Introduction to Queueing Theory and Stochastic Teletraffic Models Moshe Zukerman EE Department City University of Hong Kong email: [email protected] Copyright M. Zukerman c 2000–2020. This book can be used for educational and research purposes under the condition that it (including this first page) is not modified in any way. Preface The aim of this textbook is to provide students with basic knowledge of stochastic models with a special focus on queueing models, that may apply to telecommunications topics, such as traffic modelling, performance evaluation, resource provisioning and traffic management. These topics are included in a field called teletraffic. This book assumes prior knowledge of a programming language and mathematics normally taught in an electrical engineering bachelor program. The book aims to enhance intuitive and physical understanding of the theoretical concepts it introduces. The famous mathematician Pierre-Simon Laplace is quoted to say that “Probability is common sense reduced to calculation” [18]; as the content of this book falls under the field of applied probability, Laplace’s quote very much applies. Accordingly, the book aims to link intuition and common sense to the mathematical models and techniques it uses. It mainly focuses on steady-state analyses and avoids discussions of time-dependent analyses. arXiv:1307.2968v23 [math.PR] 21 Feb 2020 A unique feature of this book is the considerable attention given to guided homework assign- ments involving computer simulations and analyzes. By successfully completing these assign- ments, students learn to simulate and analyze stochastic models, such as queueing systems and networks, and by interpreting the results, they gain insight into the queueing performance effects and principles of telecommunications systems modelling. Although the book, at times, provides intuitive explanations, it still presents the important concepts and ideas required for the understanding of teletraffic, queueing theory fundamentals and related queueing behavior of telecommunications networks and systems. These concepts and ideas form a strong base for the more mathematically inclined students who can follow up with the extensive literature on probability models and queueing theory. A small sample of it is listed at the end of this Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 2 book. The first two chapters provide background on probability and stochastic processes topics rele- vant to the queueing and teletraffic models of this book. These two chapters provide a summary of the key topics with relevant homework assignments that are especially tailored for under- standing the queueing and teletraffic models discussed in later chapters. The content of these chapters is mainly based on [18, 34, 90, 95, 96, 97]. Students are encouraged to study also the original textbooks for more explanations, illustrations, discussions, examples and homework assignments. Chapter 3 discusses general queueing notation and concepts. Chapter 4 aims to assist the student to perform simulations of queueing systems. Simulations are useful and important in the many cases where exact analytical results are not available. An important learning objective of this book is to train students to perform queueing simulations. Chapter 5 provides analyses of deterministic queues. Many queueing theory books tend to exclude deterministic queues; however, the study of such queues is useful for beginners in that it helps them better understand non-deterministic queueing models. Chapters 6 – 14 provide analyses of a wide range of queueing and teletraffic models most of which fall under the category of continuous- time Markov-chain processes. Chapter 15 provides an example of a discrete-time queue that is modelled as a discrete-time Markov chain. In Chapter 16, various aspects of a single server queue with Poisson arrivals and general service times are studied, mainly focussing on mean value results as in [17]. Then, in Chapter 17, some selected results of a single server queue with a general arrival process and general service times are provided. Chapter 18 focusses on multi access applications, and in Chapter 19, we extend our discussion to queueing networks. Finally, in Chapter 20, stochastic processes that have been used as traffic models are discussed with special focus on their characteristics that affect queueing performance. Throughout the book there is an emphasis on linking the theory with telecommunications ap- plications as demonstrated by the following examples. Section 1.19 describes how properties of Gaussian distribution can be applied to link dimensioning. Section 6.11 shows, in the con- text of an M/M/1 queueing model, how optimally to set a link service rate such that delay requirements are met and how the level of multiplexing affects the spare capacity required to meet such delay requirement. An application of M/M/ queueing model to a multiple access ∞ performance problem [17] is discussed in Section 7.5. Then later in Chapter 18 more multi- access models are presented. In Sections 8.8 and 9.5, discussions on dimensioning and related utilization issues of a multi-channel system are presented. Especially important is the empha- sis on the insensitivity property of models such as M/M/ , M/M/k/k, processor sharing and multi-service that lead to practical and robust approximations∞ as described in Chapters 7, 8, 13, and 14. Section 19.3 guides the reader to simulate a mobile cellular network. Section 20.6 describes a traffic model applicable to the Internet. Last but not least, the author wishes to thank all the students and colleagues that provided comments and questions that helped developing and editing the manuscript over the years. Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 3 Contents 1 Background on Relevant Probability Topics 8 1.1 Events, Sample Space, and Random Variables . .... 8 1.2 Probability, Conditional Probability and Independence . ...... 9 1.3 Probability and Distribution Functions . .. 10 1.4 Joint Distribution Functions . 11 1.5 Conditional Probability for Random Variables . ... 12 1.6 Independence between Random Variables . ..... 13 1.7 Convolution...................................... 13 1.8 Selected Discrete Random Variables . ... 18 1.8.1 Non-parametric ................................ 18 1.8.2 Bernoulli.................................... 18 1.8.3 Geometric................................... 18 1.8.4 Binomial.................................... 19 1.8.5 Poisson .................................... 21 1.8.6 Pascal ..................................... 24 1.8.7 DiscreteUniform ............................... 25 1.9 Continuous Random Variables and their Distributions . ..... 25 1.10 Selected Continuous Random Variables . .... 29 1.10.1 Uniform .................................... 29 1.10.2 Exponential .................................. 30 1.10.3 Relationship between Exponential and Geometric Random Variables . 33 1.10.4 Hyper-Exponential .............................. 34 1.10.5 Erlang..................................... 34 1.10.6 Hypo-Exponential............................... 35 1.10.7 Gaussian.................................... 35 1.10.8 Pareto ..................................... 36 1.11 MomentsandVariance ............................... 36 1.11.1 Mean(orExpectation) ............................ 36 1.11.2 Moments.................................... 40 1.11.3 Variance.................................... 41 1.11.4 Conditional Mean and the Law of Iterated Expectation . ..... 43 1.11.5 Conditional Variance and the Law of Total Variance . .... 44 1.12 Mean and Variance of Specific Random Variables . .... 51 1.13 SampleMeanandSampleVariance . 55 1.14 CovarianceandCorrelation . .. 56 1.15Transforms ...................................... 58 1.15.1 Z-transform .................................. 61 1.15.2 LaplaceTransform .............................. 63 1.16 Multivariate Random Variables and Transform . ..... 66 1.17 Probability Inequalities and Their Dimensioning Applications . .... 66 1.18LimitTheorems.................................... 68 1.19 Link Dimensioning . 69 1.19.1 Case 1: Homogeneous Individual Sources . ... 70 1.19.2 Case 2: Non-homogeneous Individual Sources . .... 71 1.19.3 Case 3: Capacity Dimensioning for a Community . 72 Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 4 2 Relevant Background on Stochastic Processes 73 2.1 GeneralConcepts.................................. 73 2.2 Two Orderly and Memoryless Point Processes . .... 76 2.2.1 BernoulliProcess ............................... 77 2.2.2 PoissonProcess ................................ 78 2.3 MarkovModulatedPoissonProcess . ... 87 2.4 Discrete-timeMarkovchains . .. 87 2.4.1 Definitions and Preliminaries . 87 2.4.2 Transition Probability Matrix . 88 2.4.3 Chapman-KolmogorovEquation. 88 2.4.4 Marginal Probabilities . 89 2.4.5 Properties and Classification of States . 89 2.4.6 Steady-state Probabilities . 91 2.4.7 BirthandDeathProcess ........................... 93 2.4.8 Reversibility . 94 2.4.9 Multi-dimensional Markov Chains . 95 2.5 ContinuousTimeMarkovchains. 96 2.5.1 Definitions and Preliminaries . 96 2.5.2 BirthandDeathProcess ........................... 97 2.5.3 FirstPassageTime .............................. 98 2.5.4 Transition Probability Function . 99 2.5.5 Steady-State Probabilities . 99 2.5.6 Multi-Dimensional Continuous-time Markov Chains . 101 2.5.7 Solutions by Successive Substitutions . 101 2.5.8 The Curse of Dimensionality . 102 2.5.9 Simulations ..................................102