Difference and Differential Equations with Applications in Queueing Theory

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, delayed-service, and delayed feedback for single-server, multiple-server, 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 upper-undergraduate and graduate-level 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 First-Order ODE by Substitution 163

3.7. Applications of the First-order ODEs 169

3.8. Second-Order Homogeneous Differential Equations 175

3.9. The Second-Order Nonhomogeneous Linear ODE with Constant Coefficients 186

3.10. Miscellaneous Methods for Solving ODE 200

3.11. Applications of the Second-Order 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. Differential-Difference 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. Birth-and-Death (B-D) Processes. 299

5.4. Introduction of Queueing Theory 302

5.5. Single-server Markovian Queue, M/M/1 304

5.6. Finite Buffer Single-Server Markovian Queue, M/M/1/N 321

5.7. M/M/1 Queue with Feedback 326

5.8. Single-server Markovian Queue with State-Dependent Balking 327

5.9. Multi-server Parallel Queue 331

5.10. Many-Server Parallel Queue with Feedback 345

5.11. Many-Server Queue with Balking and Reneging 347

5.12. Single-server 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 Editor-in-Chief 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.