A newly updated and authoritative exploration of differential and difference equations used in queueing theory
In the newly revised second edition of Differential and Difference Equations with Applications in Queueing Theory, a team of distinguished researchers delivers an up-to-date discussion of the unique connections between the methods and applications of differential equations, difference equations, and Markovian queues. The authors provide a deep exploration of first principles and a wide variety of examples in applied mathematics and engineering and stochastic processes.
This book demonstrates the wide applicability of queuing theory in a range of fields, including telecommunications, traffic engineering, computing, and facility design. It contains brand-new information on partial differential equations as a prerequisite for solving queueing models, as well as sample MATLAB code for addressing these models.
Readers will also find:
- A large collection of new examples and enhanced end-of-chapter problems with included solutions
- Comprehensive explorations of single-server, multiple-server, parallel, and series queue models
- Practical discussions of splitting, delayed-service, and delayed feedback
- Enhanced treatments of concepts queueing theory, accessible across engineering and mathematics
Perfect for junior and up undergraduate, as well as graduate students in electrical and mechanical engineering, Differential and Difference Equations with Applications in Queueing Theory will also benefit students of computer science, mathematics, and applied mathematics.
Table of Contents:
About the Authors xiii
Preface to the Second Edition xv
1 Introduction 1
1.1 Introduction 1
1.2 Functions of a Real Variable 1
1.3 Some Properties of Differentiable Functions 3
1.4 Functions of More Than One Real Variable 3
1.5 Function of a Complex Variable 7
1.6 Differentiation of Functions of Complex Variables 12
1.7 Vectors 15
2 Transforms 31
2.1 Introduction 31
2.2 Fourier Series 32
2.3 Convergence of Fourier Series 39
2.4 Fourier Transform 40
2.5 Laplace Transform 50
2.6 Integral Transform 68
2.7 Z-Transform 69
3 Ordinary Differential Equations 81
3.1 Introduction and History of Ordinary Differential Educations 81
3.2 Basics Concepts and Definitions 81
3.3 Existence and Uniqueness 87
3.4 Separable Equations 89
3.5 Linear Ordinary Differential Equations 98
3.6 Exact Ordinary Differential Equations 102
3.7 Solution of the First ODE by Substitution Method 112
3.8 Applications of the First-Order ODEs 117
3.9 Second-Order Homogeneous Ordinary Differential Equation 122
3.10 The Second-Order Nonhomogeneous Linear Ordinary Differential Equation with Constant Coefficients 138
3.11 Laplace Transform Method 150
3.12 Cauchy–Euler Equation Differential Equation 157
3.13 Elimination Method to Solve Differential Equations 160
3.14 Solution of Linear ODE Using Power Series 163
4 Partial Differential Equations 173
4.1 Introduction 173
4.2 Basic Terminologies for Partial Differential Equations 174
4.3 Some Particular Functions Used in Partial Differential Equations 176
4.4 Types of Boundary Conditions for a Partial Differential Equation 178
4.5 Solution for a Partial Differential Equation 181
4.6 Linear, Semi-linear, and Quasi-linear Partial Differential Equations 184
4.7 Solution of Wave Partial Differential Equation, First and Second Orders, with Different Methods 197
4.8 A One-Dimensional, Second-Order Heat (or Parabolic) Equations 211
5 Differential Difference Equations 223
5.1 Introduction 223
5.2 Basic Terms 225
5.3 Linear Homogeneous Difference Equations with Constant Coefficients 228
5.4 Linear Nonhomogeneous Difference Equations with Constant Coefficients 235
5.5 System of Linear Difference Equations 247
5.6 Differential-Difference Equations 255
5.7 Nonlinear Difference Equations 260
6 Probability and Statistics 269
6.1 Introduction and Basic Definitions and Concepts of Probability 269
6.2 Discrete Random Variables and Probability Distribution Functions 275
6.3 Moments of a Discrete Random Variable 283
6.4 Continuous Random Variables 287
6.5 Moments of a Continuous Random Variable 291
6.6 Continuous Probability Distribution Functions 293
6.7 Random Vector 307
6.8 Continuous Random Vector 312
6.9 Functions of a Random Variable 314
6.10 Basic Elements of Statistics 317
6.11 Inferential Statistics 331
6.12 Hypothesis Testing 338
6.13 Reliability 341
7 Queueing Theory 355
7.1 Introduction 355
7.2 Markov Chain and Markov Process 357
7.3 Birth and Death Process 369
7.4 Introduction to Queueing Theory 371
7.5 Single-Server Markovian Queue, M/M/1 374
7.6 Finite Buffer Single-Server Markovian Queue: M/M/1/N 390
7.7 M/M/1 Queue with Feedback 394
7.8 Single-Server Markovian Queue with State-Dependent Balking 395
7.9 Multiserver Parallel Queue 398
7.10 Many-Server Parallel Queues with Feedback 411
7.11 Many-Server Queues with Balking and Reneging 414
7.12 Single-Server Markovian Queueing System with Splitting and Delayed Feedback 420
Appendix 443
The Poisson Probability Distribution 443
The Chi-Square Distribution 447
The Standard Normal Probability Distribution 449
The (Student) t Probability Distribution 451
Bibliography 453
Answers/Solutions to Selected Exercises 461
Index 469
