Differential and Difference Equations with Applications in Queueing Theory - Aliakbar Montazer Haghighi

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Résumé :
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Biographie:

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

Exercises 437
Appendix ...

Sommaire:

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 ?-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 = e^t or t = ln