vectors, as usual, are understood as column vectors, the prime is used for the transposition operation, while derivatives are indicated by the upper point. We introduce the following notation: . = [λc, b ]—transition rate matrix (TRM) of the process of changes in the external environment; . .c = (λc, 1 , λc, 2 , ..., λc, m )—a row vector of intensities of transitions of the external .λ environment from the state .c, .c = 1, m; m . .λc = λc, b —the intensity of changes in the .cth state of the external envi.

b=1,b.=c

ronment; . A c —TRM of dimension .(k + 1) × (k + 1) of the process of system reliability during its operation in the .cth environment, .c = 1, m,

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

⎡ −wc 0 ⎢ ⎢ ⎢ . Ac = ⎢ ⎢ ⎣

wc 0 0 0 −wc 1 wc 1 0 0 −wc 2 ... ... ... 0 0 0 0 0 0 0 0 0 ... ... ... 0 0 0 βc 0 0 0

... 0 0 0 ... 0 0 0 ... 0 0 0 ... ... ... ... ... −wc k0 −1 (n−(k0 +1))αc k0 −1 0 ... 0 −wc k0 (n−(k0 +2))αc k0 ... 0 0 −wc k0 +1 ... ... ... ... ... 0 0 0 ... 0 0 0

... 0 ... 0 ... 0 ... ... ... 0 ... 0 ... 0 ... ... ... −wc k−1 ... 0

45 0 0 0 ...

2αc k0 −1 2αc k0 2αc k0 +1 ... wc k−1 −βc 0

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

Under the assumptions made, the process . X (t) = (Z 0 (t), Z (t)) is a two-dimensional Markov process with a state space. E and a block TRM. Q = [Q x,y ], the diagonal blocks . Q x,x of which (at . y = x) have the form of matrices . Q x,x = Ac − λc I (c = 1, m), and outside the diagonal blocks . Q x,y (at . y .= x) have the form . Q x,y = λc, b I (c . = b), where . I is the identity matrix. Further, all the vectors are implied as the vector-row, and .. means the transposition. 2.3.2.2

Kolmogorov Equations

Denote by: the . pc i (t) = P{Z 0 (t) = c, Z (t) = i}, pc i (0) = l c δi,0 , (c = 1, m, i = 0, k) probabilities of the states of the process . X (t) and its initial distribution when the system operates in the .cth state of the external environment, where .δ is the Kronecker symbol; .π . . (t) = (π.1. (t), π.2. (t), ..., π.m. (t))—vector of state probabilities, where sub-vectors . .π .c (t) = ( pc 0 (t), pc 1 (t), ..., pc k (t)), (c = 1, m) describe the system state probabilities during its operation in the .cth environment; . .l. = (l 1 , ..., l m )—the initial distribution of the external environment; .e .0. = (1, 0, ..., 0)—a vector, the first component of which is equal to .1, and the remaining .0, corresponding to a completely serviceable state of the system. The system of Kolmogorov differential equations for the state probabilities of the Markov process . X (t) with the initial condition .π. . (0) in vector form takes the form: . π.˙ (t) = π. (t)Q, π. . (0) = (l1 e.0. , ..., lm e.0. ).

.

(2.47)

From here, applying the Laplace transform (LT) for it, we get a solution to the system. Hereinafter, the LT will be denoted by a tilde sign. . . π.˜ (s) = s π.˜ (s) − π. . (0), or π.˜ (s) = (s I − Q)−1 π. . (0).

.

(2.48)

Taking into account the structure of the matrix . Q, we can represent (2.47) in the form of a system of equations corresponding to the operation of the system in various environments: . π.˙ c (t) = π.c. (t)(Ac − λc I ) +

m .

.

π.b. (t)λb,c I, π.c. (0) = Ic e.0. , (c = 1, m).

b=1,b.=c

(2.49)

46

2 Reliability Function of a Complex .k-Out-of-.n Model

2.3.2.3

Stationary Characteristics

The stationary probabilities of the states of the system must satisfy the system of equilibrium equations and the normalization condition, which can be written in vector form: .π . . Q = 0, π. . 1. = 1. Similarly, taking into account the structure of the matrix . Q, we obtain a system of equations for the stationary probabilities of the states of the system in the form: π. . (Ac − λc I ) +

m .

π.b. λb,c I = 0, (c = 1, m)

. c

(2.50)

b=1,b.=c

which, with the normalization condition .

m .

c=1

Q π.c. =

k m . .

pc i = 1, allows us to find its

c=1 i=0

unique solution.

2.3.2.4

System Reliability Function

As expected, the subsets of the system’s healthy and failed states are denoted by E 0 and . E 1 , respectively. Also denoted by .T , the uptime of the system until its first failure, .T = inf{t : Z (t) ∈ E 1 }. We will calculate the reliability function through the cumulative distribution function (c.d.f.) of its uptime

.

.

F(t) = P{T ≤ t};

R(t) = 1 − F(t);

by studying the corresponding process with the failure set . E 1 as an absorbing set of states. Representing the matrix of transition intensities . Q, the state probability vector .π . . (t) and the initial state vector in block form: . .

Q=

. Q 0,0 Q 0,1 . . .0, , π. . (t) = (π. E. 0 (t), π. E. 1 (t)), e.0. = (.e0, E 0 (t), e E 1 (t)), Q 1,0 Q 1,1

where the matrix blocks with indices .0 and .1 correspond to the transitions of the process from the set of states . E 0 to the set . E 1 and vice versa, and assuming . Q 1,0 = e.0, E1 = 0 , we reduce the system of equations (2.47) to the form: .

. . π.˙ E0 (t) = π. E. 0 (t)Q 0,0 , π.˙ E1 (t) = π. E. 0 (t)Q 0,1 .

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

47

In terms of LT, considering the initial condition, this system can be represented as: .

.

s π.˜ E0 − e.0, E0 = π. E. 0 (s)Q 0,0 , s π.˜ E1 − e.0, E1 = π. E. 0 (s)Q 0,1 ,

.

(2.51)

and has a solution: . . −1 .˜ . (s) = s −1 (I s − Q 0,0 )−1 e.. Q 0,1 . π.˜ E0 (s) = e.0, E1 E 0 (I s − Q 0,0 ) , π 0, E 0

.

Since c.d.f. of the system uptime looks like: .

F(t) = P{T ≤ t} =

.

. π Z (t) = π. E1 (t)1.

z∈E 1

˜ . the generating function of the And, therefore, its LT is equal to . F(s) = π.˜ E1 (s)1, ˜ can be represented as: system uptime . f˜(s) = s F(s) .

. . . f˜(s) = s π.˜ E1 (s)1. = (I s − Q 0,0 )−1 e.0, E 0 Q 0,1 1.

The last expression has the form of a fractional rational function with respect to the variable .s, its inversion allows us to find the c.d.f. of the system uptime and the system reliability function, respectively.

2.3.3 Numerical Example For numerical analysis, consider the system of the flight module of an unmanned tethered high-altitude platform based on a hexacopter, consisting of six homogeneous rotors, operating in a random environment that takes two states (.m = 2) under the same assumption that the system is not operational when it fails two adjacent rotors or when any three rotors fail (.(2, 3)-out-of-.6 model). All assumed operating conditions of the system and designations are retained with the following parameter values:.k0 = 2, k = 3, n = 6, m = 2, l1 = 1, l2 = 0. The transition graph of the process . X (t) in this case is shown in Fig. 2.13. In this case, the transition intensity matrix . Q has the form: ⎡

−γ1,0 6α10 0 ⎢ 0 −γ1,1 3α11 ⎢ ⎢ 0 0 −γ1,2 ⎢ ⎢ β10 0 0 .Q = ⎢ ⎢ λ2,1 0 0 ⎢ ⎢ 0 0 λ 2,1 ⎢ ⎣ 0 0 λ2,1 0 0 0

0 λ1,2 0 0 2α11 0 λ1,2 0 4α12 0 0 λ1,2 −γ1,3 0 0 0 0 −γ2,0 6α20 0 0 0 −γ2,1 3α21 0 0 0 −γ2,2 λ2,1 β20 0 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ λ1,2 ⎥ ⎥, 0 ⎥ ⎥ 2α21 ⎥ ⎥ 4α22 ⎦ −γ2,3

48

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.13 Transition graph of the process . X (t) = (Z 0 (t), Z (t))

where the values .γi, j are equal to the sum of all other elements of the corresponding row of the matrix. In this case, states (1, 3) and (2, 3) are states of system failure, respectively, when the system operates in the first and second states of a random environment. We will calculate the non-stationary probabilities of the states of the process. X (t). In this case, the system of equations (2.48) will take the form: . . (π.˙ 1 (t), π.˙ 2 (t)) = (π.1. (t), π.2. (t)) · Q,

.

(2.52)

where .π.c. (t) = ( pc 0 (t), pc 1 (t), pc 2 (t), pc 3 (t)), (c = 1, 2). It needs to be solved with the initial conditions: . p10 (0)

= 1, p11 (0) = p12 (0) = p13 (0) = p20 (0) = p21 (0) = p22 (0) = p23 (0) = 0.

Similarly, to calculate the stationary probabilities of system states, we represent the system of equations (2.50) in the form: (π.1. (t), π.2. (t)) · Q = 0.

.

(2.53)

Its only solution can be found using the normalization condition: .

p10 + p11 + p12 + p13 + p20 + p21 + p22 + p23 = 1.

To calculate the system reliability function, we modify the process by prohibiting the exit from the failure states (1, 3) and (2, 3). The system of equations (2.52) in this case is modified correspondingly and is being solved in terms of the LT. The solution of the obtained system of linear algebraic equations are the system state probabilities in terms of LT, knowing which we can find all the probabilities of the states of the system, including . p13 (t) and . p23 (t).

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

49

Then, c.d.f. of the system uptime is determined by the sum of probabilities, and from here we can already find the system reliability function: .

F(t) = p13 (t) + p23 (t);

R(t) = 1 − F(t).

The solutions of the above systems of equations were found using the inverse Laplace transform using a software module developed in the MATLAB environment. Next, we will conduct a comparative analysis of the reliability characteristics of the system operating in random and non-random (stable) environments in order to identify the influence of the randomness of the environment and the dynamics of its change on the reliability of the system. To do this, it is necessary to agree on the corresponding parameters of failures and recovery of system elements. Let all the parameters for the random environment be given. We calculate the average values for the stable environment using the following formulas: .

λ2,1 λ1,2 λ2,1 λ1,2 α10 + α20 , β0 = β10 + β20 , λ1,2 + λ2,1 λ1,2 + λ2,1 λ1,2 + λ2,1 λ1,2 + λ2,1 λ1,2 λ2,1 λ1,2 λ2,1 α11 + α21 , α2 = α12 + α22 . α1 = λ1,2 + λ2,1 λ1,2 + λ2,1 λ1,2 + λ2,1 λ1,2 + λ2,1

α0 =

Thus, the stationary state probabilities and the reliability function of a system operating in a stable environment have the following form [58]: π =

. 0

π1 = π2 = π3 = R(t) = +

10β0 α1 α2 ; β0 (10α1 α2 + 12α0 α2 + 9α0 α1 ) + 60α0 α1 α2 12β0 α0 α2 ; β0 (10α1 α2 + 12α0 α2 + 9α0 α1 ) + 60α0 α1 α2 9β0 α0 α1 ; β0 (10α1 α2 + 12α0 α2 + 9α0 α1 ) + 60α0 α1 α2 60α0 α1 α2 . β0 (10α1 α2 + 12α0 α2 + 9α0 α1 ) + 60α0 α1 α2 12α0 (α1 − 2α2 )e−5α1 t 4α1 (3α0 − 5α2 )e−6α0 t + + − (6α0 − 5α1 )(6α0 − 4α2 ) (6α0 − 5α1 )(5α1 − 4α2 ) 18α0 α1 e−4α2 t . (6α0 − 4α2 )(5α1 − 4α2 )

The results of calculations for systems operating in stable and random environments, for each special case as an example, are given in tables and presented on the corresponding graphs for various values of the parameter .v, which determines the influence of the external environment on the failure rate of the system’s elements. The following values of the failure rates are used: α

. 10

= 1, α11 = 1.4, α12 = 1.6, α20 = v · α10 , α21 = v · α11 , α22 = v · α12 .

50

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.14 Non-stationary probabilities of system uptime .(1 − π f ail (t)) in a stable environment Table 2.9 Stationary system uptime probabilities for case 1 .v In a stable environment In a random environment .1 − π f ail = 1 − π3 .1 − π f ail = 1 − (π13 + π23 ) 0.1 1.0 5.0 10.0

0.4230 0.2874 0.1185 0.0683

0.4854 0.2874 0.1678 0.1446

Case 1. Transition rates of the external environment: .λ1,2 = λ2,1 = 1 and the recovery rates are commensurable .β10 = β20 = 1 (Fig. 2.14 and Table 2.9). The obtained results show that for commensurate transition rates of the external environment and recovery rates, the reliability measures of the system operating in a stable and a random environment are quite close, and for .v = 1 they coincide. With an increase in .v, which characterizes the aggressiveness of the random environment, the time-dependent reliability measures of the system decrease, both in stable and random environments (Figs. 2.15, 2.16 and 2.17). Case 2. Transition rates of the external environment .λ1,2 = λ2,1 = 1 are commensurable and there is a “fast recovery” of the elements (.β10 = β20 = 100). In this case, the dynamics of the system behavior in stable and random environments also turns out to be quite similar. Under a quick recovery, the reliability of the system predictably increases. However, with an increase in the value of the parameter .v, which characterizes the aggressiveness of the random environment, the probability of a failure-free operation of the system decreases, but it is still greater

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

51

Fig. 2.15 Non-stationary probabilities of system uptime .(1 − π f ail (t)) in a random environment

Fig. 2.16 System reliability function for a stable environment

52

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.17 System reliability function for a random environment Table 2.10 Stationary system uptime probabilities for case 2 In a stable environment In a random environment .v .1 − π f ail = 1 − π3 .1 − π f ail = 1 − (π13 + π23 ) 0.1 1.0 5.0 10.0

0.9865 0.9758 0.9308 0.8800

0.9867 0.9758 0.9327 0.8884

than in case 1 due to the sufficient rate of recovery. The system reliability function in this case has not changed compared to case 1, since only the repair rates have been changed, which does not affect the system reliability function as a whole (Fig. 2.18 and Table 2.10). Case 3. Slow change in the external environment: .λ1,2 = λ2,1 = 0.01; repair rates are commensurable: .β10 = β20 = 1 (Fig. 2.19 and Table 2.11). It can be seen from the figures that with a slow change in the external environment, the parameter .v, which characterizes the aggressiveness of the random environment, does not greatly affect the reliability function of the system. The nature of the convergence of non-stationary probabilities of failure-free operation of the system to stationary ones and the behavior of the reliability function in a random environment differ significantly from the corresponding characteristics for systems operating in a stable environment.

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

53

Fig. 2.18 Reliability measures of the system operating in a stable and a random environment— special case 2

Case 4. Rapid non-homogeneous change in the external environment: .λ1,2 = 1000, λ2,1 = 10; repair rates are commensurable: .β10 = β20 = 1 (Fig. 2.20 and Table 2.12). With a sufficiently rapid non-homogeneous change in the external environment (i.e., deterioration of external conditions), the characteristics of the non-stationary probabilities of the system when working in stable and random environments are the same. And the system reliability function tends to zero faster when working in a random environment than when working in a stable environment. In the current section, we solved the relevant problem of studying an analytical model for the reliability analysis of the flight module of a tethered multi-rotor highaltitude platform in the form of a homogeneous hot standby system consisting of .n elements operating in a random environment. The model takes into account the layout of the failed components. A general Markov reliability model of the considered system operating in a random Markov environment is proposed. Relations for calculating stationary and non-stationary reliability measures for such a system are given. A numerical study and comparison of the reliability characteristics for a sys-

54

2 Reliability Function of a Complex .k-Out-of-.n Model

Fig. 2.19 Reliability measures of the system operating in a stable and a random environment— special case 3 Table 2.11 Stationary system uptime probabilities for case 3 In a stable environment In a random environment .v .1 − π f ail = 1 − π3 .1 − π f ail = 1 − (π13 + π23 ) 0.1 1.0 5.0 10.0

0.4230 0.2874 0.1185 0.0683

0.5434 0.2874 0.1808 0.1629

2.3 On Reliability Analysis of a k-Out-of-n Model in a Random Environment

55

Fig. 2.20 Reliability measures of the system operating in a stable and a random environment— special case 4 Table 2.12 Stationary system uptime probabilities for case 4 In a stable environment In a random environment .v .1 − π f ail = 1 − π3 .1 − π f ail = 1 − (π13 + π23 ) 0.1 1.0 5.0 10.0

0.7874 0.2874 0.0752 0.0391

0.7874 0.2874 0.0752 0.0391

tem operating in a stable and random environment with two states is carried out. The results of the numerical study, presented in the form of tables and graphs, showed both common features and differences in the operation of systems in a random and a stable environment.

Chapter 3

Reliability Characteristics for Repairable k-Out-of-n Model

.

.

In this chapter, we consider time-dependent and stationary characteristics of a reparable .k-out-of-.n models. The preliminaries Sect. 3.1 provides a definition of the types of system recovery after a complete failure, as well as additional notations. The time-dependent probabilities for two repair scenarios are considered in Sect. 3.2. The stationary characteristics in Sect. 3.3 are presented. This chapter is based on the papers [14, 42, 121].

3.1 Preliminaries According to the system’s structure, failed components and the whole system are repaired, and for this, there is only one facility. Dealing with the reparable system, we need to consider some procedures of system restoration after failure. There are at least two possibilities: • partial repair means a transition from a subsequent state of the system to a previous one when any .i element (.i = 1, k) fails during a random time; • full repair occurs when after the .kth element failure system is repaired during random time, after which the system works like a new one, that is, it goes into state .0; in this case, the rest of the failed elements (from .1 to .k − 1) are repaired as in a partial repair. Suppose that the lifetimes of the systems’ components are i.i.d. r.v.’s that are exponentially distributed with parameter .α, thus .a = α −1 is the mean lifetime of system components. Repair times are also i.i.d. r.v.’s . Bi = B (i = 1, 2, . . . ) for partial and . Fi = F (i = 1, 2, . . . ) for full repair respectively with the common c.d.f.’s . B(t) = P{Bi ≤ t} and . F(t) = P{Fi ≤ t}. Suppose that the instantaneous repairs are impossible, and their mean times are finite:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 V. M. Vishnevsky et al., Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9445-8_3

57

58

3 Reliability Characteristics for Repairable k-Out-of-n Model

∫∞ ∫∞ . B(0) = F(0) = 0, b = (1 − B(x))d x < ∞, f = (1 − F(x))d x < ∞. 0

0

Corresponding LTs of p.d.f.’s .b(t) and . f (t) are denoted by ˜ b(s) =

∫∞

.

e−st b(t)dt, f˜(s) =

0

∫∞

e−st f (t)dt.

0

Denote by . E = {0, 1, 2, ..., k − 1, k} the set of system states, where • .0 means that all .n components operate, • .i means that .i components out of .n (1 ≤ i ≤ k − 1) fail, one of them is repaired, and others .(n − i) operate, • .k means that .k components fail, the whole system fails and is repaired. We also introduce a random process . J = {J (t), t ≥ 0}, where .

J (t) = j, if in the time t the system is in the state, j ∈ E.

It is supposed that . J (0) = 0. In this section, we deal with the calculation of time-dependent system state probabilities (t.d.s.s.p.’s) .π j (t) = P{J (t) = j}, j ∈ E, steady-state probabilities (s.s.p.’s) π j = lim P{J (t) = j}, j ∈ E,

.

t→∞

as well as the availability coefficient .

K av =

∑

πi = 1 − πk .

0≤i≤k−1

3.2 Time-Dependent Characteristics Analogously to the calculation of reliability function in Sect. 2.2.1 for calculating t.d.s.s.p.’s the method of supplementary variables’ introduction (markovization method) in Sects. 3.2.1 and 3.2.2 is used for partial and full repair scenarios, respectively. It is also supposed that components’ lifetimes have exponential distribution and repair time is arbitrary distributed. The Kolmogorov forward partial differential equations for the time-dependent state probabilities of the process describing the system behavior are proposed. The algorithms for this system solution are presented.

3.2 Time-Dependent Characteristics

59

To study the time-dependent characteristics as denoted above also by .

Z (t) = {J (t), X (t)}t≥0

a two-dimensional process, where . J (t) is the system state at time .t, and . X (t) represents elapsed repair time of the failed component or the whole system. The conditional partial and full repair intensities given elapsed repair times .x are .β(x) and .φ(x), respectively. They are denoted as β(x) =

.

f (x) b(x) , φ(x) = . 1 − B(x) 1 − F(x)

Denote by • .π0 (t) = P{J (t) = 0}—the probability of a “good” state of all of .k components at time .t; • .πi (t; x)d x = P{J (t) = i; x < X (t) ≤ x + d x}–the joint probability that there are .i failed components at time .t and the component is being repaired with the elapsed repair time between .x and .x + d x, .i = 1, k.

3.2.1 Partial Repair Scenario First, consider the investigation of a .k-out-of-.n system in case of partial repair scenario.

3.2.1.1

Kolmogorov Partial Differential Equations for t.d.s.s.p.

The state transition graph of a process . Z (t) described above in case of partial repair scenario is represented in the Fig. 3.1.

Fig. 3.1 Transition graph of the process . Z (t)

60

3 Reliability Characteristics for Repairable k-Out-of-n Model

By the usual method of comparing the process in the closed times .t and .t + δ, the following Kolmogorov forward system of partial differential equations for the process probabilities has been proved in [125]. Theorem 3.1 The system of Kolmogorov forward partial differential equations for the process . Z (t) t.d.s.s.p.’s in case of partial repair scenario has the form .

∫ t d β(x)π1 (t, x)d x, (3.1) π0 (t) = −λ0 π0 (t) + dt 0 ) ( ∂ ∂ + π1 (t, x) = −(λ1 + β(x))π1 (t, x), ∂t ∂x ) ( ∂ ∂ + πi (t, x) = −(λi + β(x))πi (t, x) + λi−1 πi−1 (t, x), i = 2, k − 1, ∂t ∂x ) ( ∂ ∂ + πk (t, x) = −β(x)πk (t, x) + λk−1 πk−1 (t, x), ∂t ∂x

jointly with the initial π (0) = 1, πi (0; x) = 0, i = 1, k ∀x ≥ 0,

. 0

(3.2)

and boundary conditions

.

∫ t π1 (t, 0) = λ0 π0 (t) + β(x)π2 (t, x)d x, 0 ∫ t πi (t, 0) = β(x)πi+1 (t, x)d x, i = 2, k − 1,

.

πk (t, 0) = 0.

.

(3.3)

0

Proof To obtain this system of differential equations, we first focus our attention on the transition graph of the process, shown in Fig. 3.1. To construct the system of finite difference equations by the usual method of comparison of the corresponding process state probabilities changes on an infinitesimal small-time epochs .t and .t + ∆t. Then, passing to the limit .∆t → 0, we obtain the system of differential equations (3.1). The initial conditions (3.2) follow from the assumption that, in the very beginning, the system is in the state with all components in the good state. Also by comparing the process state probabilities at close time epochs.t and.t + ∆t, when the supplementary variable takes values close to zero, we obtain the first two boundary conditions (3.3). The last boundary condition follows from the fact that the process . Z (t) never occurs ⃞ in the state .k with the elapsed repair time equal to zero. .

3.2 Time-Dependent Characteristics

3.2.1.2

61

Algorithm for t.d.s.s.p. Calculation

The derived system (3.1) is a hyperbolic system of partial differential equations. For this type of system, the solutions are also obtained by the method of characteristics. According to the method description in Sect. 2.2.2, the following algorithm is proposed to use for t.d.s.s.p.’s calculation. Algorithm 3.1 Step 1. For the solution of the second equations (3.1), rewrite the equations for characteristics (2.31) and obtain the first of equations (2.32) in symmetric form as dπ1 . (3.4) .dt = d x = − (λ1 + β(x))π1 The first integrals of system (3.4) are dt = d x

⇒

t − x = C,

dπ1 = −(λ1 + β(x))π1 d x

⇒

π1 = D1 e−λ1 x (1 − B(x)).

.

(3.5)

The general solution of a partial differential equation can be represented as a continuous function of its first integrals [97]. Thus, following this rule, the general solution of equation (3.4) along the characteristics .C = t − x is .

D1 = h 1 (C)

⇒

π1 (t, x) = h 1 (C)e−λ1 x (1 − B(x)),

(3.6)

where .h 1 is an arbitrary continuous function, which can be found from the boundary conditions. Step 2. The equations for the characteristics (2.31) and for the other functions .π j of (2.32) (for . j = 2, k − 1) in symmetric form are dt = d x = −

.

dπ j , ( j = 2, k − 1). (λ j + β(x))π j + λ j−1 π j−1

(3.7)

Step 2.1. One of the first integrals of these equations for . j = 2 is the same as before .dt = d x ⇒ t − x = C. The other first integrals of the equations (3.7) for . j = 2, along the characteristic t − x = C are using the expression for .π1 (t, x) from (3.5). They can be found from the following equation:

.

.

dπ2 = −(λ2 + β(x))π2 + λ1 h 1 (C)e−λ1 x (1 − B(x)). dx

(3.8)

For its solution, one should use the constant variation method. The general solution of the homogeneous part of this equation

62

3 Reliability Characteristics for Repairable k-Out-of-n Model

.

has the form

d πˇ 2 = −(λ2 + β(x))πˇ 2 dx

πˇ = D2 e−λ2 x (1 − B(x)).

. 2

Changing the constant . D2 by function . D2 (x), the constant variation method for unknown function . D2 (x) gives the following equation .

D2' (x) = λ1 h 1 (C)e−(λ1 −λ2 )x .

Its particular solution is .

D2 (x) = −

λ1 h 1 (C)e−(λ1 −λ2 )x . λ1 − λ2

Thus, the general solution of the heterogeneous equations (5.19) has the form ) λ1 −(λ1 −λ2 )x e−λ2 x (1 − B(x)) = h 1 (C)e .π2 (t, x) = D2 − λ1 − λ2 ) ( λ1 −λ2 x −λ1 x (1 − B(x)). h 1 (C)e − = D2 e λ1 − λ2 (

Putting . D2 = h 2 (C), where .h 2 is an arbitrary smooth function, one can obtain the general solution along the characteristic of the corresponding partial differential equation in the form ( π (t, x) = h 2 (t − x)e−λ2 x − h 1 (t − x)

. 2

) λ1 e−λ1 x (1 − B(x)). λ1 − λ2

Step 2.2. (for cases when .k ≥ 3). The same approach can be used for the other equations. One of the first integrals for any of equations of (3.7) for . j = 3, k − 1 is the same as before .dt = d x ⇒ t − x = C. The construction of general solution for equations .

dπ j = −(λ j + β(x))π j + λ j−1 π j−1 ( j = 3, k − 1) dx

gives the following result: ( π j (t, x) = h j (t − x)e

.

−λ j x

) + H j−1 (t − x; x) (1 − B(x)),

3.2 Time-Dependent Characteristics

63

where .h 1 , . . . h j are arbitrary smooth functions, and the following notation is used

.

H j (t − x; x) =

j ∑ (−1)i i=1

λ j−1 · · · λ j−i h i (t − x)e−λi x . (λi − λi+1 ) · · · (λi − λ j )

(3.9)

Step 3. The equations for the characteristics (2.31) and function .πk (2.32) in symmetric form are .

dx dπk dt = =− . 1 1 β(x)πk + λk−1 πk−1

One of the first integrals of these equations is the same as before dt = d x

.

⇒

t − x = C.

Step 3.1. (.k = 2) The general solution of the last heterogeneous equation of system (3.1) with .k = 2 and with expression for .π1 (t, x) from (3.6) takes the form .

dπ2 = −β(x)π2 + λ1 h 1 (C)e−λ1 x (1 − B(x)). dx

Similar to Step 2.1, we find the general solution ] [ π (t, x) = h 2 (t − x) − h 1 (t − x)e−λ1 x (1 − B(x)).

. 2

Step 3.2. (.k > 2) The last equation of the considered system for.k > 2 is calculated similarly to Step 2.2. As a result, it holds ( π (t, x) = h k (t − x) − h k−1 (t − x)e

. k

−λk−1 x

) λk−1 − Hk−1 (t − x, x) (1 − B(x)), λk−2

where . Hk−1 (t − x, x) is defined from (3.9). Step 4. From the first equation of (3.1), passing to the LP of the probability .π0 (t), substituting the solution (3.6) for the probability.π1 (t, x) and the first initial condition from (3.2), we get ˜ + λ1 ), s π˜ 0 (s) − 1 = −λ0 π˜ 0 (s) + h˜ 1 (s)b(s ˜ + λ1 ) 1 + h˜ 1 (s)b(s π˜ 0 (s) = . s + λ0

.

⃞

Step 5. To complete the solution, it is necessary to calculate the functions .h i . We find them from the boundary conditions from (3.3) in terms of Laplace transforms. Using the substitution (3.9), one can find recursive equations for functions.h˜ i (s) (i = 1, k − 1) calculation.

64

3 Reliability Characteristics for Repairable k-Out-of-n Model

Step 6. Substitution of the obtained solution into expressions for functions .h˜ j (s) and then into LT of functions .π j (t) ends the solution of the system. Stop. . ⃞ 3.2.1.3

Numerical Results

According to the Algorithm 3.1 the t.d.s.s.p.’s in terms of LT of repair time for a 2-out-of-.n system are

.

˜ + b(s ˜ + λ1 ) 1 − b(s) , ˜ ˜ ˜ s(1 − b(s) + b(s + λ1 )) + λ0 (1 − b(s)) ˜ + λ1 )) λ0 (1 − b(s [ ], π˜ 1 (s) = ˜ + b(s ˜ + λ1 )) + λ0 (1 − b(s)) ˜ (s + λ1 ) s(1 − b(s) [ ] ˜ ˜ − b(s ˜ + λ1 )) λ0 λ1 (1 − b(s)) − s(b(s) 1 [ ]. π˜ 2 (s) = · s (s + λ ) s(1 − b(s) ˜ + b(s ˜ + λ1 )) + λ0 (1 − b(s)) ˜ 1

π˜ (s) =

. 0

(3.10)

Consider some numerical examples for t.d.s.s.p.’s of the .2-out-of-.n system. Suppose that .n = 3 mean lifetime .a = 1, the repair time has Erlang distribution (see appropriate notations in Sect. 2.2) with mean .b = 1. Suppose that the coefficient of variation of repair time takes the following values, .v = 1, 0.7, 0.5. Note that as .v = 1 Erlang distribution turns to exponential one with the mean.b. Figure 3.2 demonstrates t.d.s.s.p.’s .πi (t), i = 0, 1, 2 of .2-out-of-.3 model with preset values above.

Fig. 3.2 T.d.s.s.p.’s of the .2-out-of-.3 model under partial repair scenario

3.2 Time-Dependent Characteristics

65

Here the legend of the figure denotes the type of line of different probabilities: a simple line corresponds to probability .π0 (t), a dashed line corresponds to probability .π1 (t), and a dot-dashed line corresponds to probability .π2 (t). Black, green, and red colors correspond to the chosen value .v = 1, 0.7, 0.5 respectively. This example shows that the different values of .v do not influence on t.d.s.s.p.’s behavior as .t → ∞. Moreover, as .t → ∞ t.d.s.s.p.’s tend to their stationary values and turn to s.s.p.’s.

3.2.2 Full Repair Scenario Further analogously, consider the investigation of a .k-out-of-.n system in case of full repair scenario.

3.2.2.1

Kolmogorov Partial Differential Equations for t.d.s.s.p.

The state transition graph of a process . Z (t) in case of full repair scenario is represented in the Fig. 3.3. The t.d.s.s.p.’s follow from the Theorem below, which, as in the previous case, is obtained by the usual method of comparing the process in the closed times .t and .t + δ. Theorem 3.2 The system of Kolmogorov forward partial differential equations for the process . Z (t) t.d.s.s.p.’s has the form .

.

∫ t ∫ t d β(x)π1 (t, x)d x + φ(x)πk (t, x)d x, π0 (t) = −λ0 π0 (t) + dt 0 0 ) ( ∂ ∂ + π1 (t, x) = −(λ1 + β(x))π1 (t, x), ∂t ∂x

Fig. 3.3 Transition graph of the process . Z (t)

(3.11)

66

3 Reliability Characteristics for Repairable k-Out-of-n Model

) ∂ ∂ + πi (t, x) = −(λi + β(x))πi (t, x) + λi−1 πi−1 (t, x), i = 2, k − 1, ∂t ∂x ) ( ∂ ∂ + πk (t, x) = −φ(x)πk (t, x), ∂t ∂x

( .

.

jointly with the initial π (0) = 1, πi (0; x) = 0, i = 1, k ∀x ≥ 0,

. 0

(3.12)

and boundary conditions .

∫ t π1 (t, 0) = λ0 π0 (t) + β(x)π2 (t, x)d x, 0 ∫ t πi (t, 0) = β(x)πi+1 (t, x)d x, i = 2, k − 2,

(3.13)

0

πk−1 (t, 0) = 0,

∫

πk (t, 0) = λk−1

t

πk−1 (t, x)d x.

0

Proof The proof of the theorem is done by the same argumentation as in the proof ⃞ of the previous Theorem. .

3.2.2.2

Algorithm for t.d.s.s.p. Calculation

The Algorithm 3.1 for calculation of t.d.s.s.p.’s in case of partial repair scenario should be expanded to the case of full system repair. So further present only those steps which are different from the algorithm above. Algorithm 3.2 Steps 1–2.2. are repeated. Step 3. For the last equation with . j = k one of the first integrals is the same as before, dt = d x

.

⇒

t − x = C.

The construction of general solution for equations .

dπk (x) = −γ (x))πk (x) dx

gives the following result: π (t, x) = h k (t − x)(1 − G(x)),

. k

where .h k is an arbitrary smooth function.

(3.14)

3.2 Time-Dependent Characteristics

67

Step 4. Similarly, passing to the LT of probability .π0 (t) and substituting the corresponding expressions gives the following result: ˜ + λ1 ) + h˜ 3 (s) f˜(s), s π˜ 0 (s) − 1 = −λ0 π˜ 0 (s) + h˜ 1 (s)b(s ˜ ˜ 1 + h 1 (s)b(s + λ1 ) + h˜ 3 (s) f˜(s) . π˜ 0 (s) = s + λ0

.

Step 5. Initial (3.12) and boundary (3.13) conditions are applied to calculate functions .h˜ j (s). From the last boundary conditions, it follows: ∫

t

π (t; 0) = h k (t) = λk−1

. k

πk−1 (t; x)d x.

0

The substitution in expression (3.9) gives h˜ (s) = λk−1

∑

. k

1≤i≤k−1

[

] ˜ + λi ) 1 − b(s h˜ i (s) . s + λi

Step 6. Substitution of the obtained solution into expressions for functions .h˜ j (s) and then into LT of functions .π j (t) ends the solution of the system. Stop. . ⃞

3.2.2.3

Numerical Results

According to the Algorithm 3.2 the t.d.s.s.p.’s in terms of LT of repair time for a 2-out-of-.n system are

.

s + λ1 , ˜ s(s + λ1 ) + λ0 (1 − b(s + λ1 ))(s + λ1 (1 − f˜(s))) ˜ + λ1 )) λ0 (1 − b(s π˜ 1 (s) = , ˜ + λ1 ))(s + λ1 (1 − f˜(s))) s(s + λ1 ) + λ0 (1 − b(s ˜ + λ1 ))(1 − f˜(s)) λ0 λ1 (1 − b(s 1 π˜ 2 (s) = · . ˜ + λ1 ))(s + λ1 (1 − f˜(s))) s s(s + λ1 ) + λ0 (1 − b(s

π˜ (s) =

. 0

(3.15)

Consider some numerical examples for t.d.s.s.p.’s of the .2-out-of-.n system. Suppose, that .n = 3, mean lifetime .a = 1, the repair time has Erlang distribution (see appropriate notations in Sect. 2.2). Due to different distributions of partial and full repair time, consider various combinations of Erlang distribution with different coefficients of variation. Figure 3.4 demonstrates t.d.s.s.p.’s .πi (t), i = 0, 1, 2 of .2-outof-.3 model. Denote by .vb and .v f the coefficient of variation of partial and full repair, respectively. Thus, the description of Fig. 3.4 is the following:

68

3 Reliability Characteristics for Repairable k-Out-of-n Model

Fig. 3.4 T.d.s.s.p.’s of the .2-out-of-.3 model under full repair scenario (. f = 1)

• the legend of the figure denotes the type of line of different probabilities; a simple line corresponds to probability .π0 (t), a dashed line corresponds to probability .π1 (t), and a dot-dashed line corresponds to probability .π2 (t); • black color corresponds to .b = f = 1, .vb = v f = 1; • red color corresponds to .b = f = 1, .vb = 0.5, .v f = 1; • green color corresponds to .b = f = 1, .vb = 1, .v f = 0.5. According to the Fig. 3.4, the curves’ behavior is analogous and values of t.d.s.s.p.’s are very close despite the different values of .vb and .v f . Consider further the case, when the full repair time is twice the partial repair time, . f = 2. This situation is presented in Fig. 3.5. All notations are the same from the previous case. These two examples show that the different values of .vb and .v f do not affect t.d.s.s.p.’s behavior as .t → ∞. Moreover, as .t → ∞ t.d.s.s.p.’s tend to their stationary values and turn to s.s.p.’s.

3.3 Stationary Characteristics This section is aimed to study the stationary characteristics of a .k-out-of-.n model using the method of markovization. The Sects. 3.3.1 and 3.3.2 present the systems of Kolmogorov differential equations and corresponding solutions, calculated with the method of constant variations, for partial and full repair scenarios, respectively.

3.3 Stationary Characteristics

69

Fig. 3.5 T.d.s.s.p.’s of the .2-out-of-.3 model under full repair scenario (. f = 2)

The s.s.p.’s of the process . Z (t) can be also obtained by passing to .πi = lim s π˜ i (s) s→0

from t.d.s.s.p.’s (see this approach in [14]). Here we present the explicit view of s.s.p.’s obtained with transition .t → ∞ in the systems above and subsequent solution of the system of Kolmogorov differential equations (3.1)–(3.3) for partial and (3.11)–(3.13) for full repair scenario, respectively.

3.3.1 Partial Repair Scenario 3.3.1.1

Balance Equations

For both the partial and the full repair scenario, the process . Z (t) is a Harris one with a positive atom in the state .0 and accordingly to the theory of Harris Markov processes it has a stationary regime and therefore for .t −→ ∞ its t.d.s.s.p.’s tends to its s.s.sp.’s. It means that the process . Z (t) has a stationary distribution for which the stationary regime differential equations (balance equations) hold. For the partial repair scenario, the Eqs. (3.1) from Theorem 3.1 are transformed to the following form: ∫ ∞ .λ0 π0 = π1 (x)β(x)d x, (3.16) 0

π˙ (x) = −(λ1 + β(x))π1 (x),

. 1

π˙ (x) = −(λi + β(x))πi (x) + λi−1 πi−1 (x), i = 2, k − 1, .π ˙ k (x) = −β(x)πk (x) + λk−1 πk−1 (x), . i

70

3 Reliability Characteristics for Repairable k-Out-of-n Model

jointly with initial .π0 (0) = 1 and boundary conditions from (3.3) ∫

∞

π (0) = λ0 π0 + π2 (x)β(x)d x, 0 ∫ ∞ .πi (0) = πi+1 (x)β(x)d x,

. 1

0

π (0) = 0.

. k

(3.17)

Using the method of constant variation, the solution of these systems has been obtained and, in the paper [44], the following theorem has been proved. Theorem 3.3 The s.s.p.’s of the process . Z (t) under partial repair scenario in terms of LT of repair time have the form 1 ˜ 1 ), C1 b(λ λ0 ˜ 1) 1 − b(λ .π1 = C 1 , λ1 ˜ i) ˜ i−1 ) 1 − b(λ 1 − b(λ .πi = C i + S(i − 1) , i = 2, k − 1, λi λi−1 ˜ k−1 ) λk−1 ˜ k−2 ) 1 − b(λ 1 − b(λ − S(k − 2) , .πk = C k · b − C k−1 λk−1 λk−2 λk−2 π =

. 0

(3.18)

where ˜ 2) C2 b(λ ), ( ˜ 1 ) 1 − λ1 1 − b(λ λ1 − λ2 ˜ i ) − S(i − 1), i = 2, k − 2, ˜ .C i = C i+1 b(λi+1 ) + S(i)b(λ λk−1 S(k − 2), .C k = C k−1 + λk−2 ⎛ ⎞ i i ∑ ∏ λ m ⎠ Cj, .S(i) = (−1)i− j+1 ⎝ λ − λm+1 j j=1 m= j C1 =

.

(3.19)

∑ and .Ck−1 is calculated according to the normalization condition . i∈E πi = 1. Proof The solution of the second equation (3.16) gives the probability .π1 (x) in the form π (x) = C1 e−λ1 x (1 − B(x)),

. 1

with the help of which we find the solution of the first equation from (3.16)

3.3 Stationary Characteristics

71

π =

. 0

1 ˜ 1 ). C1 b(λ λ0

The solution of the following two equations of the system (3.16) is calculated by the method of constants’ variation: π (x) = Ci (x)e−λi x (1 − B(x)), i = 2, k − 1, πk (x) = Ck (x)(1 − B(x)),

. i

where the functions .Ci (x) (i = 1, k) are calculated recursively ⎞ λm ⎠ C j e−(λi−1 −λi )x , λ − λ j m+1 m= j ⎛ ⎞ k−2 k−2 ∏ λ λk−1 ∑ m ⎠ C j e−λk−2 x . − (−1)k−1− j ⎝ λk−2 j=1 λ − λ j m+1 m= j

⎛ i−1 i ∑ ∏ .C i (x) = C i + (−1)i− j+1 ⎝ j=1

Ck (x) = Ck − Ck−1 e−λk−1 x

The boundary conditions (3.17) allow us to find the constants .Ci (i = 1, k). The first boundary condition gives the representation of .C1 in terms of .C2 in terms of LT ˜ 1 ) + C2 b(λ ˜ 2) − C1 = C1 b(λ

.

λ1 ˜ 1 ). C1 b(λ λ1 − λ2

The second boundary condition gives a recurrent expression for .Ci in terms of .Ci+1 for .i = 2, k − 2 ˜ i ) − S(i − 1), ˜ i+1 ) + S(i)b(λ Ci = Ci+1 b(λ

.

where ⎛ i i ∑ ∏ .S(i) = (−1)i− j+1 ⎝ j=1

m= j

⎞ λm ⎠ Cj. λ j − λm+1

The last boundary condition gives a representation for .Ck in terms of .Ck−1 Ck = Ck−1 +

.

λk−1 S(k − 2). λk−2

The constant .Ck−1 is calculated from the normalization equation ∑ .

i∈E

πi = 1.

72

3 Reliability Characteristics for Repairable k-Out-of-n Model

∫∞ Simply calculating .π = 0 π(x)d x and replacing it with .S(i) where possible com⃞ pletes the proof of the theorem. .

3.3.1.2

An Example and Numerical Analysis

According to Theorem 3.3, the s.s.p.’s of the process states for .3-out-of-.n model are calculated as follows: .

˜ ˜ ˜ 1 ˜ 1 ), π1 = C1 1 − b(λ1 ) , π2 = C2 1 − b(λ2 ) + S(1) 1 − b(λ1 ) , C1 b(λ λ0 λ1 λ2 λ1 ˜ ˜ 1 − b(λ2 ) λ2 1 − b(λ1 ) π3 = C3 · b − C2 − S(1) , (3.20) λ2 λ1 λ1

π0 =

where C1 = C2

.

˜ 2 )(λ1 − λ2 ) b(λ λ2 λ1 , C3 = C2 + S(1), S(1) = − C1 , ˜ 1 )) λ1 λ1 − λ2 λ1 − λ2 (1 − b(λ

Substituting .S(1) and .C1 into the expression for .C3 , we obtain the dependence of the constant .C3 on .C2 , ( ) ˜ 2) λ2 λ2 b(λ .C 3 = C 2 − C1 = C2 1 − . ˜ 1 )) λ1 − λ2 λ1 − λ2 (1 − b(λ Thus, substituting the obtained constants into the expressions for s.s.p.’s and applying C = C2 , we obtain the final form of the expressions (3.20). For the case of .n = 6, it holds

.

.

˜ ˜ ˜ − 5b(4α) 3 1 + 4b(5α) 6 1 − b(5α) · π0 , π2 = · π0 , ˜ ˜ ˜ 5 2 b(5α) b(5α) b(4α) ˜ ˜ ˜ − 80αb + 4b(5α)) − 5(1 − 4αb)(1 + 4b(5α))) 3 b(4α)(21 · π0 , π3 = ˜ ˜ 10 b(5α) b(4α)

π1 =

π0 =

˜ ˜ b(4α) b(5α) , ˜ ˜ ˜ 6αb(1 + 4b(5α)) − b(4α)(24αb − b(5α))

(3.21)

Remark 3.1 Consider an exponential repair time distribution . B(x) = 1 − e− b with ˜ i ) = (1 + λi b)−1 , i = 1, 2 and the mean .b. This result the corresponding LT .b(λ will match the corresponding probabilities obtained by a simple birth and death process x

3.3 Stationary Characteristics

73

1 6αb , π1 = , 2 2 3 3 + 30α b + 6αb + 1 120α b + 30α 2 b2 + 6αb + 1 30α 2 b2 120α 3 b3 , π . π2 = = 3 120α 3 b3 + 30α 2 b2 + 6αb + 1 120α 3 b3 + 30α 2 b2 + 6αb + 1

π =

. 0

120α 3 b3

The s.s.p.’s (3.21) are presented in terms of LT of the repair time of the system elements. From where we observe the obvious dependence of these probabilities on the shape of the repair time distribution function. On the other hand, papers [40, 44, 109] show that with fixed mean repair time .b and its coefficient of variation .v, the shape of repair time distribution does not affect the reliability measures, that is, there is insensitivity. Consider some numerical examples to show the asymptotic insensitivity of the system stationary availability to the shape of its components’ repair time distributions under rare failures. For numerical analysis, consider the behavior of the availability coefficient . K av = 1 − π3 , which corresponds to the stationary probability of system operation. In our experiments, the following distributions are used for the repair time: • Erlang (. Erl(m, θ )) b = m · θ −1 , v =

.

√ ˜ m/m, b(s) =

(

θ s+θ

)m ;

• Gnedenko-Weibull (.GW (k, λ)) √ ) λ2 · Γ(1 + 2/k) − b2 1 , v= , .b = λ · Γ 1 + k b (

k ˜ b(s) = k λ

∫

∞

.

e−st−(t/λ) t k−1 dt; k

0

ˆ • Uniform (.U (a, ˆ b)) 1 aˆ + bˆ · , v= .b = 2 aˆ + bˆ

/

ˆ

ˆ e−as (bˆ − a) ˆ 2 − e−bs ˜ , b(s) = 3 s(bˆ − a) ˆ

Suppose that .b = 1, the coefficient of variation .v takes values .0.5, 1, 2, according to which distribution parameters are selected. Figures 3.6 and 3.7 present the behavior of availability coefficient. K av with different repair time distribution and its coefficient variation. Figures are shown in dependence of the mean lifetime of system elements −1 .a. Here, .a = 0, 20 which corresponds to the failure intensity .α = a . According to the graphs, despite the dependence of expressions (3.21) on the shape of repair

74

3 Reliability Characteristics for Repairable k-Out-of-n Model

Fig. 3.6 . K av of .3-out-of-.6 system Fig. 3.7 . K av of .3-out-of-.6 system, . B ∼ U

time distribution, the availability coefficient is asymptotically insensitivity to this distribution. Also, the coefficient of variation does not affect . K av in case of rare elements failures.

3.3.2 Full Repair Scenario Further, consider the s.s.p.’s of a .k-out-of-.n model under full repair scenario.

3.3.2.1

Balance Equations

Analogously to Sect. 3.3 for .t −→ ∞ the system of Kolmogorov differential equations holds

3.3 Stationary Characteristics

∫

∞

λ π0 =

. 0

75

∫∞ β(x)π1 (x)d x +

0

φ(x)πk (x)d x,

(3.22)

0

π˙ 1 (x) = −(λ1 + β(x))π1 (x), π˙ i (x) = −(λi + β(x))πi (x) + λi−1 πi−1 (x), i = 2, k − 1, π˙ k (x) = −φ(x)πk (x), jointly with initial .π0 (0) = 1 and boundary conditions from (3.13) ∫ ∞ π (0) = λ0 π0 + β(x)π2 (x)d x, 0 ∫ ∞ .πi (0) = β(x)πi+1 (x)d x, i = 2, k − 2,

. 1

(3.23)

0

π

. k−1

(0) = 0,

∫

π (0) = λk−1

. k

∞

πk−1 (x)d x.

0

The following Theorem gives the result of s.s.p.’s calculation using the method of constant variation [42]. Some special cases of this model have been considered in [121]. Theorem 3.4 The s.s.p.’s of the process . Z (t) under full repair scenario in terms of LT of partial repair time have the form [ ( π = λ−1 C1 1 + 0

. 0

) ] λ1 ˜ ˜ 2) , b(λ1 ) − C2 b(λ λ 1 − λ2

(3.24)

˜ 1) 1 − b(λ , λ1 ˜ i) ˜ i−1 ) 1 − b(λ 1 − b(λ .πi = C i + S(i − 1) , i = 2, k − 1, λi λi−1 .πk = C k · f, π = C1

. 1

where . .

.

.

˜ i+1 ) + S(i)b(λ ˜ i ) − S(i − 1), i = 2, k − 2, Ci = Ci+1 b(λ Ck−1 = −S(k − 2), ( ) ˜ k−2 ) ˜ k−1 ) 1 − b(λ 1 − b(λ Ck = λk−1 Ck−1 − , λk−1 λk−2 ⎛ ⎞ i i ∑ ∏ λ m ⎠ Cj, S(i) = (−1)i− j+1 ⎝ λ − λ j m+1 j=1 m= j

∑ and .C1 is calculated according to the normalization condition . i∈E πi = 1.

(3.25)

76

3 Reliability Characteristics for Repairable k-Out-of-n Model

Proof The solution of the second and the last equations of system (3.22) gives the probabilities .π1 (x) and .πk (x) in the form π (x) = C1 e−λ1 x (1 − B(x)),

. 1

πk (x) = Ck (1 − F(x)). The solution of the third equation is found using the method of constants variation π (x) = Ci (x)e−λi x (1 − B(x)), i = 2, k − 1,

. i

where Ci (x) = Ci +

i−1 ∑

.

⎛ (−1)i− j+1 ⎝

j=1

i ∏

m= j

⎞ λm ⎠ C j e−(λi−1 −λi )x . λ j − λm+1

The boundary conditions (3.23) allow finding constants .Ci , (i = 1, k) for .k > 2. The first boundary condition gives the representation of probability .π0 via .C1 and .C 2 , ˜ 2 ) − λ1 C1 b(λ ˜ 1 ), C1 = λ0 π0 + C2 b(λ λ1 − λ2 ( ) λ1 ˜ ˜ 2 ). λ0 π0 = C1 1 + b(λ1 ) − C2 b(λ λ1 − λ2 .

The second boundary condition gives recursive expression for .Ci via .Ci+1 for i = 2, k − 2

.

˜ i ) − S(i − 1), ˜ i+1 ) + S(i)b(λ Ci = Ci+1 b(λ

.

where ⎛ i i ∑ ∏ .S(i) = (−1)i− j+1 ⎝ j=1

m= j

⎞ λm ⎠ Cj. λ j − λm+1

The additional condition for probability .πk−1 (0) gives the representation for .Ck−1 (.k > 2), Ck−1 = −S(k − 2).

.

The final constant .Ck is found from the last equation of (3.23),

3.3 Stationary Characteristics

77

Ck = λk−1 πk−1 , (

.

Ck = λk−1 Ck−1

) ˜ k−2 ) ˜ k−1 ) 1 − b(λ 1 − b(λ − . λk−1 λk−2

Thus, all the s.s.p.’s are expressed in terms of constants .Ci , (i = 1, k), which are recursively computed in terms of themselves. The last one .C1 is found from the normalization condition: ∑ ∫ ∞ . πi (x)d x = 1. 0≤i≤k

0

∫∞ The simple calculation .πi = 0 πi (x)d x, i = 1, k, and the substitution by .S(i), ⃞ where it is possible, end the proof of the theorem. .

3.3.2.2

An Example and Numerical Analysis

As before, consider as an example a .3-out-of-.6 system with arbitrary distributed repair time. From Theorem 3.4, the s.s.p.’s in the case of full repair scenario are

.

˜ ˜ ˜ 1 − b(5α) − 5b(4α) 3 1 + 4b(5α) 6 · π0 , π2 = · π0 , ˜ ˜ ˜ ˜ 5 1 + 5b(5α) 2 1 + 5b(5α) − 5b(4α) − 5b(4α) ˜ ˜ 6α f (1 + 4b(5α) − 5b(4α)) π0 , π3 = ˜ ˜ 1 + 5b(5α) − 5b(4α) ˜ ˜ 10(1 + 5b(5α) − 5b(4α)) , π0 = (3.26) ˜ ˜ 37 + 60α f − 25b(4α)(5 + 12α f ) + 2b(5α)(49 + 120α f )

π1 =

Remark 3.2 Analogously to the partial repair scenario when . B(x) = 1 − e− b the result corresponds with the probabilities obtained by a simple birth and death process x

.π0

=

π2 =

20α 2 b2 + 4αb + 1 120α 3 b2 f + 74α 2 b2 + 10αb + 1 30α 2 b2 120α 3 b2 f + 74α 2 b2 + 10αb + 1

,

π1 =

,

π3 =

24α 2 b2 + 6αb 120α 3 b2 f + 74α 2 b2 + 10αb + 1 120α 3 b2 f 120α 3 b2 f + 74α 2 b2 + 10αb + 1

, .

The obtained expressions (3.26) show that the s.s.p.’s depend on the distribution of the partial repair time. In addition, for full system repair time, it is sufficient to know only its average value, regardless of the shape of its distribution. For numerical analysis, consider also the availability coefficient . K av . Suppose that mean repair time .b = 1, f = 2 for partial and full, respectively. Mean lifetime belongs to the interval .a = 0, 20.

78

3 Reliability Characteristics for Repairable k-Out-of-n Model

Fig. 3.8 . K av of .3-out-of-.6 system Fig. 3.9 . K av of .3-out-of-.6 system, . B ∼ U

Figures 3.8 and 3.9 shows the dependence of the coefficient of the steady-state availability . K av from the mean lifetime of system elements for different repair time distributions, as well as the case of rare failures. According to the figure, over the entire interval .a under consideration all curves become very close to each other despite the different values of .v.

Chapter 4

Preventive Maintenance for k-Out-of-n Model .

.

This chapter is devoted to the development of mathematical methods for Preventive Maintenance (PM) organization for .k-out-of-.n model using a different optimization criteria. In Sect. 4.1 we consider PM organization with respect to maximization of the availability criteria. Section 4.2 continues these investigations for a cost-type criterion. The PM strategies for the system, whose failures depend not only on number of its filed components but also on its positions in the system, will be under our attention. The investigation is based on order statistics distribution. Most research on this topic assumes that any system PM and repair lead to its full renovation. This means that after each complete repair the system becomes “as anew one”. As a result of this assumption, the mathematical formulation of the problem can be conducted within the framework of decision-making theory for regenerative or semi-regenerative processes. Using the decision-making theory for regenerative processes we propose a general procedure for comparing different PM strategies based on the system state observation, aiming to choose the best one. Numerical examples that can be used for the system of tethered high-altitude unmanned telecommunication platforms will be proposed. In the chapter, we use the general notation that has been introduced in the Introduction.

4.1 Preventive Maintenance of the . k-Out-of-.n Model with Respect to Availability Criterion In this section, we consider the PM organization for .k-out-of-.n model with respect to maximization of the system availability factor when the system failure depends not only on the number of its failed components but also on their position at the system. It is based on the paper [127].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 V. M. Vishnevsky et al., Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9445-8_4

79

80

4 Preventive Maintenance for k-Out-of-n Model

In the next section, the problem set and the notations are specified. The properties of the process that have described the model behavior and the general algorithm for the PM comparison are proposed in Sect. 4.1.2. At last, two final subsections are devoted to the numerical study models when the system failure depends not only on the number of its failed components (Sect. 4.1.3) but also on their position at the system (Sect. 4.1.4). Usually, the detailed initial information about lifetimes of the system units is not available, and only one or two of their moments are known. Thus, in the investigation, we focus on the problems of the decision-making about PM sensitivity to the input information about system components’ lifetime distributions.

4.1.1 The Problem Set and Notations 4.1.1.1

Notations and Assumptions

Consider a .k-out-of-.n model whose failure can depend not only on the number of its failed components but also on their position in the system. Following to general notations and assumptions denoted by . Ai : i = 1, 2, . . . the sequence of the system components’ random lifetimes. After any system failure it is repaired with a single facility and their repair times are r.v.’s . Bi(0) : i = 1, 2, . . . . To increase the reliability of the system, the possibility of PM, based on the system states observation, is assumed. Let .L = {0, 1, . . . , L} be a set of possible PM strategies including running to the system failure for .l = 0. For investigation of reliability of the complex system, whose failure depends not only on the number of its failed components but also on their position in the system, the system state is denoted by .j = ( j1 , j2 , . . . jn ), where . ji = 1, if the .ith component is in “DOWN” state, and . ji = 0, if the .ith component is in “UP” state. Thus . j = j1 + · · · + jn means the number of failed components of the system. Let’s denote also by . E = {j = ( j1 , j2 , . . . jn ) : ( ji ∈ (0, 1))} the system set of states and by . E 0 and . E¯ 0 subsets of its “DOWN” and “UP” states accordingly. Denote also by . El the system subset for the .lth strategy using (the system “prefailure” subset of states, where .lth type of maintenance should begin). Note that the description of these sets is a special problem of concrete applications and should be considered for any special case. The duration of PM is r.v.’s . Bi(l) . It is supposed that • in the very beginning the system is absolutely reliable, i.e., it is in zero state .j = (0, . . . , 0);

4.1 Preventive Maintenance of the k-Out-of-n Model …

81

• all sequences of r.v.’s (components’ lifetimes . Ai (i = 1, 2, . . . ), repair times (0) . Bi (i = 1, 2, . . . ), and PM times . Bil (l = 1, 2, . . . , L)) are i.i.d. for each type of r.v.’s; • their common c.d.f.’s are .

A(t) = P{Ai ≤ t}, Bl (t) = P{Bi(l) ≤ t}, (l ∈ L);

• their mean values are supposed to be finite ∫ a = E[Ai ] =

.

0

∞

∫ (1 − A(t))dt < ∞, bl = E[Bil ] =

∞

(1 − Bl (t))dt < ∞;

0

• the mean PM time .bl is supposed to be less than the mean repair time .b0 , bl ≤ b0 , but may depend or not on the type of maintenance; • after any repair and PM completion the system becomes “as a new one”, i.e., goes to the zero state.

4.1.1.2

The Problem Set

The section’s aim is to compare different PM strategies .l ∈ L (including running to the system failure for .l = 0) with respect to the system availability . K av.,l , .

1 {the system working time during time t under strategy l}. t

K av.,l = lim

t→∞

(4.1)

For the stated problem solution, let’s define a random process .J = {J(t) : t ≥ 0} with the set of space . E by the relation J(t) = j, if at time t system is in the state j ∈ E,

.

and denote by . Sl (l ∈ L) time to the subset . El destination, S = inf{t : J(t) ∈ El }.

. l

Thus, the value . S0 represents lifetime of the system and . Sl (l = 1, L) is the time till the .lth-type maintenance beginning. In the paper, we are interested in calculation of different characteristics of the system, such as • the system reliability function ∫∞ .

R(t) = P{S0 > t}

and its mean value M0 =

R(t)dt; 0

(4.2)

82

4 Preventive Maintenance for k-Out-of-n Model

• distributions of time before starting different maintenance and their mean values,

.

∫∞ Ml = (1 − Fl (t))dt;

Fl (t) = P{Sl ≤ t},

(4.3)

0

• the system availability . K av.,l (4.1) for different PM strategies .l ∈ L (including running to the system failure for .l = 0). Because the initial information about system components’ lifetime is usually very limited and available only up to one or two moments, we focus on the study of how sensitive is a decision on the PM quality to the shape of their distributions.

4.1.2 Process J and the General Procedure of the PM Quality Calculation 4.1.2.1

Process J

Let us note, first of all, that due to our assumptions for any PM strategy, including the “work until the system fails” strategy, the process .J is a regenerative one, which regenerative epochs . Si(l) (l ∈ L. i = 1, 2, . . . ) are the time moments of the end of maintenance or repair. For the controllable regenerative processes, we need the ergodic theorem. A version of such a theorem for calculating the process steadystate probabilities has been proposed in [53]. We propose here this theorem in the following form. Theorem 4.1 For any admissible decision.l ∈ L of the controllable (in regeneration points) regenerative process .J = {J(t), t ≥ 0} with a finite expected regeneration period .E[S1(l) ] < ∞ and any integrable function .g, defined on the process set of states . E, the following limit property holds

.

lim

t→∞

1 t

⎡

∫t [

g(J(u))du = E

0

1 S1(l)

⎢ ]E ⎣

⎤

(l)

∫S1

⎥ g(J(u))du ⎦ .

(4.4)

0

Proof For regeneration points . Si(l) of the process .J, we denote by .

N (l) (t) =

∑

1{S (l) ≤t} i

i≥0

its renewal process and represent the left-hand part of equality (4.4) as follows:

4.1 Preventive Maintenance of the k-Out-of-n Model …

.

1 lim t→∞ t

∫t g(J(u))du = 0

= lim

t→∞

83

⎡

N (l) (t) 1 t N (l) (t)

∑ 1≤i≤N (l) (t)

⎢ ⎢ ⎣

⎤

(l)

∫Si

(l) Si−1

⎥ 1 g(J(u))du ⎥ ⎦+ t

∫t g(J(u))du. S (l)(l) N

(t)

Taking into account that, due to our assumptions, the last term in this equality tends to zero, the proof follows from the limit theorem for the renewal process and the law of large numbers for i.i.d. r.v.’s used under the sign of sum. ⃞ Denote by .∏0 and .∏l the process regeneration periods for the cases of the system working up to failure (for .l = 0) or under the .lth-type .l = 1, L of maintenance. Thus, for the function .g(j) = 1 El (j) the right-hand side of the equality (4.4) (the system availability (4.1) takes the form .

K av.,0 =

E[S0 ] , E[∏0 ]

K av.,l =

E[Sl ] for (l = 1, L). E[∏l ]

(4.5)

Therefore, due to the properties of regenerative processes for availability . K av.,l calculation, we need only the mean value .E[∏l ] of the regeneration period .∏l and the mean value . Ml = E[Sl ] of the working time . Sl in it. Since for any PM strategy .l ∈ L, the regeneration period equals to ∏l = Sl + Bl ,

.

and the mean repair and PM times.bl = E[Bl ] are supposed to be known and measured in the same scale, for the problem solution we need only to calculate the distributions .

Fl (t) = P{Sl ≤ t},

or only their mean values ∫∞ .

Ml = E[Sl ] =

(1 − Fl (t))dt 0

of the system working times . Sl for the case when it works to failure (for .l = 0) and for a system that operates under the .lth maintenance strategy. Corollary 4.1 In terms of dimensionless indexes, the .lth strategy is preferred over the . jth one (.l ≽ j) if and only if bl Ml < . (4.6) . bj Mj

84

4 Preventive Maintenance for k-Out-of-n Model

Proof As it has been shown above in terms of mean times to the destination of the respective subsets, the system availability can be represented as .

K av.,l =

Ml E[Sl ] = E[∏l ] Ml + bl

and because the .lth PM strategy is preferred over the . jth one if . K av.,l > K av., j from the inequality Mj Ml . > Ml + bl Mj + bj it follows that the .lth strategy is preferred over the . jth one (.l ≽ j) if and only if .

Ml b j > M j bl ,

or in terms of dimensionless indexes it can be rewritten as (4.6), that ends the proof. ⃞

4.1.2.2

The General Procedure of the PM Quality Calculation

For the solution of the declared problem, we need to calculate system availability K av.,l for different preventive maintenance strategies .l ∈ L, including running to the system failure (for .l = 0). To do that, we have to calculate c.d.f.’s (4.2), (4.3) of the subset’s . El (l ∈ L) destination times. Note that the time . Sl (l ∈ L) of the subset’s. El destination coincides with the corresponding order statistics of the system components’ failure times. Therefore, for the solution of the stated problem, the following general algorithm is supposed to use.

.

Algorithm 4.1 Start. Determine: Integers .n, k, distribution . A(t) of components lifetime, subsets . El (l ∈ L) for the PM beginnings or of the system failure for .l = 0, mean PM and system repair times .bl (l ∈ L). Step 1. Describe the duration of the subset’s . El (l ∈ L) destination in terms of order statistics, . A (1) , . . . , A ( j) , . . . , A (n) of the system components’ failure times (i.i.d r.v.) . A1 , . . . , A j , . . . , An that bring the system to the subset . El . Step 2. Calculate distributions of the respective members of the variation series .

A( j) (t) = P{A( j) ≤ t} =

∑ (n ) Ai (t)(1 − A(t))n−i . i j≤i≤n

(4.7)

Step 3. Calculate the distributions . Fl (t) of the subset’s . El destination times in terms of distributions of respective series members and their expectations,

4.1 Preventive Maintenance of the k-Out-of-n Model …

85

∫∞ .

Ml = E[Sl ] =

(1 − F(l) (t))dt. 0

Step 4. Compare different PM strategies with respect to maximizing the system availability given by (4.6). Step 5. Print results in terms of mean operational times . Ml (l ∈ L) and their M ratio . Mlj as advice to a Decision-Maker (DM) in order to choose the best strategy accordingly to inequality (4.6). Stop. Remark 4.1 The algorithm also can be used to solve other different problems, for example, to analyze if the preference of one strategy over another is sensitive to the shape of the system components’ lifetime distributions. In the next two subsections, based on the algorithm, some numerical experiments from the paper [127] are done. It can be used for the tethered high-altitude unmanned telecommunication platforms.

4.1.3 The PM of the . k-Out-to-.n Model When Its Failure Depends Only on Number of Its Failed Components 4.1.3.1

Preliminaries

For a .k-out-of-.n model, whose failure depends only on the number of its failed components, the subsets of UP and DOWN states are: .

E¯ 0 = {0, 1, 2, . . . , k − 1} E 0 = {k, k + 1, . . . , n}.

The subset . El for the PM .l ∈ L beginning in this case is a single state . El = {l} with l ≤ k − 1. Thus, in this case, we can investigate .k strategies .l = {0, 1, . . . , .k − 1}, where .0-strategy means allow the system to operate up to its failure. At that, the general Algorithm 4.1 gets essentially simpler because the time to the subset . El destination coincides with the respective member . A(l) of the variation series of the times to the system components’ failures . Ai : (i = 1, n). The analytical expressions for mean values. Ml are not always accessible. However, their numerical calculation in accordance with the above algorithm is not too difficult, and it has been done in the paper [127] for the special case of a .4-out-of-.6 system.

.

86

4.1.3.2

4 Preventive Maintenance for k-Out-of-n Model

Numerical Analysis

For the model of .4-out-of-.6-system, only four PM strategies are possible. • Strategy 0 is that the system operates up to its failure. • Strategy .l (.l = 1, 2, 3) is to begin the PM when the system occurs in the state .l. In order to compare Strategy .l with the Strategy 0 (to work without any PM up to the system failure), we need to know the ratio . bb0l . It is supposed that the values of mean repair and PM times .b0 , bl as well as their ratios are known to a DM. Therefore, to make a decision about the preference of one strategy over the other, one only needs Ml for them. to know the ratios of the mean working time of the system . M 0 The main aim of the numerical analysis is the sensitivity analysis of the DM choice to the shape of the system components’ lifetime distributions and their coefficient of variation. To do that in the paper [127] several numerical experiments have been fulfilled for four types of distributions: exponential with parameter .α, . E x p(α), Gamma distribution, .Γ(Θ, k), Gnedenko–Weibull distribution, .GW (λ, k), and log-normal distribution . Ln N (μ, σ 2 ). The parameters of all distributions in experiments have been chosen such that their expectations coincide for different distributions and equal to 1 (it means that we scale it with respect to mean components’ lifetime), while the coefficient of variation .v = μσ is varied in the interval .v ∈ [0.3, 5.0]. The results of the experiments are presented in Fig. 4.1 and Tables 4.1, 4.2. In Fig. 4.1, the ratios of mean working times . Ml under different PM strategies .l = 1, 2, 3 to mean working time . M0 for the system operating up to its failure (.l = 0) for different distributions of components lifetime versus the coefficient of variation are shown. Bold dashed horizontal lines correspond to the ratios of the mean PM time .bl for any strategy .l to mean repair time .b0 . Intersections of these lines with the curves . Ml /M0 for different distributions determine the boundary values .v∗ of the coefficient of variation, where the preference of appropriate strategy .l is changed to the preference for “the system working up to the failure”. If the coefficient of variation exceeds the boundary value .v > v∗ “the system working up to the failure” strategy (.l = 0) is preferable for a given distribution, otherwise the appropriate PM strategy should be used. The boundary values .v∗ of the coefficient of variation when the PM strategy .l = 1, 2, 3 is preferable to strategy .l = 0 (running to the system failure) for two values of the ratios mean PM duration .bl to mean repair time .bl /b0 = 0.5 (upper horizontal line in Fig. 4.1) and .bl /b0 = 0.2 (lower horizontal line in Fig. 4.1) for different distributions of system components’ lifetime are represented in Table 4.1. Figure 4.1 demonstrates an almost evident fact that the highest value of the ratios Ml is achieved for .l = 3, which means that the strategy .l = 3 is preferable over other . M0 strategies in the case when all mean PM times are equal,.bl = b for.l = 1, 2, 3. Moreover, this strategy will be better than the strategy .l = 0 “the system runs to failure” until the coefficient of variation is less than the boundary value .v∗ for any specified ratio . bb0 . If the coefficient of variation .v > v∗ strategy .l = 0 will be preferable to all others.

4.1 Preventive Maintenance of the k-Out-of-n Model …

87

4−out−of−6 : F − system

0.9

M1/M0

0.8

M2/M0

0.7

M3/M0

0.6

bi/b0=0.2

bi/b0=0.5

0.5 0.4 0.3 0.2 0.1 0

0.5

1

1.5

2 2.5 3 3.5 Coefficient of variation

4

4.5

5

Fig. 4.1 The dependence of the ratios. Ml /M0 for different distributions of system components lifetime versus their coefficient of variation for .4-out-of-.6 system. Solid lines—.Γ-distribution, dashed lines—.GW -distribution, dotted lines—log-normal distribution, circles—exponential distribution Table 4.1 Boundary values for the coefficient of variation .v ∗ for .4-out-of-.6 system Distribution .bl /b0 = 0.5 .bl /b0 = 0.2 .l = 1 .l = 2 .l = 3 .l = 1 .l = 2 .Γ-distribution

0.44 0.38 .GW -distribution Log-normal distribution 0.51

0.76 0.71 0.99

1.51 1.76 4.08

0.93 0.91 1.66

1.57 1.96 .>5

.l

=3

3.30 .>5 .>5

However, depending on the coefficient of variation, the decision about the choice of the PM is sensitive to distributions of the system components’ lifetime. Increasing of the coefficient of variation leads to decreasing of the ratio . Ml /M0 , but the difference between distributions grows. Suppose the repair time is twice longer than PM time (the violet line in Fig. 4.1). Assuming an exponential distribution of components lifetime, .l = 3 is preferable for .4-out-of-.6 model. The decision for other distributions of components lifetime depends on the coefficient of variation. If .v > 1.51 for .Γ-distribution or .v > 1.76 for . GW -distribution, the strategy .l = 0 should be chosen. In case components’ lifetime follows a log-normal distribution, the strategy .l = 0 should be chosen if .v > 4.08.

88

4 Preventive Maintenance for k-Out-of-n Model

Table 4.2 Boundary values for the coefficient of variation .v ∗∗ to compare strategies .l = 2 and .l = 3 for .4-out-of-.6 system .bl /b0 = 0.5 .bl /b0 = 0.2 Distribution .Γ-distribution .GW -distribution

Log-normal distribution

1.275 1.427 3.771

2.749 .>5 .>5

If the repair time is five times longer than PM time (the cyan line in Fig. 4.1) and the coefficient of variation is no greater than 3.3, the strategy .l = 3 will be the best regardless of the type of components’ lifetime distribution. However, as it is possible to see from Fig. 4.1, for .v < 3.3 the choice of the strategy significantly depends on the components’ lifetime distribution. In case of different mean PM times .bl , strategies .l = 1, 2, 3 can be compared one with another. We compare the strategies .l = 2 and .l = 3 for the studied distributions, and present the results in Table 4.2. The table provides the boundary values for the coefficient of variation .v∗∗ , where the preference for the strategy .l = 2 is changed to the preference for the strategy.l = 3 for.b2 /b3 = 0.5 or.b2 /b3 = 0.2. If the coefficient of variation exceeds the boundary value .v > v∗∗ , the strategy .l = 3 “to start PM after 3 components failure” for given distribution is preferable. Otherwise, the strategy .l = 2 “to start PM after 2 components failure” should be used. So according to Table 4.2 if the mean PM time for .l = 3 is twice as much as the mean PM time for .l = 2 and .Γ-distribution with the coefficient of variation .v > 1.275 is taken, the strategy .l = 3 is better than the strategy .l = 2. The same conclusion can be made if components’ lifetime follows .GW -distribution and .v > 1.427, and for log-normal distribution if .v > 3.771. If the mean PM time for .l = 3 is five times longer than the mean PM time for .l = 2 assuming .GW - or log-normal distribution, the strategy .l = 2 is preferable for the whole interval .v ∈ [0.3, 5.0], and for a .Γ-distribution it will be the best choice if .v < 2.749. The numerical experiments show an essential sensitivity of the decision-making about PM preference over the strategy .l = 0 “working up to the system failure” to the shape of the system components’ lifetime distributions and their coefficient of variation.

4.1.3.3

Special Case: Exponential Distribution of Components Lifetime

A special case when the lifetimes of the system components . Ai have an exponential distribution −αt . A(t) = P{Ai ≤ x} = 1 − e ,

4.1 Preventive Maintenance of the k-Out-of-n Model …

89

due to the independence of the residual lifetimes of all survived components of the failure time of any one of them allow an analytical solution. The results, obtained by both methods in the paper [127], coincide. For the details, see [127].

4.1.4 The PM of the . k-Out-of-.n Model, Which Failure Depends on the Location of Its Failed Components 4.1.4.1

Preliminaries

If the system failure depends on the location of the failed components, the comparison of strategies, including “running to the system failure”, and the decision about the choice of PM are system specific and depend on the exploitation conditions. Thus, it is impossible to solve these problems in general setting and in this section we consider this problem for the concrete case of.(3 + 1, 5)-out-of-.6 model with specific conditions of its exploitation.

4.1.4.2

Example: Model .(3 + 1, 5)-Out-of-.6

Let us turn back to the investigation of the .k-out-of-.n model that has been proposed in the Introduction for .n = 6 under the condition that the system fails, when four (moreover three from one side and one from the other side) or five motors fail. In other words, it means that the system operates if any three or at least one from one side and one from the other side of its motors operate. Thus, this model could be considered as a combination of .(3 + 1)-out-of-.6 and .5-out-of-.6 models. For the simplicity in further for such kind of model, a special notation .(3 + 1, 5)-out-of-.6 model is proposed. For the convenience, a binary code is used to indicate system states, namely, the number of the state .j = ( j1 , j2 , . . . , j6 ) is given in accordance with the formula .

j = |j| =

∑

ji 26−i .

0≤i≤6

Then the subset of failure states . E 0 includes the states with the numbers .

E 0 = {15, 23, 31, 39, 47, 55, 57, 58, 59, 60, 61, 62, 63},

where the states with three failures on the same side and one on the other are highlighted in bold. By analogy with how it is defined in Sect. 4.1.2 consider four strategies:

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4 Preventive Maintenance for k-Out-of-n Model

• Strategy 0 is to run to the system failure (do not use any PM). It means that the repair begins when 4 failures occur at that 3 of them on one side and one on the other or 5 failures occur. The subset of the states for the repair beginning is . E 0 . • Strategy .l (l = 1, 2, 3) is to begin the PM after the failure of any .l components. In this case, the order statistics do not determine uniquely the distribution time to the corresponding subset of states destination. Thus, Algorithm 4.1 takes the following form: Algorithm 4.2 Start is repeated from Algorithm 4.1. Step 1. Determine the times to set of states . El destination in terms of order statistics . A( j) . Because the system failure occurs when 4 .(3 + 1) or 5 motors fail, so the time . S0 to the subset . E 0 destination has the following form: ⎧ A(4) , if four motors fail, at that three from one side and one from ⎪ ⎪ ⎪ ⎨the other side (3 + 1), . S0 = ⎪ A(5) , if four motors fail, at that two from one side and two from the ⎪ ⎪ ⎩ other side (2+2), and the system failure occurs after the fifth failure. The times . Sl to the subset’s . El (l = 1, 2, 3) destination coincide with the relevant variation series members, namely: S = A(l) , the PM under strategy l begins after the failure of l motors.

. l

Step 2. Calculate the distributions of times to the corresponding subset’s destination in terms of the order statistics distributions. Since the system failure occurs when .(3 + 1) or 5 motors fail, and .(3 + 1) failures state contains 6 of the 15 states from the complete subset . E 4 of states with 4 failures, taking into account that the probabilities of any component failures are equal, so the probability of time to the destination of subset . E 0 has a form .

F0 (t) =

2 3 A(4) (t) + A(5) (t). 5 5

The distributions of the subset of states . El (l = 1, 2, 3) destination are .

Fl (t) = A(l) (t), (l = 1, 2, 3),

where according to (4.23) distributions . A( j) (t) are .

A( j) (x) = P{X ( j) ≤ x} =

∑ (n ) Ai (x)(1 − A(x))n−i . i j≤i≤n

Step 3. Calculate the expectations times to destinations of the subsets . El .

4.1 Preventive Maintenance of the k-Out-of-n Model …

.

M0 =

3 2 E[A(4) ] + E[A5) ], 5 5

91

Ml = E[A(l) ], (l = 1, 2, 3).

Step 4. With the help of obtained values, compare different PM strategies using the necessary and sufficient condition to prefer the . jth strategy over the .lth one in the form of inequality (4.6). Step 5. Print results in terms of mean operational times . Ml (l ∈ L) and their ratio Mj . as advice to a DM in order to choose the best strategy. Ml Stop. Based on the algorithm, numerical experiment is proposed below:

4.1.4.3

Numerical Analysis

As in the previous section, four failure distributions: Gamma (.Γ) distribution, Gnedenko–Weibull (.GW ) distribution, log-normal (. L N ) distribution, and exponential (. E x p) distribution are examined. The results of the numerical experiments are Ml of presented in Fig. 4.2 and Table 4.3. As in the previous case in Fig. 4.2 the ratios . M 0 mean system working times . Ml under different strategies .l = 1, 2, 3 to mean system (3+1, 5)−out−of−6 : F − system

0.9

M1/M0

0.8

M2/M0

0.7

M3/M0

0.6

bi/b0=0.2

bi/b0=0.5

0.5 0.4 0.3 0.2 0.1 0

0.5

1

1.5

2 2.5 3 3.5 Coefficient of variation

4

4.5

5

Fig. 4.2 The dependence of the ratios. Ml /M0 for different distributions of system components’ lifetime versus their coefficient of variation for .(3 + 1, 5)-out-of-.6 model. Solid lines—.Γ-distribution, dashed lines—.GW -distribution, dotted lines—log-normal distribution, circles—exponential distribution

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4 Preventive Maintenance for k-Out-of-n Model

Table 4.3 Boundary values for the coefficient of variation .v ∗ for .(3 + 1, 5)-out-of-.6 model .bl /b0 = 0.5 .bl /b0 = 0.2 Distribution .l = 1 .l = 2 .l = 3 .l = 1 .l = 2 .l = 3 .Γ-distribution .GW -distribution

Log-normal distribution

0.37 0.34 0.42

0.6 0.56 0.68

0.98 0.98 1.27

0.82 0.78 1.2

1.26 1.36 2.49

2.05 2.85 .>5

working time . M0 up to its failure (.l = 0) for different distributions of system components’ lifetime versus the coefficient of variation for .(3 + 1, 5)-out-of-.6 model are given. Bold dashed horizontal lines correspond to the ratios of the mean PM time .bl for any PM strategy .l to the mean repair time .b0 . The intersections of these lines with curves . Ml /M0 for different distributions determine the boundary values .v∗ of the coefficient of variation, where the preference of appropriate strategy .l is changed to the preference for the strategy “the system running to the failure” (.l = 0). If the coefficient of variation exceeds the boundary value .v > v∗ , the strategy .l = 0 for given distribution is the preferable one. Otherwise, appropriate PM strategy .l should be chosen. The boundary values.v∗ of the coefficient of variation when PM strategy.l = 1, 2, 3 is preferable to strategy .l = 0 for .bl /b0 = 0.5 (upper horizontal line in Fig. 4.2) and .bl /b0 = 0.2 (lower horizontal line in Fig. 4.2) for the two values .bl /b0 = 0.5 and .bl /b0 = 0.2 numerically calculated for different distributions of system components lifetime are represented in Table 4.3. Analogously to the previous section, the numerical study shows that the choice of the PM strategy significantly depends on the distribution of components’ lifetime and their coefficient of variation.

4.1.4.4

Special Case

In the special case of exponential system components’ lifetime distribution, due to the same arguments as before, the problem can be solved analytically. The calculations fulfilled by analytical and numerical methods demonstrate their coincidence.

4.2 PM of the . k-Out-of-.n Model with Respect to Cost-Type Criterion In this section, we prolong the investigations of the previous one with respect to a cost-type criterion. The material of the section is based on the paper [128].

4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion

93

In the next section, the problem set and the notations are specified. The main result and the general procedure for the PM comparison are proposed in Sect. 4.2.2 and the numerical study of the PM choice for some examples is represented in the last two subsections.

4.2.1 The Problem Setup, Assumptions, and Notations 4.2.1.1

Assumptions and Notations

In this section, we use the same notations and assumptions as in the previous one. Remind that by . Ai : i = 1, 2, . . . a sequence of system components’ random lifetimes are denoted, which is supposed to be i.i.d. r.v.’s with common c.d.f . A(t) = P{Ai ≤ t} and finite expectation .a = E[A]. After any system failure, it is repaired with a single facility, and the repair times are i.i.d. r.v.’s . Bi(0) : i = 1, 2, . . . with common c.d.f. . B0 (t) = P{Bi(0) ≤ t} and finite expectation .b0 = E[B (0) ]. To increase the productivity of the system, the possibility of PM based on the system state observation is considered. By .L = {0, 1, . . . , L} the set of possible PM strategies, including running to the system failure for .l = 0, and by . El the subset of system “pre-failure” set of states, in which the .lth type of PM must be started, are denoted. The times of PM are i.i.d. r.v.’s . Bi(l) with c.d.f. . Bl (t) = P{Bi(l) ≤ t} and finite expectation .bl = E[B (l) ]. It is also supposed that the system gets a reward .$ c during the unit of its operating time, and pays the cost .$c0 for its repair, and the cost .$cl for its PM per unit time. The quality of any PM strategy .l ∈ L during time .t is evaluated by the value V (t) = {the income of the system operating during time t for l ∈ L}.

. l

(4.8)

The system states are denoted by .j = ( j1 , j2 , . . . jn ), where . ji = 1, if the .ith unit is in a failed state, and . ji = 0, if it is in an operational state. Thus, . j = j1 + · · · + jn is the number of failed system units. By .

E = {j = ( j1 , j2 , . . . jn ) : ( ji ∈ (0, 1))}

the set of all system states are denoted, and by . E 0 and . E¯ 0 —subsets of its failed and operational states, accordingly. Note that the description of these sets is an application-specific problem, and should be considered in detail for each particular case. It is also supposed that • at the very beginning the system is absolutely reliable, i.e., it is in zero state .j = (0, . . . , 0); • the mean PM duration is less than the mean repair time, i.e., .bl ≤ b0 , but may or may not depend on the type of PM;

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4 Preventive Maintenance for k-Out-of-n Model

• the mean lifetime, as well as the mean repair and PM times .b0 , bl are known to the decision-maker (DM); • after any repair and PM completion, the system starts working “as a new one”, i.e., returns into the zero state. In other words, the model of the perfect PM is considered.

4.2.1.2

The Problem Setup

The section aims to investigate various PM strategies .l ∈ L, and the goal is to choose the best one with respect to the system average income (4.8) per unit time to be maximized. To solve the problem, we define a random process .J = {J(t) : t ≥ 0} with a set of space . E = {j = ( j1 , j2 , . . . jn ) : ( ji ∈ (0, 1))} by the relation J(t) = j, if at time t the system is in the state j ∈ E.

.

(4.9)

Note that, due to our assumptions, the process .J is regenerative under any PM strategy. Its regeneration epochs . S0(l) = 0, Si(l) : i = 1, 2, . . . are the times of the PM completion for .l = {1, 2, . . . , L} and the repair completion for .l = 0. In other words, the values . Si(l) (l ∈ L) are the times until the process reaches the set of states . El , (l) (l) . Si+1 = inf{t : t > Si , J(t) ∈ E l }. (4.10) (l) Since the intervals. Si(l) − Si−1 are i.i.d. r.v.’s, it is enough to study the process behavior only on one of them, for example, within the first one, . S1(l) . In the section, we are interested in calculating various characteristics of the system, such as:

• System reliability function

.

R(t) =

P{S1(0)

∫∞ > t}

and its mean value M0 =

R(t)dt.

(4.11)

0

• Distributions of the times up to various PM starts and their mean values

.

Fl (t) =

P{S1(l)

≤ t},

∫∞ Ml = (1 − Fl (t))dt;

(4.12)

0

• System quality measure .Vl of different PM strategies defined for .l ∈ L as V = lim

. l

t→∞

1 E[Vl (t)], t

(4.13)

4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion

95

where the income .Vl (t) of the system operating during time .t is given by the equality (4.8). As a result, we compare different strategies with respect to the system quality .Vl and choose the best one. Because the initial information about the system components’ lifetime is usually very limited and available only up to knowledge of one, or two moments, our main focus is the study of the sensitivity of any decision about the PM quality, with respect to the shape of the system components’ lifetime distributions.

4.2.2 The Problem Solution and the General Procedure for Comparing the Quality of PM Strategies 4.2.2.1

The Problem Solution

Because due to our assumptions the process .J is a regenerative one for any PM strategy .l ∈ L, to calculate the long run average income, we use here the version of this theorem for any additive functional on trajectories of regenerative processes that has been done in Sect. 4.1.2 (see Theorem 4.1). The theorem implies the following corollary: Corollary 4.2 For any PM .l ∈ L, the long run average income takes the form V = lim

. l

t→∞

1 cMl − cl bl Vl (t) = , t Ml + bl

(4.14)

where . Ml = E[S1(l) ] is the mean time to the set of states . El destination under PM strategy .l ∈ L. Proof The analytical expression of the income .Vl (t) (4.8) from the system operating during some time .t under the PM strategy .l ∈ L can be represented as ∫t V (t) =

[c1{J(u)∈ E¯ 0 } − cl 1{J(u)∈El } ]du,

. l

(4.15)

0

where .1{A} is the indicator function of the event . A and . E¯ 0 is the subset of the system operating states. From this representation, the expression (4.14) immediately follows. ⃞ Let us introduce the following dimensionless indicators: M0 Ml c0 cl , ml = , c¯0 = , c¯l = , b0 bl c c + 1 + 1 c ¯ m l l ∗ and cl∗ = . .m l = m0 + 1 c¯0 + 1 m0 =

.

(4.16) (4.17)

96

4 Preventive Maintenance for k-Out-of-n Model

In these notations, the following theorem holds. Theorem 4.2 The.l-strategy is preferable to the 0-strategy: “to work up to the system failure” iff ∗ ∗ .m l ≥ cl . (4.18) Proof Based on the quality of strategy.l ∈ L given by Formula (4.14) in Corollary 4.2 the .lth strategy will be preferable over the 0-strategy if and only if .

cM0 − c0 b0 cMl − cl bl ≤ . M0 + b0 Ml + bl

(4.19)

Using simple algebra, the last expression can be represented equivalently as c(M0 bl − Ml b0 ) ≥ b0 bl (c0 − cl ) + c0 b0 Ml − cl bl M0 .

.

Dividing by .cb0 bl both sides, and then by .m 0 , in terms of dimensionless indicators introduced in (4.16), the last relation can be represented as .

c¯l + 1 ml + 1 ≥ . m0 + 1 c¯0 + 1

Hence, due to (4.17), Formula (4.18) follows, which completes the proof.

(4.20) ⃞

Remark 4.2 This result can be extended to compare any two strategies PM .i, j ∈ {1, 2, . . . , L}(i /= j). For this, it is sufficient, by analogy with (4.16) and (4.17), to consider the dimensionless indicators Mj cj ci Mi , mj = , c¯i = , c¯ j = , bi bj c c c¯i + 1 mi + 1 and ci∗j = . m i∗j = mj +1 c¯ j + 1 mi =

.

Thus, the .ith strategy is preferable over the . jth strategy iff m i∗j ≥ ci∗j .

.

(4.21)

Further, Theorem 4.2 will be used to propose an algorithm for comparing any PM strategy with the strategy “to work up to the system failure” in order to choose the preferable one. Since, for any PM strategy .l ∈ L, the mean value of regeneration period equals .E[∏(l) ] = Ml + bl , and mean repair time .b0 and mean PM time .bl , as well as reward .c, repair .c0 , and PM .cl costs are supposed to be known to a DM, to solve the problem, it is enough to calculate the mean times . Ml of the set . El destination. An algorithm for this is proposed in the next section.

4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion

4.2.2.2

97

The General Procedure for Comparing the Quality of PM Strategies

To solve the stated problem according to Theorem 4.2, it is necessary to calculate and compare the dimensionless indexes (4.17). To do this, we propose to use the order statistics distributions and the following general algorithm is proposed. Algorithm 4.3 Beginning. Determine: - Integers .n, k, set of PM strategies .L. - Set of system states . E. - Subsets of states . El (l = 1, L) for the PM beginnings or of the system failure for .l = 0. - Distribution . A(t) of the system unit lifetime, which is defined in Sect. 4.2. - Mean PM and system repair times .bl (l ∈ L). - Rewards .c, cost of repair .c0 , cost of .lth PM .cl . Calculate value .cl∗ according to (4.17). Step 1. Describe the connection between subsets of the PM and repair beginning states in terms of order statistics. Step 2. Represent the time . Sl of the subsets . El (l ∈ L) destinations in terms of the order statistics . A (1) , . . . , A ( j) , . . . , A (n) (4.22) of the system components’ failure times (i.i.d r.v.) . A1 , . . . , A j , . . . , An , which bring the system to the subset . El . Step 3. Calculate distributions of the respective ordered statistics .

A( j) (t) = P{A( j)

∑ (n ) Ai (t)(1 − A(t))n−i . ≤ t} = i j≤i≤n

(4.23)

Step 4. Calculate the distributions .

Fl (t) = P{S1(l) ≤ t} (l ∈ L)

of the subset’s . El destination times and their expectations,

.

Ml =

E[S1(l) ]

∫∞ =

(1 − Fl (t))dt

(4.24)

0

in terms of distributions of respective ordered statistics. Step 5. Using Formulas (4.16) and (4.17) calculate .m 0 , m l , m l∗ . Step 6. Compare the calculated values .m l∗ with the indicator .cl∗ , (l ∈ L) as advice to a DM in order to choose the best strategy according to inequality (4.18). Stop.

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4 Preventive Maintenance for k-Out-of-n Model

Remark 4.3 The algorithm can also be used to solve other problems, for example, to compare PM strategies with each other, according to inequality (4.21), and also to analyze the sensitivity of the decision-making to the shape of the system unit lifetime distributions.

4.2.3 Numerical Experiments To demonstrate the applicability of Algorithm 4.3, we propose two numerical examples. In the experiments, we use the same .4-out-of-.6-model that has been presented in Sect. 4.1. The same two cases are considered here. • The system failure occurs when any four motors fail, regardless of their location. This situation is modeled as a .4-out-of-.6 model. • The system failure depends on the location of its failed components, and occurs when three motors from one side and one motor from the other side fail (this situation is usually denoted as .(3 + 1)-out-of-6: . F model), but the system does not fail when two motors fail on one side, and two motors on the other. Any next component failure leads to the system failure (this situation is usually denoted as a .5-out-of-.6 model). Therefore, from the point of view of mathematical modeling, the last situation can be considered as a combination of the above two special types of .k-out-of-.n systems, where failure depends on the position of its failed components. For such kind of systems, a special notation .(3 + 1, 5)-out-of-.6 model has been proposed, and it is considered in Sect. 4.1. This model is the starting point of our numerical study. Since the initial information is very limited, as in the previous section, we focus on the sensitivity analysis of decision-making to the shape of the system components’ lifetime distributions. For the numerical experiments we consider as before four types of popular distributions: (i) exponential, . E x p(α); (ii) Gamma distribution, .Γ(Θ, k); (iii) Gnedenko– Weibull distribution, .GW (λ, k); and (iv) log-normal distribution, . Ln N (μ, σ 2 ). The parameters of all distributions in the experiments are chosen so that their expectations remain constant and coincide for all of these different distributions, being equal to 1, .a = E[Ai ] = 1. This means that we scaled the time with respect to the mean components’ lifetime .a. At that, the second parameters are chosen so that the coefficient of variation .v = σa varies in the interval .v ∈ [0.3, 5.0]. Analogously, we choose the pricing scale in such a way that the reward equals to one, i.e., .c = 1. Let us see how change of the parameters .c0 , cl affects the value .cl∗ . Due to representation of .cl∗ by (4.17), it is clear that in any case .cl∗ ≤ 1. Moreover, if .c >> max{c0 , cl }, then .cl∗ ≈ 1. On the contrary, if .c is negligible relative to the values .c0 and .cl , then cl ∗ . For further numerical analysis, the parameter .c3∗ is essential, and it is shown .cl ≈ c0 in Fig. 4.3 with respect to the value .c0 and the ratio . cc03 . Further, two cases are considered.

4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion

99

Fig. 4.3 The cost criterion .c3∗ versus .c0 and the ratio . cc03

4.2.3.1

PM of a System, Whose Failure Does Not Depend on the Location of Its Failed Components

First, we consider a.4-out-of-.6 model, in which failure does not depend on the location of its failed components. In this case, only four strategies of PM are possible: • .0-strategy is that the system operates up to its failure; • .l-strategy (.l = 1, 2, 3) is to begin the PM when the system reaches the state .l. For our numerical experiment, we restrict ourselves to comparing the 0-strategy and .3-strategy. In this case, it is true that . E 3 = {3} and . E 0 = {4}. The analytical expressions for the mean values. Ml are not always accessible. However, their numerical calculation in accordance with Algorithm 4.3 is not too difficult, and it is proposed below. In this case, this algorithm is essentially simplified, since the time to the subset . El destination coincides with respective order statistics . A(l) of the times to the system components’ failures . Ai : (i = 1, 6). Therefore, Steps 1 and 3 of Algorithm 4.3 look like Step 1. . E 3 = {3}, E 0 = {4}. Step 3. . F3 (t) = A(3) (t), F0 (t) = A(4) (t). All other steps remain unchanged. The results of the calculations performed in accordance to the above algorithm are presented in the graphs shown in Fig. 4.4. Figure 4.4 represents 3D graphs of surface.m ∗3 jointly with their level curves versus to the. bb03 ratio and coefficient of variation.v for different values of.c3∗ . The upper graphs show the surfaces .m ∗3 for the Gamma distribution of the model components’ lifetime for the values .b0 = 0.001 (left) and .b0 = 0.1 (right) and their intersection with the

100

4 Preventive Maintenance for k-Out-of-n Model

Fig. 4.4 The surfaces.m ∗3 for.b0 = 0.001 and.b0 = 0.1 versus coefficient of variation.v and fraction b ∗ . 3 (upper graphs) and level curves of the surface .m 3 (lower graphs) for the .4-out-of-.6 model b0

plane corresponding to the value of the indicator.c3∗ = 0.7. The red line on the surfaces corresponds to an exponential distribution, at which the surface turns into a curve. Above the plane, the 3-strategy gains an advantage over the 0-strategy. Below the plane, it is preferable to use the 0-strategy. As can be seen from the graphs, for the exponential distribution of the model components’ lifetime, the 3-strategy will be preferable over 0-strategy, regardless of the . bb03 ratio. For the Gamma distribution, the decision on the choice of the PM strategy depends on the value of the . bb03 ratio and the coefficient of variation .v. The lower graphs show the level curves of the .m ∗3 -surface for different values of .c3∗ for four types of distributions: .Γ(Θ, k) (solid curves), .GW (λ, k); (dashed curves), . Ln N (μ, σ 2 ) (dotted curves), and . E x p(α) (red circle), where the axis labels correspond to the upper graph. Above the level curve, preference should be given to the 3-strategy, below—to the 0-strategy. The red circle on the solid magneto color line corresponds to the exponential distribution of the system components’ lifetime at the value of the cost criterion.c3∗ = 0.9. In that case, for an exponential distribution, the value of the . bb03 ratio will influence the decision on choosing the best strategy.

4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion

101

The results can be used as a DM adviser and demonstrate the possibility to study the sensitivity of the decision about PM to the shape of distribution of the system components’ lifetime distributions.

4.2.3.2

Special Case: Exponential Distributions of Components’ Lifetimes

Under the assumption about exponential distributions of the system components’ lifetimes . A(t) = P{Ai ≤ x} = 1 − e−αt , due to its properties, the analytical calculation of the parameter.m ∗ is possible for any.k and.n. It has been proposed in [128]. The calculations, performed according to the necessary and sufficient condition (4.18), show that the 3-strategy will be preferable to the 0-strategy, if ∗

.m 3

=

37 + 0.06 bb03 57.06 bb03

> c3∗ for b0 = 0.001, and

m ∗3 =

37 + 6 bb03 63 bb03

> c3∗ for b0 = 0.1.

Following the equation .m ∗3 = c3∗ = 0.9, one can calculate . bb03 ≈ 0.7213 for .b0 = 0.001 and . bb03 ≈ 0.7298 for .b0 = 0.1. These values coincide with the coordinates of the red circles shown in Fig. 4.4.

4.2.3.3

PM of a System, Whose Failure Depends on the Location of Its Failed Components

If the system failure depends on the location of its failed components, the comparison of strategies and the decisions about the choice of PM are system specific and depend on the exploitation conditions. In this case, we consider the model described in Sect. 4.1, which is denoted as the .(3 + 1, 5)-out-of-.6 system. Further, for convenience, a binary code is used to indicate system states, namely, the number of the state .j = ( j1 , j2 , . . . j6 ) is given in accordance to the formula .

j = |j| =

∑

ji 26−i .

(4.25)

0≤i≤6

Then, the set of failure states . E 0 consists of a combination of states with four failures and five failures. By analogy with Sect. 4.1, we consider two strategies: • 0-strategy: run up to the system failure. The subset of the states for the repair beginning is . E 0 . • 3-strategy: start the PM after the failure of any three units. The subset of the states for the PM beginning is . E 3 .

102

4 Preventive Maintenance for k-Out-of-n Model

In this case, the order statistics do not uniquely determine the time to the corresponding subset of state . E 0 destination. Thus, to apply Algorithm 4.3 it is necessary to specify some of its steps. Algorithm 4.4 Step 1. To form a subset of the system failure states, consider the set of states with four failed units, .

E 4 = {15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60}.

(4.26)

In this set, the bold numbers are associated with states having three failed components on one side and one on the other side, denoted as .(3 + 1), when the system failure occurs. The pale numbers represent the states with two units failed, one side and two on the other, denoted as .(2 + 2), which leads to the system failure after any next component failure. The 3-strategy starts after the failure of any three components, the subset for it is . E3. Step 2. Accordingly, to the step 1, the times . S0 to the system failure coincide with ordered statistics . A(4) for bold states of subset . E 4 , and coincide with ordered statistics . A(5) for the pale states of subset . E 4 . { S =

. 0

A(4) , if the system is in “bold” states of the set E 4 , A(5) , if the system is in “pale” states of the set E 4 ,

(4.27)

The time . S3 to the set . E 3 destination coincides with the relevant order statistics, namely: . S3 = A(3) . Step 3. Does not change: the distributions . A( j) (t) of the . jth ordered statistics are calculated according to (4.23), .

A( j) (x) = P{X ( j)

∑ (n ) Ai (x)(1 − A(x))n−i . ≤ x} = i j≤i≤n

Step 4. The distribution of the time . S0 to the subset . E 0 destination according to its determination by (4.27) equals to .

F0 (t) =

6 9 2 3 A(4) (t) + A(5) (t) = A(4) (t) + A(5) (t). 15 15 5 5

The distribution of the subset of states . E 3 destination is . F3 (t) = A(3) (t). Appropriate expectations of times to the subsets . El for .l = 0 and .l = 3 destinations are .

M0 =

3 2 E[A(4) ] + E[A(5) ], 5 5

M3 = E[A(3) ].

Step 5. Does not change but specified: following to step 5 of Algorithm 4.3, calculate .m 0 , m 3 , m ∗3 .

4.2 PM of the k-Out-of-n Model with Respect to Cost-Type Criterion

103

Fig. 4.5 Level curves for .m ∗3 versus variation coefficient .v and fraction . bb03 for .(3 + 1, 5)-out-of6.: F model

Step 6. Compare the calculated values .m ∗3 with indicator .c3∗ , according to the rule given by (4.18) as advice to a DM in order to choose the best strategy. Stop. The results of the numerical experiments are presented in Fig. 4.5. The graphs show the level curves of the surface .m ∗3 for the values .b0 = 0.001 (on the left) and .b0 = 0.1 (on the right). The color of curves corresponds to the values of the cost criterion .c3∗ : 0.6 (cyan), 0.7 (blue), 0.9 (magneto). The graphs show the results for four different distributions of system components’ lifetimes: 2 .Γ(Θ, k) (solid curves), . GW (λ, k) (dashed curves), . Ln N (μ, σ ) (dotted curves), and . E x p(α), where the axis labels correspond to Fig. 4.4. In contrast to the model, whose failure does not depend on position of its failed units, the influence of the parameter .b0 is much less (especially for a gamma distribution), in comparison to the ratio . bb03 . The shape of the lifetime distribution, as well as the coefficient of variation .v, has a particular influence on the choice of the preferred strategy. Compared to the previous case, the area where the 0-strategy is preferable has increased significantly. Both considered cases show that, for making decisions on the choice of the best PM strategy, it is not enough to take into account only the expectation of system components’ lifetimes and the costs of PM and repair.

4.2.3.4

Special Case: Exponential Distribution of Components’ Lifetimes

In the special case of exponential distribution of system components’ lifetimes, it is possible to use the same approach as in Sect. 4.2.3.2 and the problem can be solved analytically. 37 57 In this case, as before, it holds . M3 = 60α , M4 = 60α and therefore . M5 = M4 + 87 1 = . From here, it follows 2α 60α

104

4 Preventive Maintenance for k-Out-of-n Model

.

M0 =

5 2 3 2 3 375 E[A(4) ] + E[A(5) ] = M4 + M5 = = . 5 5 5 5 300α 4α

(4.28)

Substitution of the calculated . M3 and . M0 for the exponential distribution makes it possible to obtain .m ∗3 as a function of . bb03 and to formulate a necessary and sufficient condition (4.18) for the 3-strategy preference over the 0-strategy for this system in the form ∗

.m 3

=

37 + 0.06 bb03 75.06 bb03

> c3∗ , for b0 = 0.001, and m ∗3 =

37 + 6 bb03 81 bb03

> c3∗ for b0 = 0.1

The coordinates of the red circles presented in Fig. 4.5 for .c3∗ = 0.9 can be calculated analytically:. bb03 ≈ 0.5482 for.b0 = 0.001 (Fig. 4.5, graph on the left) and. bb03 ≈ 0.5531 for .b0 = 0.1 (Fig. 4.5, graph on the right). These results are in complete agreement with the numerical calculations.

Chapter 5

New Approaches to Reliability Investigation of k-Out-of-n Models .

.

This chapter aims to demonstrate some new approaches to investigation of .k-out-ofn models. Section 5.1 considers the capabilities of decomposable semi-regenerative processes. In Sects. 5.2 and 5.3, some simulation methods and machine learning techniques are considered to assess the probabilistic and time characteristics of the reliability of repairable and non-repairable .k-out-of-.n models. The material of this chapter is based on papers [43, 44, 120].

.

5.1 Application of Decomposable Semi-regenerative Processes to the Study of Repairable . k-Out-of-.n Models Proposed in 1955 by W. Smith, the idea of regeneration has come a long way of development and has found wide application. Section 5.1.1 recalls the history of development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Some new and already familiar notations and assumptions are presented in Sect. 5.1.3. Then in Sect. 5.1.2 the methods of the theory of decomposable semi-regenerative processes are used for study of a .k-out-of-.n renewable system with exponentially distributed life and generally distributed repair times of its components. As previously, two scenarios of the system repair after its failure are considered (partial and full repair scenarios). For both scenarios, the t.d.s.s.p.’s are calculated in terms of their LT’s. The closed-form representation of the s.s.p.’s for both scenarios is also presented.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 V. M. Vishnevsky et al., Reliability Assessment of Tethered High-altitude Unmanned Telecommunication Platforms, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9445-8_5

105

106

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

5.1.1 Preliminaries As a generalization of classical independence, the regeneration idea has been proposed by Smith [143]. For a stochastic process . X = {X (t) : t ∈ R} with its flow of .σ -algebras .F tX it is supposed that there exist a sequence of points of time— regeneration times .{Sn : n = 1, 2, . . . }—in which the process forgets its past, i.e., for any .Γ it holds that P{X (Sn + t) ∈ Γ |F SXn } = P{X (Sn + t) ∈ Γ |Sn } = P{X (S1 + t) ∈ Γ}.

.

At that the process. X is known as a regenerative process (RP), the intervals.[Sn−1 , Sn ) and their lengths .Tn = Sn − Sn−1 are called regeneration periods. The state probabilities .π(t; Γ) of any RP can be calculated in terms of its state probabilities in a separate regeneration period .π (1) (t, Γ) = P{X (Sn + t) ∈ Γ, t < Tn } in the form ∫t π(t; Γ) ≡ P{X (t) ∈ Γ} =

P{X (t − u) ∈ Γ, t − u < T }H (du) ≡

.

0

∫t ≡

π (1) (t − u, Γ)H (du),

(5.1)

0

where . F(t) = P{Tn ≤ t} and the renewal function of the process [ .

H (t) = E

Σ n≥0

] 1{Sn ≤t} =

Σ

P{Sn ≤ t} =

n≥0

Σ

F ★n (t)

n≥0

satisfies the equation below, where the symbol “.★” means convolution, ∫t .

H (t) = F(t) + F ★ H (t) ≡ F(t) +

F(du)H (t − u).

(5.2)

0

Moreover, this approach allows also to prove the existence of stationary probabilities and gives its closed-form representation in terms of the process distribution in a separate regeneration period, 1 .π(Γ) = lim π(t; Γ) = t→∞ E[Tn ]

∫∞ 0

π (1) (t, Γ)dt.

(5.3)

5.1 Application of Decomposable Semi-regenerative Processes …

107

However, if the process behavior in a separate regeneration period is complex enough and the probability distributions in it cannot be analytically represented, the more detailed investigation of the process can be performed by using Markov type of dependency that leads to the construction of the theory of semi-Markov processes (see Cinlar [10], Jacod [45], Korolyuk and Turbin [55], and other). The combination of these notions led to the introduction of semi-regenerative processes, which first appeared under different names as semi-Markov processes with additional trajectories in 1966 (Klimov [54]), regeneration processes with several types of regeneration points in 1971 (Rykov and Yastrebenetsky [134]) before they were named semi-regenerative processes (SRP) thanks to Nummeline [83]. The difference between SRP and RP is in the assumption that in its semiregeneration points of time . Sn , the future of the process does not depend on its past but depends on its present—the state of the process at the last semi-regeneration point of time . X (Sn ). We will refer to them as the regeneration states, and denote their set by . E P{X (Sn + t) ∈ Γ |F SXn } = P{X (Sn + t) ∈ Γ |X (Sn )} = P{X (S1 + t) ∈ Γ |X (S1 )}.

.

The main characteristic of the SRP [55, 83, 101] is its semi-Markov matrix (SMM) Q(t) = [Q i j (t)]i j∈E with components

.

.

Q i j (t) = P{X (Sn+1 ) = j, Tn < t, | X (Sn ) = i}.

For semi-regenerative processes with denumerable set of regeneration states. E that start from regeneration states with an initial distribution .α = {αi , i ∈ E} formula (5.1) takes the following form: π(t, Γ) ≡ P{X (t) ∈ Γ} = ⎤ ⎡ ∫t Σ ⎦ αi ⎣δi j π (1) Hi j (du)π (1) = j (t, Γ) + j (t − u, Γ) ,

.

i∈E

(5.4)

0

where .π (1) j (t, Γ) = P{X (Sn + t) ∈ Γ, t < Tn | X (Sn + 0) = j} is the process state probability distribution at SRP of type . j, and . H (t) = [Hi j (t)] is its Markov renewal matrix (MRM) with

.

Hi j (t) = E

[ Σ

] 1{Sn ≤t, X (Sn )= j} | X (0) = i =

n≥1

[ Σ n≥1

] Q ∗n (t)

, i j∈E

which satisfies the following equation: .

H (t) = Q(t) + Q ★ H (t).

(5.5)

108

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

On the other hand, formula (5.3) now takes the following form: 1 .π(Γ) = lim π(t, Γ) = t→∞ m

∫∞ Σ 0

α¯ j π (1) j (t, Γ)dt,

(5.6)

j∈E

.α ¯ = {α¯ i , (i ∈ E)} is invariant probabilities of embedded Markov chain, and whereΣ m = i∈E α¯ i Ei [Tn ] is an expectation of stationary regeneration period. The next step in the generalization of the regeneration idea consists in finding within the main regeneration period some new regeneration time points and construction of the so-called decomposable semi-regenerative process (DSRP). If the process behavior in some separate regeneration period .Tn is complex enough and its distribution cannot be analytically represented, sometimes it is possible to find (1) . Sk (k within this period some embedded regeneration points of time( ) = 1, 2, . . . ), in

.

which the process forgets its past up to the present state . X (1) Sk(1) conditionally to (1) . The subset of its behavior in the embedded regeneration period .Tk(1) = Sk(1) − Sk−1 (1) embedded regeneration states of the process will be denoted by . E . Being extended to any regeneration periods .Tn , this procedure leads to construction of DSRP. The strong definitions and details can be found in Rykov [110] (see also [101, 126]). While analyzing the DSRP, the role of the ordinary MRM plays appropriate embedded Markov renewal matrix (EMRM) . H (1) (t) = [Hi(1) j (t)] with components

[ (1) . Hi j (t)

= Ei

Σ

( 1{[0, t), j}

Sk(1) ,

X k(1)

)

] 1{S (1) t},

5.1 Application of Decomposable Semi-regenerative Processes …

111

and the steady-state probabilities (s.s.p.’s) π j = lim π j (t)

.

t→∞

are calculated.

5.1.3 Partial Repair Regime Consider first a partial repair regime, when after the system failure (when the system enters state .k) the repair of the component being repaired continues and after its end the system goes to state .k − 1.

5.1.3.1

Semi-regenerative Process

In this case, we consider the process . X as a semi-regenerative one (see Fig. 5.1). Its regeneration time points . Sn of the type . j are the repair completion times when the system enters state . j, X (Sn + 0) = j, semi-regeneration periods are .Tn = Sn − Sn−1 , and the regeneration set of states is . E 1 = { j : ( j = 0, k − 1)}. Recall first that the time-dependent state probabilities of the process . X depend on the initial process distribution .α→ (0) = {αi(0) : i ∈ E}. Thus, denoting by .π→ (t) = {π j (t) : j ∈ E} the vector of the process state probabilities, and by .Π(t) = [πi j (t)]i j∈E the probability transition matrix of the process . X , where

Fig. 5.1 Trajectory of the process . X for the system with partial repair

112

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

π (t) = P{X (t) = j | X (0) = i} ≡ Pi {X (t) = j}, (i, j ∈ E)

. ij

in matrix form it holds that

π→ (t) = α→ (0) Π(t).

.

On the other hand, according to the theory of semi-regenerative processes as it was mentioned in the introduction section the process transition probabilities.Π(t) = [πi j (t)]i j∈E can be represented in the following form: Π(t) = Π(1) (t) + H ★ Π(1) (t)

(5.8)

.

in terms of corresponding state probabilities .Π(1) (t) = [πi(1) j (t)]i∈E 1 , j∈E in separate regeneration periods,1 where π (1) (t) = P{X (Sn−1 + t) = j, t ≤ Tn | X (Sn−1 + 0) = i)}, (i ∈ E 1 , j ∈ E),

. ij

[ ] and the Markov renewal matrix . H (t) = Hi j (t) i j∈E1 with [ .

Hi j (t) = E

Σ

] 1{Sn ≤t, X (Sn )= j} | X (0) = i ≡ Ei

n≥1

[

Σ

] 1{Sn ≤t, X (Sn )= j} .

n≥1

It is known from the introduction that the MRM in matrix form satisfies the following equation: . H (t) = Q(t) + Q ★ H (t), (5.9) where . Q(t) = [Q i j (t)]i j∈E1 is the semi-Markov matrix of the process . X with .

Q i j (t) = P{X (Sn + 0) = j, Tn ≤ t |X (Sn−1 + 0) = i}.

Equations (5.8) and (5.9) show that the best way to calculate the process timedependent probabilities is using Laplace (LT) and Laplace–Stieltjes Transforms (LST). Thus, further, along with all time-dependent characteristics of the process, we will present their LT or LST. Denote by ˜ .Π(s) =

∫∞ e 0

−st

˜ (1) (s) = Π(t)dt, Π

∫∞ e 0

−st

˜ Π (t)dt, and h(s) = (1)

∫∞

e−st H (dt)

0

LT of matrices .Π(t), Π(1) (t), and LST of matrix . H (t). Following this approach, the transition to the LT in relations (5.8) and LST in (5.9) gives

1

Note that the representation (5.8) is the solution of the equation .Π(t) = Π(1) (t) + Q ★ Π(t).

5.1 Application of Decomposable Semi-regenerative Processes …

˜ ˜ ˜ (1) (s), and Π(s) = (I + h(s)) Π

.

113

˜ ˜ h(s) = q(s) ˜ + q(s) ˜ h(s).

−1 ˜ The solution of the last equation is .h(s) q(s). ˜ = (I − q(s)) ˜ It leads to the following result: −1 ˜ ˜ ˜ (1) (s) = [I + (I − q(s)) ˜ (1) (s) = Π(s) = (I + h(s)) Π ˜ q(s)] ˜ Π ) ] [ ( Σ −1 ˜ (1) ˜ (1) (s) = (I − q(s)) Π Π (s). q˜ n (s) q(s) ˜ ˜ = I+

.

(5.10)

n≥0

Thus, our goal is to calculate the matrix of the process state probabilities.Π(1) (t) in separate semi-regenerative periods and the SMM. Q(t) = [Q i j (t)]i j∈E1 of the process . X . In order to calculate the last matrix denoted by . pi j (t) the probability that the nonrepairable .k-out-of-.n system during time .t passes from state .i to state . j, i.e., that during time .t . j − i components of the system out of .n − i that are working in the beginning of the interval fail. Denote by . Pik (t) the probability that starting from state .i, the system leaves the subset of working states during time .t, i.e., the probability that during time .t at least .k − i system components fail out of .n − i that are working in the beginning of the time interval. These probabilities are obtained as follows: ( ) n−i . pi j (t) = (1 − e−αt ) j−i e−(n− j)αt , j −i and .

Pik (t) =

Σ

Σ

pi j (t) = 1 −

j≥k

pi j (t).

i≤ j≤k−1

Note that the latter probability is the lifetime c.d.f. of a non-repairable .k-out-of-.n model. To simplify further calculations, we represent these probabilities by Newton’s binomial formula as the sum and with the substitution .λi = (n − i)α (note that the symbol .α is also possible), we obtain ( ) ( ) j−i n−i Σ m j −i e−λ j−m t . pi j (t) = (−1) m j − i m=0 and .

Pik (t) =

j−i Σ (n − i ) Σ j≥k

j −i

m=0

( (−1)

m

) j − i −λ j−m t e . m

(5.11)

(5.12)

Using these notations for the differentials of the SMM the following lemma holds.

114

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

Lemma 5.1 The differentials of the SMM of the process . X are ∫t .

Q 0 j (dt) =

λ0 e−λ0 u du p1 j+1 (t − u)B(dt − u), j = 0, k − 2;

0

∫t Q 0k−1 (dt) =

λ0 e−λ0 u du P1k (t − u)B(dt − u);

0

Q i j (dt) = pi j+1 (t)B(dt), (i = 1, k − 2, j = i − 1, k − 2); Q ik−1 (dt) = Pik (t)B(dt), and their LST .q˜i j (s) =

∫∞ 0

(5.13)

e−st Q i j (dt) equal

( ) j ( ) λ0 n−1 Σ j ˜ (−1) j−l b(s + λl+1 ), j = 0, k − 2; s + λ0 j l l=0 ( ) j ( ) λ0 Σ n − 1 Σ j −1 ˜ b(s + λl ); q˜0k−1 (s) = (−1) j−l s + λ0 j≥k j − 1 l=1 j −l q˜ (s) =

. 0j

(

q˜i j (s) = q˜ik−1 (s) =

)Σ ( ) i−1 n−i j−l j − i + 1 ˜ b(s + λl+1 ); (−1) j − i + 1 l= j j −l

i Σ (n − i ) Σ j≥k

j −i

l= j

(

(−1)

j−l

) j −i ˜ b(s + λl ). j −l

(5.14)

Proof Indeed, in order for the system in some regeneration epoch to occur in state j ≤ k − 2 being in the previous regeneration epoch in state .0 it is necessary and sufficient that at some time point .u before time point .t one of the system components fails with probability .λ0 e−λ0 u du, then during the remaining time .t − u another . j components fail with probability . p1 j+1 (t − u) and the originally failed component is repaired in the small vicinity of the time point .t with probability . B(dt − u). Similarly, in order for the system in some regeneration epoch to occur in state .k − 1 being in the previous one in state .0 it is necessary and sufficient that at some time point .u before .t one of the system components fails with probability .λ0 e−λ0 u du, then during the remaining time.t − u it must reach state.k with probability. P1k (t − u) and the originally failed component is repaired in the small vicinity of the time point .t with probability . B(dt − u). Analogously, in order for the system in some regeneration epoch to occur in state . j being in the previous one in state .i it is necessary and sufficient that the repair of the component under repair ends with probability . B(dt), while other . j + 1 − i components fail during time .t with probability . pi j+1 (t). Finally, in order for the system in some regeneration epoch to occur in state .k − 1 being in the previous one in state .i, it must reach state .k with probability . Pik (t) and .

5.1 Application of Decomposable Semi-regenerative Processes …

115

return to state .k − 1 due to the end of repair of the component under repair with probability . B(dt). The proof of expression (5.14) follows from passing to LST in formula (5.13) with the help of expressions (5.11) and (5.12) ∫

∞

e−st Q 0 j (dt) = 0 ∫ ∞ ∫ t = λ0 e−st e−λ0 u p1 j+1 (t − u)B(dt − u)du = 0 ∫ ∞ 0 λ0 e−sv p1 j+1 (v)B(dv) = = s + λ0 0 ( ) j ( ) n−1 Σ j ˜ λ0 (−1)m b(s + λ j+1−m ); = s + λ0 j m m=0

q˜ (s) =

. 0j

∫

q˜

. 0k−1

∞

e−st Q 0k−1 (dt) = 0 ∫ ∞ ∫ t = λ0 e−st e−λ0 u P1k (t − u)B(dt − u)du = 0 ∫ ∞ 0 λ0 e−sv P1k (v)B(dv) = = s + λ0 0 ( ) j−1 ( ) λ0 Σ n − 1 Σ m j −1 ˜ (−1) b(s + λ j−m ); = s + λ0 j≥k j − 1 m=0 m

(s) =

∫

∞

q˜ (s) =

e

. ij

−st

∫

0

( =

) j−i+1 Σ

n−i j −i +1 ∫

q˜

. ik−1

∞

(s) =

∞

Q i j (dt) =

e

e−st pi j+1 (t)B(dt) =

0

(

) j −i +1 ˜ b(s + λ j+1−m ); m

(−1)

m

m=0

−st

∫

0

=

j−i Σ (n − i ) Σ j≥k

∞

Q ik−1 (dt) =

j −i

m=0

0

( (−1)

m

e−st Pik (t)B(dt) =

) j −i ˜ b(s + λ j−m ). m

By substituting .l = j − m, we get formula (5.14).

⃞

In the next section, the behavior of the process . X in a separate semi-regeneration period is considered.

116

5.1.3.2

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

Behavior of the Process in a Separate Semi-regeneration Period

The system time-dependent state probabilities in separate regeneration periods are given in the following lemma. Lemma 5.2 The system time-dependent semi-regeneration periods are

state

probabilities

in

separate

π (1) (t) = e−λ0 t ; ∫t (1) π0 j (t) = λ0 e−λ0 u p1 j (t − u)(1 − B(t − u))du, ( j = 1, k − 1);

. 00

0 (1) (t) = π0k

∫t

λ0 e−λ0 u P1k (t − u)(1 − B(t − u))du;

0

πi(1) j (t) (1) πik (t)

= pi j (t)(1 − B(t)), (1 ≤ i ≤ j ≤ k − 1); = Pik (t)(1 − B(t)), (i = 1, k − 1).

(5.15)

(1) The LT .π˜ i(1) j (s) of the process state probabilities .πi j (t) in the separate embedded regeneration period are

1 ; s + λ0 ( ) j ( ) ˜ λ0 n−1 Σ j−l j − 1 1 − b(s + λl ) π˜ 0(1) (s) = (−1) , ( j = 1, k − 1); j s + λ0 j − 1 l=0 s + λl l −1 ( ) j ( ) ˜ + λl ) λ0 Σ n − 1 Σ j − 1 1 − b(s (1) π˜ 0k (s) = (−1) j−l ; s + λ0 j≥k j − 1 l=0 s + λl l −1 π˜ (1) (s) =

. 00

) j+1 ( ) ˜ + λl ) n−i Σ j − i 1 − b(s (−1) j−l , (1 ≤ i ≤ j ≤ k − 1); j − i l=i s + λl l −i ( ) j+1 Σ (n − i ) Σ ˜ + λl ) j − i 1 − b(s π˜ ik(1) (s) = (−1) j−l . (5.16) s + λl j − i l=i l −i j≥k

π˜ i(1) j (s) =

(

Proof In fact, the semi-regeneration periods of the zero type are .Λ0 + B, and the semi-regeneration periods of all other types, equal to the repair times . B. Thus, in order for the process to be in state .0 at time .t within a separate zero-type regeneration period it should not leave this state before the first component’s failure during time −λ0 t .t with probability .e π (1) (t) = P0 {X (t) = 0, t < Λ0 + B} = P0 {t < Λ0 } = e−λ0 t .

. 00

5.1 Application of Decomposable Semi-regenerative Processes …

117

Analogously, in order for the system to occur in state . j within a separate semiregeneration period of zero type it is necessary and sufficient that at some time .u before time .t one of the system’s components fails with probability .λ0 e−λ0 u du and during the remaining time .t − u exactly . j − 1 new components of the system fail with probability . p1 j (t − u) before the repair of the first one ends with probability .(1 − B(t − u)), (1) .π0 j (t)

∫t = P0 {X (t) = j, t < Λ0 + B} =

λ0 e−λ0 u du p1 j (t − u)(1 − B(t − u)).

0

Similarly, in order for the process to occur in state .k during a separate semiregeneration period of zero type it is necessary and sufficient that at some time −λ0 u .u before .t one of the system’s components fails with probability .λ0 e du and during the remaining time .t − u the process reaches state .k with probability . P1k (t − u) before the repair of the first one ends with probability .(1 − B(t − u)), (1) .π0k (t)

∫t = P0 {X (t) = j, t < Λ0 + B} =

λ0 e−λ0 u du P1k (t − u)(1 − B(t − u)).

0

Analogously, in order for the process to be in state. j at time .t in a separate .ith-type semi-regeneration period it is necessary and sufficient that during this time period the repair doesn’t end with probability .1 − B(t) and during this time the system passes from state .i to state . j with probability . pi j (t) that is equal to π (1) (t) = Pi {X (t) = j, t < B} = pi j (t)(1 − B(t)).

. ij

The same arguments lead to the last of formulas (5.15), π (1) (t) = Pik (t)(1 − B(t)).

. ik

The proof of formulas (5.16) follows from passing to LT in expression (5.15) with the help of representations (5.11) and (5.12)

(1) .π ˜ 00 (s)

∫t = 0

e−t (s+λ0 ) dt =

1 ; s + λ0

118

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

∫t

.π ˜ 0(1) j (s)

=

e

−st

0

=

∫t

λ0 e−λ0 u du p1 j (t − u)(1 − B(t − u)) =

0

λ0 s + λ0

∫∞

e−st p1 j (t)(1 − B(t))dt =

0

λ0 π˜ (1) (s) s + λ0 1 j

(

) j−1 ( ) ˜ n−1 Σ λ0 m j − 1 1 − b(s + λ j−m ) (−1) ; = s + λ0 j − 1 m=0 s + λ j−m m (1)

.π ˜ 0k

∫t

e−st

(s) = 0

=

∫t

λ0 e−λ0 u du P1k (t − u)(1 − B(t − u)) =

0

λ0 s + λ0

∫∞

e−st P1k (t)(1 − B(t))dt =

0

⎡

˜ λ0 ⎣ 1 − b(s) − = s + λ0 s

.π ˜ i(1) j (s)

∫∞ =

Σ 1≤ j≤k−1

⎤ ) j−1 ( ) ˜ n−1 Σ m j − 1 1 − b(s + λ j−m ) ⎦ (−1) ; s + λ j−m j −1 m

(

m=0

e−st pi j (t)(1 − B(t))dt =

0

( ) ) j−i ˜ n−i Σ m j − i 1 − b(s + λ j−m ) (−1) ; = s + λ j−m m j − i m=0 (

.π ˜ ik(1) (s)

∫∞ =

e−st Pik (t)(1 − B(t))dt =

0

=

( ) j−i Σ (n − i ) Σ ˜ + λ j−m ) ˜ j − i 1 − b(s 1 − b(s) (−1)m . − s s + λ j−m j − i m=0 m i≤ j≤k−1

By substituting .l = j − m, we get formula (5.16).

5.1.3.3

⃞

Time-Dependent and Stationary Probabilities

By combining the results obtained above, the following theorem holds: Theorem 5.1 The LT of the time-dependent state probabilities of the process . X in matrix form is given by equality (5.10)

5.1 Application of Decomposable Semi-regenerative Processes …

119

−1 ˜ (1) ˜ Π(s) = (I − q(s)) ˜ Π (s),

.

˜ (1) where components .π˜ i(1) j (s) of the matrix .Π (s) are given by formulas (5.16) from Lemma 5.2, and components .q˜l j (s) of the matrix .q(s) ˜ given by formula (5.14) from Lemma 5.1. The steady-state probabilities of the process could be calculated by passing to the limit in the last equality. But it would be preferable to use the limit theorem for transition probabilities of semi-regenerative processes. Theorem 5.2 The stationary regime of the considered system under partial repair scenario exists and the steady-state probabilities equal πj =

.

1 m

Σ

αl π˜ l(1) j (0),

( j = 0, k),

(5.17)

0≤l≤ j∧(k−1)

where .α→ = {αl : l ∈ E 1 } satisfies the system of equations Σ

α→ ' = α→ ' q(0), ˜

.

αl = 1,

(5.18)

0≤l≤k−1

m = λ−1 ˜ i(1) 0 (α0 + λ0 b), and .π j (0) can be found from formula (5.16) of Lemma 5.2.

.

Proof The embedded Markov chain of the process . X is finite and irreducible. Thus, its invariant probabilities exist and according to the limit theorem for transition probabilities of semi-regenerative processes and according to Introduction from Eq. (5.6) independently of the initial state for any . j = 0, k it holds that 1 .π j = lim πi j (t) = t→∞ m

Σ 0≤l≤ j∧(k−1)

∫∞ αl 0

πl(1) j (t)dt =

1 m

Σ

αl π˜ l(1) j (0),

0≤l≤ j∧(k−1)

where vector.α→ = {αl : l ∈ E 1 } is the invariant probability measure of the embedded Markov chain that satisfies Σ to the system of Eq. (5.18) and .m is the mean semiαl El [T ]. Taking into account that the zero-type regenerative period .m = 0≤l≤k−1

semi-regeneration period is equal to .Λ0 + B and all others equal to . B it is possible to show that 1 −1 (α0 + λ0 b). .m = α0 (λ0 + b) + (1 − α0 )b = λ0 ⃞

120

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

5.1.3.4

An Example

As an example, consider a special case of a .2-out-of-.n system. To find the s.s.p. of the ˜ system, it is necessary to calculate LST of SMM . Q(s) as well as LT of the process ˜ ˜ (1) (s). From Lemma 5.1 we find . Q(s) using formula state probabilities in the SRP .Π (5.11), (5.12). ⎡

⎤ ˜ − b(λ ˜ 1 + s)) ˜ 1 + s) λ0 (b(s) λ0 b(λ ˜ ⎦. . Q(s) = ⎣ λ0 + s λ0 + s ˜b(λ1 + s) ˜b(s) − b(λ ˜ 1 + s) ˜ (1) (s) Lemma 5.2 gives the representation of .Π ( )⎤ ˜ 1 + s)) λ0 ˜ ˜ 1 + s) λ0 (1 − b(λ 1 − b(s) 1 − b(λ 1 − ⎢ ⎥ ⎥ s + λ0 (λ0 + s)(λ1 + s) λ0 + s s λ1 + s ˜ (1) (s) = ⎢ .Π ⎢ ⎥. ⎣ ⎦ ˜ 1 + s) ˜ ˜ 1 + s)) 1 − b(λ 1 − b(s)) 1 − b(λ 0 − λ1 + s s λ1 + s ⎡

To find the stationary regime of the considered system, the invariant probability measure of the embedded Markov chain .α→ ' and the mean semi-regenerative period .m should be calculated. From Eq. (5.18) and Theorem 5.2, we get ˜ ˜ 1 ), α1 = 1 − b(λ ˜ 1 ), m = b + b(λ1 ) . α = b(λ λ0

. 0

Finally using (5.17) we get the s.s.p. of the .2-out-of-.n system: π =

. 0

˜ 1 )) ˜ 1 ))) ˜ 1) b(λ λ0 (1 − b(λ λ0 (λ1 b − (1 − b(λ , π1 = , π2 = . ˜ ˜ ˜b(λ1 ) + λ0 b λ1 (b(λ1 ) + λ0 b) λ1 (b(λ1 ) + λ0 b)

1 ˜ = 1+λb In case of exponentially distributed repair time with LST .b(λ) the steadystate probabilities coincide with those, obtained for the birth and death process,

π =

. 0

1 λ0 b λ0 λ 1 b 2 , π = , π = . 1 2 1 + λ0 b + λ0 λ1 b2 1 + λ0 b + λ0 λ1 b2 1 + λ0 b + λ0 λ1 b2

Note that the analytical results for the special case of a .3-out-of-.n system were also verified, but due to their cumbersomeness, we do not present them. These results also coincide with those, obtained for the simple Markov model. Also note that the result above coincides with the previous one obtained with the construction of two-dimensional Markov process and solution of a system of Kolmogorov partial differential equations with method of characteristics (see Sect. 3.2.1.3).

5.1 Application of Decomposable Semi-regenerative Processes …

121

Fig. 5.2 Trajectory of the process . X for the system with full repair scenario

5.1.4 Full Repair Regime To study the system’s behavior under the full repair regime, we consider the main process . X as a regenerative one, whose regeneration points . Sn (n = 1, 2, . . . ), S0 = 0 are the full repair completion times of the system after its failure. The regeneration periods .Θn = Sn − Sn−1 of the process . X consist of two terms: the system lifetimes (times to the system failure after its repair) . Fn and the system repair times after its failure .G n : Θn = Fn + G n (see Fig. 5.2). Denote by . F(t) = P{Fn ≤ t} and .Γ(t) = P{Θn ≤ t} the common distribution function of r.v.’s . Fn and .Θn (n = 1, 2, . . . ), respectively.

5.1.4.1

The Main Regenerative Process

From the theory of regenerative processes in the sense of Smith, it follows that the time-dependent probabilities of the system can be represented in terms of the corresponding process state distribution .π (Θ) j (t) = P{X (t) = j, t < Θ} in a separate regeneration period .Θ as follows2 : π j (t) =

.

π (Θ) j (t)

∫t +

π (Θ) j (t − u)H (du).

(5.19)

0

Remind, that we denote by . F and .Θ the representatives of the corresponding sequences of i.i.d r.v. . Fn and .Θn .

2

122

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

Here . H (t) is the corresponding renewal function, which is generated by the c.d.f. Γ(t) = P{Θn ≤ t} of r.v’s..Θn , and according to the Smith theorem can be calculated as Σ Σ . H (t) = P{Sn ≤ t} = Γ (∗n) (t),

.

n≥1

whose LST equals .

˜ h(s) =

∫∞ 0

n≥1

e−st H (dt) =

γ˜ (s) , 1 − γ˜ (s)

(5.20)

where .γ˜ (s) is LST of the c.d.f. .Γ(t). Thus, to study the process we need first to calculate the distribution of the regeneration periods .Θn , the corresponding renewal function . H (t), and the process distribution .π (Θ) j (t) at them. Denote by . Fn (n = 1, 2, . . . ) time to the .nth system failure after its previous full repair, and by . F(t) their common distribution function . F(t) = P{Fn ≤ t}. Remind that the r.v. .Θn is the sum of two independent r.v.’s . Fn and .G n , the distribution of the second one is supposed to be known and the distribution of the first one will be obtained jointly with its LST later in Lemma 5.5 (see Remark 5.1 to it). Now pass on to the calculation of the process state distribution.π (Θ) j (t) in a separate regeneration period .Θn = Fn + G n . The process behavior in the separate regeneration period can be divided into two parts: the process behavior within a separate system lifetime . Fn and its behavior during the repair time .G n , {

X (Sn−1 + t) = j : for t ≤ Fn , j /= k k: for Fn < t ≤ Θn , j = k. (5.21) Thus the process time-dependent state probabilities in a separate regeneration period are given in the following lemma. {X (Sn−1 + t) = j : t ≤ θn } =

.

Lemma 5.3 The system state probabilities in a separate regeneration period .Θ in terms of the corresponding probabilities within a separate system lifetime . F are (F) π (Θ) j (t) = (1 − δ jk )π j (t) + δ jk P{F ≤ t < Θ}

.

with their Laplace transform .

˜ ˜ 1 − g(s) π˜ (Θ) ˜ (F) . j (s) = (1 − δ jk )π j (s) + δ jk f (s) s

(5.22)

Proof The first equality of the lemma follows directly from relation (5.21). In terms of LT from this equality it follows that

5.1 Application of Decomposable Semi-regenerative Processes …

.π ˜ (Θ) j (s)

∫∞ =

e

−st

π (Θ) j (t)dt

= (1 −

δ jk )π˜ (F) j (s)

123

∫∞ + δ jk

0

P{F ≤ t < Θ}dt = 0

= (1 −

δ jk )π˜ (F) j (s)

= (1 −

δ jk )π˜ (F) j (s)

∫∞ + δ jk

e

−st

0

+ δ jk

∫t (1 − G(t − u))F(du) =

dt 0

1 − g(s) ˜ , f˜(s) s

which is the second equality of the lemma.

⃞

The last equality shows that we need the process state distribution within a separate system lifetime . F, which is considered in the following subsection.

5.1.4.2

Embedded Semi-regenerative Process

Since the process behavior within the system lifetime . F is rather complicated, the process . X during this time will be considered as an embedded semi-regenerative process (ESRP) . X n(1) = {X n(1) (t) : t ≥ 0} with .

X n(1) (t) = X (Sn−1 + t), t ≤ Fn .

Its semi-regeneration times. Sn(1) of the type. j are the same as for the semi-regenerative process in Sect. 5.1.3.1 (see Fig. 5.2) except for the fact that now the process . X n(1) is considered in a separate main regeneration period and therefore never enters state .k − 1 after the system’s repair completion. Thus, there are only .k − 1 embedded regeneration states, . E (1) = {0, 1, . . . , k − 2}. In order to study the behavior of the process before it enters state .k, denoted by (1) • .Tl(1) = Sl(1) − Sl−1 , l = 1, 2, . . . the intervals between embedded semi-regeneration times of the ESRP . X (1) (time points between of the components’ repair completions). • . Q (1) (t) = [Q i(1) j (t)]i j∈E (1) the embedded semi-Markov matrix (ESMM) whose components are the process transition probabilities between semi-regeneration times,

.

(1) (1) (1) (1) Q i(1) ≤ t |X (1) (Sl−1 + 0) = i}. j (t) = P{X (Sl + 0) = j, Tl

(1) → (1) (t) = [Q ik • .Q (t)] the vector-function, components of which are c.d.f.’s of the first passage time from state .i to the absorbing state .k by the ESRP along a monotone trajectory. → = [Fik (t)] the vector-function, components of which are c.d.f.’s of the • . F(t) absorbing state .k destination time by the ESRP starting from state .i (i = 0, k − 2).

124

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

• . H (1) (t) = [Hi(1) j (t)]i j∈E (1) the embedded Markov renewal matrix whose components are the conditional embedded renewal functions in separate lifetime period [ (1) . Hi j (t)

=E

Σ

] 1{X (1) (S (1) +0)= j, S (1) ≤t} | X l

l

(1)

(S0(1) )

=i .

l≥1

We start with the calculation of the ESMM . Q (1) (t) = [Q i(1) j (t)]i j∈E (1) of the (1) ESRP . X . To calculate it, we note that its components coincide with those from Sect. 5.1.3.1 that have been determined in Lemma 5.1 except for the fact that now the process . X (1) never enters state .k − 1 after the completion of the repair. Thus, they are defined only for . j ≤ k − 2 and in terms of notations (5.11), (5.12) are represented in differential form in Lemma 5.1. At that, since the set of embedded regeneration states of the process . X (1) is a proper subset of the set of states of the process . X , its transition matrix is a degenerate matrix in contrast to the matrix of the previous section. Since the corresponding expressions are used later, we repeat them jointly (1) → (1) (t) of probabilities of reaching state (t) of the vector . Q with the components . Q ik .k, together with their LST in the following lemma. Lemma 5.4 (1) The differentials of the ESMM components of the process . X (1) are (1) . Q 0 j (dt)

∫t =

λ0 e−λ0 u du p1 j+1 (t − u)B(dt − u), ( j = 0, k − 2),

0

Q i(1) j (dt) = pi j+1 (t)B(dt), (i = 1, k − 2, j = i − 1, k − 2),

(5.23)

(1) → (1) (dt) = [Q ik while the differentials of the components of the vector . Q (dt)]i=0,k−2 are

.

Q (1) 0k (dt) =

∫t

λ0 e−λ0 u du P1k (dt − u)(1 − B(t − u)),

0 (1) (dt) Q ik

= Pik (dt)(1 − B(t)).

(2) The corresponding LST for the components of matrix . Q (1) (t) is equal to: ( ) j ( ) λ0 n−1 Σ j ˜ (−1)m b(s + λ j+1−m ), ( j = 0, k − 2); s + λ0 j m m=0 ( ) j−i+1 ( ) Σ n−i m j −i +1 ˜ (s) = (−1) q˜i(1) b(s + λ j+1−m ), j j − i + 1 m=0 m q˜ (1) (s) =

. 0j

(i = 1, k − 2, j = i − 1, k − 2),

(5.24)

5.1 Application of Decomposable Semi-regenerative Processes …

125

→ (1) (t) is while the corresponding LST for the components of vector . Q ( ) j Σ (n − 1) Σ ˜ j−l j − 1 1 − b(s + λl ) (−1) , j − 1 l=0 l −1 s + λl 1≤ j≤k−1

(1) .q ˜0k (s)

λ0 = s + λ0

q˜ik(1) (s)

( ) j+1 Σ (n − i ) Σ ˜ j−l j − i 1 − b(s + λl ) = (−1) . s + λl j − i l=i l −i 1≤ j≤k−1

(5.25)

Proof Formulas (5.23) and (5.24) are derived in the same way as in Lemma 5.1. ' (t). Based on (5.12) for In order to calculate .q˜ik(1) (s), we need first to calculate . P1k its derivative, it holds ( ) Σ (n − i ) Σ ' m j −i λ j−m e−λ j−m t . . Pik (t) = (−1) j − i m i≤ j≤k−1 0≤m≤ j−i → (1) Regarding the components . Q (1) 0k (dt) of the vector . Q (dt), for the process to be in state .k in the vicinity of the time point .t, leaving state .0, it is necessary and sufficient that during time .u before time .t one of the components fails with probability .λ0 e−λ0 u du and in the remaining time .t − u its repair was not completed with probability .(1 − B(t − u)) and there were at least .k − 1 failures in the neighboring of point .t − u with probability . P1k (dt − u). Thus, (1) .q ˜0k (s) =

∫ ∞ 0

(1) e−st Q 0k (dt) = λ0

∫ ∞

e−st

∫ t

0

0

e−λ0 u P1k (dt − u)(1 − B(t − u))du =

∫ ∞ λ0 ' (v)(1 − B(v))dv = e−sv P1k s + λ0 0 ( ) ˜ + λ j−m ) Σ (n − 1) Σ 1 − b(s j −1 λ0 m λ j−m = (−1) . s + λ0 s + λ j−m j −1 m

=

1≤ j≤k−1

0≤m≤ j−1

In order for the process in the vicinity of the point .t to hit state .k from state .i, it is necessary and sufficient that up to the time .t the repair was not completed with probability .(1 − B(t)) and there were at least .k − i new failures in the neighbor this time .t with probability . Pik (dt). .q ˜ik(1) (s)

∫ =

∞

e 0

−st

(1) Q ik (dt)

∫

∞

= 0

e−st Pik' (t)(1 − B(t))dt =

( ) Σ (n − i ) Σ ˜ + λ j−m ) 1 − b(s m j −i λ j−m (−1) . = j − i 0≤m≤ j−i m s + λ j−m i≤ j≤k−1 By substituting .l = j − m, we get formula (5.25). → For the vector . F(t), the following representation holds.

⃞

126

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

→ Lemma 5.5 The vector-function of differentials . F(dt) satisfies the following equation3 : → → (1) (dt) + Q (1) ★ F(dt), → =Q . F(dt) (5.26) whose unique solution in terms of LST is (1) f→˜(s) = (I − q˜ (1) (s))−1 q→˜ (s).

.

(5.27)

→ satisfy the almost evident Proof The differentials of components of the vector . F(t) equation t Σ ∫ (1) (1) . Fik (dt) = Q ik (dt) + Q il (du)Flk (dt − u). l∈E (1) 0

Truly, starting from any state.i the process. X (1) reaches the absorbing state.k in a small interval .dt around .t if it reaches it along the monotone trajectory with probability (1) . Q ik (dt) (the first term in the above equality) or if the process first enters some other non-absorbing state .l in time .u before time .t with probability . Q il(1) (du) and then reaches the absorbing state .k in a small interval .dt during the remaining time .t − u with probability . Fik (dt − u) (the second term in the above equality). In terms of LST equation (5.26) takes the following form: .

˜f→(s) = q→˜ (1) (s) + q˜ (1) (s) ˜f→(s),

whose unique solution due to the degeneracy of matrix .q˜ (1) (s) is (5.27).

⃞

Remark 5.1 Further, as it was noted in Sect. 5.1.4.1, the first components . F0k (t) of → are the c.d.f. of time to the first (and between) failure for system starting vectors . F(t) from state .0. Its moment generating function (m.g.f.) is the first component . f˜0k (s) of the vector . ˜f→(s), therefore they will be denoted without indices, .

F0k (t) ≡ F(t),

f˜0k (s) ≡ f˜(s).

Taking into account that ∫t .

R(t) = 1 − F(t) = 1 −

f (u)du 0

the following corollary holds.

3

The vectors are as usually vector-columns.

5.1 Application of Decomposable Semi-regenerative Processes …

127

Corollary 5.1 The LT of the reliability function of the system is .

1 ˜ R(s) = (1 − f˜(s)). s

Corollary 5.2 Since the process regeneration cycle .Θ is equal to the sum of two independent r.v.’s: time to the system failure . F and its repair time .G, its m.g.f. equals ] [ ˜ γ (s) ≡ E e−sΘ = f˜(s)g(s).

.

Now, expression (5.20) leads to the following corollary. ˜ Corollary 5.3 The LST .h(s) of the renewal function . H (t) of the system operating in the full repair regime equals to

.

5.1.4.3

˜ h(s) =

f˜(s)g(s) ˜ . ˜ 1 − f (s)g(s) ˜

(5.28)

Process State Probabilities in a Separate Lifetime Period

To study the process behavior during a separate system lifetime we consider • matrix .Π(F) (t) = [Πi(F) j (t)]i j∈E (1) , whose components are transition probabilities of the process within a separate lifetime, (1) (1) Πi(F) j (t) = P{X n (t) = j, t < Fn | X n (0) = i};

.

• matrix.Π(1) (t) = [Πi(1) j (t)]i j∈E (1) , whose components are transition probabilities of the process in a separate embedded semi-regeneration period (between successive repair completions), { } (1) (1) (1) (1) Πi(1) | X n(1) (Sl−1 + 0) = i . j (t) = P X n (Sl−1 + t) = j, t < Tl

.

In terms of these notations and according to the DSRP theory the following representations hold: (F) .Π (t) = Π(1) (t) + H (1) ★ Π(1) (t); and in terms of LT for matrices .Π(F) (t), Π(1) (t) and LST for matrices . H (1) (t) this equation takes the following form: ˜ (F) (s) = Π ˜ (1) (s) + h˜ (1) (s) · Π ˜ (1) (s). Π

.

In our case .

H (1) (dt) = Q (1) (dt) + Q (1) ★ H (1) (dt)

128

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

and therefore .

From here .

h˜ (1) (s) = (I − q˜ (1) (s))−1 q˜ (1) (s).

I + h˜ (1) (s) = I +

Σ

q˜ (1)∗l (s) = (I − q˜ (1) (s))−1

l≥1

and .

˜ (1) (s) + h˜ (1) (s) · Π ˜ (1) (s) = (I − q˜ (1) (s))−1 · Π ˜ (1) (s). ˜ (F) (s) = Π Π

(5.29)

The system state transition probabilities within a separate embedded repair time Π(1) (t) coincide for the embedded set of states . E (1) with the corresponding probabilities in Sect. 5.1.3.2 and presented in formula (5.16) of Lemma 5.2. We repeat (1) for further here the corresponding LT .π˜ i(1) j (s) for the subset of embedded states . E references.

.

(1) Lemma 5.6 The LT .π˜ i(1) j (s) of the process state probabilities .πi j (t) in a separate embedded regeneration period are

1 ; s + λ0 ( ) j ( ) ˜ λ0 n−1 Σ j−l j − 1 1 − b(s + λl ) π˜ 0(1) (s) = (−1) ( j = 1, k − 2); j s + λ0 j − 1 l=0 s + λl l −1 ( ) j Σ (n − 1) Σ ˜ + λl ) λ0 j − 1 1 − b(s (1) π˜ 0k−1 (s) = (−1) j−l ; s + λ0 j≥k−1 j − 1 l=0 s + λl l −1 π˜ (1) (s) =

. 00

) j+1 ( ) ˜ + λl ) n−i Σ j − i 1 − b(s (−1) j−l , (1 ≤ i ≤ j ≤ k − 2); j − i l=i s + λl l −i ( ) j+1 Σ (n − i ) Σ ˜ + λl ) j − i 1 − b(s (1) π˜ ik−1 (s) = (−1) j−l . (5.30) s + λl j − i l=i l −i j≥k−1 π˜ i(1) j (s) =

(

Proof The proof repeats the arguments of Lemma 5.2 taking into account the new ⃞ set of regeneration states . E (1) .

5.1.4.4

Process Time-Dependent and Stationary State Probabilities

In this subsection we unite obtained above results in order to represent t.d.s.p.’s and s.s.p.’s of the process . X (and appropriate system) in terms of LST.

5.1 Application of Decomposable Semi-regenerative Processes …

129

Theorem 5.3 The LT .π˜ j (s) of the time-dependent state probability processes . X , starting from the state zero, equals to { 1 π˜ (F) j (s) .π ˜ j (s) = ˜ f˜(s) 1−sg(s) 1 − f˜(s)g(s) ˜

for j = 0, k − 1, for j = k,

(5.31)

where . f˜(s) is defined as the first component of vector .f→˜(s) from Remark 5.1, which is represented by formula (5.27), and the LT of time-dependent process state probabil˜ (F) (s), ities in a separate process lifetime period .π˜ (F) j (s) is the first row of matrix .Π which can be calculated from (5.29), (5.30), (5.24). Proof Expression (5.31) follows passing to LT in formula (5.19) from Sect. 5.1.4.1 with the help of expressions (5.22) and (5.28) from Corollary 5.3. ⃞ Remark 5.2 Really, because any main regeneration period begins with the state .0 for calculation of the process t.d.s.p.’s as well as s.s.p.’s we need only in probabilities with the initial zero state. Remark 5.3 The representation of the final results in the initial system information in general is enough cumbersome and will be presented in examples. From this theorem, by using Smith’s key renewal theorem, one can obtain the stationary process probabilities. Theorem 5.4 The s.s.p.’s .π j of the process . X , starting from any state, equal to 1 .π j = f +g

{

π˜ (F) j (0) g

for j = 0, k − 1, for j = k,

(5.32)

where .g = E[G], . f = E[F] = − f˜' (0) is the mean system lifetime that can be found from formula (5.27), and the values .π˜ (F) j (0) are the components of the first row of (F) ˜ (0), which can be calculated from (5.29), (5.30), (5.24). the matrix .Π

5.1.4.5

An Example

Consider as before an example of a .2-out-of-.n system. Here we calculate the reliability function and the s.s.p.’s of the process. According to Corollary 5.1 the LT of the reliability function is connected with m.g.f. . f˜(s) of the system’s lifetime . F. From Remark 5.1 one can obtain ˜ + λ1 )) λ0 λ1 (1 − b(s , ˜ + λ1 ))) (s + λ1 )(s + λ0 (1 − b(s (5.33) (1) (1) where .q˜00 (s) and .q˜02 (s) are given from (5.24), (5.25). Thus, .

(1) (1) (s))−1 q˜02 (s) = f˜(s) ≡ f˜02 (s) = (1 − q˜00

130

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

[ ] ˜ + λ1 )) λ (1 − b(s 1 λ 1 0 1 ˜ . R(s) 1− = (1 − f˜(s)) = = ˜ + λ1 ))) s s (s + λ1 )(s + λ0 (1 − b(s =

˜ + λ1 )) + λ1 s + λ0 (1 − b(s . ˜ + λ1 ))) (s + λ1 )(s + λ0 (1 − b(s

The mean time to the system failure can be calculated as . f = E[F] = R(0) which allows obtaining the following expression for it: .

f = R(0) =

1 1 . + ˜ 1 )) λ1 λ0 (1 − b(λ

(5.34)

According to Theorem 5.4, the s.s.p.’s .π j of the process . X for . j = 0, 1 are presented in terms of .π˜ (F) j (0), i.e., mean sojourn time of the process in state . j within a separate lifetime . F. For . j = 2, these probabilities are presented in terms of mean repair time in a separate regeneration period (see formula (5.32)). The corresponding expressions are obtained using formula (5.29) for . j = 0, 1 with .s = 0, 1

(1) (1) π˜ (F) (0) = (1 − q˜00 (0))−1 π˜ 00 (0) =

. 0

˜ 1 )) λ0 (1 − b(λ 1 (1) (1) (0))−1 π˜ 01 (0) = , π˜ 1(F) (0) = (1 − q˜00 λ1

,

(1) (1) ˜ 1 ), and where transition probability .q˜00 (0) follows from (5.24) as .q˜00 (0) = b(λ (1) .π ˜ 0 j (0) are obtained from (5.30) of Lemma 5.6 as

{ .π ˜ 0(1) j (0)

=

1 λ0 ˜ 1) 1−b(λ λ1

for

j = 0,

for

j = 1.

Thus, the s.s.p.’s .π j from (5.32) are equal to π =

. 0

1 ˜ 1 )) ( f + g)λ0 (1 − b(λ

, π1 =

1 g , π2 = , ( f + g)λ1 f +g

where the mean time to failure of the system . f is done in (5.34). Consider now a special case of the corresponding model with exponential distribution of repair times of the components and the whole system, where .b and .g are mean times of partial and full repair, respectively, B(t) = 1 − e− b , G(t) = 1 − e− g . t

.

t

5.2 Application of Simulation and ML Methods to the Study of Non-repairable …

131

In this case both the LT of the reliability function .

˜ R(s) =

1 + b(s + λ0 + λ1 ) s(1 + bs) + λ0 λ1 b + bs(λ0 + λ1 )

and the s.s.p.’s 1 + λ1 b λ0 b , π1 = , 1 + b(λ0 + λ1 + λ0 λ1 g) 1 + b(λ0 + λ1 + λ0 λ1 g) λ0 λ1 bg π2 = 1 + b(λ0 + λ1 + λ0 λ1 g)

π =

. 0

coincide with those, obtained with the simple birth and death process. Note again that the results above coincide with the previous ones obtained with the construction of two-dimensional Markov process and solution of a system of Kolmogorov partial differential equations with method of characteristics (see Sects. 2.2.2 and 3.2.2.3).

5.2 Application of Simulation and ML Methods to the Study of Non-repairable . k-Out-of-.n Models In this section, several machine learning (ML) methods are used to assess reliability characteristics of .k-out-of-.n models for arbitrary input data based on practically significant parameters. Section 5.2.2 contains numerical examples and sensitivity analysis of a simple homogeneous .k-out-of-.n system and a homogeneous .(2, k)-outof-.n system, the failure of which depends on the location of the failed elements (see corresponding description in Sects. 2.1.2, 2.1.3, 2.3). In Sect. 5.2.3, various ML techniques for predicting the reliability will discuss. Methods and data are also presented, which are implemented in Sect. 5.2.4.

5.2.1 Preliminaries In queuing and reliability theories, machine learning (ML) methods are usually used for studying various probabilistic and time characteristics of complex systems. They are also useful in those cases when it is impossible to obtain results either analytically or using simulation [36]. The application of ML techniques for analyzing the reliability of an unmanned high-altitude module arises due to the following factors: 1. From a practical point of view, the system service time is often estimated by its average value, while the shape of the lifetime distribution is unknown and can only be assumed based on some statistical data. ML model can operate based on

132

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

the mean value without considering a specific distribution function of the lifetime of system elements. 2. Some parameters inside the system can significantly impact its reliability. However, from practice, this information may also be absent. The sensitivity analysis helps identify these weaknesses, after which they will be included in the ML model. 3. A model built and trained using ML techniques can predict the system reliability characteristics faster than a simulation model. In addition, it allows making accurate predictions on many data simultaneously, while simulation can only give a similar result after a lot of iterations. 4. Trained model can be useful and used by engineers at the development stage of such modules for many aims: to determine a highly reliable system architecture (parameters .k, .n), select the module components, the characteristics of which will support reliability and long-term operation of HAP (mean lifetime .a and the coefficient of variation .v), and also predict how long this unmanned module will operate with a satisfactory level of reliability. There are many machine learning techniques. Here, supervised learning for some types of regressions and neural networks using a Python programming language are presented. For this, scikit-learn and TensorFlow libraries are used.

5.2.2 Numerical Example and Sensitivity Analysis To represent the possibility of applying ML methods, consider the models according to the definitions from Sects. 2.1.2 and 2.1.3. As an example, consider again the case of .3-out-of-.6 model. It is supposed that the lifetime of the system’s components has the following distributions: [ ( )] • Gamma . Γ 1/v2 , av[2 . ( )] a . • Gnedenko–Weibull . GW θ, Γ(1+1/θ) / ( [ ( ))] a , ln 1 + v2 , • Log-normal . Ln N ln √1+v 2 where .a is the mean lifetime of the system components and .v is its coefficient of variation. .θ is the shape parameter of .GW distribution and selected based on the value of .v. In our experiments, we choose .a = 1 and .v = [0.1, 0.5, 1, 5, 10]. First, consider the simple case of homogeneous .3-out-of-.6 model. Figure 5.3 shows the dependence of system reliability function on the time .t. Black, red, and blue colors correspond the .Γ, .GW , and . Ln N distributions, respectively. As it can be seen from the curves, the reliability function of the system is asymptotically insensitive to the form of the lifetime distribution at fixed mean and coefficient of variation .v ≤ 1. At the same time, with .v > 1, this insensitivity disappears, and the system loses its reliability very quickly. We can conclude that the system behavior depends on the value .v.

5.2 Application of Simulation and ML Methods to the Study of Non-repairable …

133

Fig. 5.3 Reliability function . R(t) of homogeneous .3-out-of-.6 model

Fig. 5.4 Reliability function . R(t) of homogeneous .(2, 3)-out-of-.6 system

Further, consider a .3-out-of-.6 model, taking into account the location of the failed units. Suppose that the system is failed iff at least .3 out of .6 engines are failed, and two failed motors should not be located next to each other. Denote such a system as a .(2, 3)-out-of-.6 system. In other words, the system fails when two adjacent motors stop to operate, or when any three engines fail. Figure 5.4 presents this example, where the examples of both the same system and parameters as before are used. As

134

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

Table 5.1 MTTF of .3-out-of-.6 model . A(t) .v = 0.1 .v = 0.5

.v

.Γ

.0.9775

.0.8445

.0.6174

.0.0077

.GW

.0.9886

.0.8593

.0.6176

.0.0697

. Ln N

.0.9760

.0.8349

.0.6509

.0.2047

=1

.v

=5

Table 5.2 MTTF of .(2, 3)-out-of-.6 model . A(t) .v = 0.1 .v = 0.5

.v

.Γ

.0.9608

.0.7747

.0.5154

.0.0041

.GW

.0.9693

.0.7777

.0.5177

.0.0503

. Ln N

.0.9598

.0.7743

.0.5714

.0.1621

=1

.v

=5

.v

= 10

∗ 10−5 .0.0197 .0.1154 .8

.v

= 10

∗ 10−5 .0.0136 .0.0874 .3

can be seen from the numerical examples, the behavior of .3-out-of-.6 and .(2, 3)-outof-.6 system reliability functions is very similar. To see the difference between them, consider corresponding MTTF (.E[T ]) (Tables 5.1 and 5.2).

5.2.3 Methods and Data As ML methods, we consider the following from scikit-learn (for regressions) and TensorFlow (for neural network) libraries: • • • •

Linear regression (LinReg). Polynomial regression (degree .= 4) (PolyReg). K-nearest neighbors regression (n_neighbors = 5) (KNN). Multi-output regression with cross-validation (scoring = MSE) based on Ridge regression (MultiReg). • Artificial neural network with three hidden layers (optimizer = RMSprop(1e-3), loss = MSE, batch size = 96) (ANN). As it was noted in the preliminaries, the purpose of machine learning application is to predict the reliability and time characteristics of a tethered unmanned module. Therefore, the output parameters are. R(t), E[T ] (Table 5.3). The set of parameters, as well as their ranges, is associated with the following. The previous section concludes some hidden parameters of the system, namely, the coefficient of variation, have a significant impact on its behavior and performance. Moreover, the system is insensitive to the shape of the lifetime distribution with .v < 1. In addition, from a practical point of view, we assume that the system is at a satisfactory level of reliability if . R(t) ≥ 0.5. We have generated two datasets for training the models.

5.2 Application of Simulation and ML Methods to the Study of Non-repairable … Table 5.3 Variables for machine learning models and their ranges Type Variables Symbol Input

Output

Total number of system’s elements Needed number of failed components to system failure Mean lifetime Coefficient of variation Time to system acceptable level of R Reliability MTTF

135

Range

.n

.4−10

.k

.2

.a

.0.1−1

.v

.0.01−1

.t

.>0

. R(t)

.0.5−1

.E[T ]

.>0

− (n − 1)

1. To train the model, which describes the behavior of THAP by homogeneous .k-out-of-.n system, the dataset was generated using formula from Sect. 2.1.2, in which . A(t) ∼ Γ. 2. For the second case, in which system failure depends on the location of the failed elements, simulation results were used, here also . A(t) ∼ Γ. This data supposes that a system failure occurs either when .2 adjacent or any .k elements have failed (according to Sect. 2.1.3). Denote such a system as .(2, k)-out-of-.n system. The architecture of the selected ML models is different. Some can predict several outputs simultaneously, while others can operate with only one outcome. The whole process contains two phases—training and testing. Before training, we divide the initial dataset into train and test sets with a ratio of .70% and .30%, respectively. The learning process for LinReg, PolyReg, and KNN is structured as follows. The first step is to predict reliability . R using parameters .n, k, a, v, t. The next cycle ends with a forecast of .E[T ] based on the set .n, k, a, v, R, t. After each round, the accuracy of the trained model is assessed, and testing begins on a new data sample. For MultiReg and ANN, there is one training cycle, in which the model predicts . R, E[T ] simultaneously based on .n, k, a, v. These models provide an additional phase for monitoring training, the so-called cross-validation. In this way, the initial set is divided into .70%, .20% and .10% for train, validation, and final test.

5.2.4 Training and Testing Results Now move on to the results of ML technique applications for analyzing the reliability of a tethered unmanned high-altitude module.

136

5.2.4.1

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

A Simple Homogeneous . k-Out-of-.n Model

First, consider the .k-out-of-.n system. Table 5.4 shows the mean square error (MSE) for the predicted values on the training set. The table results show the smallest prediction error was achieved using PolyReg and KNN. The greatest error corresponds to MultiReg. The closest prediction in the training phase among all methods was made for MTTF .E[T ]. Table 5.5 demonstrates MSE, mean absolute error (MAE) as well as the coefficient variation (. R 2 ) for the test set. Analyzing the results obtained, we can note that MSE estimate for all cases lies in an acceptable interval. MAE estimate shows the relative value of the prediction error. In our task, MAE .≥ 0.05 is considered unsatisfactory. Therefore, only the K-nearest neighbors regression shows the obtained accuracy result among all the considered cases. . R 2 estimate indicates how well the constructed model adequately describes the initial data. The best result for this indicator is again shown by the KNN method. Note that all methods are suitable for predicting the meantime .E[T ]. The estimates MSE and MAE are quite small, and . R 2 is high, which confirms the high dependence between the input and output parameters. Consider prediction results on the test set graphically. Figures 5.5, 5.6, 5.7, 5.8, and 5.9 show the scatter diagrams for ML methods described above. For each of these figures, 500 samples were taken at random. In reality, the test sample contains about .200.000 values. LinReg and MultiReg demonstrate similar results for all predicted parameters, but their accuracy is quite low. PolyReg and ANN show acceptable prediction accuracy of .E[T ]. For . R(t), the prediction error is too high. These methods present insufficient prediction accuracy. It suggests that models do not reflect the relationship between input and output data. Predictions for . R and .E[T ] using KNN are close enough to their exact values. Conclude that the

Table 5.4 Accuracy of training LinReg MSE

PolyReg

KNN

. R(t)

.0.0094

.0.0028

.10

.E[T ]

.0.0093

.10

Table 5.5 Accuracy of testing LinReg MSE MAE .R

2

−4

.4

−4

∗ 10−4

PolyReg

KNN

. R(t)

.0.0094

.0.0028

.2

.E[T ]

.0.0107

.0.0056

. R(t)

.0.0708

.0.0365

.E[T ]

.0.0690

.0.0121

. R(t)

.0.3804

.0.8134

.E[T ]

.0.8934

.0.9295

∗ 10−4 −6 .2 ∗ 10 .0.0033 −4 .3 ∗ 10 .0.9894 .0.9999

MultiReg

ANN

.0.0313

.0.0094

.0.0123

.10

−4

MultiReg

ANN

.0.0239

.0.0131

.0.0105

.0.0043

.0.0761

.0.0746

.0.0687

.0.0273

.0.1343

.0.1370

.0.8942

.0.9571

5.2 Application of Simulation and ML Methods to the Study of Non-repairable …

137

Fig. 5.5 Scatter plots for LinReg

Fig. 5.6 Scatter plots for PolyReg

Fig. 5.7 Scatter plots for KNN

application of the KNN method obtains the most accurate prediction result for all metrics among the considered ML techniques.

5.2.4.2

A Homogeneous .(2, k)-Out-of-.n Model

The application of ML techniques to the task at hand has shown that KNN most accurately predicts the reliability of an unmanned high-altitude module, the failure of which occurs after the failure of .k its elements. Therefore, for the second case

138

5 New Approaches to Reliability Investigation of .k-Out-of-.n Models

Fig. 5.8 Scatter plots for MultiReg

Fig. 5.9 Scatter plots for ANN Table 5.6 Accuracy of training (MSE) . R(t) KNN

.E[T ]

−4 .10

.10

−4

. R(t)

.E[T ]

−4 .1.6 ∗ 10

∗ 10−6 −4 .3.3 ∗ 10 .0.9999

Table 5.7 Accuracy of testing MSE MAE 2 .R

−3 .3.6 ∗ 10 .0.9904

.1.9

(.(2, k)-out-of-.n model) of dependence of the system failure on the location of the failed elements, we will consider only the KNN method. Consider the learning accuracy results (Table 5.6). MSE is small enough and takes the desired value. The results on the test set are presented in Table 5.7 and Fig. 5.10. The results of the prediction accuracy take acceptable values. MSE and MAE are small enough, and the coefficient of determination . R 2 is high.

5.3 Application of Simulation and ML Methods to the Study of Repairable …

139

Fig. 5.10 Scatter plots for KNN (.(2, k)-out-of-.n model)

The graphical results show similar prediction accuracy to the .k-out-of-.n system. The KNN model accurately reflects the dependence of . R and .E[T ] on the initial data.

5.3 Application of Simulation and ML Methods to the Study of Repairable . k-Out-of-.n Models Analogously to Sect. 5.2 machine learning techniques can be used for s.s.p.’s calculation of a .k-out-of-.n model with component-wise and system repairs. However, the repaired system is more complex, therefore, before applying the ML methods, it is necessary to carry out a deeper numerical and sensitivity analysis (Sect. 5.3.2). After some numerical analysis, in Sect. 5.3.3, the full description of neural network construction as well as data collection for training and testing is presented. The training results with some tests are shown in Sect. 5.3.4. Section 5.3.5 presents comparative analysis of the results of analytics, simulation, and neural network prediction.

5.3.1 Preliminaries In this section, we consider the same model as in Sect. 3.3.1 so the model description as well as its notations are also the same. However, just to simplify the definition of model under consideration, introduce the following. Here and further, a modified Kendall’s notation.