Introduction to Queueing Theory and Stochastic Teletra±c Models by Moshe Zukerman Copyright M. Zukerman °c 2000{2012 Preface The aim of this textbook is to provide students with basic knowledge of stochastic models that may apply to telecommunications research areas, such as tra±c modelling, resource provisioning and tra±c management. These study areas are often collectively called teletra±c. This book assumes prior knowledge of a programming language, mathematics, probability and stochastic processes normally taught in an electrical engineering course. For students who have some but not su±ciently strong background in probability and stochastic processes, we provide, in the ¯rst few chapters, a revision of the relevant concepts in these areas. 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" [13]; as the content of this book falls under the ¯eld 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. A unique feature of this book is the considerable attention given to guided projects involving computer simulations and analyzes. By successfully completing the programming assignments, students learn to simulate and analyze stochastic models, such as queueing systems and net- works, and by interpreting the results, they gain insight into the queueing performance e®ects and principles of telecommunications systems modelling. Although the book, at times, pro- vides intuitive explanations, it still presents the important concepts and ideas required for the understanding of teletra±c, 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 book. As mentioned above, the ¯rst two chapters provide a revision of probability and stochastic processes topics relevant to the queueing and teletra±c models of this book. The content of these chapters is mainly based on [13, 24, 70, 75, 76, 77]. These chapters are intended for students who have some background in these topics. Students with no background in probability and stochastic processes are encouraged to study the original textbooks that include far more explanations, illustrations, discussions, examples and homework assignments. For students with background, we provide here a summary of the key topics with relevant homework assignments that are especially tailored for understanding the queueing and teletra±c models discussed in Queueing Theory and Stochastic Teletra±c Models °c Moshe Zukerman 2 later chapters. Chapter 3 discusses general queueing notation and concepts and it should be studied well. 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 teletra±c 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 Chapters 16 and 17, various aspects of a single server queue with Poisson arrivals and general service times are studied, mainly focussing on mean value results as in [12]. Then, in Chapter 18, some selected results of a single server queue with a general arrival process and general service times are provided. Next, in Chapter 19, we extend our discussion to queueing networks. Finally, in Chapter 20, stochastic processes that have been used as tra±c models are discussed with special focus on their characteristics that a®ect queueing performance. Throughout the book there is an emphasis on linking the theory with telecommunications applications as demonstrated by the following examples. Section 1.19 describes how properties of Gaussian distribution can be applied to link dimensioning. Section 6.6 shows, in the context 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 a®ects the spare capacity required to meet such delay requirement. An application of M/M/1 queueing model to a multiple access performance problem [12] is discussed in Section 7.6. In Sections 8.6 and 9.5, discussions on dimensioning and related utilization issues of a multi-channel system are presented. Especially important is the emphasis on the insensitivity property of models such as M/M/1, M/M/k/k, processor sharing and multi-service that lead to practical and robust approximations as described in Sections 7, 8, 13, and 14. Section 19.3 guides the reader to simulate a mobile cellular network. Section 20.6 describes a tra±c 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 Teletra±c Models °c Moshe Zukerman 3 Contents 1 Revision of 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 . 11 1.6 Independence between Random Variables . 12 1.7 Convolution . 12 1.8 Selected Discrete Random Variables . 16 1.8.1 Bernoulli . 16 1.8.2 Geometric . 16 1.8.3 Binomial . 17 1.8.4 Poisson . 19 1.8.5 Pascal . 22 1.9 Continuous Random Variables and their Probability Functions . 22 1.10 Selected Continuous Random Variables . 25 1.10.1 Uniform . 25 1.10.2 Exponential . 26 1.10.3 Relationship between Exponential and Geometric Random Variables . 27 1.10.4 Hyper-Exponential . 28 1.10.5 Erlang . 28 1.10.6 Hypo-Exponential . 29 1.10.7 Gaussian . 30 1.10.8 Pareto . 30 1.11 Moments . 31 1.12 Mean and Variance of Speci¯c Random Variable . 33 1.13 Sample Mean and Sample Variance . 37 1.14 Covariance and Correlation . 37 1.15 Transforms . 40 1.15.1 Z-transform . 43 1.15.2 Laplace Transform . 45 1.16 Multivariate Random Variables and Transform . 47 1.17 Probability Inequalities and Their Dimensioning Applications . 47 1.18 Limit Theorems . 49 1.19 Link Dimensioning . 51 1.19.1 Case 1: Homogeneous Individual Sources . 51 1.19.2 Case 2: Non-homogeneous Individual Sources . 52 1.19.3 Case 3: Capacity Dimensioning for a Community . 53 2 Relevant Background in Stochastic Processes 55 2.1 General Concepts . 55 2.2 Two Orderly and Memoryless Point Processes . 58 2.2.1 Bernoulli Process . 59 2.2.2 Poisson Process . 60 2.3 Markov Modulated Poisson Process . 66 Queueing Theory and Stochastic Teletra±c Models °c Moshe Zukerman 4 2.4 Discrete-time Markov-chains . 67 2.4.1 De¯nitions and Preliminaries . 67 2.4.2 Transition Probability Matrix . 67 2.4.3 Chapman-Kolmogorov Equation . 68 2.4.4 Marginal Probabilities . 68 2.4.5 Properties and Classi¯cation of States . 69 2.4.6 Steady-State Probabilities . 71 2.4.7 Birth and Death Process . 73 2.4.8 Reversibility . 74 2.4.9 Multi-Dimensional Markov-chains . 75 2.5 Continuous Time Markov-chains . 76 2.5.1 De¯nitions and Preliminaries . 76 2.5.2 Birth and Death Process . 77 2.5.3 First Passage Time . 78 2.5.4 Transition Probability Function . 79 2.5.5 Steady-State Probabilities . 79 2.5.6 Multi-Dimensional Continuous Time Markov-chains . 81 2.5.7 The Curse of Dimensionality . 81 2.5.8 Simulations . 81 2.5.9 Reversibility . 82 3 General Queueing Concepts 84 3.1 Notation . 84 3.2 Utilization . 85 3.3 Little's Formula . 85 3.4 Work Conservation . 88 3.5 PASTA . 88 3.6 Queueing Models . 89 4 Simulations 90 4.1 Con¯dence Intervals . 90 4.2 Simulation of a G/G/1 Queue . 91 5 Deterministic Queues 94 5.1 D/D/1 . 94 5.2 D/D/k ......................................... 95 5.3 D/D/k/k ....................................... 97 5.3.1 The D/D/k/k process and its cycles . 97 5.3.2 Blocking probability, mean queue-size and utilization . 98 5.3.3 Proportion of time spent in each state . 98 5.4 Summary of Results . 100 6 M/M/1 101 6.1 Steady-State Queue Size Probabilities . 101 6.2 State Transition Diagram of M/M/1 . 103 6.3 Delay Statistics . 103 6.4 Mean Delay of Delayed Customers . 105 6.5 Using Z-Transform . 107 Queueing Theory and Stochastic Teletra±c Models °c Moshe Zukerman 5 6.6 Multiplexing . 107 6.7 Dimensioning Based on Delay Distribution . ..