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# Solutions Manual to accompany Fundamentals of Queueing Theory, 4e

ISBN: 978-0-470-07796-2 July 2008 88 Pages

## Description

Presents the basic statistical principles that are necessary to analyze the probabilistic nature of queues

Thoroughly revised and expanded to reflect the latest developments in the field, the fourth edition of Fundamentals of Queueing Theory illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research. It takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics include retrial queues, approximations for queueing networks, numerical inversion of transforms, and determining the appropriate number of servers to balance quality and cost of service.

Dedication v

Preface xi

Acknowledgments xiii

Introduction 1

Description of the Queueing Problem 2

Characteristics of Queueing Processes 3

Notation 7

Measuring System Performance 8

Some General Results 9

Simple Data Bookkeeping for Queues 12

Poisson Process and the Exponential Distribution 16

Markovian Property of the Exponential Distribution 20

Stochastic Processes and Markov Chains 24

Introduction to the QtsPlus Software 40

Problems 41

Simple Markovian Queueing Models 49

Birth-Death Processes 49

Single-Server Queues (M/M/1) 53

Multiserver Queues (M/M/c) 66

Choosing the Number of Servers 73

Queues with Truncation (M/M/c/K) 76

Erlang's Loss Formula (M/M/c/c) 81

Queues with Unlimited Service (M/M/[infinity]) 84

Finite-Source Queues 85

State-Dependent Service 91

Queues with Impatience 95

Transient Behavior 97

Busy-Period Analysis 102

Problems 103

Bulk Input (M[superscript X]/M/1) 117

Bulk Service (M/M[superscript Y]/1) 123

Erlangian Models 128

Priority Queue Disciplines 141

Retrial Queues 157

Problems 171

Networks, Series, and Cyclic Queues 179

Series Queues 181

Open Jackson Networks 187

Closed Jackson Networks 195

Cyclic Queues 209

Extensions of Jackson Networks 210

Non-Jackson Networks 212

Problems 214

General Arrival or Service Patterns 219

General Service, Single Server (M/G/1) 219

General Service, Multiserver (M/G/c/[infinity], M/G/[infinity]) 254

General Input (G/M/1, G/M/c) 259

Problems 270

General Models and Theoretical Topics 277

G/E[subscript k]/1, G[superscript k]/M/1, and G/PH[subscript k]/1 277

General Input, General Service (G/G/1) 284

Poisson Input, Constant Service, Multiserver (M/D/c) 294

Semi-Markov and Markov Renewal Processes in Queueing  296

Other Queue Disciplines 301

Design and Control of Queues 306

Statistical Inference in Queueing 317

Problems 325

Bounds and Approximations 329

Bounds 330

Approximations 343

Network Approximations 356

Problems 367

Numerical Techniques and Simulation 369

Numerical Techniques 369

Numerical Inversion of Transforms 385

Discrete-Event Stochastic Simulation 398

Problems 421

References 427

Symbols and Abbreviations 439

Tables 447

Transforms and Generating Functions 455

Laplace Transforms 455

Generating Functions 462

Differential and Difference Equations 467

Ordinary Differential Equations 467

Difference Equations 483

QtsPlus Software 489