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• Introduction to Queueing Theory andStochastic Teletraffic Models

by Moshe Zukerman

Preface

The aim of this textbook is to provide students with basic knowledge of stochastic models thatmay apply to telecommunications research areas, such as traffic modelling, resource provisioningand traffic management. These study areas are often collectively called teletraffic. This bookassumes prior knowledge of a programming language, mathematics, probability and stochasticprocesses normally taught in an electrical engineering course. For students who have some butnot sufficiently strong background in probability and stochastic processes, we provide, in thefirst few chapters, a revision of the relevant concepts in these areas.

The book aims to enhance intuitive and physical understanding of the theoretical concepts itintroduces. The famous mathematician Pierre-Simon Laplace is quoted to say that Probabilityis common sense reduced to calculation [13]; as the content of this book falls under the fieldof applied probability, Laplaces quote very much applies. Accordingly, the book aims to linkintuition 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 involvingcomputer 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 effectsand principles of telecommunications systems modelling. Although the book, at times, pro-vides intuitive explanations, it still presents the important concepts and ideas required for theunderstanding of teletraffic, queueing theory fundamentals and related queueing behavior oftelecommunications networks and systems. These concepts and ideas form a strong base forthe more mathematically inclined students who can follow up with the extensive literatureon probability models and queueing theory. A small sample of it is listed at the end of thisbook.

As mentioned above, the first two chapters provide a revision of probability and stochasticprocesses topics relevant to the queueing and teletraffic models of this book. The contentof these chapters is mainly based on [13, 24, 70, 75, 76, 77]. These chapters are intended forstudents who have some background in these topics. Students with no background in probabilityand stochastic processes are encouraged to study the original textbooks that include far moreexplanations, illustrations, discussions, examples and homework assignments. For students withbackground, we provide here a summary of the key topics with relevant homework assignmentsthat are especially tailored for understanding the queueing and teletraffic models discussed in

• Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 2

later chapters. Chapter 3 discusses general queueing notation and concepts and it should bestudied 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 notavailable. An important learning objective of this book is to train students to perform queueingsimulations. Chapter 5 provides analyses of deterministic queues. Many queueing theory bookstend to exclude deterministic queues; however, the study of such queues is useful for beginnersin that it helps them better understand non-deterministic queueing models. Chapters 6 14provide analyses of a wide range of queueing and teletraffic models most of which fall underthe category of continuous-time Markov-chain processes. Chapter 15 provides an example ofa discrete-time queue that is modelled as a discrete-time Markov-chain. In Chapters 16 and17, various aspects of a single server queue with Poisson arrivals and general service timesare studied, mainly focussing on mean value results as in [12]. Then, in Chapter 18, someselected results of a single server queue with a general arrival process and general service timesare provided. Next, 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 withspecial focus on their characteristics that affect queueing performance.

Throughout the book there is an emphasis on linking the theory with telecommunicationsapplications as demonstrated by the following examples. Section 1.19 describes how propertiesof Gaussian distribution can be applied to link dimensioning. Section 6.6 shows, in the context ofan M/M/1 queueing model, how optimally to set a link service rate such that delay requirementsare met and how the level of multiplexing affects the spare capacity required to meet such delayrequirement. An application of M/M/ queueing model to a multiple access performanceproblem [12] is discussed in Section 7.6. In Sections 8.6 and 9.5, discussions on dimensioningand related utilization issues of a multi-channel system are presented. Especially important isthe emphasis on the insensitivity property of models such as M/M/, M/M/k/k, processorsharing and multi-service that lead to practical and robust approximations as described inSections 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 providedcomments and questions that helped developing and editing the manuscript over the years.

• Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 3

Contents

1 Revision of Relevant Probability Topics 81.1 Events, Sample Space, and Random Variables . . . . . . . . . . . . . . . . . . . 81.2 Probability, Conditional Probability and Independence . . . . . . . . . . . . . . 91.3 Probability and Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 101.4 Joint Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Conditional Probability for Random Variables . . . . . . . . . . . . . . . . . . . 111.6 Independence between Random Variables . . . . . . . . . . . . . . . . . . . . . 121.7 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Selected Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8.1 Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.2 Geometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.3 Binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.8.4 Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8.5 Pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.9 Continuous Random Variables and their Probability Functions . . . . . . . . . . 221.10 Selected Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . 25

1.10.1 Uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.10.2 Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.10.3 Relationship between Exponential and Geometric Random Variables . . 271.10.4 Hyper-Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.10.5 Erlang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.10.6 Hypo-Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.10.7 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.10.8 Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.11 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.12 Mean and Variance of Specific Random Variable . . . . . . . . . . . . . . . . . . 331.13 Sample Mean and Sample Variance . . . . . . . . . . . . . . . . . . . . . . . . . 371.14 Covariance and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.15 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.15.1 Z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.15.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.16 Multivariate Random Variables and Transform . . . . . . . . . . . . . . . . . . . 471.17 Probability Inequalities and Their Dimensioning Applications . . . . . . . . . . 471.18 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.19 Link Dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.19.1 Case 1: Homogeneous Individual Sources . . . . . . . . . . . . . . . . . . 511.19.2 Case 2: Non-homogeneous Individual Sources . . . . . . . . . . . . . . . 521.19.3 Case 3: Capacity Dimensioning for a Community . . . . . . . . . . . . . 53

2 Relevant Background in Stochastic Processes 552.1 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2 Two Orderly and Memoryless Point Processes . . . . . . . . . . . . . . . . . . . 58

2.2.1 Bernoulli Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2.2 Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3 Markov Modulated Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . 66

• Queueing Theory and Stochastic Teletraffic Models c Moshe Zukerman 4

2.4 Discrete-time Markov-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.4.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 672.

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