An Introduction to Cellular Network Analysis Using Stochastic Geometry 3031297423, 9783031297427

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Synthesis Lectures on Learning, Networks, and Algorithms

Jeffrey G. Andrews · Abhishek K. Gupta · Ahmad Alammouri · Harpreet S. Dhillon

An Introduction to Cellular Network Analysis Using Stochastic Geometry

Synthesis Lectures on Learning, Networks, and Algorithms Series Editor Lei Ying, ECE, University of Michigan–Ann Arbor, Ann Arbor, USA

The series publishes short books on the design, analysis, and management of complex networked systems using tools from control, communications, learning, optimization, and stochastic analysis. Each Lecture is a self-contained presentation of one topic by a leading expert. The topics include learning, networks, and algorithms, and cover a broad spectrum of applications to networked systems including communication networks, data-center networks, social, and transportation networks.

Jeffrey G. Andrews · Abhishek K. Gupta · Ahmad Alammouri · Harpreet S. Dhillon

An Introduction to Cellular Network Analysis Using Stochastic Geometry

Jeffrey G. Andrews The University of Texas at Austin Austin, TX, USA

Abhishek K. Gupta Indian Institute of Technology Kanpur Kanpur, India

Ahmad Alammouri The University of Texas at Austin Austin, TX, USA

Harpreet S. Dhillon Virginia Tech Blacksburg, VA, USA

ISSN 2690-4306 ISSN 2690-4314 (electronic) Synthesis Lectures on Learning, Networks, and Algorithms ISBN 978-3-031-29742-7 ISBN 978-3-031-29743-4 (eBook) https://doi.org/10.1007/978-3-031-29743-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my talented and dedicated graduate students: past, present, and future. —Jeffrey G. Andrews To Manu, Saloni and our beloved family. —Abhishek K. Gupta To Mohammad, Halimeh, and our beloved family. —Ahmad Alammouri To Harnaaz, Donia, and our beloved family —Harpreet S. Dhillon

Preface

The high-speed global cellular communication network is one of humanity’s most impressive and important technologies, and it continues to rapidly evolve and improve with each new generation. In fact, as we wrote parts of this book from home during the great selfquarantine of 2020–2021, most human interaction is taking place over communication networks with at least some wireless links. However, until fairly recently, mathematical performance analysis of cellular networks was not possible without resorting to extremely simple spatial models, such as the well-known Wyner and other similar models [1, 2, 3, 4]. The alternative to such simplified approaches has been to exhaustively simulate the networks to average out the many sources of randomness, such as the base-station (BS) and user locations and fading distributions, as well as the noise. These simulations can be extremely time-consuming and error prone. Although system-level simulations will continue to be indispensable for cellular network analysis and design, the need for an analytical approach for the purposes of benchmarking and comparison has long been compelling. This book has been written to provide an accessible introduction to just such an analytical approach. The approach is based on stochastic geometry as applied to cellular networks, including both the downlink and uplink, and to the dense small cell networks that are becoming increasingly relevant. We intend this book to be an approachable first reference for graduate students or anyone else curious about the analysis of cellular networks. While this book’s results largely draw from the research efforts of the authors over the past decade, the book itself is intended for use in teaching and education. In fact, the roots of this book are from JGA’s graduate class on wireless communications at UT Austin. The other three authors have each attended this class and then subsequently served as Teaching Assistants while pursuing their doctoral degrees at UT Austin under JGA’s guidance. Specifically, JGA conceived the idea of developing a single self-contained reference to this topic for his students in the 2015 offering when AKG was assisting him with his course, which led to a tutorial article (posted on arXiv) [5] that has been subsumed into this book. Interestingly, HSD’s Ph.D. dissertation on Heterogeneous Networks (the topic of Chap. 4) started as a course project in the same course in 2010, two years before he assisted JGA with the course. AA was the course’s Teaching Assistant in 2018. vii

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Preface

With regards to the composition of the book, we have included topics from our own research that form a coherent and digestible introduction for someone wishing to rigorously study and analyze cellular networks, in order to provide insights beyond what system-level simulations can provide (simulations will of course remain indispensable indefinitely). Analysis can illuminate key dependencies in the system and provide guidance on what features and trends to more closely inspect. An obvious analogy is the computation of bit error probability (BEP) in noisy (and possibly fading) channels for different modulation formats: although this too can be simply simulated, it has been indispensable to have formulas that characterize the expected BEP. Generally, these expressions are in the form of a Gaussian Q-function integral, a special function so common that it is often considered closed-form. As will be evident in this book, powerful tools from stochastic geometry can be used to derive functions almost this simple that can describe the outage probability of the entire cellular network under fairly general models, despite the presence of an enormous (even infinite) number of random variables such as the BS locations and all their channel fading gains. Wherever necessary, we have also included results from the other recent articles to provide the most updated exposition (especially in Chap. 3). The key aspect of the approach developed in this book is that all the BSs are located according to a Poisson point process (PPP), which intuitively means that they are randomly scattered in the plane with independent locations. We focus on the calculation of the coverage probability which gives the complementary cumulative distribution function (ccdf) of signal-to-interference-and-noise ratio (SINR) and is the complement of the outage probability. Either one gives the entire SINR distribution. In particular, this book covers the following topics. 1. Chapter 1 provides a concise background on the key tools used in the subsequent chapters, which includes the most essential stochastic geometry Definitions and Theorems. The authors also recommend full length texts focusing on stochastic geometry for wireless networks, such as [6], [7], [8], for a more comprehensive and deeper treatment, including [9] which is focused on cellular networks specifically. 2. Chapter 2 summarizes the downlink model and methodology to computing the coverage probability (SINR ccdf), which are amongst the most tractable and fundamental results in stochastic geometry as applied to wireless communication. In JGA’s course, which includes many other topics and is just meant to provide an introduction to stochastic geometry, it is often not feasible to go much further than the results in this chapter. There have been some recent advances in the downlink analysis of the so-called typical cell, which are also included in this chapter. 3. Chapter 3 focuses on an uplink cellular system model including transmit power control at the mobile user (i.e., handset). This problem is more difficult than the downlink due to the coupling between the handset point process and the BS point process when it is assumed (realistically) that only a single handset can be active per cell (in a given

Preface

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time/frequency resource). Using some recent results, this chapter carefully models this coupling and provides a comprehensive treatment of uplink coverage. 4. Chapter 4 generalizes the downlink results to the case of a heterogeneous cellular network (HetNet), where different classes of base-stations are present in the network. Deploying small cells overlaid on an existing wide area macrocell network is a key direction in ongoing and future cellular network deployments, so this is an important but nontrivial generalization, and one that has gained considerable popularity for the stochastic geometric approach. Overall, the outlook for densification with small cells is quite bright in light of the results of Chap. 4. 5. Chapter 5 asks the question, “is there a limit to how much density a cellular network can tolerate?” We show that the answer to that question is “yes”, and the precise answer of “how much” hinges on the path loss model in particular, as well as several other network parameters. In particular, the path loss model should change for short range communications, and the improved propagation paradoxically decreases the network throughput once a certain density is reached. Overall, this chapter provides a more sobering view of densification than the preceding one and shows the importance of revisiting models and analyses in the light of new information and circumstances. 6. Chapter 6 briefly overviews other key directions of research, although given the continual and rapid evolution of the cellular network, the number of extensions is unlimited. The authors would like to acknowledge that there are many important related works by their colleagues that they have learned a great deal from but are not included in this book. Given that only a handful of people were interested in this topic when JGA started working on this topic, we have been delighted to see the global explosion of interest in these tools, with many new outstanding results being published by other scholars in the last several years.

Austin, USA Kanpur, India Austin, USA Blacksburg, USA January 2023

Jeffrey G. Andrews Abhishek K. Gupta Ahmad Alammouri Harpreet S. Dhillon

Acknowledgements

The authors gratefully acknowledge the indispensable contributions of our colleague F. Baccelli to several of these results, and more generally to inspiring us with his seminal contributions to stochastic geometry for networks and his ability to see the beauty in a new result. We admire his ceaseless pursuit of new knowledge and the joy and boundless energy he brings to his research. We also recognize with gratitude our colleagues R. K. Ganti, T. Novlan, S. Singh, H. S. Jo, P. Xia and X. Zhang for their contributions to the original works described in this book. We also thank P. Mankar, M. Afshang and P. Parida for their technical inputs and careful proofreading of the manuscript. January 2023

Jeffrey G. Andrews Abhishek K. Gupta Ahmad Alammouri Harpreet S. Dhillon

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Contents

1 Key Background on Stochastic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Point Process Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Intensity Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Mean Summation Over a PP and Campbell’s Theorem . . . . . . . . . . 1.1.3 Probability Generating Functional (PGFL) . . . . . . . . . . . . . . . . . . . . . 1.1.4 Marked Point Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Palm Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Campbell-Mecke Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Pair Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Poisson Point Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Campbell’s Theorem for a PPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 PGFL of a PPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Slivnyak’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Campbell-Mecke Theorem for PPP . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Properties of the PPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Poisson Voronoi Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Useful Distance Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 3 4 5 7 8 8 9 9 11 11 12 13 13 14

2 Downlink Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Downlink Model and Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Distance to the Nearest Base Station (Serving Link Distance) . . . . 2.2 Interference Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Coverage Probability (SINR Distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Noise Still Present, α = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Interference-Limited, Any Path Loss Exponent . . . . . . . . . . . . . . . . . 2.4.3 Interference-Limited, α = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Incorporating Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Coverage with a General Path Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 19 19 20 22 22 23 23 23 25 26 xiii

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2.8 Coverage Analysis for the Typical Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Area Spectral Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 32

3 Uplink Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Characterization of the PP of Interfering Users . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Distribution of Distances R and Ri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Interference Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Coverage Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Special Cases in Terms of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Full Channel Inversion ( = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Fixed Transmit Power ( = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Approximation for  = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Validation and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 38 39 40 41 42 42 42 43 44

4 Heterogeneous Cellular Network Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 HetNet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cell Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Analysis for Average Power-Based Cell Association . . . . . . . . . . . . . . . . . . . 4.3.1 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Analysis for Instantaneous Power-Based Cell Selection . . . . . . . . . . . . . . . . 4.4.1 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Interpretations and Impact on Network Throughput . . . . . . . . . . . . . . . . . . . .

47 47 48 50 53 54 58 59

5 Dense Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Standard Power-Law Path Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Physically Feasible Path Loss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 SINR and ASE Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 63 63 65 67 70

6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 General Fading Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Advanced Cell Selection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 General Spatial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Multiple-Input Multiple-Output (MIMO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Spectrum and Resource Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Millimeter-Wave and TeraHertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Modern Communication Paradigms Including Beyond-5G and 6G . . . . . . . 6.8 Parting Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 74 75 76 76 77 78 78

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Jeffrey G. Andrews is the Truchard Family Endowed Chair in Engineering and Director of 6G@UT at the University of Texas at Austin. He received the B.S. in Engineering with High Distinction from Harvey Mudd College, and the M.S. and Ph.D. in Electrical Engineering from Stanford University. He developed CDMA systems at Qualcomm, and has served as a consultant to Samsung, Nokia, Qualcomm, Apple, Verizon, AT&T, Intel, Microsoft, Sprint, and NASA. He is co-author of the books Fundamentals of WiMAX (Prentice-Hall, 2007) and Fundamentals of LTE (Prentice-Hall, 2010). He was the Editor-in-Chief of the IEEE Transactions on Wireless Communications from 2014–2016, Chair of the IEEE Communications Society Emerging Technologies Committee from 2018 to 2019, and the founding Chair of the Steering Committee for the IEEE Journal on Selected Areas in Information Theory, among other IEEE leadership roles. Dr. Andrews is an IEEE Fellow and ISI Highly Cited Researcher and has been co-recipient of 16 paper awards including the 2016 IEEE Communications Society and Information Theory Society Joint Paper Award, the 2014 IEEE Stephen O. Rice Prize, the 2014 and 2018 IEEE Leonard G. Abraham Prize, the 2011 and 2016 IEEE Heinrich Hertz Prize, and the 2010 IEEE ComSoc Best Tutorial Paper Award. He received the 2015 Terman Award, the 2021 IEEE Communications Society Joe LoCicero Award, the 2021 Gordon Lepley Memorial Teaching Award, and the 2019 IEEE Kiyo Tomiyasu technical field award.

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About the Authors

Abhishek K. Gupta received his B.Tech.-M.Tech dual degree in Electrical Engineering from IIT Kanpur in 2010 and Ph.D. degree in the Department of Electrical and Computer Engineering at the University of Texas at Austin in 2016. He is currently an Assistant Professor in the Department of Electrical Engineering at Indian Institute of Technology Kanpur. Dr. Gupta was recipient of Young Faculty Fellowship (2022) by IIT Kanpur, IEI Young Engineer Award (2021–2022) by Institute of Engineers (India), GE-FS Leadership Award by General Electric (GE) Foundation, and Institute of International Education in 2009 and IITK Academic Excellence Award for four consecutive years. He is author of the books MATLAB by Examples (Finch, 2010) and Numerical Methods using MATLAB (Springer Apress, 2014). In the past, he has worked at Samsung Research America (TX), Applied Microelectronics Circuit Corporation (Pune), Futurewei Technologies (NJ), and Nokia Networks (IL). Ahmad Alammouri received his B.Sc. degree (Hons.) from the University of Jordan, Amman, Jordan, in 2014, his M.Sc. degree from King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, in 2016, and his Ph.D. degree from The University of Texas at Austin, Texas, in 2020, all in Electrical Engineering. He is currently a Senior Research Engineer with the Samsung Research America, Plano, TX, USA. He has held summer internships at Samsung Research America, Richardson, TX, in 2017 and 2018, and was a visiting researcher at INRIA, Paris, in 2019 and 2020. He was awarded the Chateaubriand Fellowship from the French Embassy in the USA and the Professional Development Award from UT Austin, both in 2019, and the WNCG Student Leadership Award in 2020.He was recognized as an Exemplary Reviewer by the IEEE Transactions on Communications in 2017 and by IEEE Transactions on Wireless Communications in 2017 and 2018. His research interests include statistical modeling and performance analysis of wireless networks.

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Harpreet S. Dhillon is a Professor of electrical and computer engineering, the chair of the communications area, the Associate Director of Wireless@VT, and the Elizabeth and James E. Turner Jr. ’56 Faculty Fellow at Virginia Tech. He received the B.Tech. degree in electronics and communication engineering from IIT Guwahati in 2008, the M.S. degree in electrical engineering from Virginia Tech in 2010, and the Ph.D. degree in electrical engineering from the University of Texas at Austin in 2013. He has received six best paper awards including the 2014 IEEE Leonard G. Abraham Prize, the 2015 IEEE ComSoc Young Author Best Paper Award, and the 2016 IEEE Heinrich Hertz Award. He has also received Early Achievement Awards from three IEEE ComSoc Technical Committees, namely, the Communication Theory Technical Committee (CTTC) in 2020, the Radio Communications Committee (RCC) in 2020, and the Wireless Communications Technical Committee (WTC) in 2021. He was named the 2017 Outstanding New Assistant Professor, the 2018 Steven O. Lane Junior Faculty Fellow, the 2018 College of Engineering Faculty Fellow, and the recipient of the 2020 Dean’s Award for Excellence in Research by Virginia Tech. His other academic honors include the 2008 Agilent Engineering and Technology Award, the UT Austin MCD Fellowship, the 2013 UT Austin WNCG leadership award, and the inaugural IIT Guwahati Young Alumni Achiever Award 2020. He is an IEEE Fellow, an AAIA Fellow, and a Clarivate Analytics (Web of Science) Highly Cited Researcher.

1

Key Background on Stochastic Geometry

The main theme of this book is to develop from first principles easy-to-use analytical expressions for the signal-to-interference-and-noise ratio (SINR) distribution in cellular networks. A critical aspect of this approach is to endow random distributions on the locations of both the base stations (BSs) and users and then use powerful tools from stochastic geometry to derive the coverage probability, which is the complementary cumulative density function (ccdf) for SINR. Owing to the central role played by stochastic geometry in this analysis, we begin our book with a concise introduction to basic concepts. Thus, this chapter introduces key definitions and results for the Poisson point process (PPP), along with simple illustrative examples. These results will be directly applied to the coverage analyses presented in the subsequent chapters. Notation: Throughout this book, we denote the random variables by upper case letters and their realizations or other deterministic quantities by lower case letters. We use bold font to denote vectors and normal font to denote scalar quantities. Therefore, a random vector (e.g., a random location) will be denoted by an upper case letter in bold font. Similarly, lower case letters in the bold font will denote deterministic vectors. For instance, X and X denote a one-dimensional (scalar) random variable and a random vector (containing more than one element), respectively. Similarly, x and x denote scalar and vector of deterministic values, respectively. The upper case letters in the san-serif font, such as A and L, will be used to denote sets. The set of natural numbers will be denoted by N, and B (x, r ) is a ball of radius r centered at point x.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. G. Andrews et al., An Introduction to Cellular Network Analysis Using Stochastic Geometry, Synthesis Lectures on Learning, Networks, and Algorithms, https://doi.org/10.1007/978-3-031-29743-4_1

1

2

1.1

1 Key Background on Stochastic Geometry

Point Process Essentials

A point process (PP)  = {Xi , i ∈ N} is a random collection of points residing in a measure space, which for cellular networks is the d dimensional Euclidean space Rd . One way to interpret  is in terms of the so-called random set formalism, where  = {Xi } ⊂ Rd is a countable random set with each element Xi being a random variable. An equivalent and usually more convenient interpretation is in terms of the random counting measure, where the idea is to simply count the number of points falling in any set A ⊂ Rd . It is mathematically defined as  (A) = 1(Xi ∈ A), (1.1) Xi ∈

where 1(·) ∈ {0, 1} is the indicator function. Note that (A) is a random variable whose distribution depends upon . Clearly, if (A) is exhaustively considered for all possible sets A, it can completely describe the PP . While this insight is sufficient to understand this text, we note here for completeness that (A) cannot be defined for every subset A of Rd . Such “measurability” questions are treated in the area of measure theory [10]. The idea is to construct a rich set of subsets of Rd for which (A) is defined. One such set (and arguably the most popular in this context) is the Borel σ-algebra of Rd . As a matter of choice, we will avoid such pedantic arguments in the interest of keeping this text accessible to the wireless communications community. Before proceeding further, let us look at a simple example of a PP and random counting measure below. Example 1.1 Consider a PP S in R2 which admits just two possible realizations: (i) S = φ1 with probability 41 , which consists of two points at x1 = (1, 0) and x2 = (0, 1) and (ii) S = φ2 with probability 43 which consists of three points at x1 = (0, 0), x2 = (1, 1) and x3 = (2, 2) (See Fig. 1.1). This PP can be equivalently characterized in terms of the random counting measure (·), which for any set A can take on two possible values: ψ1 (A) = 1((1, 0) ∈ A) + 1((0, 1) ∈ A), ψ2 (A) = 1((0, 0) ∈ A) + 1((1, 1) ∈ A) + 1((2, 2) ∈ A). Clearly, (A) = ψ1 (A) with probability 41 , and (A) = ψ2 (A) with probability 34 . Now consider the set A = B ((1, 1), 1.1) which is a ball of radius r = 1.1 around the location (1, 1), which happens to be the location of a point in ψ2 . Then ψ1 (A) = 2 and ψ2 (A) = 1. If the distribution of the PP is invariant to the shift of the origin, it is said to be a stationary PP. Similarly, if the distribution of the PP is invariant to the rotation of the axes, it is said to be isotropic. A PP that is both stationary and isotropic is termed motion-invariant. Now, we discuss some of the most important statistical measures of a PP.

1.1

Point Process Essentials

3

φ1

φ2 (2, 2) (1, 1)

(0, 1)

(1, 0)

(0, 0)

Fig. 1.1 Two realizations φ1 (left) and φ2 (right) of the PP S in Example 1.1

1.1.1

Intensity Measure

The intensity measure of a PP gives the average number of points in any set A. It is defined as the mean of (·): μ(A) = E [(A)] .

(1.2)

For the PP S in Example 1.1, the intensity measure can be computed as μ(A) =

3 1 ψ1 (A) + ψ2 (A). 4 4

The intensity measure is said to admit a density function λ(·) (also called intensity function of the PP) if, for all sets A,  μ(A) = λ(x)dx. A

If the PP is stationary, it is easy to argue that λ(x) = λ, ∀x, which can be interpreted as the point density or the average number of points of the PP per unit volume.

1.1.2

Mean Summation Over a PP and Campbell’s Theorem

Let f : Rd → R+ be some function that maps points in the PP to a positive real number: for example, f could be a norm or a path loss function. The sum of the function values evaluated over each point of the PP can be expressed as  f (Xi ). (1.3) F= Xi ∈

4

1 Key Background on Stochastic Geometry

Here, F is a function of the PP and hence, a random variable. Example 1.2 Let f (x) = x2 for a given point x ∈ R2 . For the PP S in Example 1.1, E[F] can be computed as ⎤ ⎡  1 3 Xi 2 )⎦ = (1 + 1) + (0 + 2 + 8) = 8. E[F] = E ⎣ 4 4 Xi ∈

Campbell’s theorem can be used to compute the expectation of any F in the form (1.3) for a general PP. It states that ⎤ ⎡   f (Xi )⎦ = f (x)μ(dx). (1.4) E[F] = E ⎣ Rd

Xi ∈

Campbell’s theorem thus provides a key tool: it allows us to convert a sum over the points in a PP into a (hopefully computable) integral over its intensity measure.

1.1.3

Probability Generating Functional (PGFL)

Let f : Rd → [0, 1] be a measurable function such that 1 − f (x) has a bounded support. Then the PGFL of the PP with respect to this function is defined as the mean of the product of the function’s values at each point of the PP, ⎡ ⎤  P ( f ) = E ⎣ f (Xi )⎦ . (1.5) Xi ∈

Thus, whereas Campbell’s theorem provides a tool for converting the expected value of a sum of a function over the points in the PP, the PGFL provides a tool for representing the expected value of a product of a function over the points in the PP. Example 1.3 Let f (x) = x2 for a given point x ∈ R2 . For the PP S in Example 1.1, the PGFL of f can be computed as

P ( f ) =

1 3 · 1 · 1 + · 0 · 2 · 8 = 0.25. 4 4

The PGFL can also be used to compute the Laplace transform L F (s) of a random variable F defined in (1.3) as follows:

1.1

Point Process Essentials

5

⎡ ⎛ ⎞⎤ ⎡ ⎤

  L F (s) = E e−s F =E ⎣exp ⎝−s f (Xi )⎠⎦ = E ⎣ e−s f (Xi ) ⎦ . Xi ∈

(1.6)

Xi ∈

For those unaccustomed to Laplace transforms of random variables, we note that analogously to deterministic functions y(t) that have transform  ∞ L y (s) = Y (s) = y(t)e−st dt, (1.7) 0

the Laplace transform of a non-negative random variable X with probability density function (pdf) f X (x) is essentially the Laplace transform of f X (x) defined as

 ∞ L X (s) = E e−s X = f X (x)e−sx dx. (1.8) 0

The PGFL is particularly important in many wireless applications, where the Laplace transform of the interference is usually an intermediate step in the characterization of the SINR distribution. The following example demonstrates how the PGFL can be used to derive the Laplace transform of the interference in a downlink cellular network. Example 1.4 Assume that cellular base stations (BSs) are randomly deployed in an infinite 2D space R2 and the PP BS = {Xi , i ∈ N} denotes their locations. The sum downlink interference at any point y is a random variable that can be written as the summation of signals transmitted from each BS attenuated according to the standard power-law path loss model, meaning that the received power attenuates with distance r = x − y as r −α , thus I =

 Xi ∈BS

p Xi − yα

where α is the path loss exponent and p is the transmit power of BSs. The form is the same as (1.3), with f (Xi ) = p||Xi − y||−α . Hence the Laplace transform of the interference is given by (1.6) with respect to this function f (Xi ).

1.1.4

Marked Point Process

A marked point process is a PP where a random variable Q Xi (termed a mark) is associated with each point Xi . This mark can be an independent random variable with some distribution or it can be a function of the PP as seen from the point Xi (e.g., a feature of a PP measured at Xi ). For example, let PP  = {Xi , i ∈ N} denote the locations of the BSs. We can now assign an independent random transmit power PXi ∼ exp(1) to each point. The combined PP M = {(Xi , PXi )} is then a marked PP.

6

1 Key Background on Stochastic Geometry

Since marks are associated with points, conditioning on the occurrence of a point at the location x is required to define the distribution of a mark of a point at x. Let us consider a   marked PP M = { Xi , Q Xi }. Let L be an event from sample space of the mark Q x . Given that there is a point Xi at the location x, the probability that its mark lies in the set L, is denoted as ν Q x (L) = P [Q x ∈ L|x ∈ ] = Px [Q x ∈ L].

(1.9)

Notice the new notation Px [E] which denotes the conditional probability of an event E conditioned on the occurrence of a point at x. This distribution is also known as the Palm distribution. In the case of the stationary marked PP, the location of the point x does not matter. Hence, we can take the point at the origin (or in fact anywhere in Rd ) and compute the distribution of its mark conditioned on the occurrence of that point. In this case, the notation Po [Q o ∈ ·] represents the distribution of the mark at the origin conditioned on the fact that there is a point at the origin i.e., ν Q (L) = P [Q o ∈ L|o ∈ ] = Po [Q o ∈ L].

(1.10)

In a stationary setting, one interpretation of ν Q (L) is to go to each point and check if its mark lies in L and then determine the fraction of points that have the desired marks, as given below:      E 1 (Xi ∈ A) 1 Q Xi ∈ L ν Q (L) = Po [Q o ∈ L] =

Xi ∈

E





Xi ∈



.

(1.11)

1 (Xi ∈ A)

Finally, this is also interpreted as the distribution of the mark as seen from the typical point of . In the wireless literature, the concept of typicality is often misinterpreted as an outcome of some selection strategy in which a point is first arbitrarily selected in a given realization and then placed at the origin (often justified by stationarity). This is, however, not true. For starters, selecting one point arbitrarily out of potentially infinite points and calling it typical may seem intuitive but is not completely rigorous. Instead of thinking of the typical point as an outcome of a selection strategy, typicality should be thought of as a useful concept that quantifies the typical (or “average”) properties of the process as observed from a specific location, say x. In the stationary setting, one can consider this location to be the origin without loss of generality. In this case, the performance of the typical point is then studied by holding a point of the PP constant at the origin and then averaging over the PP. The origin then becomes the typical point in the sense that it captures the ensemble average in the same way as one would get a spatial average in a large realization of a PP

1.1

Point Process Essentials

7

by observing the quantity of interest at each point and then averaging over all the points. Of course, in an ergodic setting, these two notions of averages are the same. In a non-stationary setting, the typical performance observed at each location x will be different. Therefore, it will not be of interest to us in this text.

1.1.5

Palm Distribution

As noted above, the Palm distribution for a PP at a given location is its conditional distribution conditioned on the presence of a point at that location. Therefore, the Palm distribution represents how the PP would look when viewed from one of its atoms (points). It is useful in studying the properties of a PP as observed from one of its points, such as the distance of a point of a PP to its nearest point, or the average number of points in a ball of radius r with its center at a point of the PP. For example, let P be a property of the PP, then the Palm probability Po [ ∈ P] denotes the probability that conditioned on the presence of a point of this PP at the origin, the PP has the property P. Since such properties can be interpreted as marks of the points where they are observed, they can be formally studied using the Palm distribution of the corresponding marks. As a concrete example, we consider the nearest-neighbor distance distribution of a stationary PP next. Example 1.5 Consider a stationary PP . For each point Xi ∈ , let us assign the distance of its closest point from itself as its mark RXi i.e., RXi =

min

X j ∈,X j =Xi

X j − Xi .

The PP {(Xi , RXi )} is a marked PP. As discussed in the context of (1.10) and (1.11), the distribution of this mark can be studied by considering the typical point at the origin (since  is stationary). In other words, the cdf of the mark R is given as FR (r ) = Po [Ro ≤ r ], which is also termed the nearest-neighbor distance distribution of . Before proceeding further, we introduce a related notion, termed the reduced Palm measure, which represents how the PP would look when viewed from one of its points while excluding that point from the view. For a stationary PP, it is denoted by P!o [ ∈ P] which represents the probability that conditioned on the presence of a point at the origin, the set of points excluding the point at the origin, which is denoted as  \ {o}, has the property P, i.e., P!o [ ∈ P] = P [ \ {o} ∈ P | o ∈ ] .

8

1.1.6

1 Key Background on Stochastic Geometry

Campbell-Mecke Theorem

We saw that Campbell’s theorem can be used to compute the expectation of any F of the form (1.3). In Chap. 4, we will come across random variables expressed in the following form  F= f (Xi ,  \ {Xi }). (1.12) Xi ∈

Here, f is a function of the location and a PP realization. Since function f (·) is dependent on both Xi and PP  \ {Xi }, we cannot directly apply Campbell’s theorem. To evaluate the expectation of this F, we need to use the closely related Campbell-Mecke theorem, which for a stationary PP is given as ⎤ ⎡   ⎦ ⎣ f (Xi ,  \ {Xi }) = E!o [ f (x, )] μ(dx), E (1.13) Rd

Xi ∈

Note that E!o denotes the expectation with respect to the reduced Palm probability measure i.e., E!o [ f (x, )] = Eo [ f (x,  \ {o})] .

1.1.7

Pair Correlation Function

For a motion-invariant PP  on R2 , the pair correlation function (pcf) is defined as g(r ) =

1 d K (r ), for r > 0, 2πr dr

where

(1.14)

1 !o E [(B (o, r ))]. λ The function K (·) is termed Ripley’s K -function [6, Definition 6.8]. Note that λK (r ) is equal to the number of points of  lying within B (o, r ) (excluding o) conditioned on o ∈ . If  is a homogeneous Poisson Point Process (PPP), defined next in Sect. 1.2, it is easy to check that K (r ) = πr 2 and hence g(r ) = 1 (see Example 1.10). If a PP exhibits clustering, we expect more points to fall in B (o, r ), which implies g(r ) > 1. Likewise, if a PP exhibits repulsion, we expect fewer points to fall in B (o, r ), which implies g(r ) < 1. An important use of pcf in this text is to approximate intractable PPs with more tractable PPs (such as PPPs) with the same intensity functions. This will play a key role in our analysis in Chap. 3. K (r ) =

1.2 The Poisson Point Process

1.2

9

The Poisson Point Process

A point process is a PPP with intensity measure μ(·) if 1. (A) is Poisson distributed with mean μ(A) for every set A. 2. For any m disjoint sets A1 , . . . , Am , the random variables (A1 ), . . . , (Am ) are independent. We will be most interested in a homogeneous PPP, which is a PPP with uniform density λ such that μ(A) = λ(A),

(1.15)

where (A) is the Lebesgue measure (i.e. size) of A. An important property of a homogeneous PPP is that conditioned on the number of points in A, which is Poisson distributed with mean λ(A), all the points are independently and uniformly distributed in A. Example 1.6 Consider a PPP in R2 with density λ having units of points/area. If A is a set denoting a circle of radius r , we would have (A) = πr 2 and μ(A) = λπr 2 . The probability that there are n points in A is given by P [(A) = n] = exp(−λπr 2 )

(λπr 2 )n . n!

(1.16)

We now list some of the important properties and statistical measures for the homogeneous PPP.

1.2.1

Campbell’s Theorem for a PPP

Recall that Campbell’s theorem can be used to compute the expectation of a random variable F of the form (1.3) for a general PP. For a homogeneous PPP, it can be expressed as ⎤ ⎡   f (Xi )⎦ = λ f (x)dx. (1.17) E[F] = E ⎣ Xi ∈

Rd

While Campbell’s theorem is stated in the context of a homogeneous PPP above, readers should note that (1.17) holds for any stationary PP. In the following example, we will see how Campbell’s theorem can be used to compute the mean interference in a cellular system.

10

1 Key Background on Stochastic Geometry

Example 1.7 Goal: find the mean interference at origin from the BSs located in the annular region A = {x : a ≤ x < b}. Assuming a homogeneous PPP  with density λ, power-law path loss and constant transmit power p, the interference is 

IA =

p Xi α

(1.18)

p 1(a ≤ Xi  < b). Xi α

(1.19)

Xi ∈,a≤Xi  2. This is intuitive: the average number of interferers grows quadratically with b per Example 1.6, so the interference each one contributes needs to reduce by more than an inverse square law if the sum interference is to remain finite. From a practical point of view, it means that free space propagation (α = 2) is not quite sufficient in a PPP network that extends infinitely in the plane. This simple example illustrates why virtually all stochastic geometry results require α > 2 and a > 0 in R2 , or a modified path loss model like (r + 1)−α that avoids the singularity as r → 0. Note that the Campbell theorem can be extended to the independently marked homogeneous PPP  = {(Xi , Q Xi )} as ⎡ ⎤   E⎣ f (Xi , Q Xi )⎦ = λ E Q x [ f (x, Q x )] dx. (1.24) (Xi ,Q Xi )∈

Rd

1.2 The Poisson Point Process

1.2.2

11

PGFL of a PPP

The PGFL of a homogeneous PPP is given as ⎡ ⎤     ⎣ ⎦ P ( f ) = E f (Xi ) = exp −λ (1 − f (x))dx . Rd

Xi ∈

(1.25)

As discussed earlier, the Laplace transform of a random variable F (defined in (1.3)) can be evaluated using the PGFL as ⎡ ⎛ ⎞⎤

 sg(Xi )⎠⎦ (1.26) E e−s F = E ⎣exp ⎝− Xi ∈



= P e

 −sg

  = exp −λ

 R2

1−e

−sg(x)



 dx .

(1.27)

Example 1.8 Consider a PPP  with intensity λ and assume power-law path loss with α = 4 and transmit power p. The Laplace transform of the sum interference at the origin can be computed as    ∞

 −4 E e−s I = exp −2πλ 1 − e−spx xdx (1.28) 0   √ (1.29) = exp −πλ πsp . Similarly, the PGFL of a independently marked homogeneous PPP  = {(Xi , Q Xi )} is given as ⎡ ⎤       1 − E Q x [ f (x, Q x )] dx . P ( f ) = E ⎣ f (Xi , Q Xi )⎦ = exp −λ (Xi ,Q Xi )∈

Rd

(1.30)

1.2.3

Slivnyak’s Theorem

Slivnyak’s theorem states that for a PPP , because of the independence between all of the points, conditioning on a point at x does not change the distribution of the rest of the process. This can be thought of as removing an infinitesimally small area corresponding to a ball B (x, ) for → 0, since the distributions of points in all non-overlapping regions are independent for a PPP. This means that any property seen from a point x is the same whether or not we condition on having a point at x in .

12

1 Key Background on Stochastic Geometry

Even though this result may appear simple at the first glance, it is quite profound and is a key enabler of our analysis. For instance, it allows us to add a node to the PPP at any location we like, such as the origin or at a fixed distance from the origin, without changing its statistical properties. In the context of a cellular downlink network, it allows us to treat the interference as coming from a PPP despite removing the serving BS from the PPP. We will discuss this in detail in Chap. 2. Example 1.9 Consider a PPP in R2 with intensity λ. The goal is to compute the probability that a randomly chosen point X0 is farther than r from its nearest-neighbor point. The desired probability can be expressed as   (1.31) Po min(Xi − X0 ) > r , i =0

where we have used the Palm probability since the PP is observed from one of its points. Using Slivnyak’s theorem       (1.32) Po min(Xi − X0 ) > r = P min(Xi ) > r = exp −λπr 2 , i =0

i

which is simply the void probability of the PPP. Example 1.10 Ripley’s K -function from (1.1.7) for a homogeneous PPP in R2 with density λ can be evaluated as K (r ) =

1 !o (a) 1  nP[(B (o, r )) = n] = πr 2 E [(B (o, r ))] = λ λ n

(1.33)

where step (a) follows from Slivnyak’s theorem which implies P!x = P for a PPP, i.e., conditioning on a point at x does not change the distribution of PPP. Therefore, using (1.14), the pcf for the homogeneous PPP is g(r ) =

1 d K (r ) = 1. 2πr dr

(1.34)

In other words, a homogeneous PPP can be considered a baseline or neutral case for the correlation between points as measured by the K -function.

1.2.4

Campbell-Mecke Theorem for PPP

For a homogeneous PPP, the Campbell-Mecke theorem (1.13) can be further simplified. From Slivnyak’s theorem, we can write

1.2 The Poisson Point Process

13

E!o [ f (x, )] = E [ f (x, )] . Using this result in (1.13), the Campbell-Mecke theorem takes the following form for the homogeneous PPP with density λ: ⎤ ⎡   f (Xi ,  \ {Xi })⎦ = λ E [ f (x, )] dx. (1.35) E⎣ Xi ∈

1.2.5

Rd

Properties of the PPP

We now list a few other important properties of the PPP. (i) Independent thinning of a PPP results in a different PPP. For example, if we independently assign random binary {0, 1} marks Q Xi with P[Q Xi = 1] = q to each point Xi in a PPP and collect all the points which are marked as 1, this new PP will also be PPP now with density qλ. (ii) Superposition of independent PPPs results in a PPP. Thus if we combine m independent homogeneous PPPs characterized by densities λi , i = 1, 2, . . . , m to form a new m λi . PP, this new PP will also be a PPP, now with density i=1 (iii) Displacement of a PPP results in a different PPP. This means that if we displace each point of a PPP independently by some random law, for example by adding independent and identically distributed (iid) 2D Gaussian random variables to each point, the PP consisting of these new random points will also be a PPP.

1.2.6

Poisson Voronoi Tessellation

For a given PPP  in R2 , the set VX ⊂ R2 defined as   VX = y ∈ R2 : X − y ≤ Z − y, ∀Z ∈ 

(1.36)

is termed the Poisson Voronoi (PV) cell associated with point X ∈ . In words, the PV cell VX is a set of points in R2 that are closer to the point X than any other point in . The collection of these PV cells {VX }X∈ is called as the PV tessellation (See Fig. 1.2) [7, 8]. Now, if the BS locations in a cellular network follow a PPP, the PV cell of a given point can be interpreted as the service region (or cell) for the BS located at that point under the nearest BS association policy.

14

1 Key Background on Stochastic Geometry

Fig. 1.2 Illustration showing Voronoi tessellation for a PPP deployment

VX o y

X

Z

1.2.7

Useful Distance Distributions

Let R be the distance from any fixed point, say y, in R2 to the nearest point of the PPP , i.e., R = minX∈ X − y. Given the stationarity of PPP , the distribution of distance R can be determined using the void probability (see Example 1.9) as   FR (r ) = 1 − P No point within B (y, r ) = 1 − exp(−πλr 2 ), for 0 ≤ r .

(1.37)

Distance R is also called the contact distance. The contact distance distribution is useful for modeling the serving link distance distribution in cellular networks where the BS locations are modeled using a PPP independent of the user locations. The PV cell containing the origin (or any fixed point) is termed the 0-cell. It is worth noting that the 0-cell is bigger in distribution as compared to the typical cell in the PV tessellation. This general idea appears very frequently in probability and is known by many names, such as the waiting bus paradox and Feller’s paradox. It is an outcome of another well-known concept in probability, termed length-biased sampling. While interested readers are referred to [11, 12] for more details, we will attempt to explain the basic idea through a simple example below. Example 1.11 Consider a unit interval [0, 1] that is partitioned into two sub-intervals of unequal length, say [0, 0.7) and [0.7, 1]. Let’s select one of the sub-intervals using two different random experiments. In the first experiment, we simply select one of them such that either is equally likely to be selected. In this case, the probability of selecting a shorter or longer sub-interval is the same by construction. However, in the second experiment, we first select a point uniformly at random in the original interval [0, 1] and then select the sub-internal in which that point lies. Clearly, the probability of selecting the longer subinterval, in this case, is 0.7 and that of the shorter sub-interval is 0.3. This is because the probability of the randomly selected point lying in the longer sub-interval is higher. This is

1.2 The Poisson Point Process

15

termed length-biased sampling. Because of exactly the same reason, 0-cell is larger than the typical cell since the origin (or any fixed point) is biased to lie in the larger cells. In [13], it was shown that the origin is uniformly distributed within the 0-cell. Therefore, the distance between the nucleus of the 0-cell and a point chosen uniformly at random in that cell is equivalent in distribution to the contact distance. As will be discussed in detail in Sect. 2.8, a similar experiment in which we select a point uniformly at random in the typical cell will be useful for the performance analysis from the typical cell perspective (as opposed to the 0-cell perspective). More concretely, let R˜ = X − y be the distance from X ∈  to a uniformly random point y in the PV cell VX . The exact distribution of R˜ has been recently characterized in [13]. While the exact distribution is unwieldy due to a multiintegral expression, [13] also provides the following closed-form approximation using the contact distance distribution of the PPP with a correction factor FR˜ (r ) = 1 − exp(πc1 λr 2 ), for 0 ≤ r ,

(1.38)

where c1 = 97 is the correction factor which is equal to the ratio of the mean areas of the √ ˜ 0-cell and the typical cell. From √ (1.37) and (1.38), one can observe that E[ R] = 1/(2 c1 λ) is smaller than E[R] = 1/(2 λ), which is consistent with intuition because the typical cell is smaller on average than the 0-cell.

2

Downlink Analysis

We now apply the mathematical tools introduced in the previous chapter to the analysis of a downlink cellular network, following the framework and approach as first given in [14]. Even though we will not trace the rich history of this area, we would be remiss not to mention two particularly impactful prior works. First is [15] that proposed the very first stochastic geometric framework for the analysis of cellular networks and second is [16] that analyzed the SINR-based random coverage model for a stochastic geometry setting. Within this context, [14] specialized [16] to a wireless cellular network setting by including the cellular structure. As we will discuss shortly, this involves carefully handling the connection of the typical user with its closest BS, which results in an exclusion zone (empty ball) in the interfering PP.

2.1

Downlink Model and Metrics

The goal of this chapter is to carefully derive the probability of coverage in a downlink cellular network. The coverage probability is defined as pc (τ , λ, α)  P[SINR > τ ],

(2.1)

which is the ccdf of SINR over the entire network, since the cdf gives P[SINR ≤ τ ]. The parameters τ , λ, and α will be defined next. The coverage probability can be thought of equivalently as: 1. the probability that a randomly chosen user can achieve a target SINR τ , 2. the average fraction of users who at any time achieve SINR τ , or

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. G. Andrews et al., An Introduction to Cellular Network Analysis Using Stochastic Geometry, Synthesis Lectures on Learning, Networks, and Algorithms, https://doi.org/10.1007/978-3-031-29743-4_2

17

18

2 Downlink Analysis

3. the average fraction of the network area that is in “coverage” (relative to some particular SINR τ ) at any time. The key aspects of the downlink cellular system model are as follows. • BS locations. The BSs are randomly located according to a homogeneous PPP  of density λ in R2 . • User locations. Mobile users are independently located according to some other stationary PP u . • BS transmit power. The BSs transmit constantly with fixed power p to a single desired mobile user on any particular time-frequency resource, i.e., we assume orthogonal multiple access within a cell. Therefore, the mobile user sees interference from all other BSs in the network but not from its own desired BS. • Path loss model. As seen in (2.2), all signals attenuate with distance according to the standard power-law path loss propagation model with path loss exponent α > 2. Specifically, we assume that the average received power at distance r is prx (r ) = pr −α . Necessary propagation constants—notably the path loss experienced at the reference distance r = 1—are excluded for simplicity but can easily be incorporated into either the transmit power p or the additive noise power σ 2 . As we demonstrate in Sect. 2.7, the main result of this chapter can also be extended to more general path loss models with some loss of tractability. • SNR. The SNR = σp2 is defined to be the average received signal-to-noise ratio (SNR) at a distance of r = 1, which is our way of handling the aforementioned missing constants. • Fading and shadowing. Random channel effects are incorporated by a multiplicative random value H for the desired signal and G i for interferer i as will be clear from (2.2) and (2.3). For simplicity, we now assume these all correspond to Rayleigh fading with mean 1, so H and {G i } all are i.i.d. and follow an exponential distribution with mean 1. We note that more general fading/shadowing distributions can be accommodated as in [14] and many subsequent papers, but with a loss of tractability and without much change to the results and system design insights. We will specifically discuss shadowing in Sect. 2.6. • User-BS association rule. All analysis is for the typical mobile user at the origin. This user connects to the BS that provides the highest average SNR which is equivalently the one providing the highest average SINR, as well as being the closest BS. Note that a different BS could provide a higher instantaneous SINR depending on the fading values. The instantaneous SINR-based association will be discussed in the context of heterogeneous cellular networks in Chap. 4. For this setup, we now express the SINR for the typical mobile user located at the origin. As noted above, it is assumed to connect to its closest BS, which is denoted by Bo . The distance between the typical user and Bo is denoted by R, which is also called the serving

2.2

Interference Characterization

19

link distance. Since BS i is located at Xi , the serving BS’s location is X0 . Hence, the serving link distance is defined as R = X0 . The distribution of R will be provided shortly in (2.4). Now, the SINR for the typical user at the origin is SINR = where IR =

H p R −α , σ 2 + IR



G i pXi −α

(2.2)

(2.3)

Xi ∈\{X0 }

is the cumulative interference, which is the received power from all the BSs except the serving one.

2.1.1

Distance to the Nearest Base Station (Serving Link Distance)

An important quantity in the SINR expression given in (2.2) is the random distance R separating the typical user from its serving BS. Since each user communicates with the closest BS, no other BS can be closer than R, so all interfering BSs must be farther than R. The pdf of R can be derived using the simple fact that the null probability of a 2D Poisson process 2 in an area a is exp(−λa). Therefore, the cdf becomes FR (r ) = P[R ≤ r ] = 1 − e−λπr , as given in (1.37), and the pdf can be found as f R (r ) =

dFR (r ) 2 = 2πλr e−λπr , r ≥ 0. dr

(2.4)

Therefore, R is simply the contact distance whose distribution was derived in (1.37). For a PPP, this is also the same as the nearest-neighbor distance distribution, which is a special case of the kth nearest-neighbor distance (with k = 1), as derived in [17].

2.2

Interference Characterization

The interference IR is what is known as a standard shot noise [18] created by a PPP of density λ outside a disc centered at the origin o and of radius R. For such a sum over a function of the points in a PPP, we can invoke some of the results we overviewed in Chap. 1. To characterize the interference IR , we first express its Laplace transform conditioned on the random distance R = r to the closest BS from the origin, which we denote as L IR (s). The Laplace transform definition yields

20

2 Downlink Analysis

⎡ ⎛   L IR (s) = E e−s IR = E,{G i } ⎣exp ⎝−s ⎡ = E,{G i } ⎣

⎞⎤



G i pXi −α ⎠⎦

Xi ∈\{X0 }

⎤

exp(−sG i pXi −α )⎦ .

(2.5)

Xi ∈\{X0 }

Now using the independence of the G i ’s, we can move the expectation with respect to G i inside the multiplication to get ⎡ ⎤

L IR (s) = E ⎣ EG [exp(−sGpXi −α )]⎦ . (2.6) Xi ∈\{X0 }

Using the PGFL of PPP with respect to the function f (x) = EG [exp(−sGpx−α )], we get      1 − EG [exp(−sGpx−α )] dx . L IR (s) = exp −λ (2.7) R2 \B(0,r )

Employing a transformation to polar coordinates x = (x, θ), we get    ∞   1 − EG [exp(−sGpx −α )] xdx . L IR (s) = exp −2πλ

(2.8)

r

The integration range excludes a ball centered at 0 and radius r since the closest interferer has to be farther than the desired BS, which is at distance r . Since G i ∼ exp(1), the moment generating function of an exponential random variable can be used to simplify the EG [·] term, to yield     ∞ 1 1− xdx (2.9) L IR (s) = exp −2πλ 1 + spx −α r     ∞ 1 = exp −2πλ xdx . (2.10) −1 x α 1 + (sp) r Armed now with an expression for the Laplace transform of the interference, we proceed to the main result.

2.3

Coverage Probability (SINR Distribution)

We now proceed to derive our first main result. Conditioning on the nearest BS being at a distance r from the typical user, the probability of coverage relative to an SINR threshold τ can be written as

2.3

Coverage Probability (SINR Distribution)

21







pc (τ , λ, α) = E R P[SINR > τ | R = r ] =

r >0

P[SINR > τ | R = r ] f R (r )dr

Using the distribution of f R (r ) from (2.4), we get     H p R −α 2  pc (τ , λ, α) = P 2 > τ  R = r e−πλr 2πλr dr σ + I R r >0  −πλr 2 e P[H > τ p −1 R α (σ 2 + IR ) | R = r ]r dr . = 2πλ r >0

Using the fact that H ∼ exp(1), the inner probability term can be further simplified as   P[H > τ p −1 R α (σ 2 + IR ) | R = r ] = E IR P[H > τ p −1 R α (σ 2 + IR ) | R = r , IR ]   = E IR exp(−τ p −1r α (σ 2 + IR ))   (2.11) = exp − p −1 τr α σ 2 L IR (τ p −1r α ), where L IR (s) is the interference Laplace transform we just computed. This gives a coverage expression  pc (τ , λ, α) = 2πλ

e−πλr e−τ p 2

r >0

−1 r α σ 2

L IR (τ p−1r α )r dr .

(2.12)

From (2.10) we have   L IR (τ p−1r α ) = exp −2πλ

 τ xdx , τ + (x/r )α

∞

r 2

and employing a change of variables u = (x/r )2 τ − α results in the expression   L IR (τ p−1r α ) = exp −πr 2 λρ(τ , α) , 

where ρ(τ , α) = τ 2/α

∞

τ −2/α

1 du. 1 + u α/2

(2.13)

(2.14)

The following theorem provides the final expression for the coverage probability, which is found by plugging (2.13) into (2.12) and simplifying, with a final substitution of v = r 2 . Theorem 2.1 The probability of coverage of the typical mobile user is  ∞   pc (τ , λ, α) = πλ exp −πλv(1 + ρ(τ , α)) − τ SNR−1 v α/2 dv.

(2.15)

0

This fairly simple integral expression already hints at some of the key dependencies on the SINR distribution in terms of the network parameters. It is worth emphasizing that we have

22

2 Downlink Analysis

now averaged over all sources of randomness, including countably infinite BS locations and channel gains. However, the coverage probability can be further simplified in three special cases that we explore next.

2.4

Special Cases

We now consider three special cases where the expression in Theorem 2.1 can be further simplified. These correspond to exploring the high SNR regime SNR → 0—equivalently referred to as the “no noise” or “interference-limited” case—and to the case where the path loss exponent is constrained to be α = 4, which is a fairly typical value for terrestrial propagation at moderate to large distances [19]. There are three such combinations of these simplifications that we consider.

2.4.1

Noise Still Present, α = 4

In this case, the probability of coverage can be written as  ∞   exp −πλv(1 + ρ(τ , 4)) − τ p −1 σ 2 v 2 dv, pc (τ , λ, α) = πλ

(2.16)

0

where ρ(τ , 4) can be computed as ρ(τ , 4) =

√ τ



∞ √

τ

√ √ 1 du = τ arctan τ . 2 1+u

(2.17)

Let us define κ(τ ) = 1 + ρ(τ , 4). Also, note that    2   ∞ a π a 2 Q √ , (2.18) e−ax e−bx dx = exp b 4b 2b 0 ∞ where Q(x) = √1 x exp(−y 2 /2)dy is the standard Gaussian tail probability (ccdf). 2π Using the above result, we get for α = 4 in Theorem 2.1:     3 λπκ(τ ) π2λ (λπκ(τ ))2 Q √ pc (τ , λ, 4) = √ exp , 4τ /SNR τ /SNR 2τ /SNR

(2.19)

where SNR = p/σ 2 . This expression is practically closed form, requiring only the computation of a simple Q(x) value which is similar to the bit error rate (BER) expression for a single link in AWGN.

2.5 Validation

2.4.2

23

Interference-Limited, Any Path Loss Exponent

The coverage probability for the noiseless case can be easily obtained from Theorem 2.1 by considering SNR → ∞ and evaluating the resulting simple integral over an exponential function. The result of that exercise is given by the following simple expression: pc (τ , λ, α) =

2.4.3

1 . 1 + ρ(τ , α)

(2.20)

Interference-Limited, α = 4

When the path loss exponent α = 4, using (2.17), the no noise coverage probability can be further simplified to 1 pc (τ , λ, 4) = (2.21) √ √ . 1 + τ arctan τ This is a remarkably simple expression for coverage probability that depends only on the SIR threshold τ , and as expected it goes to 1 for τ → 0 and to 0 for τ → ∞. For example, if τ = 1 (0 dB, which would allow a maximum rate of 1 bps/Hz), the probability of coverage in this fully loaded network is 0.56. Cellular engineers will notice that this value of coverage probability seems a bit too low for τ = SINR = 1. This is for several reasons, most notably that we have ignored sectoring and other antenna gains which can increase both the received power as well as reduce interference (as in sectoring). Also, we assume that all BSs are transmitting at all times to the mobiles in their own cells, which results in a worst-case interference environment referred to as a fully loaded frequency reuse 1 system. These are caveats that can easily be handled with tweaks to the model.

2.5

Validation

A natural question to ask is whether these mathematical results reasonably describe realworld cellular networks, which do not typically have a Poisson BS distribution. Real cellular networks are obviously deployed in a more strategic manner than just random independent dropping, and for this reason a regular grid—either square or hexagonal—has been used most frequently. However, this is idealized in the other direction and is too perfectly regular. Thus, for simulation-based studies the grid-based BS locations are sometimes perturbed by a random variable, for example, a zero mean 2D Gaussian or a 2D uniform random variable [20], to account for the imperfections relative to the regular grid. A representative illustration can be seen in Fig. 2.1 that shows a 40 km ×40 km section of a real-world LTE network in a large flat urban American city, and a sample of BSs from a PPP of the same density. One can also easily imagine a hexagonal or square grid. Subjectively,

24

2 Downlink Analysis

Fig. 2.1 Poisson BSs (left) compared with Actual LTE BSs over a 40 × 40 km area

it is straightforward to observe that this real-world LTE network lies somewhere between the two extremes of perfect regularity (hexagonal or square grid) and complete randomness (PPP). Thus, we would expect that the SINR coverage probability of a real-world LTE-like cellular network to also be bounded by these two extremes, as pointed out as early as 2000 by [21]. Indeed, the coverage probability does quantitatively lie between these two extremes in general. A representative plot is given in Fig. 2.2, where the coverage probability for the actual BS locations is seen to lie roughly between a square grid and the PPP. The curves have the same shape and it can be observed that the Poisson curve is pessimistic by about 2 dB over nearly the entire SINR range. The gap depends on the actual BS layout used, as well as the path loss exponent. 1 5x5 Square Grid Actual BS Locations Poisson

0.8 Probability of Coverage

Fig. 2.2 A comparison of the interference-limited coverage probability with α = 3 for 3 different cellular layouts: a 5 × 5 square grid (simulated), an actual LTE network layout in a large urban area (simulated), and a Poisson layout, computed by (2.20)

0.6

0.4

0.2

0 −10

−5

0 5 10 SINR Threshold (dB)

15

20

2.6

Incorporating Shadowing

25

This strong similarity between the SINR coverage probabilities for different BS layouts has been explored rigorously in subsequent work. Haenggi and his collaborators have shown that for BSs drawn from any motion-invariant PP, it is possible to rigorously characterize the SINR coverage probability using the corresponding expression for a PPP layout, and adding a horizontal SINR shift in dB [22, 23]. For example, these works show that the shift from the PPP to a hexagonal lattice is 3.4 dB which is the maximum for any layout. This is similar to the shift we can empirically observe in Fig. 2.2 for the square grid, which Haenggi showed had an average shift of exactly 3 dB [22]. Therefore, one can use the PPP for analysis, and then add a calibrated horizontal SINR shift of a few dB depending on the regularity of the desired BS layout. Another interesting and relevant theoretical result shows that even completely regular networks like the hexagonal grid effectively appear to be Poisson when there is sufficient shadowing [24]. That is, instead of modeling the network as hexagonal grid BSs with lognormal shadowing, one could instead simply perturb the BS locations (and hence the path loss from each one) to account for the randomness introduced by shadowing. With sufficiently large shadowing (about 10 dB, a realistic value), the perturbations are large enough to make the resulting PP effectively Poisson. Thus, the randomness in the PPP layout (and its resulting loss in SINR vs. a regular layout) can be interpreted as being due to lognormal shadowing, which provides further justification for using a PPP for the BS locations. The same basic idea can be used to incorporate the effect of shadowing in the downlink analysis, which we discuss next.

2.6

Incorporating Shadowing

We now discuss a simple way of incorporating the effect of shadowing in the coverage analysis. We enrich the system model described in Sect. 2.1 slightly by assuming that each link from the BS Xi additionally suffers from shadowing with gain χi . We assume shadowing gains {χi } to be i.i.d. and also independent of the locations of the BSs {Xi }. Now, the instantaneous received power at the user from the ith BS is given as pri,inst = Hi pXi −α χi .

(2.22)

To be consistent with the discussion so far, we consider the maximum average powerbased cell association rule. Since shadowing is a long-term effect, it will also appear in the association rule while deciding the serving BS for each user. Therefore, the average power received from the BS located at Xi is pri = pXi −α χi .

(2.23)

While the presence of additional random variable χi naturally seems to complicate the analysis, we will show that it is possible to absorb it in the location term Xi to define

26

2 Downlink Analysis

an equivalent PPP  D , which has the same coverage probability, but does not have any shadowing. Therefore, the typical user will connect to the closest BS in the equivalent PPP  D . More precisely, let us define a PP  D as −1/α

 D = {Yi : Yi = Xi χi

, Xi ∈ }.

(2.24)

It can be seen that the association, serving power, and interference for the derived BS PP is the same as the setup considered above (without shadowing) for the coverage derivation. Therefore, the coverage probability of the derived PP is the same as that of the original setting (without shadowing). From the displacement theorem [7], we know that the derived PP is a PPP with intensity measure given as    μ D (A) = λP xχ−1/α ∈ A dx 2 R    = λE 1 xχ−1/α ∈ A dx R2    −1/α  = λE 1 xχ ∈ A dx . (2.25) R2

Now, let A = {z : z ≤ r }, then    1 x ≤ r χ1/α dx R2  2 2/α    = λE πr χ = λπr 2 E χ2/α . 

μ D (A) = λE

(2.26) (2.27)

The necessary and sufficient condition for the last term in the above equation to exist is that   E χ2/α < ∞ [25]. Now, the density of the derived PPP can be obtained from the intensity measure as λ D (r ) =

  1 d μ(A) = λE χ2/α . 2πr dr

(2.28)

Therefore, the coverage probability of the derived PPP (equivalently the coverage in the above setup with shadowing) can be obtained from (2.15) with λ replaced by λ D . Curious readers are directed to the following representative set of works that use this general idea: [24–30].

2.7

Coverage with a General Path Loss Model

The main purpose of this section is to show that even though our main coverage probability result presented in Theorem 2.1 was derived under a standard power-law path loss model, it can be easily extended to a general path loss model, where the signal power at distance r is given by a generic function l(r ). In this more general case, the received SINR at the typical

2.8

Coverage Analysis for the Typical Cell

27

user located at the origin when it connects to its closest BS (located at distance R = X0  from the origin) is SINR =



p H0 l(X0 ) , p Hi l(Xi ) + σ 2

(2.29)

Xi ∈\{X0 }

where { \ {X0 }} is the set of interfering BSs. Using the machinery that we developed so far in this chapter, we can derive an expression for the coverage probability under path loss model l(r ) as follows: ⎡ ⎛ ⎞⎤ ∞ 2/ p rl(r ) τ σ − 2πλτ dr ⎠⎦ , (2.30) pc (τ , λ) = E ⎣exp ⎝− l(R) l(R) + τl(r ) R

where the expectation is over the serving link distance R whose pdf is given by (2.4). We can get this form by following the same steps used to derive (2.12), with the exception that the path loss is given by l(r ) instead of r −α . As we already saw above for the power-law path loss model, this general expression can be simplified further for specific forms of l(r ). For simplicity, we will keep focusing on the power-law model in the subsequent chapters, knowing that all the expressions can be derived for the general path loss model l(r ) at the expense of computational complexity. Then in Chap. 5, we will investigate the effect of changing the path loss model on the performance of downlink cellular networks in detail.

2.8

Coverage Analysis for the Typical Cell

From the discussion in Sect. 2.3, it is evident that placing users independently of the BS PPP  (as an independent PPP or another stationary PP) provides mathematical tractability to the downlink coverage analysis. This is a result of the stationarity and independence of the user PP because of which the concept of coverage of the typical user and coverage of an arbitrary fixed location are identical. Consequently, it is not needed to explicitly consider Palm conditioning on the user PP and the analysis can just focus on the origin as the location of the typical user. The subsequent analysis is facilitated by two observations: (i) the link distance (distance between the typical user and its serving BS) is equivalent to the contact distance, and (ii) the interference field forms a homogeneous PPP outside the ball centered at the typical user and radius equal to the serving link distance. This has been the popular approach in the literature and will henceforth be referred to as the typical user approach in this chapter. As discussed in Sect. 1.2.7 (and Example 1.11), the typical user selected in the above approach falls in the 0-cell, which is bigger on average than the typical cell. This implies that the typical user (as modeled in Sect. 2.1) is served by the BS located in the 0-cell because of which its performance differs from the service provided by the typical BS. However, it is

28

2 Downlink Analysis

Fig. 2.3 Illustration of the typical user (left) and the typical cell (right) viewpoints. The red dots and blue circles denote the locations of the BS and the typical user, respectively

often the typical BS performance that is important from the network designer’s perspective. Taking a cue from the above discussion, one way of characterizing the performance of the typical cell is to consider a user distribution model in which the typical user lies in the typical cell. One way of doing that is to distribute an equal number of users uniformly at random independently from each other in every PV cell. Since practical cellular networks are dimensioned such that each cell serves roughly the same load, this user distribution model is at least as meaningful as the one used earlier in this chapter. Now if we focus on the users scheduled in a given time-frequency resource, they form the so-called Type I user PP, introduced in [31], wherein a single user is placed at a uniformly random location in each PV cell. Now the selection of the typical user in this setting is equivalent to the selection of the typical cell because each cell contains exactly one user. The focus of this section is on the downlink analysis of the typical user of this setup, which will be termed the typical cell approach. Figure 2.3 depicts the difference between the typical user and typical cell viewpoints. We will now proceed with the technical analysis. Interested readers are advised to refer to [32] for more details, where this typical cell approach appeared for the first time. In this analysis, our focus will be on the typical BS located at the origin. Since by Slivnyak’s theorem, conditioning on a point is the same as adding a point to a PPP, we will focus on the typical cell of the point process  ∪ {o} located at o. For this setup, the Type I user PP is defined in [31] as Iu = {U (Vx ) : x ∈  ∪ o}.

(2.31)

where U (A) represents a uniformly random location selected from a set A ⊂ R2 and Vx is the PV cell with nucleus x ∈  ∪ o. Note that the location of the typical user of Iu can

2.8

Coverage Analysis for the Typical Cell

29

be modeled using a uniformly distributed point Y within Vo , i.e., Y ∼ U (Vo ). Thus,  becomes the PP of interfering BSs to the typical user at Y ∈ Vo . Figure 2.3 (right) shows that the typical user (indicated by blue circle) placed in the typical PV cell by conditioning the location of the corresponding BS at o. The serving link distance R for the typical cell setup (depicted in Fig. 2.3 (right)) is statistically different from that of the typical user setup (depicted in Fig. 2.3 (left)). The exact distribution of R is derived in [13]. Unfortunately, the final multi-integral expression is unwieldy, which makes it less useful for this downlink analysis. That said, [13] also provides a tight closed-form approximation for the distribution of R, which was presented in (1.38) and the corresponding pdf is f R (r ) = 2πλc1r exp(−πc1 λr 2 ),

(2.32)

where c1 = 97 (refer to Sect. 1.2.7 for more details). Having characterized the serving distance distribution, we now need the knowledge of interfering BS PP as seen by the typical user at Y ∼ U (Vo ). From the complexity of the serving link distance distribution [13], it is reasonable to presume that the exact characterization of interfering BS PP w.r.t to the user located at Y ∼ U (Vo ) is very challenging. That said, using the relation between (1.37) and (1.38), one can deduce that the interfering BSs are closer to the user at Y ∼ U (Vo ) as compared to the homogeneous PPP case. To efficiently capture this, [32] explicitly considers the interference from the dominant BS, located at Xd ∈  (closest interfering BS to Y), and then approximates the PP of the remaining interfering BSs as the homogeneous PPP beyond Rd = Xd − Y. This is a reasonable approximation because most of the interference power is contributed by the dominant interfering BS. In addition, a pcf-based approach is developed in [33] for a more accurate approximation of this interfering BS PP. However, we will stick to the approach of [32] for the simplicity of exposition. Under this modified system model, the downlink SINR at Y ∈ Vo is given by (2.2) such that  G i pXi − Y−α , (2.33) IR = Xi ∈

is the net received interference power received by the user placed at Y ∈ Vo . Now, we derive the downlink coverage probability for the typical cell. For the simplicity of exposition, we focus on the interference-limited scenario. The interference power received by the typical user can be expressed as IR = G d p Rd−α + I˜R ,  ˜ =  \ {Xd }. First, we obtain the Laplace transwhere I˜R = Xi ∈˜ G i pXi − Y−α and  form of I˜R given Rd as below

30

2 Downlink Analysis

  L I˜R (s|Rd ) = E exp(−s I˜R ) | Rd ⎡ ⎤

   EG exp −GpXi − Y−α | Rd ⎦ = E˜ ⎣ ⎡

˜ Xi ∈

⎤ 1 = E˜ ⎣ | Rd ⎦ 1 + spXi − Y−α ˜ Xi ∈    1 (a) = exp −λ dx −1 α R2 \B(Y,Rd ) 1 + (sp) x − Y    ∞ 1 (b) = exp −2πλ r dr −1 α Rd 1 + (sp) r    ∞ 2 1 , = exp −πλ(sp) α α du 2 (sp)− α Rd2 1 + u 2

(2.34)

˜ beyond Rd from Y and then using where step (a) follows using the PPP approximation of  PGFL of the PPP. Step (b) follows through the substitution of x − Y = z and then from the Cartesian to polar coordinates conversion. Averaging over the distribution of the distances R and Rd , the coverage probability can be written as   pc (τ , λ, α) = E R,Rd P[SINR > τ | R = u, Rd = v]  ∞ ∞ P[SINR > τ | u, v] f Rd |R (v|u) f R (u)dvdu = 0 ∞ u ∞   P H > τ u α p −1 (G d pv −α + I˜R ) f Rd |R (v|u) f R (u)dvdu = 0 ∞ u ∞     = LG d τ (u/v)α L I˜R τ u α p−1 | v f Rd |R (v|u) f R (u)dvdu. (2.35) 0

u

The f R (r ) is given in (2.32) and the pdf of Rd conditioned on R is well approximated in [32] as   (2.36) f Rd |R (r | R) = 2πc1 λr exp −πc1 λ(r 2 − R 2 ) , for R ≤ r . From (2.34), we have   2 L I˜R (τ u α p−1 | v) = exp −πλu 2 τ α

∞ 2

τ − α (v/u)2

Finally, substituting (2.32), (2.36), (2.37) and LG (τ (u/v)α ) =

1 1+t

α 2

 dt .

1 1+τ (u/v)α

(2.37)

in (2.35), we get

2.8

Coverage Analysis for the Typical Cell

31

   ∞ 1 1 2 α2 exp −πλu τ α dt 2 1 + τ (u/v)α u τ − α (v/u)2 1 + t 2   2πc1 λv exp −πc1 λ(v 2 − u 2 ) 2πλc1 u exp(−πc1 λu 2 )dvdu. (2.38)

∞ ∞

 pc (τ , λ, α) = 0

On further simplifying the above, we obtain the final expression for the coverage probability, which is given in the following theorem. Theorem 2.2 The probability of coverage of the user located uniformly at random in the typical cell is pc (τ , λ, α) = where β(t) = t

∞

1 t −1 1+u α2

2 c12 τ − α



2

τα 0

(c1 + β(t))−2 α

1+t2

dt,

(2.39)

du.

Figure 2.4 verifies the approximate coverage probability derived in Theorem 2.2 by comparing it with the true coverage obtained from simulations. The figure also shows that the coverage probabilities from the viewpoints of the typical user and typical cell are fairly similar despite significant differences in the means of the signal and interference powers received under these two viewpoints (which differ by almost 3 dB as discussed in [32]). This is because both the signal and interference powers scale by similar factors under these two approaches because of which their absolute differences do not appear prominently in their corresponding ratios.

Fig. 2.4 Downlink coverage probability in an interference-limited cellular network for α = 3 and α = 4

32

2.9

2 Downlink Analysis

Area Spectral Efficiency

We have focused so far on characterizing the ccdf of SINR (or coverage probability) in this chapter. This result is not only important because it provides useful insights into the network performance but also because it facilitates the analysis of other key performance metrics. One of these performance metrics is the average area spectral efficiency (ASE), which represents the average network throughput per unit area and has been widely used to study different trade-offs in cellular networks that cannot be studied by only looking at the coverage probability alone. The definition in [3], where ASE was proposed as a performance metric, is based on the sum of the spectral efficiency (Shannon limit) of the users per unit area  1   E log2 (1 + SINRk ) , |A| N

E [E (N )] =

(2.40)

k=1

where A ⊂ R2 is the considered region, |A| is the area of the region A, and N is the number of users within A, and log2 (1 + SINRk ) is the spectral efficiency in bps/Hz. Then by averaging over different fading realizations, we get the expression in (2.40). In our model presented in Sect. 2.1, there is randomness in the BS/user locations and in the small-scale fading. However, since the SINR distribution seen by the typical user at the origin is the same as the SINR distribution seen by any arbitrary user (owing to the stationarity of this setup), the definition (2.40) simplifies to the following after averaging over the network and fading realizations. Definition 2.1 (Unconstrained ASE) The average unconstrained ASE is defined as:   E [E (λ)] = λE log2 (1 + SINR) .

(2.41)

Note that λ appears in (2.41) because the density of active users in a given time-frequency resource is the same as the density of BSs. Also note that the average ASE in the previous form assumes that the system can work with any arbitrarily small SINR, which is possible in theory, but may not be feasible in practice, hence the term unconstrained. Although it is used in the literature [34–38], a second more conservative definition, called the average potential throughput and denoted by R(·, ·), has been used in [37–42]. It is defined next. Definition 2.2 (Potential throughput) The average potential throughput is defined as E [R(λ, τ0 )] = λ log2 (1 + τ0 )P [SINR ≥ τ0 ] .

(2.42)

2.9

Area Spectral Efficiency

33

The potential throughput captures the case where accurate channel state information is not available at the transmitter. Hence, it transmits with a constant rate and the messages are only decodable at the receiver if the SINR is larger than some threshold τ0 . This means that high SINR values are not exploited, and if the SINR is small, the link is in a complete outage. The two different ASE definitions are related as follows:   E [R(λ, τ0 )] = E λ log2 (1 + τ0 )1 (SINR ≥ τ0 )   ≤ E λ log2 (1 + SINR)1 (SINR ≥ τ0 )   ≤ E λ log2 (1 + SINR) = E [E (λ)] . Hence E [R(λ, τ0 )] ≤ E [E (λ)] for all τ0 ≥ 0. Now, let us express these two versions of ASE in terms of coverage probability. First the unconstrained ASE can be written as  ∞ λ P [ln(1 + SINR) > y] dy, (2.43) E [E (λ)] = ln(2) 0  ∞ λ pc (t, λ) = dt, (2.44) ln(2) 0 t +1 ∞ where (2.43) holds since E[X ] = 0 P [X > y] dy for any positive random variable X and (2.44) follows by substituting e y − 1 with t. Similarly, the potential throughput can be expressed in terms of the coverage probability as E [R(λ, τ0 )] = λ log2 (1 + τ0 )pc (τ0 , λ).

(2.45)

Now we can easily plug the coverage probability expressions into (2.44) and (2.45) to find the potential throughput and ASE for the network setup that we desire. For example, we can plug the expression we derived in (2.19) for the power-law path loss model to find the potential throughput and ASE. However, the form in (2.19) does not exactly reveal how the ASE or potential throughput scale with λ. For the special case of interference-limited network we derived in (2.20), we can clearly see that the coverage probability is independent of λ, which is commonly referred to as the SIR-invariance property. Looking back at the SINR expression in (2.2), we can see that increasing λ has two effects: (1) it reduces the serving link distance between the typical UE and its serving BS which increases the received power of the desired signal, and (2) it increases the interference term because of network densification. The SIR-invariance property shows that the rate at which the desired received power increases with λ is exactly the same rate at which the interference increases under the assumption of the power-law path loss model. Another implication of the SIR-invariance property is that the potential throughput and ASE only depend on λ through the pre-log factor of λ, and hence, we can conclude that the potential throughput and ASE scale linearly with λ. This suggests that cellular operators can keep adding more BSs to the network and harvest linear gains in the ASE with no limits.

34

2 Downlink Analysis

In Chap. 5, we revisit this result and study scaling laws under different assumptions regarding the path loss model. More precisely, we will show that the aforementioned scaling law is a result of the power-law path loss model, and more practical path loss models will result in different scaling laws. Hence, as is the case in any work involving mathematical models, one should always remain cognizant of the limitations of specific models and assumptions.

3

Uplink Analysis

In this chapter, we build on the previous chapter to present an uplink analysis of cellular networks. There are fundamental differences in the downlink and uplink analyses because of which the downlink analysis from the previous chapter cannot be directly applied to the uplink case. First and foremost, in the uplink case, the receiving entity is now a BS. Because of irregularities in the cell shapes of the BSs, an interfering user (now a mobile user) can be quite close to the BS of interest, even closer than the user that this BS is serving. For example, the desired user could be in the corner of the cell of the typical BS, while the interfering user is near the cell edge. This is illustrated in Fig. 3.1 where the interfering user Y1 is closer to the typical BS than the user Y that this BS is serving. This is in stark contrast to the downlink analysis of Chap. 2 in which the serving link was always shorter than all of the interfering links. From an analysis standpoint, the uplink case is significantly more challenging because of the need to characterize a more complicated PP of transmitters, which are mobile users. This is because, in the uplink setting for a data-centric system, we should assume that each BS schedules at most one user in its cell for uplink transmission in a given time-frequency resource.1 This restriction results in a dependent thinning of the original user PP that is dependent on the BS PPP. Consequently, even if we start with a homogeneous PPP of user locations, the locations of the users scheduled at a given time will not form a PPP. Characterizing the resulting PP and its spatial coupling with the BS PPP is very challenging because of the correlation between the PV cells of the underlying PV tessellation (of the BS PPP). In order to understand this correlation, again refer to Fig. 3.1 and note that the cell boundary of each PV cell is basically a perpendicular bisector of the line joining two proximate 1 For a CDMA-type system as used in 3G and earlier, many or even all users may be allowed to

simultaneously transmit on a given time-frequency resource. This simplifies the uplink analysis, see, e.g., [43], but is not terribly relevant for modern data systems which employ orthogonal or nearly orthogonal resource allocation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. G. Andrews et al., An Introduction to Cellular Network Analysis Using Stochastic Geometry, Synthesis Lectures on Learning, Networks, and Algorithms, https://doi.org/10.1007/978-3-031-29743-4_3

35

36

3 Uplink Analysis

Fig. 3.1 An illustration of the typical cell in a cellular network. Y represents the user served by the typical BS (desired user) whereas Y1 is one of the uplink interfering users. This interfering user is closer to the typical BS than its own desired user

O O O

O O

Y O

O Typical BS O

O

Y1

O

BSs. Therefore, the shape of a given PV cell is not only defined by the BS at its nucleus but also by several neighboring BSs. As a result, the location of one BS ends up contributing to the shapes of several adjoining PV cells, which introduces the aforementioned correlation. Now, if we thin the user PP based on the underlying PV tessellation, the locations of the retained points will naturally be correlated because of the underlying correlation in the PV cells based on which they were retained. One way to rigorously proceed with an uplink analysis is to capture the aforementioned coupling by considering the Type I user PP introduced in [31] to model users scheduled on a given time-frequency resource and then focusing on the uplink analysis in the typical cell. In this sense, the analysis of this chapter can be thought of as the uplink counterpart of the typical cell downlink analysis presented in Sect. 2.8, where the Type I user PP has already appeared. Our exposition in this chapter will closely follow that of [44], which is the first to provide a comprehensive uplink analysis for the typical cell setup. There have also been numerous works focusing on the uplink analysis of the typical user located in the 0-cell using various approximations (the so-called typical user viewpoint discussed in the previous chapter). We will briefly discuss this line of work at the end of the chapter.

3.1

Model and Preliminaries

Many of the downlink modeling assumptions from Sect. 2.1 are adopted here, including the BS locations following a homogeneous PPP with density λ; the use of power-law path loss, Rayleigh fading, and Gaussian noise with power σ 2 ; and the maximum average received power-based association rule (a mobile user would attach to the BS to which it has the

3.1

Model and Preliminaries

37

smallest average path loss). The other modeling assumptions for the uplink case are outlined below. • Each BS has a single active uplink user scheduled on a given time-frequency resource which is randomly chosen from all the users located in its PV cell. Let the locations of the active users in the reference time-frequency resource be given by a PP Iu . We assume, quite reasonably, that Iu is a Type I user PP Iu , which places exactly one user uniformly at random in each PV cell of the BS PV tessellation (see (2.31)). Roughly, Iu can be seen as the outcome of the dependent thinning (dependent on ), of the user PPP u that was initially considered in Chap. 2 (under the assumption that the density of u is much larger than λ so that almost no PV cells are empty). • As discussed earlier, we perform the uplink analysis from the viewpoint of the typical cell. By Slivnyak’s theorem, conditioning on a point is the same as adding a point to the PPP. Therefore, similar to Sect. 2.8, we condition the typical BS at the origin o of PPP  ∪ {o}. For this setting, the user PP becomes Iu = {U (Vx ) : x ∈  ∪ {o}}. The typical user is placed at Y ∼ U (Vo ) (whose SINR is computed at the BS placed at o), and the PP of interfering users with respect to the BS at o becomes ˜ Iu = {U (Vx ) : x ∈ }.  In both the above expressions, Vx is the PV cell with nucleus x ∈  ∪ o with the Voronoi tessellation corresponding to the PP  ∪ o. Figure 3.2 illustrates this system setup. • As shown in Fig. 3.2, we denote the locations of the typical user and interfering users by Y and Yi , respectively. Since the SINR is measured at the typical BS placed at o, the serving and interfering link distances are respectively defined as R = Y and Di = Yi . • The mobiles utilize a general form of power control, which is distance-proportional fractional power control of the form Riα , where  ∈ [0, 1] is known as the power control factor or exponent. Thus, as a user moves closer to the desired BS, the transmit power required to maintain the same received signal power decreases. This is similar to the uplink operation in LTE and 5G and includes fixed transmit power ( = 0) and complete channel inversion ( = 1) as special cases.

38

3 Uplink Analysis

Fig. 3.2 Uplink system model focusing on the performance of the typical cell located at the origin o. The pink circle, blue circles, and red dots denote the locations of the typical user, the interfering users, and the BSs, respectively

3.2

Characterization of the PP of Interfering Users

Because of the correlation introduced by the underlying PV tessellation, the exact charac˜ Iu is challenging. In [31], the pcf of  ˜ Iu w.r.t. the typical terization of the distribution of  BS at o is characterized as g(r ) = 1 − exp(−πc2 λd 2 ), for d > 0,

(3.1)

˜ Iu and the location where c2 = 12/5. This pcf does not incorporate dependence between  of the typical user Y. However, it is reasonable to expect that the location of the typical user Y will not have a large impact on this pcf because the interfering link distance Di can be arbitrarily small for any given serving link distance R = Y. Besides, the above pcf also ˜ Iu w.r.t the origin. Interested readers are advised to ignores inherent clustering present in  refer to [31] for more details on this. Despite these apparent shortcomings, the above pcf enables remarkably accurate analysis of the uplink coverage probability, as described in detail in the rest of this chapter. ˜ Iu w.r.t. Using the above pcf, [31] has provided an approximate intensity function of  the typical BS placed at o, which is given as λIu (d) = λ(1 − exp(−πc2 λd 2 )),

(3.2)

˜ Iu , a reasonable way forward is to model  ˜ Iu In the absence of further information about  as a non-homogeneous PPP with the above intensity (owing to the unparalleled tractability of the PPP). Under this system model, the uplink SINR at the typical BS located at o is

3.3

Distribution of Distances R and Ri

SINR =

39

H p R α(−1) , σ2 + I

(3.3)

where interference I is given by I =



p Riα G i Yi −α .

(3.4)

˜ Iu Yi ∈

If  = 1, the numerator of (3.3) becomes H p, with the path loss completely inverted by power control, and if  = 0 no channel inversion is performed and all the mobiles transmit with the same power p. Recall also from the above discussion that the PP of interferers ˜ Iu and hence the interference power distribution turns out to be independent of the desired  signal power from the typical user (given by the numerator term H p R α(−1) of (3.3)). This is an outcome of the way the interference field is modeled and will be useful in the coverage probability analysis presented later in this chapter.

3.3

Distribution of Distances R and Ri

Recall that Ri = Yi − Xi  is the link distance between the interfering user Yi and its serving BS Xi , which is determined by the smallest average path loss rule. Since the analysis is focused on the typical cell, it is easy to argue that the random variables {Ri } and R are identically distributed. However, because of the correlation in the structures of the PV cells, they are not independent. As discussed in Sects. 1.2.7 and 2.8, the exact distribution of R has been characterized in [13]. In addition, [13] also provides an approximate closedform expression for the pdf of R, which was provided in (2.32) and is given below for completeness: (3.5) f R (r ) = 2πλc1r exp(−πc1 λr 2 ), where c1 = 97 . Since Ri is identically distributed to R, its pdf is also given by the above expression. As a part of the interference distribution analysis in the next subsection, we will need to capture the dependence between the link distance Ri = Yi − Xi  and the distance between the interfering user Yi and the typical BS, given by Di = Yi . While deriving the exact joint distribution of these random variables is challenging, one can capture the dependence between them to some extent by noticing that Ri is always upper bounded by Di . Note that Ri cannot be bigger than Di because the distance of an interfering user to its serving BS has to be smaller than its distance to the typical BS (otherwise the typical BS at o will become its serving BS). Accounting for this fact, the distribution of Ri conditioned on Di can be expressed as f Ri (r |Di ) =

2πc1 λr exp(−πc1 λr 2 ) , 0 ≤ r ≤ Di , 1 − exp(−πc1 λDi2 )

(3.6)

40

3 Uplink Analysis

which is a truncated version of the Rayleigh distribution. As noted earlier, the above distribution is an approximation. Using these distance distributions, we characterize interference next.

3.4

Interference Characterization

The net interference received at the typical BS placed at o is the sum of powers from all ˜ Iu ). Under the power the transmitting mobiles (modeled by the non-homogeneous PPP  control model described in the previous subsection, this power depends upon the distance of a mobile to its serving BS and the power control factor  ∈ [0, 1]. For this setup, we now compute the Laplace transform of interference distribution observed at the tagged BS. Recalling that the interference is denoted by I , we get ⎡ ⎛ ⎞⎤    L I (s) = E I e−s I = E I ⎣exp ⎝− sp Riα G i Yi −α ⎠⎦ (3.7) ˜ Iu Yi ∈

⎡

= E{Ri ,G i ,Yi } ⎣



exp −sp Riα G i Yi 

 −α

⎤ ⎦.

(3.8)

˜ Iu Yi ∈

˜ Iu , we get Using the independence of G i ’s and Ri ’s across  ⎡ ⎤

   L I (s) = E˜ I ⎣ E Ri ,G i exp −sp Riα G i Yi −α ⎦ , u

(3.9)

˜ Iu Yi ∈

where the inner expectation is conditional on Yi . Since G i ∼ exp(1), ⎡ ⎤  

1 ⎦. L I (s) = E˜ I ⎣ E Ri u 1 + sp Riα Yi −α

(3.10)

˜ Iu Yi ∈

˜ Iu is a non-homogeneous Now using the PGFL of a PPP and the fact that PP of interferers  I PPP with distance-dependent density function λu (d) = λ(1 − exp(−πc2 λd 2 )) relative to the typical BS, we get     1  L I (s) = exp −2πλ (1 − exp(−πc2 λx )) 1 − E Ri = x xdx D  i 1 + sp Riα Di−α  0       ∞  1  2 = exp −2πλ (1 − exp(−πc2 λx )) E Ri  Di = x xdx , (3.11) 1 + (sp)−1 Ri−α Diα  0 



∞



2



3.5

Coverage Probability

41

where Di = Yi . Now using the conditional pdf of Ri (given Di ) from (3.6), the Laplace transform of interference can be expressed as ⎛ ⎞ ∞ x 2 1 − exp(−πc2 λx 2 ) πc1 λe−c1 λπu ⎜ ⎟ L I (s) = exp ⎝−2πλ duxdx ⎠ . (3.12) 2 1 − exp(−πc1 λx ) 1 + (sp)−1 u −α/2 x α 0

0

Recall that in this approach the interference distribution I is not a function of the link distance R from the typical user at Y ∈ Vo to its serving BS at o. Therefore, we have all the ingredients to derive the uplink coverage probability, which we do next.

3.5

Coverage Probability

We will now compute the uplink coverage probability which is defined as the ccdf of uplink SINR, i.e., the probability that the uplink SINR at the typical BS is greater than the target SINR τ . Starting from the definition of uplink coverage probability and SINR, we get  ∞ P (SINR > τ ) f R (r )dr (3.13) pc (τ , λ, α, ) = 0      ∞ H p r α(−1) 2 = P (3.14) > τ 2πc1 λr e−πc1 λr dr 2+I σ 0   ∞  τ (σ 2 + I ) 2 2πc1 λr e−πc1 λr dr = P H> (3.15) α(−1) pr 0 Since H ∼ exp(1), we get  ∞   2 −1 α(1−) σ 2 −1 α(1−) I pc (τ , λ, α, ) = dr 2πc1 λr e−πc1 λr e−τ p r E I e−τ p r 0 ∞   2 −1 α(1−) σ 2 = 2πc1 λr e−πc1 λr e−τ p r L I τ p−1r α(1−) dr .

(3.16) (3.17)

0

Now the Laplace transform of interference at s = τ p −1 r α(1−) is given as L I (τ p −1r α(1−) ) =    ∞  x2 1 − exp(−πc2 λx 2 ) πc1 λe−πc1 λu duxdx . (3.18) exp −2πλ 1 − exp(−πc1 λx 2 ) 1 + τ −1r −α(1−) u −α/2 x α 0 0 Using the expression of Laplace transform of interference, we now provide the final coverage expression in Theorem 3.1.

42

3 Uplink Analysis

Theorem 3.1 The uplink coverage probability is given by  ∞ 2 −1 α(1−) σ 2 pc (τ , λ, α, ) = 2πc1 λ r e−πc1 λr −τ p r ν(r , λ, α, )dr ,

(3.19)

0

where ν(r , λ, α, ) is given by 



ν(r , λ, α, ) = exp −2πλ 0

3.6

∞  x2 0

 1 − exp(−πc2 λx 2 ) πc1 λe−πc1 λu duxdx . (3.20) 1 − exp(−πc1 λx 2 ) 1 + τ −1 r −α(1−) u −α/2 x α

Special Cases in Terms of 

The coverage probability expression derived above can be simplified for the cases of full channel inversion power control ( = 1) and fixed transmit power ( = 0) in the interferencelimited scenario as follows.

3.6.1

Full Channel Inversion ( = 1)

The uplink coverage probability for the full channel inversion case assuming no noise (σ 2 = 0) is given by  ∞   2 (3.21) 2πc1 λr e−πc1 λr L I τ p −1 dr pc (τ , λ, α,  = 1) = 0

where L I (s) is given by (3.12) with  = 1. Note that the argument of L I (s) is independent of r and can hence be moved out of the integral to get   −1  ∞ 2 2πc1 λr e−πc1 λr dr pc (τ , λ, α,  = 1) = L I τ p 0   = L I τ p −1 (3.22) ⎛ ⎞ 2 ∞ x 2 −πc λu 1 − exp(−πc2 λx ) πc1 λe 1 ⎜ ⎟ = exp ⎝−2πλ duxdx ⎠ . 1 − exp(−πc1 λx 2 ) 1 + τ −1 u −α/2 x α 0

0

(3.23)

3.6.2

Fixed Transmit Power ( = 0)

The uplink coverage probability for fixed transmit power assuming no noise (σ 2 = 0) is given by  ∞   2 pc (τ , λ, α,  = 0) = (3.24) 2πc1 λr e−πc1 λr L I τ p −1r α dr 0

3.6

Special Cases in Terms of 

43

Now, using (3.12), the Laplace transform of interference at s = τ p −1r α can be obtained as ⎛ ⎞ ∞ 2) 1 − exp(−πc λx 2 L I (τ p−1r α ) = exp ⎝−2πλ xdx ⎠ . (3.25) 1 + τ −1r −α x α 0

Therefore, we get 

∞

pc (τ , λ, α,  = 0) =

⎛ 2πc1 λr e−πc1

0

λr 2

exp ⎝−2πλ

∞ 0

⎞ 1 − exp(−πc2 λx 2 ) xdx ⎠ dr . 1 + τ −1r −α x α (3.26)

3.6.3

Approximation for  = 1

The coverage probability expression can be further simplified for the full inversion control case ( = 1) (in the interference-limited scenario) by employing the following approximation: 1 − exp(−πc2 λx 2 ) ≈ 1, 1 − exp(−πc1 λx 2 )

(3.27)

which is especially accurate for higher BS density λ. Under this approximation, (3.12) can be rewritten as ⎛ ⎞ ∞ ∞ 1 L I (s) ≈ exp ⎝−2πλ ı(u < x 2 ) πc1 λe−πc1 λu duxdx ⎠ . 1 + (sp)−1 u −α/2 x α 0

0

Interchanging the order of integration, we get ⎛ ⎞ ∞ ∞ 2) ı(u < x L I (s) ≈ exp ⎝−2πλ xdxπc1 λe−πc1 λu du ⎠ . 1 + (sp)−1 u −α/2 x α 0

0

Now using the variable substitution x = u 1/2 v and solving the inner integral we get

44

3 Uplink Analysis

⎛

∞ ∞

⎞

ı(1 < v) vdvπc1 λe−πc1 λu du ⎠ 1 + (sp)−1 v α 0 0 ⎞ ⎛ ∞ ∞ 1 −πc λu vdv πc1 λe 1 du ⎠ = exp ⎝−2πλ 1 + (sp)−1 v α 1 0 ⎞ ⎛ ∞ 1 1 vdv ⎠ . = exp ⎝−2πλ 1 + (sp)−1 v α λπ

L I (s) ≈ exp ⎝−2πλ

1

Substituting this expression of L I (s) in (3.22), we get the approximate probability coverage for the full power control as follows: pc (τ , λ, α,  = 1) = L I (τ p −1 ) ⎛ ∞ = exp ⎝−2

⎞ 1 vdv ⎠ 1 + τ −1 v α 1    ∞ 1 2/α dv = exp −τ α/2 τ −2/α 1 + v = exp (−ρ(τ , α)) ,

(3.28)

where the last step is due to the definition of ρ(τ , α) in (2.14). In addition to providing a useful closed-form expression, this also allows us to compare the coverage probabilities for uplink full power control and downlink cases in the interference-limited regime. The downlink coverage probability, given by (2.20), decreases as 1/(1 + ρ(τ , α)) with τ while the uplink coverage decreases as exp (−ρ(τ , α)) with τ . The fall-off is faster for the uplink case due to the fact that it is basically impossible to get a very high uplink SIR with full uplink power control.

3.7

Validation and Discussion

Since the uplink analysis is performed under various simplifying assumptions, it is important to validate it by comparing it with the results obtained from Monte Carlo simulations. We perform this comparison in Fig. 3.3 for the setup with λ = 4 × 10−6 BS/m2 (4 BSs per square km), α = 4, and p = 1. We assume an interference-limited scenario, where the interference power dominates the thermal noise (i.e., σ can be assumed to be 0). We present results for three values of . The results corresponding to  = 1 and 0 are respectively given by equations (3.23) and (3.26). However, since a simpler expression is not available for  = 0.5, we use Theorem 3.1 to plot this result. In addition to these three results, we also plot the approximate full inversion power control result given by (3.28). In all cases, we

3.7 Validation and Discussion Fig. 3.3 Uplink coverage probability for several values of the power control fraction . Lines and markers correspond to the theoretical and simulation results, respectively. The “Approx.  = 1” result corresponds to the case discussed in Sect. 3.6.3

45 1 0.9 0.8 0.7

0.6 0.5 0.4

0.3 0.2 0.1 0 -15

-10

-5

0

5

10

15

get a reasonable match with the simulation results. Through extensive simulations, we have noticed that the analysis works reasonably well for all the values of simulation parameters that we considered (despite the approximations). This concludes our discussion of the uplink coverage analysis in the typical cell. From the above discussion, it is fair to say that the application of Type I user PP provides a tractable approach to the uplink analysis in this case. This is primarily because the accurate characterization of the pcf of the PP of interferers with respect to the typical BS, presented in (3.1), is possible in this case. Our exposition in this chapter closely followed that of [44], which is the first to provide the uplink coverage analysis for the typical cell. However, there have been many works prior to [44] that approached this problem from different perspectives using different models, viewpoints and approximations, e.g., see [11, 45–50]. Some of these, e.g., [46] focused on the uplink analysis from the typical user viewpoint for which the downlink analysis was performed in Chap. 2. In other words, these works focused on the uplink analysis of the typical user chosen from a homogeneous PPP that connects to its closest BS and is hence located in the 0 cell. For concreteness, we will refer on [46] for the subsequent discussion. The PP of interfering users with respect to the tagged BS (i.e., the BS of the 0 cell) was modeled as a nonhomogenous PPP in [46], which was obtained by applying the distance-dependent thinning with the function 1 − exp(−πλx 2 ) to the homogeneous PPP of the users. While the original justification of using this thinning is not precise (as discussed in detail in [44]), it can nevertheless be viewed as the pcf of the PP of the interfering users with respect to the tagged BS. Note that the same function also governs the serving link distance distribution for the typical user viewpoint (see Sect. 1.2.7). Therefore, the sole difference between these two viewpoints for the uplink case boils down to the appropriate scaling of the serving link distance distributions and the intensity functions for the PP of the interfering users. In particular, the coverage probability result for the typical user setup of [46] can be recovered

46

3 Uplink Analysis

from Theorem 3.1 by setting c1 = c2 = 1. In fact, for the full power control, the approximate coverage probability result presented in (3.28) as the special case of the typical cell analysis can in fact be directly interpreted as the coverage probability of the typical user located in the 0 cell because (3.27) will directly hold in that case.

4

Heterogeneous Cellular Network Analysis

We now focus on the modeling and analysis of downlink heterogeneous cellular networks (HCNs), commonly referred to as HetNets [51] or small cell networks. We consider k overlaid tiers of BSs that differ in terms of the transmit power, deployment density, and target SINR. These tiers model overlaid macro, micro, pico, and femtocells, as well as distributed antenna systems. Because the smaller-form BSs are much less expensive and easier to deploy and can be targeted toward network hotspots and coverage holes, the trend in cellular systems is toward HetNets. As was the case in the previous two chapters, our emphasis will be on exposing the analytical tools without worrying about covering all possible generalizations.

4.1

HetNet Model

The key aspects of the downlink HetNet model follow [52, 53] and are: • BS and user locations. The BS locations of the ith tier are modeled by an independent homogeneous PPP i with density λi . User locations are again modeled by a different independent PP. • HetNet model. There are k overlaid tiers of BSs. All the BSs of the ith tier are assumed to transmit at the same fixed power pi . As an example, we could have k = 3 with λ3 = 10λ2 = 100λ1 and p1 = 10 p2 = 100 p3 as an approximation for a conventional macrocell network (tier-1) overlaid with picocells (tier-2) and numerous end-user deployed low-power femtocells (tier-3). • Target SINR. To maintain generality, the target SINR for the ith tier is modeled as τi . Combining this with the previous two bullets, ith tier can be completely characterized by the tuple {λi , pi , τi }.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. G. Andrews et al., An Introduction to Cellular Network Analysis Using Stochastic Geometry, Synthesis Lectures on Learning, Networks, and Algorithms, https://doi.org/10.1007/978-3-031-29743-4_4

47

48

4 Heterogeneous Cellular Network Analysis

• BS-user association. We consider an open access network, which means that a given mobile user is allowed to connect to any BS in the network without any restriction. This is best-case from an SINR coverage probability point of view and corresponds to an operator-deployed network. The choice of the specific BS to connect to is a very important aspect in a HetNet and we consider two models which are given in Sect. 4.2. • Signal attenuation model. Standard power-law path loss and i.i.d. Rayleigh fading are assumed as in Chap. 2. Therefore, the received power from an ith tier BS distance r away is pi Hr −α , with H ∼ exp(1). Shadowing can be incorporated in the same way as discussed in Sect. 2.6 but is left out in this chapter. • Typical user. The analysis will again be performed on the typical mobile user assumed to be located at the origin, i.e., the so-called typical user viewpoint. As in Chap. 2, all the BSs in the network are assumed to be active, which results in worst-case interference. In view of the above model, the SINR assuming that the typical user connects to an ith tier BS located at Xi ∈ i is SINR(Xi ) = k j=1



pi HXi Xi −α X∈ j \Xi

p j HX X−α + σ 2

=

pi HXi Xi −α , I ( \ {Xi }) + σ 2

(4.1)

where I ( \ {Xi }) denotes interference and σ 2 is the constant additive noise power.

4.2

Cell Association

In the previous two chapters, we used the average power-based cell association scheme, which was equivalent to associating with the closest BS in terms of the Euclidean distance. This meant that the coverage footprint of each BS corresponded to a “cell” of the standard PV tessellation as demonstrated in Fig. 2.1. This however is not true for HetNets due to the differences in the transmit powers of the BSs across tiers. For instance, the downlink signal from a femtocell located 100 m from a mobile user will usually be much weaker compared to the downlink signal from a macrocell located at 500 m, due to the much lower transmit power of a femtocell. This is illustrated in Fig. 4.1, where the space is tessellated based on the maximum average received power, i.e., all the points lying in a “cell” receive maximum average power from the BS located at the nucleus of that cell. Due to the differences in the transmit power, this tessellation corresponds to a multiplicatively weighted Voronoi diagram, where smaller cells represent coverage footprints of low-power BSs, such as femtocells. In particular, Fig. 4.1 illustrates two-tier scenario where large circles represent macrocells and small blue circles represent femtocells. Clearly, the footprints of the femtocells are much smaller than those of the macrocells. For cell selection in HetNets, we will consider in this chapter two criteria, which are both quite popular in the literature.

4.2

Cell Association

49

Fig. 4.1 Coverage regions in a two-tier network where tier-1 “macro” BS locations (large circles) correspond to an actual 4G deployment, whereas tier-2 BSs (small blue circles) are modeled as an independent PPP

1. Average power-based cell association. This is the same as in Chap. 2 and was the approach in [53]. 2. Instantaneous received power-based cell selection. This is where each user connects to its strongest BS instantaneously, i.e., the BS that offers the highest received SINR, as in [52]. These two approaches are both similar and quite reasonable but require a slightly different analytical approach that will be useful for the readers to master. Also, the resulting expressions for each approach have their own pros and cons. The coverage probability analysis for the typical user in both association rules is performed in the following subsections. Finally, we note that real-world HetNets typically use neither of the above approaches, but rather some sort of association rule that biases toward small cells. In other words, a user connects to a (usually lightly loaded) small cell even if its received power from it is a little bit lower than from the (usually more heavily loaded) macrocell. Such an approach results in higher throughput for both the small cell user (which then can use more bandwidth) and the remaining macrocell users (who no longer have to share resources with the departed user). Nevertheless, the results in this chapter serve as a very useful baseline for HetNets, and can be extended to more complex cell association rules, as we will discuss in Sect. 6.2.

50

4.3

4 Heterogeneous Cellular Network Analysis

Analysis for Average Power-Based Cell Association

In this subsection, we derive the coverage probability for the typical mobile user in a k tier HetNet under average power-based cell association [53]. As was the case in the previous chapters, we will first compute the distribution of the distance of the typical mobile user from its serving BS. As discussed above, the closest BS may not always be the serving BS for a given mobile user. It is, however, easy to see that the serving BS will lie in the set containing closest BS from each tier. Let us denote the serving tier by S and the closest BS of the ith tier by Bi and its location by Yi . Following the discussion in Sect. 2.1.1, the distribution of the distance Ri of the closest BS Bi of the ith tier from the typical mobile user is given by f Ri (r ) = 2πλi r exp(−λi πr 2 ).

(4.2)

Now given that Bi is located at distance r , Bi can be the serving BS only if B j ’s ( j  = i) provide lower received power than Bi . This leads to the following condition on the distances R j , j  = i, pi Ri−α > p j R −α =⇒ R j > j



pj pi

1/α r.

(4.3)

Recall that in the single tier case, there was an exclusion region of radius r where no interfering BSs could lie. Similarly, in this case, there is an exclusion region of radius  1/α p r for each tier j in which no interfering BSs of the jth tier can lie. Note ei ( j, r ) = pij that ei ( j, r ) is defined conditioned on the serving BS being from the ith tier. Now, we will compute the probability ai that the typical user connects to a BS of the tier i. This is known as the association probability of the ith tier. Conditioned on Ri = r , the typical user will be connected to Bi if all the other B j ’s ( j  = i) are located farther away than ei ( j, r ):    1/α pj r , ∀ j = i P [S = i|Ri = r ] = P R j > pi   1/α  pj (a) = P Rj > r pi j=i

 2/α  pj (b) (4.4) = exp −πλ j r2 , pi j=i

where (a) follows from the independence of the k tiers, and (b) follows from the void probability of a PPP (discussed in detail in Sect. 2.1.1). Therefore, association probability ai can be computed by averaging (4.4) over the distribution of Ri (given by (4.2)):

4.3

Analysis for Average Power-Based Cell Association



∞

ai = P [S = i] = 0

51

P [S = i|Ri = r ] f Ri (r )dr .

(4.5)

After deriving the distance distributions and the association probabilities, let us derive the joint probability of the event {S = i} and the event that the serving BS is located at a distance larger than r , which will be useful in the derivation of the coverage probability:    1/α pj Ri , ∀ j  = i P [Ri > r , S = i] = P Ri > r , R j > pi   1/α ∞  pj P Rj > u, ∀ j  = i f Ri (u)du = pi r

 2/α  ∞ pj (a) = exp −πλ j u 2 f Ri (u)du (4.6) pi r j=i

 2/α  ∞ pj (b) = exp −πλ j u 2 2πλi u exp(−λi πu 2 )du, (4.7) p i r j=i

where (a) follows on the same lines as (4.4), and (b) follows from (4.2). Using this result, the distribution of the distance of the typical user from its serving BS given that the typical user is connected to ith tier is given by d d P [Ri > r , S = i] P [Ri > r |S = i] = dr dr P [S = i]

 2/α  pj 1 2 = 2πλi r exp(−λi πr ) exp −πλ j r2 . ai pi

f Ri (r |S = i) =

(4.8) (4.9)

j=i

Using these intermediate results, we are now ready to derive the coverage probability. By definition, the coverage probability in this k-tier HetNet case is (a)

pc ({τi }, {λi }, { pi }) = P [SINR > τ S ] =

k

P [S = i] P [SINR > τi |S = i]

i=1

=

k

i=1

=

k

ai P [SINR > τi |S = i]    pic (τi ,{λi },{ pi })

ai pic (τi , {λi }, { pi }),

(4.10)

i=1

where (a) follows from the total probability law. Since we already derived an expression for ai in (4.5), we just need to compute per-tier coverage probability pic (τi , {λi }, { pi }) in order to complete our derivation. We do this next:

52

4 Heterogeneous Cellular Network Analysis

pic (τi , {λi }, { pi }) = P [SINR > τi |S = i]   = E Ri P[SINR > τi | Ri = r , S = i] P[SINR > τi | Ri = r , S = i] f Ri (r |S = i)dr , = r >0

(4.11)

where the unconditioning in the last step with respect to distance Ri needs to be done using conditional distribution f Ri (r |S = i) given by (4.9). Now substituting the above expression in (4.10), we get pc ({τi }, {λi }, { pi })    2/α k 

HYi pi Ri−α 2πλi −π  j λ j pp j r2  i R = ai P = r , S = i r e dr > τ  i i σ2 + I ai r >0 i=1

=

k

i=1

2πλi

r >0

re

−π

 j

λj

 p 2/α j pi

r2

P[HYi > τi pi−1 r α (σ 2 + I ) | S = i, Ri = r ]r dr . (4.12)

Using the fact that HYi ∼ exp(1), the inner probability term can be further simplified as P[HYi > τi pi−1 r α (σ 2 + I ) | Ri = r , S = i] −1

= e− pi

τi r α σ 2

L I (τi pi−1r α |Ri = r , S = i),

(4.13)

where L I (s|Ri = r , S = i) is the conditional interference Laplace transform which is the last component that needs to be computed. For notational simplicity, we will denote this conditional Laplace transform by L I (s) with the understanding that this is conditioned on the event {Ri = r , S = i}. Note that the interference power experienced by the typical mobile user in the HetNet case, as evident from (4.1), is the summation of the interference power I j from each tier. Hence, the Laplace transform of the interference is given as ⎡ ⎤ k

  L I (s) = E exp(−I (s)) = E ⎣exp(− I j (s))⎦ j=1

=

k j=1

  E exp(−I j (s) =

k

L I j (s),

(4.14)

j=1

where L I j (s) is the Laplace transform of the interference power from the BSs belonging to the jth tier. Note that the second to the last step is due to the independence assumption among tiers. Now, similar to the interference in the single tier case, the interference I j from the jth tier is also a standard M/M shot noise created by a PPP of intensity λ outside a disc of radius ei ( j, r ) centered at the origin o. As discussed earlier in this section, the exclusion

4.3

Analysis for Average Power-Based Cell Association

53

radius ei ( j, r ) is different across tiers. Using this exclusion radius, we can derive the Laplace transform of the jth tier interference on the same lines as Sect. 2.2. The final expression is the same as (2.10) with exclusion radius being ei ( j, r ) and is given below:    ∞  1 xdx . (4.15) L I j (s) = exp −2πλ j −1 α ei ( j,r ) 1 + (sp j ) x Now substituting this result in (4.14), L I (τi pi−1r α ) can be computed as

L I (τi pi−1r α ) =

k

 exp −2πλ j

∞

ei ( j,r )

j=1



1 1 + (τi p j / pi )−1 x α /r −α



 xdx .

(4.16)

2

Employing change of variables u = (x/r )2 (τi p j / pi )− α results in

L I (τi pi−1 r α )

=

k

exp −πλ j r (τi p j / pi ) 2

2 α



j=1

∞ 2 −α

τi



1 1 + u α/2





udu .

Now using the definition of function ρ(·, ·) from (2.14), we get ⎛ ⎞ k

2 L I (τi pi−1 r α ) = exp ⎝−πλ j r 2 ( p j / pi ) α ρ(τi , α)⎠ .

(4.17)

(4.18)

j=1

Combining (4.12), (4.13) and (4.18), we get the following Theorem. Theorem 4.1 The downlink coverage probability for the typical mobile user in an open access k-tier HetNet with average power-based cell selection is pc ({τi }, {λi }, { pi }) = ⎛ ⎞ k k  

2 2πλi r exp − pi−1 τi r α σ 2 exp ⎝−π λ j r 2 ( p j / pi ) α (1 + ρ(τi , α))⎠ r dr . i=1

r >0

j=1

(4.19)

4.3.1

Special Cases

As in single tier case studied in Chap. 2, we now consider some special cases for HetNets which offer further simplifications of Theorem 4.1 and additional insights. (i) Interference-limited (No-noise) Case The general downlink coverage result of Theorem 4.1 can be specialized to a practically relevant case of interference-limited networks (σ 2 = 0) as follows.

54

4 Heterogeneous Cellular Network Analysis

Corollary 4.1 In an interference-limited network (σ 2 = 0), the downlink coverage probability of the typical mobile user in a k-tier HetNet with average power-based cell association simplifies to k pc ({λi }, {τi }, { pi }) =  k

i=1 λi ( pi )

j=1 λ j ( p j )

2 α

2 α

(1 + ρ(τi , α))

.

(4.20)

Comparing (4.20) and (2.20), the results are identical for k = 1. And so (4.20) can be viewed as a generalization of (2.20), where we allow for multiple tiers with their own parameters. (ii) Same per-tier SIR threshold If the per-tier SIR thresholds are the same (τi = τ ∀ i), which is quite reasonable, it can be observed that the coverage probability simplifies to: pc ({λi }, τ , { pi }) =

1 . 1 + ρ(τ , α)

(4.21)

This result indicates that the outage probability is now independent of the BS transmit power { pi } and BS density {λi }. It can be intuitively seen by the fact that as we change the density of BSs by some factor, it impacts the distances to the serving and all interfering BSs by the same factor, leaving the SIR invariant to the changes in density. Furthermore, not surprisingly, with k = 1, i.e., the single tier case, we also get the same result as obtained in (2.20).

4.4

Analysis for Instantaneous Power-Based Cell Selection

After discussing the average power-based cell selection in the previous subsection, we now consider instantaneous power-based cell selection in this subsection. Under this cell association rule, the typical user at the origin is in coverage if max {SINR(X)} > τi ,

X∈i

for some 1 ≤ i ≤ k, where SINR(x) is given by (4.1). The key challenge in the downlink analysis under this cell association scheme is the fact that the closest BS from each tier is not necessarily the serving BS due to the presence of fading. This does not allow us to fix the serving BS in the same way as we did in the previous subsection. Recall that we assume open access network where the typical mobile user is allowed to connect to any BS in the network. Under this assumption, the mobile user is said to be in coverage if the received SINR from at least one BS is higher than its corresponding target τi . This can be mathematically expressed as

4.4

Analysis for Instantaneous Power-Based Cell Selection

⎛

⎞



pc ({λi }, {τi }, { pi }) = P ⎝

55

SINR(Xi ) > τi ⎠

i∈{1,2,...k},Xi ∈i

⎡ ⎛

⎞⎤



= E ⎣1 ⎝

SINR(Xi ) > τi ⎠⎦ .

(4.22)

i∈{1,2,...k},Xi ∈i

As demonstrated first in [52], the analysis greatly simplifies for τi > 1 (0 dB) for all tiers. This is because under this assumption at most one BS across all k tiers can satisfy the coverage condition, i.e., at most one BS can provide SINR greater than the required SINR threshold. To understand this intuitively, consider a system with two BSs. Denote the received powers from these two BSs at the mobile user by a1 and a2 . Depending upon the choice a2 a1 1 of the serving BS, the received SINR will be either a a+σ 2 or a +σ 2 . If a +σ 2 > 1, it means 2

1

2

a1 > a2 + σ 2 , which in turn implies a1 > a2 . Therefore, aa21 < 1, which further implies that 2 the received SINR from the other BS is a a+σ 2 < 1. In short, only one of the two SINRs can 1 be larger than 1 (0 dB). Thus if we enforce the target SINRs for all the tiers to be larger than 1, the coverage condition can be met by at most one BS. As an aside, note that this result can actually be extended to show that at most m BSs can meet the coverage condition if the target SINR is greater than 1/m for any positive integer m. Interested readers are referred to [52, Lemma 1] for more details about this result. For the rest of this discussion, we assume τi > 1, ∀ i, which means that the typical mobile user can connect to at most one BS (as discussed above). Under this condition, the coverage probability can be expressed in terms of the sum of the indicator functions as follows: ⎡ ⎛ ⎞⎤  SINR(Xi ) > τi ⎠⎦ pc ({λi }, {τi }, { pi }) = E ⎣1 ⎝ i∈{1,2,...k},Xi ∈i

⎤ k

(a) = E⎣ [1 (SINR(Xi ) > τi )]⎦ , ⎡

i=1

(4.23)

Xi ∈i

where (a) is in general an upper bound (by union bound) but holds with equality if at most one of the BSs satisfy the coverage condition, which is precisely the case when we assume τi > 1, ∀ i. We first compute the inner expectation, which can be written as ⎤ ⎡

pci = E ⎣ [1 (SINR(Xi ) > τi )]⎦ Xi ∈i

⎡ = E⎣

Xi ∈

⎤ [1 (SINR(Xi ) > τi ) 1 (Xi ∈ i )]⎦ ,

(4.24)

56

4 Heterogeneous Cellular Network Analysis

k  for notational simplicity. The reason for converting the where we defined  = ∪i=1 i summation over i to  will be clear shortly. Note that the above expression is simply the expected value of the sum of the function f (Xi ,  \ {Xi }) = 1 (SINR(Xi ) > τi ) 1 (Xi ∈ i ) over all points of the PPP . Since function f (·) is dependent on both Xi and point process  \ {Xi }, we can directly use Campbell-Mecke theorem to evaluate the above expression. Please refer to (1.35) in Chap. 1 for the definition of this theorem for the homogeneous PPP. Let us first compute E [ f (xi , )] below:     pi Hxi xi −α E [ f (xi , )] = E 1 (4.25) > τ 1 ∈  (x ) i i i I () + σ 2   pi Hxi xi −α (4.26) > τ =P i P (xi ∈ i ) I () + σ 2   λi pi Hxi xi −α (4.27) = P > τ i , λ I () + σ 2

where we note that the “interference” term I () is now sum of received powers from all points of the PPP (i.e., point xi is not excluded), which follows from the Campbell-Mecke theorem given by (1.13) (and using the fact that the reduced Palm distribution for the PPP is the same as its original distribution by Slivnyak’s Theorem). Also, the λλi factor is due to the term P (xi ∈ i ) which is equal to λλi from the independent thinning theorem. Now combining (4.24), (1.13), and (4.27), we get   λi pi Hxi xi −α pci = λ > τi dxi P I () + σ 2 R2 λ   pi Hxi xi −α = λi P > τi dxi . (4.28) I () + σ 2 R2 Substituting (4.28) into (4.24), we get pc ({λi }, {τi }, { pi }) =

k



λi

R2

i=1

P

 pi Hxi xi −α > τ i dxi . I () + σ 2

(4.29)

  For notational simplicity, we denote I () = kj=1 X j ∈ j p j HX j X j −α by simply I . Now using the fact that HXi ∼ exp(1), we can express coverage probability in terms of the Laplace transform of I as follows: pc ({λi }, {τi }, { pi }) =

k

i=1



λi

R2

LI

τi pi xi −α



 exp

 −τi σ 2 dxi pi xi −α

The Laplace transform of interference can now be computed as follows:

(4.30)

4.4

Analysis for Instantaneous Power-Based Cell Selection

L I (s) = E I



⎡ = E⎣

57

⎡ ⎛ ⎞⎤ k

 exp (−s I ) = E ⎣exp ⎝−s p j HX j X j −α ⎠⎦ j=1 X j ∈ j

k

exp −sp j HX j X j 

! −α

⎤ ⎦

j=1 X j ∈ j (a)

=

k

⎡

E⎣

(b)

=

⎡

E j ⎣

j=1 (c)

=

k

exp −sp j HX j X j 

X j ∈ j

j=1 k



⎡ E j ⎣



X j ∈ j

j=1

⎤ ⎦

⎤  ! Eh exp −sp j H X j −α ⎦

X j ∈ j



! −α

⎤ 1 ⎦, 1 + sp j X j −α

(4.31)

where (a) follows from the independence of PPPs modeling different tiers, (b) follows from the independence of the fading variables HX j ’s (generic fading variable is denoted by H in this step), and (c) follows from the Rayleigh fading assumption, i.e., H ∼ exp(1). Now we can use the PGFL of PPP to convert product over the PPP to an integral as follows:

L I (s) =

k



 exp −λi

1−

R2

j=1

1 1 + sp j x j −α



 dx j .

(4.32)

Converting from cartesian to polar coordinates x j = (r , θ), we get

L I (s) =

k



∞



exp −2πλi 0

j=1

  1 1− r dr . 1 + sp j r −α

(4.33) 1

To simplify this integral further, we perform the change of variable (sp j )− α r → u, which gives

L I (s) =

k

 exp −2πλi (sp j )

0

j=1

=

k j=1

∞

2/α

∞

 exp −2πλi (sp j )2/α 0



= exp −s 2/α ζ(α)

k

i=1

1 1− 1 + u −α



 udu

  1 udu uα + 1

 2/α λi pi

,

(4.34)

58

4 Heterogeneous Cellular Network Analysis 2

where ζ(α) = 2πα csc( 2π α ). Substituting this expression in (4.30) gives the final expression for coverage probability pc ({λi }, {τi }, { pi }), which is given next. Theorem 4.2 For τi > 1, the downlink coverage probability for the typical mobile user in an open access k-tier HetNet under maximum instantaneous power-based cell selection is pc ({λi }, {τi }, { pi }) = 

  2/α  ∞ k k

τi σ 2 α 2/α 2 τi exp − 2π λi exp −x ζ(α) λm pm x xdx, pi pi 0 where ζ(α) =

4.4.1

(4.35)

m=1

i=1

2π 2 α

csc( 2π α ).

Special Cases

As in the macrocell-only case studied in Chap. 2 and the HetNet under average power cell selection rule studied in Sect. 4.3, we now consider some special cases for the instantaneous power-based cell selection which offer further simplifications of Theorem 4.2 and additional insights. (i) Interference-limited (No-noise) Case The general downlink coverage result of Theorem 4.2 can be specialized for the interferencelimited networks (σ 2 = 0) as follows. This case is particularly relevant for HetNets, which are often interference-limited due to the dense and organic deployment of small cells. Corollary 4.2 In an interference-limited network, i.e., , σ 2 = 0, the downlink coverage probability of the typical mobile user simplifies to π pc ({λi }, {τi }, { pi }) = ζ(α)

k

2/α −2/α i=1 λi pi τi , k 2/α i=1 λi pi

τi > 1.

This follows immediately from Theorem 4.2 with σ 2 = 0. This is a remarkably simple closed-form expression, which is useful in understanding how coverage probability depends upon various system parameters, such as the transmit powers of different tiers. (ii) Same per-tier SIR thresholds If the per-tier SIR thresholds are the same (τi = τ ∀ i), which is quite reasonable, it can be observed that the coverage probability simplifies to: pc (λ, τ , p) =

π . ζ(α)τ 2/α

(4.36)

4.5

Interpretations and Impact on Network Throughput

59

From the above result, we can observe that the coverage probability is independent of the density of the BSs, number of tiers, and their respective powers which indicates that the SIR distribution is invariant of density and transmit powers. Interestingly, the same conclusion was also obtained for the average power-based association case in the previous subsection (see (4.21)). From the analytical standpoint, it was useful to study the instantaneous power-based association rule since it involves additional tools (such as the Campbell-Mecke theorem) that did not explicitly appear in the previous two chapters.

4.5

Interpretations and Impact on Network Throughput

In this chapter, we studied the downlink coverage probability for a HetNet under two significantly different association rules. In the first association rule, the serving BS was selected based on the average received power. However, in the second we allowed association to the highest instantaneous SINR, including fading, and assumed that the threshold for success is τi > 1. Despite these differences, we arrived at exactly the same conclusion in both cases that the coverage probability is independent of the number of tiers, their respective powers, as well as the density of BSs in each tier, when interference dominates noise and the per-tier SIR thresholds are the same. These results appear in (4.21) and (4.36) for the two association policies. Given the widespread initial fear that unplanned deployment of small cells will create unmanageable interference and deadzones, this insight offers confidence about deploying small cells in the existing macrocellular networks. In particular, it shows that the per-user throughput will not be affected with the increasing number of tiers and the increasing density of BSs in each tier. This also implies that the area spectral efficiency will increase linearly as we add BSs since the SINR becomes SIR in the interference-limited regime, whose distribution is invariant of the network density. This generalizes our discussion about SIRinvariance from Sect. 2.9 in single tier networks. This paints a rosy picture of the future of HetNets. The obvious question to ask at this point is: is there a catch? In order to answer this question rigorously, we will need to carefully revisit some of the key modeling assumptions, especially the path loss model, as mentioned already in Sect. 2.9. We do this in the next chapter, which will yield several surprising insights about the behavior of area spectral efficiency in dense networks. Finally, please that while the discussion on HetNets in this chapter was limited to the downlink case, the analysis of uplink HetNets does not require any new additional tools, besides the ones introduced in the previous chapter for uplink analysis.

5

Dense Cellular Networks

In this chapter, we specifically focus on understanding the effect of aggressive network densification on the network performance [54]. Network densification—meaning simply adding more BSs—has been a key enabler for increasing data rates in cellular networks, as discussed in Chap. 4 on HetNets. Historically, doubling the number of BSs has also about doubled the network throughput over a particular area, since the spectrum can be reused in both sets of BSs. Looking back at the analysis we did in Chap. 2, we showed that the coverage probability is independent of the BS density λ in the interference-limited case, i.e., the expression in (2.20) is independent of λ, which is referred to as SIR-invariance property as we discussed in Sect. 2.9. A very similar conclusion was drawn in Chap. 4 for HetNets as well. If the network is noise-limited, the coverage probability increases with the BS density since shortening the communication links increases the SNR. At some point, the network becomes interferencelimited and hence the coverage probability depends just on the SIR. In this regime, the coverage probability does not change as we increase λ further. Dense networks are by their nature interference-limited, and in fact, the regime of BS density where SINR → SIR is a good high-level definition of what we mean by a “dense” network. Since the per-user throughput tracks Shannon’s limit, i.e., E[log2 (1 + SINR)], it also follows from Chaps. 2 and 4 that the per-user throughput should be agnostic to the BS density. Consequentially, doubling the density of the BSs doubles the number of supported users with no effect on the per-user throughput as we discussed in Sect. 2.9. Hence, the insights obtained from the models in the previous chapters agree with the performance gains we observed in cellular networks in the last decade. But is this the complete story? Can the network be continually densified to yield linear throughput gains, or is there some limit to the amount of profitable densification? Does

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. G. Andrews et al., An Introduction to Cellular Network Analysis Using Stochastic Geometry, Synthesis Lectures on Learning, Networks, and Algorithms, https://doi.org/10.1007/978-3-031-29743-4_5

61

62

5 Dense Cellular Networks

it saturate or reverse at some point, as some researchers or operators have empirically observed? This chapter explores these questions by more carefully considering the nature of signal attenuation (path loss) in a dense cellular network.

5.1

The Standard Power-Law Path Loss Model

A critical assumption in the baseline model in Chap. 2 and indeed up until now in this book is that signal attenuation follows the standard power-law path loss model, where the signal power homogeneously decays with the distance r as r −α , with α > 2. Despite having a long history and being the most widely used path loss model in the literature—and perhaps even in the cellular industry—the power-law model has several important shortcomings that become noticeable in very dense networks. First, the standard path loss model assumes signals attenuate uniformly over all distances as r −α . That is, α is a constant, rather than a function of r . Empirically this is known to be false, and usually, α itself is better modeled as an increasing function of r . A simple example of this can be observed mathematically for the well-known 2-ray ground bounce model, in which α = 2 for r  rc and α = 4 for r  rc [19]. Here, rc is the critical distance that is a function of the wavelength and the heights of the transmitter and receiver. By restricting the path loss to follow a single path loss exponent α, typically an average value of α is selected to model the entire cell, usually between 3 and 4. This greatly overestimates the attenuation of close-by transmitters while underestimating those of distant transmitters. Second, the power-law path loss model requires α > 2 to have a non-zero coverage probability, and in many cases the results of the previous chapters, as in the downlink case, begin to diverge for α < 2. However, for dense networks, line of sight (LOS) communication is common, for which α ≈ 2 [19]. Thus the results in the prior chapters, and indeed the vast majority of results using stochastic geometry, are not directly applicable to dense network scenarios. Third, the model is unbounded, meaning that the channel gain becomes unboundedly large as r → 0. The unbounded model allows greater tractability but is problematic in dense networks where r < 1 is possible and thus the received power is greater than the transmitted power, which is obviously impossible physically. This limitation of the standard path loss model is also observed in network simulations, which often skirt the issue by disallowing r < 1, for example, by removing any randomly placed nodes closer than a certain distance. In this chapter, we will introduce appropriate path loss models for dense networks that avoid the above issues, while being well-grounded in field measurements and electromagnetic propagation theory. We will also extend our downlink analysis tools developed thus far to handle these new path loss models and observe the expected effects of continuing network densification.

5.2

Physically Feasible Path Loss Models

5.2

Physically Feasible Path Loss Models

5.2.1

Definition

63

After establishing the shortcomings of the power-law path loss model, we start looking into more suitable models for dense networks. Instead of focusing on a specific model, we first discuss the desired properties that we need in a path loss model to be suitable in modeling dense networks. Let the path gain at distance r be denoted by l(r ). First, we assume that l(0) = l0 , where l0 is a finite constant, which we interpret as the average transmit power directly at the antenna (that needs to be finite). One of the consequences of this interpretation is that we would not need a separate transmit power term in this chapter. Second, we know from physics and the conservation of energy that the average received power at any distance r > 0 is less than the transmit power, i.e., l(r ) ≤ l0 . Third, we require that the total average received power from all BSs at any arbitrary point in the network be finite. This requirement can be mathematically distilled as ⎡ ⎤ ⎡ ⎤   Hi l(Xi )⎦ = E ⎣ l(Xi )⎦ pavg = E ⎣ Xi ∈  ∞

= 2πλ

Xi ∈

rl(r )dr = 2πλγ,

(5.1)

0

 where γ 

∞

rl(r )dr .

0

To obtain (5.1), we used the fact that the small-scale fading has unit mean followed by Campbell’s theorem, which was discussed in Sect. 1.2.1. Hence, for the path loss model to be physically feasible, γ need to be finite. In other words, the path loss function has to be integrable over R2 . Note that given the first (boundedness) and the second (non-increasing power) requirements, the third puts a restriction on how fast l(r ) drops to zero as r → 0. In fact, if the path loss model is bounded but the integral in (5.1) is not finite, then the received interference at any point in the network is infinite almost surely and not only on average [6, Theorem 4.6].  In summary, we have three conditions so far, l0 < ∞, l(r ) ≤ l0 ∀ r ≥ 0, and γ  ∞

rl(r )dr < ∞. With these three conditions, we now define a class of path loss models

0

that we call physically feasible path loss models [55]. Throughout this chapter, we denote the set of non-negative real numbers by R+ and the set of strictly positive real numbers by R∗+ . Definition 5.1 (Physically feasible path loss) Physically feasible path loss models are the family of path loss functions l(·) with the following properties

64

5 Dense Cellular Networks

Table 5.1 Examples of physically feasible path loss functions and their asymptotic ASE defined in (2.41) as lim E [E(λ)] λ→∞

Path loss function l(r )

˜ ) l(r

r0

a > 0, d0 > 0, α > 2

ar −α

 −1 max 1, c0α

α−2 2πd02 ln(2)

a > 0, r0 > 0, α > 2

a(c02 + r 2 )

α2 −3α+2 2π ln(2)d02

a > 0, d0 > 0, α > 2

a(d0 + r )−α

a > 0, d0 > 0, α > 2

a(d0 + r α )−1 d0 (α−2) 2

lim E [E(λ)] Parameters

λ→∞

2 (α−2)d0α

l1 (r ) = a min(c0 , r −α ) απ ln(2) l2 (r ) = a(d02 + r 2 )

−α 2

l3 (r ) = a(d0 + r )−α l4 (r ) = a(d0 + r α )−1

l5 (r ) = ae−κr

β

α sin



2π α

−α 2

1 1

 2

2π 2 ln(2)d0α 2

βκ β  2π ln(2) β2

a > 0, 2 ≥ β > 0, κ > 0 ae−κr

β

1

1. l0 = l(0) ∈ R+ . 2. l(r ) ≤ 0 ∀r ∈ R+ .  l∞ 3. γ = rl(r )dr ∈ R∗+ . 0

Unsurprisingly, almost all the bounded path loss models proposed in the literature are included in the class of physically feasible path loss models, from simple ones in Table 5.1 [38, 40, 56, 57] with the specified range of their parameters, to more complicated ones like the multi-slope path loss model [19, 39] and the models adopted in the 3GPP standards, both for the conventional sub-6 GHz [58] as well as in the mmWave bands [59]. Note that ˜ ) and r0 in Table 5.1 are, respectively, an arbitrary function and an arbitrary constant l(r ˜ ) and r0 for a given path loss function that will appear in Theorem 5.6. The existence of l(r (as is the case for all the choices in Table 5.1) will ensure that the lower bound on the asymptotic average ASE derived in Lemma 5.5 holds with equality. This will be discussed more rigorously in the context of Theorem 5.6. Throughout this chapter, all results are for this class of physically feasible path loss models, unless otherwise stated. Moreover, we allow for any general small-scale fading model as long as it has a finite mean. However, for simplicity, we assume unit mean. The analysis can be easily extended to the non-unit mean fading distributions by taking the mean as a common factor in the SINR expression in (2.29). Hence, it is equivalent to considering unit mean distributions with a scaled noise power.

5.3

5.3

SINR and ASE Scaling Laws

65

SINR and ASE Scaling Laws

After defining the class of path loss models that is suitable to study dense networks, we go back to answering our main questions: What is the effect of densifying the network on the SINR, potential throughput, and ASE? Thankfully, we have already derived the SINR distribution for a general path loss model in (2.30) and we know how to find the ASE and potential throughput using the coverage probability as we have shown in (2.44) and (2.45), respectively. Hence, one way to answer this question is to plug the path loss function of interest into (2.30), then numerically find the ASE (2.44) and potential throughput (2.45). For example, let us focus on l2 (·), since it has a clear physical meaning (with d0 denoting the height difference between the BSs and users), and plot the coverage probability for different coverage thresholds τ and for different BS densities in Fig. 5.1. Interestingly, the coverage probability decreases with the BS density in this case, which is very different from the SIRinvariance property that we discussed in Sect. 2.9, where the BS density does not affect the coverage probability. Note that we only changed the path loss model from the power-law model to the bounded model l2 (·), and that alone resulted in a completely different scaling law. Looking back at the SINR expression in (2.29), Fig. 5.1 shows that the increase in the interference dominates the enhancement in the received signal power of the desired signal with network densification. However, this cannot be directly generalized to the average ASE due to the multiplicative factor of λ in (2.41), and (2.42). Hence, we plot the average ASE in Fig. 5.2. First, note the different trends for the different definitions (refer to Sect. 2.9). The unconstrained ASE is increasing with the BS density, but with diminishing gains for higher densities. For example, densifying the network from 50 to 400 BS/km2 provides a gain of about 300% in terms of the ASE. However, from 400 to 750 BS/km2 only provides a gain of about

Fig. 5.1 Coverage probability as a function of SINR threshold τ in a cellular network assuming the path loss model to be l2 (·)

1 =20 BS/km 2

0.9

=500 BS/km 2

0.8

pc( , )

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -20

-15

-10

-5

0

(dB)

5

10

15

20

66

5 Dense Cellular Networks 10-4

8

Average ASE (bps/Hz/m2 )

Fig. 5.2 Average area spectral efficiency vs BS density (λ) assuming the path loss model l4 (·)

Unconstrained ASE Potential Throughput

6

4

2

0 0

200

400

600

800

1000

2

(BS/km )

30%. Hence, more and more diminishing gains for higher densities. This trend is referred to as a densification plateau. Figure 5.2 also shows that the potential throughput is maximized for a finite BS density and then decays to zero. Hence, after a specific density, there is no point in densifying the network, since it leads to a lower sum throughput and a lower per-user throughput. This observation is referred to as a densification crash. Recall that under the standard power-law model, where the SIR-invariance property holds, the ASE scales linearly with λ for the two definitions. We have shown in this section that adopting a different path loss model results in completely different scaling laws and the linear gain predicted previously vanishes by simply bounding the path loss function. But can we generalize these observations for all physically feasible path loss models and different fading models? It turns out that we can actually generalize these observations and formally prove them. However, we have to follow a slightly different approach compared to the approach that we followed to prove the SIR-invariance property in Sect. 2.9. The approach that we will follow now relies on four simple concepts: the superposition property of PPPs that was discussed in Sect. 1.2.5, the law of large numbers, Fatou’s lemma, and uniform integrability. Simple forms of the last three of these are given next without proofs [60, 61]. In these statements, N will denote the set of natural numbers. Lemma 5.2 (Law of large numbers) Let {X i }i∈N be a sequence of i.i.d. random variables with a finite mean E [X], then, n 1 X i = E[X ]. lim n→∞ n

(5.2)

i=1

Lemma 5.3 (Fatou’s lemma) Let {X i }i∈N be a sequence of random variables with a finite limit, i.e., lim X i is finite almost surely (a.s.), then, i→∞

5.3

SINR and ASE Scaling Laws

67

lim E[X n ] ≥ E lim X n .

n→∞

n→∞

(5.3)

(Uniform integrability) Furthermore, if X i , ∀i have a finite second moment, i.e., E[X i2 ], ∀i is finite, then the inequality becomes a strict equality. Fatou’s lemma and uniform integrability are usually introduced in graduate courses on the theory of probability (and courses on measure theory). For readers who have not yet come across these results, it will be useful to keep an informal interpretation of these results in mind to understand the asymptotic analysis presented in Sect. 5.3.1. First, note that both these results relate to exchanging the order of expectation and limit (over a sequence of random variables). Fatou’s lemma gives us a bound when we exchange this order, whereas uniform integrability ensures equality. Fatou’s lemma will be used in Lemma 5.5 to find a lower bound on the asymptotic average ASE and the uniform integrability will be used in Theorem 5.6 to come up with conditions that will ensure that this bound holds with equality.

5.3.1

Asymptotic Analysis

In this section, we generalize the observations we discussed so far by analyzing the asymptotic behavior of cellular networks with attenuation laws that satisfy Definition 5.1 in terms of the average ASE when the BS density is high, i.e., λ → ∞. Hence, the underlying model for the network is the same as in Chap. 2, with the exception that the path loss function l(·) is physically feasible as in Definition 5.1. Let us first look at the SINR expression in (2.29). Due to the boundedness of l(·), the numerator of the SINR is bounded on average. However, the denominator can get arbitrarily large as we increase the BS density. Hence, intuitively, the SINR approaches zero for high BS densities. To be more rigorous, let λ = kλ0 , where k ∈ N and λ0 ∈ R∗+ , and let us study lim kλ0 SINR(kλ0 ) for a given network realization. k→∞

h 0 l(r0 ) , h i l(ri ) + σ 2

(5.4)

h 0 l(r0 ) , h i l(ri ) − h 0 l(r0 ) + σ 2

(5.5)



lim kλ0 SINR(kλ0 ) = lim kλ0

k→∞

k→∞

ri ∈\B(0,r0 )

= lim kλ0  k→∞

ri ∈

where in the last step, we added and subtracted the desired power term corresponding to the 2 typical user. Note that as k → ∞, l(r0 ) → l0 , σk → 0 and h 0kl0 → 0 almost surely. With this in mind, the limit is simplified to

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5 Dense Cellular Networks

h 0 l0 lim kλ0 SINR(kλ0 ) = lim λ0 1  . k→∞ h i l(ri ) k

(5.6)

k→∞

ri ∈

Now we can exploit the superposition property of the PPP, where  that has a density of λ0 k can be decomposed as the sum of k i.i.d. PPPs, i ∀i ∈ {1, . . . , k}, each with density λ0 . Hence, the limit can expressed as lim kλ0 SINR(kλ0 ) = lim

k→∞

k→∞

1 k

k 

λ0 h 0 l 0 .  h i, j l(ri, j )

(5.7)

i=1 ri, j ∈ j

Note that we now have a sequence of i.i.d. random variables

 ri, j ∈ j

h i, j l(ri, j ) that are

independent of k. Hence, we can use the law of large number in Lemma 5.2 to find the limit. Then, by using Campbell’s theorem, it is straightforward to see that 

lim kλ0 SINR(kλ0 ) =

k→∞

E

λ0 h 0 l 0  Ri,1 ∈1



Hi,1l(Ri,1 )

=

λ0 h 0 l 0 h 0 l0  ∞ = . 2πγ 2πλ0 rl(r )dr

(5.8)

0

Finally, since the result is independent of λ0 , we can conclude that lim λSINR(λ) = which is finite a.s and hence lim SINR(λ) = 0 a.s.

λ→∞

h 0 l0 2πγ

λ→∞

In summary, we have proved that the SINR approaches zero when λ → ∞ as λ−1 . This is stated in the following theorem. Theorem 5.4 When λ → ∞, for a path loss model satisfying Definition 5.1, the SINR tends to zero a.s. Next, we study the average unconstrained ASE as defined in (2.41) since it is the more general metric. After that, we move to the potential throughput (2.42). We start with the following lemma. Lemma 5.5 The asymptotic average ASE is lower bounded by lim E [E (λ)] ≥

λ→∞

l0 . 2π ln(2)γ

(5.9)

Proof Let λ = kλ0 , where k ∈ N and λ0 ∈ R∗+ . We are interested in lim E (kλ0 ), which is k→∞

found next:

5.3

SINR and ASE Scaling Laws

69

lim E (kλ0 ) = lim λ0 k log2 (1 + SINR(kλ0 )),

k→∞

(5.10)

k→∞

λ0 kSINR(kλ0 ), ln(2) h 0 l0 = , ln(2)2πγ   l0 , E lim E (kλ0 ) = k→∞ ln(2)2πγ = lim

(5.11)

k→∞

(5.12) (5.13)

where (5.11) follows because the SINR approaches zero a.s when k → ∞ as we proved in Theorem 5.4, (5.12) follows from (5.8), and (5.13) follows by taking the expectation of (5.12) with respect to h 0 which has a unit mean. Then by using Lemma 5.3, the following is true   l0 . lim E [E (kλ0 )] ≥ E lim E (kλ0 ) = k→∞ k→∞ ln(2)2πγ Note that the result is independent of λ0 , hence we can conclude that lim E [E (λ)] ≥ λ→∞

l0 ln(2)2πγ .



The last lemma shows that the average unconstrained ASE is lower bounded by a constant and does not drop to zero. However, it is more interesting to show that it holds with equality. Then we will have an exact characterization of the limit and prove the existence of a densification plateau. Based on Lemma 5.3, we just need to show that E (kλ0 ) is uniformly integrable to replace the inequality in Lemma 5.5 with an equality. In the following theorem, we provide sufficient conditions on l(·) that yield to the uniform integrability of E (kλ0 ). The proof of the theorem is provided in [55, Appendix B], and it relies on bounding the SINR and simple mathematical manipulation of the PGFL defined in Sect. 1.1.3. Theorem 5.6 If ∃ r0 ∈ R+ , ζ ∈ R∗+ and a differentiable decreasing function l˜ : [r0 , ∞) → R+ such that: ˜ ) ≤ l(r ) ∀r ∈ [r0 , ∞)]. 1. l(r ˜ ) ≥ ζ, ∀r ∈ [r0 , ∞). 2. r l(r  −l˜ (r ) ∞ r −πλ r 2 0 dr is finite for all λ > λ , where λ ∈ R is an arbitrary constant. 3. 0 c c + ˜ 2e r0 l(r )

Then, lim E [E (λ)] =

λ→∞

l0 . 2π ln(2)γ

(5.14)

70

5 Dense Cellular Networks

Hence, for any path loss function of interest, we can check the conditions in Theorem 5.6 to determine the asymptotic behavior of the average unconstrained ASE. Under the same conditions, the scaling law of the potential throughput is given by the following theorem. Theorem 5.7 If the path loss function satisfies the condition in Theorem 5.6, then for all τ0 ∈ R+ , lim E [R(λ, τ0 )] = 0,

λ→∞

(5.15)

where R(λ) is the average potential throughput defined in (2.42). Proof This result can be proved as follows: lim E [R(λ, τ0 )] = lim λ log2 (1 + τ0 )P {SINR ≥ τ0 } ,

λ→∞

λ→∞

= lim λ log2 (1 + τ0 )E [1 {SINR ≥ τ0 }] , (5.16) λ→∞   ≤ lim λE log2 (1 + SINR)1 {SINR ≥ τ0 } , λ→∞   (5.17) = E lim λ0 k log2 (1 + SINR)1 {SINR ≥ τ0 } , k→∞    l0 H0 =E (5.18) 1 lim SINR ≥ τ0 = 0, k→∞ 2π ln(2)γ     where (5.17) follows since λ0 kE log2 (1 + SINR)1 {SINR ≥ τ0 } ≤ λ0 kE log2 (1 + SINR) which is uniformly integrable as proved in Theorem 5.6. Hence, we can push the limit inside the expectation. Further, (5.18) follows from Theorem 5.6 and the fact that 1 {x ≥ τ0 } is continuous at x = 0 under the assumption that τ0 ≥  > 0, and the last equality follows by using Theorem 5.4.  Hence, we have derived the scaling laws of the ASE and potential throughput, and the laws match the trends that we observed for l2 (·) at the beginning of this section. For the other models in Table 5.1, the conditions needed to satisfy the conditions in Theorem 5.6 are provided in the table along with the ASE limit. Other more complicated examples of path loss models are provided in [55].

5.4

Final Remarks

Theorem 5.7 shows that the potential throughput has a completely different behavior than the average unconstrained ASE: we observe a densification plateau (it saturates to a constant) for the average ASE and a densification collapse (it drops to zero) for the average potential throughput. This agrees with the observations we had in the previous sections. From a

5.4

Final Remarks

71

network throughput perspective, the unconstrained ASE result means that although for very high densities the SINR approaches zero, the average sum throughput of the users can still be higher than zero since there are many users in the network. In other words, the increase in the co-channel interference is fully balanced by the increase in the number of users using this channel. Another insight from the previous theorems is that the channel state information has to be available at the transmitter to be able to harvest gains from densifying the network, otherwise many of these users will go into outage and the sum throughput of all users approaches zero as we showed in Theorem 5.7. The approach that we have followed in this chapter can be extended to other network setups, such as multi-antenna BSs and users, mmWave cellular networks, and uplink cellular networks [62–64]. Hence, when studying scaling laws of wireless networks with a stochastic geometry framework, it is always useful to check if this approach can be extended to the desired network setup, especially if the expressions for the coverage probability and ASE are too complicated to deduce the scaling laws from them.

6

Extensions

The model and results presented in the preceding four chapters have been extended and generalized in many directions. Without any hope of discussing them all, we focus now on a few important representative classes of generalization, including generalizations in the foundational modeling aspects as well as those inspired by important emerging applications. Since this is still a highly active area of research, we look forward to seeing many new directions in the years ahead.

6.1

General Fading Models

The assumption of Rayleigh fading (exponential distribution for the channel power gain) leads to a particularly convenient form for the coverage probability in terms of the Laplace transform of interference power distribution, which is in general easier to characterize than the probability density function of interference power [6, 65]. Technically, assuming Rayleigh fading for only the serving link is sufficient to express the coverage probability in terms of the Laplace transform of interference, which means the fading distribution for the interfering links can be generalized without loss of tractability, as done in [14]. The generalization of the fading distribution for the serving link is usually much more complex [26, 27, 66]. One key exception is when the channel power gain of the serving link is chosen from the exponential family, in particular, Gamma distribution (e.g., Nakagami-m). In the case of Nakagami-m, the coverage probability can be expressed in terms of the higher-order derivative of the Laplace transform of interference, which is relatively easier to evaluate compared to a more general distribution [67, 68].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. G. Andrews et al., An Introduction to Cellular Network Analysis Using Stochastic Geometry, Synthesis Lectures on Learning, Networks, and Algorithms, https://doi.org/10.1007/978-3-031-29743-4_6

73

74

6 Extensions

In addition to the direct extension discussed above, note that the effect of general fading and shadowing distributions can also be incorporated by treating them as equivalent random perturbations in the locations of the BSs. Using the displacement theorem, their effect on the received power can be incorporated by transforming the original homogeneous PPP of the BSs into a new homogeneous PPP whose density depends upon the fractional moment of the shadowing distribution. This was discussed in Sect. 2.6 in the context of incorporating shadowing in the downlink analysis. For the application of this general idea to other scenarios, please refer to [24–30] and the references therein.

6.2

Advanced Cell Selection Strategies

There are three general classes of cell selection strategies: (i) average power based, where the receivers connect to the BSs providing maximum average received power (considered for downlink analysis of macro-only case in Chap. 2 and k-tier HetNet case in Sect. 4.3), (ii) instantaneous power based, where the receivers connect to the BSs providing maximum instantaneous received power (considered for the k-tier HetNet analysis in Sect. 4.4), and (iii) strategies with cell selection bias, where an additional parameter, termed the cell selection bias, is considered along with the received power, e.g., see [69]. While we discussed the first two cell selection strategies for HetNets, including cell selection bias in the analysis does not require any new tools. Interested readers can refer to [53, 69] for more details. We also only considered the open access case for HetNets, where users are allowed to connect to any BS in the network. This requirement of open access is often not satisfied, in particular for femtocells and if WiFi is introduced in a so-called multi-RAT (radio access technology) network. Reference [52] also handles closed-access (which reduces the coverage probability and violates the SIR-invariance property) and the multi-RAT case is considered in [69]. The rationale for cell selection bias stems from the desire to maximize the per link data rate, which is related to but is not necessarily equivalent to the SINR. Particularly in a HetNet, too many of the mobiles connect to a macrocell, because of its high power, leaving the small cells underutilized and the macrocells heavily congested. The selection bias is a simple and crude but remarkably effective tool for pushing mobiles onto the small cells and has been observed to achieve near-optimum results when the bias is chosen correctly [69, 70]. These three strategies may appear to be different at the surface but are fundamentally related. For instance, it can be shown that the maximum instantaneous power-based cell selection in the presence of general fading is equivalent to the maximum power-based cell selection in the absence of fading in an equivalent PPP (where the effect of fading has been incorporated in the density of the equivalent PPP) [26]. This general idea was introduced in the context of including the effect of shadowing in the performance analysis of downlink cellular networks in Sect. 2.6.

6.3

General Spatial Models

75

We conclude this discussion about cell selection strategies by revisiting the assumption in Sect. 4.4 that τi > 1 (0 dB) for all tiers. This assumption allowed us to use union bound in the coverage analysis with equality, thereby lending tractability to the downlink HetNet analysis. The analysis has been extended to τi < 1 in [29, 71] using two entirely different approaches. Naturally, the resulting expressions in both the cases are not as nice as the ones that we get under the assumption τi > 1, ∀i. The extension in [71] is based on deriving the joint distribution of interference and maximum signal strength using tools from [72]. On the other hand, the extension in [29] is based on deriving the k-coverage result for cellular networks.

6.3

General Spatial Models

BSs are not independently placed in practice, but the results given here can in principle be generalized to point processes that model repulsion or minimum distance, such as determinantal [73] and Matérn processes [74]. Other recent results from Haenggi indicate that a fixed positive SINR shift (e.g., about 2 or 3 dB) can be applied to the results obtained under the PPP assumption to yield accurate results for other more general point processes or even the hexagonal grid [23]. On the same lines as above, the BS locations across tiers in a HetNet also exhibit dependence. For instance, an operator will likely not deploy a picocell very close to a macrocell. This repulsion has been modeled in [75] using a Poisson Hole Process (PHP) [76, 77]. The idea is to carve out holes around the macro BS locations in which certain types of small cells, such as picocells, cannot be deployed. The small cell locations in such a setup are modeled by a PHP where the holes are driven by the macrocell locations. Finally, throughout this text, we assumed that the user locations are independent of the BS locations. While this is a preferred case for the analysis of coverage-centric deployments, as was the case in conventional macro-only networks, this is not quite accurate for the analysis of capacity-centric deployments, where the BSs are deployed at the areas of high user density (and hence high data traffic). In such cases, it is important to model correlation in the BS and user locations. This correlation has recently been modeled in [78–82] using a Poisson cluster process (PCP) model where the small cells are assumed to lie at the cluster center around which the mobile users form clusters with a general distribution. The analysis is enabled by new distance distributions and analytical tools recently presented in [42, 83–87]. On similar lines, if small cell deployments exhibit clustering, they can also be modeled using PCP, as done in [88]. Two more examples of general spatial models will appear in Sect. 6.7.

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6.4

6 Extensions

Multiple-Input Multiple-Output (MIMO)

While this book assumed single-antenna per wireless node (both BSs and users), the analysis has already been extended in several important ways in the literature to incorporate multiple antennas per BSs and/or users. One fundamental consideration in networks with multiantenna nodes is that the received power distribution from a given wireless node (BS or a user) will be different depending upon whether that node is a serving node or an interfering node (due to the beamforming or the precoding gain). This consideration alone complicates the analysis of MIMO HetNets significantly. For instance, as discussed in detail in [89] for the downlink MIMO HetNets, if a BS equipped with m antennas serves a single user per resource block, it can be shown under Rayleigh fading and zero-forcing precoding that the fading component of the effective channel power gain for the serving link is Gamma distributed with shape parameter m and scale parameter 1. The fading component of the channel power gain from similar interfering BSs can be shown to be exponentially distributed. Therefore, the effect of some MIMO transmission techniques can be effectively captured by considering different effective fading distributions for the serving and interfering links. As discussed above, under Rayleigh fading, the channel power distribution of the serving link can be modeled as Gamma, which as discussed in Sect. 6.1 will reduce the coverage probability to a higher order derivative of the Laplace transform of interference. While this derivative can be evaluated using Faa di Bruno’s rule along with Bell polynomials, as done in [89, 90], exploiting the recursive structure of the expression, the coverage probability can also be expressed as a function of lower triangular Toeplitz matrix as presented in [91]. Similar approach has been followed in the performance analysis of multi-tier cellular systems in [41].

6.5

Spectrum and Resource Sharing

Stochastic geometry can also be used to analyze systems consisting of multiple networks coexisting together and sharing various resources including spectrum and infrastructure [92]. With ultra densification of networks, growing market fluidity, and its dynamic nature, service providers are opting to share more resources besides spectrum sharing to reduce operating costs. One such example is infrastructure sharing which includes sharing of tower sites, base-station services, backhaul, and cell towers [93]. This is a common practice where sites are costly and owned by a third party and multiple operators just lease these sites. The superposition property of PPPs is the key property enabling us to model the coexistence of independent multiple networks including networks owned by different operators [94] and even networks using different communication paradigms, via superposition of these networks similar to the approach used in Chap. 4. To explore opportunity of spectrum sharing, interference characterization and the impact of mutual interference on the performance of each operator can help us quantify the spatial and temporal gaps in the utilization of the

6.6

Millimeter-Wave and TeraHertz

77

spectrum. Readers can refer to [95] for a comprehensive survey of works that use stochastic geometry to analyze cognitive radio networks. To include networks which have correlated deployments, other PPs can be used to model co-existence of these networks as described in Sect. 6.3. For example, fully or partially co-located PPPs [96] can be used to model sitesharing among multiple operators. Works [69, 94, 97] have also considered operators sharing the radio access network and network core, including the connection-access to their BSs and compared the impact of closed and open access on the user performance.

6.6

Millimeter-Wave and TeraHertz

The stochastic geometry tools in this book are also useful for analyzing wireless systems operating at mmWave and TeraHertz (THz) frequencies—roughly above 30 GHz—as first done in [98] and summarized in [99] with a recent extension to THz in [100]. These highfrequency systems exhibit two fundamental differences compared to sub-6 GHz systems. The first difference is that the blocking of wireless signals as a result of various obstacles, such as buildings and foliage, is much more severe in mmWave systems. This necessitates the separate modeling of Line-of-Sight (LOS) and non-LOS (NLOS) links. For tractability, it is often assumed that the wireless link to a given BS is either LOS or NLOS with a certain probability independent of the state of the links to other BSs. The probability with which a BS is NLOS (also termed blocking probability) is dependent on the distance between the BS and the receiver of interest. The effect of blocking can be modeled using random shape theory [101] or a LOS ball model [98, 102]. Ultimately, given the typical user, the BS locations can be modeled as a superposition of two non-homogeneous PPPs, each modeling the LOS and NLOS BSs. The second key difference is that mmWave systems will have large antenna arrays (each element being very small) which result in highly directional communication. Hence, Nakagami-m fading with a large degree of freedom is more suitable for modeling mmWave channels compared to Rayleigh fading [98]. The non-homogeneity of the BS PPP process (consequence of blocking) and general Nakagami-m fading adds significant complexity to the analysis. The analysis can be extended to multi-tier mmWave systems [103] and the related multi-operator mmWave systems [94, 104] through the superposition of more than one BS PPP. Highly directional communication also offers spatial/temporal opportunities for spectrum sharing, possibly under cognition which can be investigated using stochastic geometry approaches [104, 105]. As we increase the frequencies further and start focusing on sub-THz and THz systems, additional propagation characteristics, specifically atmospheric absorption and scattering loss, also become important [106]. Therefore, the analysis of mmWave systems can be extended to the THz frequencies by incorporating these additional effects in the analytical models [100, 107–109].

78

6.7

6 Extensions

Modern Communication Paradigms Including Beyond-5G and 6G

Apart from conventional communication systems operating at radio frequencies, stochastic geometry can also be used to analyze modern communication technologies such as visible light communication [110–113], molecular communications [114, 115], geolocation [116–119], unmanned aerial vehicles (UAV)-assisted wireless networks [120–123], multiBS communication [124–126] including broadcast networks [127], reconfigurable intelligent surface (RIS) assisted communication [128–132], and applications based on radar, joint sensing and communication [133, 134], to name a few. In addition, emerging applications have also led to significant enhancement in the basic model described in this book. For instance, the 2D PPP model that was the basis of this book cannot be directly used for modeling the positions of vehicles on an irregular layout of roads or the positions of satellites in a massive constellation. For modeling vehicular networks, there has been a significant recent interest in modeling the road layouts with a Poisson line process and the positions of the vehicles on each road (line) as a point process [135–142]. In other words, the vehicles are modeled by a Cox process whose intensity is driven by a Poisson line process modeling the roads. Many variants of this basic setup, including an ad hoc setup, a cellular setup including just roadside units, as well as a cellular setup additionally including macrocellular networks have been considered. Likewise, the analysis of massive satellite constellations requires one to consider a much larger spatial scale than the models discussed in this book. At these large spatial scales, it is not appropriate to model transmitters and receivers on a plane. Recent works have focused on modeling these networks as 3D spherical networks in which the satellites are distributed according to a point process on a larger sphere (or multiple spheres in case of multi-altitude setups), whereas the users are distributed on a smaller concentric sphere (modeling the Earth’s surface) [143–147].

6.8

Parting Remarks

As is evident from the above discussion, this area is still expanding at a remarkable rate with many outstanding results being published every year. While the purpose of this chapter was not to provide a comprehensive overview of all the results, we hope to have provided enough pointers for curious readers to start exploring cutting-edge research in the topics and applications of their interest. More importantly, we hope that the accessible but rigorous presentation of the foundational results in the previous chapters will help new readers ease into this area and develop a deeper understanding and appreciation of the tools that have facilitated these analyses.

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