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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.
Contents
Preface xi
1. Probability and Statistics 1
1.1. Basic Definitions and Concepts of Probability 1
1.2. Discrete Random Variables and Probability Distribution Functions 7
1.3. Moments of a Discrete Random Variable 16
1.4. Continuous Random Variables 20
1.5. Moments of a Continuous Random Variable 25
1.6. Continuous Probability Distribution Functions 26
1.7. Random Vector 41
1.8. Continuous Random Vector 48
1.9. Functions of a Random Variable 49
1.10. Basic Elements of Statistics 53
1.10.1. Measures of Central Tendency 58
1.10.2. Measure of Dispersion 59
1.10.3. Properties of Sample Statistics 61
1.11. Inferential Statistics 67
1.11.1. Point Estimation 68
1.11.2. Interval Estimation 72
1.12. Hypothesis Testing 75
1.13. Reliability 78
Exercises 81
2. Transforms 90
2.1. Fourier Transform 90
2.2. Laplace Transform 94
2.3. Z-Transform 104
2.4. Probability Generating Function 111
2.4.1. Some Properties of a Probability Generating Function 112
Exercises 116
3. Differential Equations 121
3.1. Basic Concepts and Definitions 121
3.2. Existence and Uniqueness 130
3.3. Separable Equations 132
3.3.1. Method of Solving Separable Differential Equations 133
3.4. Linear Differential Equations 140
3.4.1. Method of Solving a Linear First-Order Differential Equation 141
3.5. Exact Differential Equations 144
3.6. Solution of the First ODE by Substitution Method 153
3.6.1. Substitution Method 154
3.6.2. Reduction to Separation of Variables 158
3.7. Applications of the First-Order ODEs 159
3.8. Second-Order Homogeneous ODE 164
3.8.1. Solving a Linear Homogeneous Second-Order Differential Equation 165
3.9. The Second-Order Nonhomogeneous Linear ODE with Constant Coefficients 175
3.9.1. Method of Undetermined Coefficients 178
3.9.2. Variation of Parameters Method 184
3.10. Miscellaneous Methods for Solving ODE 188
3.10.1. Cauchy-Euler Equation 188
3.10.2. Elimination Method to Solve Differential Equations 190
3.10.3. Application of Laplace Transform to Solve ODE 193
3.10.4. Solution of Linear ODE Using Power Series 195
3.11. Applications of the Second-Order ODE 199
3.11.1. Spring-Mass System: Free Undamped Motion 199
3.11.2. Damped-Free Vibration 200
3.12. Introduction to PDE: Basic Concepts 203
3.12.1. First-Order Partial Differential Equations 205
3.12.2. Second-Order Partial Differential Equations 208
Exercises 213
4. Difference Equations 218
4.1. Basic Terms 220
4.2. Linear Homogeneous Difference Equations with Constant Coefficients 224
4.3. Linear Nonhomogeneous Difference Equations with Constant Coefficients 231
4.3.1. Characteristic Equation Method 231
4.3.2. Recursive Method 237
4.4. System of Linear Difference Equations 244
4.4.1. Generating Functions Method 245
4.5. Differential-Difference Equations 253
4.6. Nonlinear Difference Equations 259
Exercises 264
5. Queueing Theory 267
5.1. Introduction 267
5.2. Markov Chain and Markov Process 268
5.3. Birth and Death (B-D) Process 281
5.4. Introduction to Queueing Theory 284
5.5. Single-Server Markovian Queue, M/M/ 1 286
5.5.1. Transient Queue Length Distribution for M/M/ 1 291
5.5.2. Stationary Queue Length Distribution for M/M/ 1 294
5.5.3. Stationary Waiting Time of a Task in M/M/1 Queue 300
5.5.4. Distribution of a Busy Period for M/M/1 Queue 300
5.6. Finite Buffer Single-Server Markovian Queue: M/M/1/N 303
5.7. M/M/1 Queue with Feedback 307
5.8. Single-Server Markovian Queue with State-Dependent Balking 308
5.9. Multiserver Parallel Queue 311
5.9.1. Transient Queue Length Distribution for M/M/m 312
5.9.2. Stationary Queue Length Distribution for M/M/m 320
5.9.3. Stationary Waiting Time of a Task in M/M/m Queue 323
5.10. Many-Server Parallel Queues with Feedback 326
5.10.1. Introduction 326
5.10.2. Stationary Distribution of the Queue Length 326
5.10.3. Stationary Waiting Time of a Task in Many-Server Queue with Feedback 327
5.11. Many-Server Queues with Balking and Reneging 328
5.11.1. Priority M/M/2 with Constant Balking and Exponential Reneging 328
5.11.2. M/M/m with Constant Balking and Exponential Reneging 332
5.11.3. Distribution of the Queue Length for M/M/m System with Constant Balking and Exponential Reneging 333
5.12. Single-Server Markovian Queueing System with Splitting and Delayed Feedback 334
5.12.1. Description of the Model 334
5.12.2. Analysis 336
5.12.3. Computation of Expected Values of the Queue Length and Waiting Time at each Station, Algorithmically 341
5.12.4. Numerical Example 349
5.12.5. Discussion and Conclusion 350
Exercises 353
Appendix 358
The Poisson Probability Distribution 358
The Chi-Square Distribution 361
The Chi-Square Distribution (continued) 362
The Standard Normal Probability Distribution 363
The Standard Normal Probability Distribution (continued) 364
The (Student's) t Probability Distribution 365
References and Further Readings 366
Answers/Solutions to Selected Exercises 372
Index 379
