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
1.4.1 The Chain Rule for Real Multivariable Functions 4
1.5 Function of a Complex Variable 7
1.5.1 Complex Numbers and Their Properties 7
1.5.2 Properties of a Complex Variable z 9
1.5.3 Complex Variables and Functions of Complex Variables 10
1.5.4 Some Particular Functions of Complex Variables 12
1.6 Differentiation of Functions of Complex Variables 12
1.6.1 Partial Differentiation of Functions of Complex Variables 13
1.7 Vectors 15
1.7.1 Dot (or Scalar or Inner) Product of Vectors and Some of Its Properties 17
1.7.2 The Cross Product (or Vector Product) of Vectors and Some of Its Properties 18
1.7.3 Directional Derivatives and Gradient Vectors 19
1.7.4 Eigenvalues and Eigenvectors 24
Exercises 25
2 Transforms 31
2.1 Introduction 31
2.2 Fourier Series 32
2.3 Convergence of Fourier Series 39
2.4 Fourier Transform 40
2.4.1 Continuous Fourier Transform 44
2.4.2 Discrete Fourier Transform 48
2.4.3 Some Properties of a Fourier Transform 48
2.4.4 Fast Fourier Transform 49
2.5 Laplace Transform 50
2.5.1 Properties of Laplace Transform 51
2.5.1.1 Linearity 51
2.5.1.2 Existence of Laplace Transform 52
2.5.1.3 Uniqueness of the Laplace Transforms 53
2.5.1.4 The First Shifting or s-Shifting 54
2.5.1.5 Time Delay 54
2.5.1.6 Laplace Transform of Derivatives 56
2.5.1.7 Laplace Transform of Integral 56
2.5.1.8 The Second Shifting or t-Shifting Theorem 57
2.5.1.9 Laplace Transform of Convolution of Two Functions 59
2.5.2 Partial Fraction and Inverse Laplace Transform 63
2.6 Integral Transform 68
2.7 Ƶ-Transform 69
Notes 70
Exercises 75
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
3.4.1 Method of Solving Separable Ordinary Differential Equations 90
3.5 Linear Ordinary Differential Equations 98
3.5.1 Method of Solving a Linear First-Order Differential Equation 99
3.6 Exact Ordinary Differential Equations 102
3.7 Solution of the First ODE by Substitution Method 112
3.7.1 Substitution Method 113
3.7.2 Reduction to Separation of Variables 116
3.8 Applications of the First-Order ODEs 117
3.9 Second-Order Homogeneous Ordinary Differential Equation 122
3.9.1 Solution of the Homogenous Second-Order Homogeneous Ordinary Differential Equation with Constant Coefficients, Equation (3.9.3) 123
3.10 The Second-Order Nonhomogeneous Linear Ordinary Differential Equation with Constant Coefficients 138
3.10.1 Method of Undetermined Coefficients 140
3.10.2 Variation of Parameters Method 147
3.11 Laplace Transform Method 150
3.12 Cauchy–Euler Equation Differential Equation 157
3.12.1 The Second-Order Homogenous Cauchy–Euler Equation 157
3.12.2 Solving the Second-Order Homogeneous Cauchy–Euler Equation Using x = et or t = ln |x| 158
3.13 Elimination Method to Solve Differential Equations 160
3.14 Solution of Linear ODE Using Power Series 163
Exercises 168
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
4.5.1 Methods of Finding Solution for a Partial Differential Equation 182
4.6 Linear, Semi-linear, and Quasi-linear Partial Differential Equations 184
4.6.1 Examples and Solutions of One- and Two-Dimensional Linear and Quasi-linear Partial Differential Equations of the First, Second, and Third Order 188
4.6.2 Characteristics Equation Method with Steps 189
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
Exercises 219
5 Differential Difference Equations 223
5.1 Introduction 223
5.2 Basic Terms 225
5.3 Linear Homogeneous Difference Equations with Constant Coefficients 228
5.3.1 Recursive Method 229
5.3.2 Characteristic Equation Method 230
5.4 Linear Nonhomogeneous Difference Equations with Constant Coefficients 235
5.4.1 Characteristic Equation Method 236
5.4.1.1 Case 1: a = 1 236
5.4.1.2 Case 2: a ≠ 1 237
5.4.1.3 Case 3:a=−1 238
5.4.1.4 Case 4: a > 1 239
5.4.1.5 Case 5: 0 < a < 1 239
5.4.1.6 Case 6: −1 < a < 0 240
5.4.1.7 Case 7: a < − 1 240
5.4.1.8 Case 8: a ≠ 1, c = b/(1 − a) 240
5.4.2 Recursive Method 241
5.4.3 Solving Differential Equations by Difference Equations 245
5.5 System of Linear Difference Equations 247
5.5.1 Recursive Method 247
5.5.2 Generating Functions Method 248
5.6 Differential-Difference Equations 255
5.6.1 Recursive Method 256
5.6.2 Generating Function Method 257
5.7 Nonlinear Difference Equations 260
Exercises 265
6 Probability and Statistics 269
6.1 Introduction and Basic Definitions and Concepts of Probability 269
6.1.1 Axioms of Probabilities of Events 271
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
6.10.1 Measures of Central Tendency 323
6.10.2 Measure of Dispersion 324
6.10.3 Properties of Sample Statistics 326
6.11 Inferential Statistics 331
6.11.1 Point Estimation 331
6.11.2 Interval Estimation 335
6.12 Hypothesis Testing 338
6.13 Reliability 341
Exercises 344
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
7.5.1 Transient Queue Length Distribution for M/M/ 1 378
7.5.2 Stationary Queue Length Distribution for M/M/ 1 382
7.5.3 Stationary Waiting Time of a Task in M/M/1 Queue 387
7.5.4 Distribution of a Busy Period for M/M/1 Queue 388
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
7.9.1 Transient Queue Length Distribution for M/M/m 399
7.9.2 Stationary Queue Length Distribution for M/M/m 406
7.9.3 Stationary Waiting Time of a Task in M/M/m Queue 408
7.10 Many-Server Parallel Queues with Feedback 411
7.10.1 Introduction 411
7.10.2 Stationary Distribution of the Queue Length 412
7.10.3 Stationary Waiting Time of a Task in Many-Server Queue with Feedback 412
7.11 Many-Server Queues with Balking and Reneging 414
7.11.1 Priority M/M/2 with Constant Balking and Exponential Reneging 414
7.11.2 M/M/m with Constant Balking and Exponential Reneging 417
7.11.3 Distribution of the Queue Length for M/M/m System with Constant Balking and Exponential Reneging 418
7.12 Single-Server Markovian Queueing System with Splitting and Delayed Feedback 420
7.12.1 Description of the Model 420
7.12.2 Analysis 421
7.12.3 Computation of Expected Values of the Queue Length and Waiting Time at Each Station, Algorithmically 426
7.12.4 Numerical Example 434
7.12.5 Discussion and Conclusion 436
Exercises 437
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
About the Author :
Aliakbar Montazer Haghighi, PhD, is Regent Professor, Professor, and former Head of the Department of Mathematics at Prairie View A&M University. He’s the Co-founder and Founding Editor-in-Chief of Applications and Applied Mathematics: An International Journal (AAM).
Dimitar P. Mishev, PhD, is Professor in the Department of Mathematics at Prairie View, A&M University. His research is focused on differential and difference equations and queueing theory.
