Difference and Differential Equations with Applications in Queueing TheoryISBN: 9781118393246
424 pages
July 2013

A Useful Guide to the Interrelated Areas of Differential Equations, Difference Equations, and Queueing Models
Difference and Differential Equations with Applications in Queueing Theory presents the unique connections between the methods and applications of differential equations, difference equations, and Markovian queues. Featuring a comprehensive collection of topics that are used in stochastic processes, particularly in queueing theory, the book thoroughly discusses the relationship to systems of linear differential difference equations.
The book demonstrates the applicability that queueing theory has in a variety of fields including telecommunications, traffic engineering, computing, and the design of factories, shops, offices, and hospitals. Along with the needed prerequisite fundamentals in probability, statistics, and Laplace transform, Difference and Differential Equations with Applications in Queueing Theory provides:
 A discussion on splitting, delayedservice, and delayed feedback for singleserver, multipleserver, parallel, and series queue models
 Applications in queue models whose solutions require differential difference equations and generating function methods
 Exercises at the end of each chapter along with select answers
The book is an excellent resource for researchers and practitioners in applied mathematics, operations research, engineering, and industrial engineering, as well as a useful text for upperundergraduate and graduatelevel courses in applied mathematics, differential and difference equations, queueing theory, probability, and stochastic processes.
Dedication ix
Preface xii
ONE Probability and Statistics, Outlines 1
1.1. Basic Definitions and Concepts of Probability 1
1.2. Discrete Random Variable and Probability Distribution Functions 8
1.3. Moments of a Discrete Random Variable 18
1.4. Continuous Random variables 22
1.5. Moments of a Continuous Random Variable 28
1.6. Continuous Probability Distribution Functions 29
1.7. Random Vector 47
1.8. Continuous Random Vector 53
1.9. Functions of Random variables 54
1.10. Basic Elements of Statistics 59
1.11. Inferential Statistics 72
1.12. Hypothesis Testing 80
1.13. Reliability 82
Chapter One Exercises 87
TWO Transforms 96
2.1. Fourier Transform 96
2.2. Laplace Transform 100
2.3. Transform 111
2.4. Probability Generating Function 118
Chapter Two Exercises 124
THREE Differential Equations 129
3.1. Basics Concepts and Definitions 129
3.2. Existence and Uniqueness 138
3.3. Separable Equations 140
3.4. Linear Differential Equation 150
3.5. Exact Differential Equations 154
3.6. Solution of the FirstOrder ODE by Substitution 163
3.7. Applications of the Firstorder ODEs 169
3.8. SecondOrder Homogeneous Differential Equations 175
3.9. The SecondOrder Nonhomogeneous Linear ODE with Constant Coefficients 186
3.10. Miscellaneous Methods for Solving ODE 200
3.11. Applications of the SecondOrder ODE 211
3.12. Introduction to the Partial Differential Equations: Basic Concepts 216
Chapter Three Exercises 227
FOUR Difference Equations 234
4.1. Basic Terms 236
4.2. Linear Homogeneous Difference Equations with Constant Coefficient 239
4.3. Linear Nonhomogeneous Difference Equations with Constant Coefficient 247
4.4. System of Linear Difference equations 260
4.4.A. Generating Function Method 262
4.5. DifferentialDifference Equations 269
4.6. Nonlinear Difference Equations 275
Chapter Four Exercises 281
FIVE Queueing Theory 284
5.1. Introduction 284
5.2. Markov Chain and Markov Process 285
5.3. BirthandDeath (BD) Processes. 299
5.4. Introduction of Queueing Theory 302
5.5. Singleserver Markovian Queue, M/M/1 304
5.6. Finite Buffer SingleServer Markovian Queue, M/M/1/N 321
5.7. M/M/1 Queue with Feedback 326
5.8. Singleserver Markovian Queue with StateDependent Balking 327
5.9. Multiserver Parallel Queue 331
5.10. ManyServer Parallel Queue with Feedback 345
5.11. ManyServer Queue with Balking and Reneging 347
5.12. Singleserver Markovian Queuing System with Splitting and Feedback 354
Chapter Five Exercises 373
Appendix 379
References and Further Readings 386
Answers/Solutions to Selected Exercises 390
Authors Index 397
Subject Index 400
ALIAKBAR MONTAZER HAGHIGHI, PhD, is Professor and Head of the Department of Mathematics at Prairie View A&M University, as well as founder and EditorinChief of Applications and Applied Mathematics: An International Journal (AAM). Dr. Haghighi's research interests and publications are in the areas of probability, statistics, stochastic processes, and queueing theory.
DIMITAR P. MISHEV, PhD, is Professor in the Department of Mathematics at Prairie View A&M University. The author of numerous research papers and three books coauthored with Dr. Haghighi, Dr. Mishev's areas of research interest include differential and difference equations and queueing theory.