Applied Probability and Stochastic Processes (eBook)

Richard M. Feldman is a Professor of Industrial and Systems Engineering at Texas A&M University. He received a B.A. degree in mathematics from Hope College, an M.S. degree in mathematics from Michigan State University, an M.S. degree in Industrial and Systems Engineering from Ohio University, and a Ph.D. in Industrial Engineering from Northwestern University. His teaching interests include simulation, applied probability, and queueing theory. His consulting and funded research activities have involved modeling and simulation within manufacturing, transportation, and biological contexts. He has received several teaching awards, published papers in applied probability and queueing theory, and co-authored four books. Ciriaco Valdez-Flores is senior risk assessment consultant at Sielken & Associates Consulting, Inc. He received a bachelor’s degree from the Tecnológico at Cd. Victoria in México and master’s and Ph.D. degrees from Texas A&M University, all in Industrial Engineering. He has taught graduate courses at Texas A&M University focusing in the areas of operations research and applied stochastic processes. As a consultant, he applies his background to the development of new methods of quantitative health risk assessment that incorporate simulation and decision tree theory. He has published in health risk assessment and stochastic processes, co-authored one book and has contributed to books in engineering economy and risk assessment.

Preface 6
Contents 9
Basic Probability Review 14
Basic Definitions 14
Random Variables and Distribution Functions 17
Mean and Variance 23
Important Distributions 26
Multivariate Distributions 36
Combinations of Random Variables 44
Fixed Sum of Random Variables 45
Random Sum of Random Variables 45
Mixtures of Random Variables 47
Appendix 48
Problems 50
References 56
Basics of Monte Carlo Simulation 57
Simulation by Hand 58
Generation of Random Numbers 61
Multiplicative Linear Congruential Generators 62
A Multiple Recursive Generator 64
Composite Generators 65
Generation of Random Variates 67
Discrete Random Variates 67
Continuous Random Variates 73
Bivariate Continuous Random Variates 76
Random Variates from Empirical Distributions 78
Appendix 79
Problems 81
References 84
Basic Statistical Review 85
Collection of Data 85
Preliminary Definitions 86
Graphical Representations 87
Parameter Estimation 90
Method of Moments 92
Maximum Likelihood Estimation 93
Confidence Intervals 94
Means 95
Proportions 99
Variances 99
Correlation Coefficient 101
Fitting Distributions 102
The Chi-Square Test 103
The Kolmogorov-Smirnov Test 107
Fitting Models 111
Ordinary Least Squares Regression 112
Maximum Likelihood Estimates for Linear Models 117
Appendix 117
Problems 121
References 125
Poisson Processes 126
Basic Definitions 126
Properties and Computations 129
Extensions of a Poisson Process 132
Compound Poisson Processes 132
Non-stationary Poisson Process 134
Poisson Regression 135
Poisson Regression with One Independent Variable 135
Poisson Regression with Several Independent Variables 139
Appendix 144
Problems 147
References 150
Markov Chains 151
Basic Definitions 152
Multistep Transitions 155
Classification of States 159
Steady-State Behavior 167
Computations 172
Appendix 179
Problems 181
References 188
Markov Processes 190
Basic Definitions 190
Steady-State Properties 193
Revenues and Costs 197
Time-Dependent Probabilities 200
Appendix 204
Problems 205
Queueing Processes 209
Basic Definitions and Notation 209
Single Server Systems 211
Infinite Capacity Single-Server Systems 211
Finite Capacity Single Server Systems 218
Multiple Server Queues 221
Approximations 225
Appendix 227
Problems 228
References 233
Queueing Networks 234
Jackson Networks 234
Open Jackson Networks 235
Closed Jackson Networks 239
Network Approximations 242
Deterministic Routing with Poisson Input 243
Deterministic Routing with non-Poisson Input 249
Appendix 252
Problems 253
References 256
Event-Driven Simulation and Output Analyses 257
Event-Driven Simulations 257
Statistical Analysis of Output 268
Terminating Simulations 269
Steady-State Simulations 272
Comparing Systems 281
Problems 285
References 290
Inventory Theory 291
The News-Vendor Problem 292
Single-Period Inventory 295
No Setup Costs 295
Setup Costs 298
Multi-Period Inventory 301
Problems 305
References 309
Replacement Theory 310
Age Replacement 310
Discrete Life Times 311
Continuous Life Times 314
Minimal Repair 317
Minimal Repairs without Early Replacements 318
Minimal Repairs with Early Replacements 319
Block Replacement 321
Problems 324
Markov Decision Processes 327
Basic Definitions 328
Expected Total Discounted Cost Criterion 330
Average Long-Run Cost Criterion 331
Stationary Policies 331
Discounted Cost Algorithms 333
Value Improvement for Discounted Costs 335
Policy Improvement for Discounted Costs 336
Linear Programming for Discounted Costs 340
Average Cost Algorithms 341
Policy Improvement for Average Costs 344
Linear Programming for Average Costs 348
The Optimal Stopping Problem 350
Problems 353
Advanced Queues 359
Difference Equations 360
Batch Arrivals 362
Quasi-Birth-Death Processes 364
Batch Arrivals (continued) 366
Phase-Type Distributions 368
Systems with Phase-Type Service 372
The M/Ph/1 Queueing System 372
The M/Ph/c Queueing System 376
Systems with Phase-Type Arrivals 378
Problems 379
References 383
Matrix Review 384
Matrix Addition and Subtraction 385
Matrix Multiplication 385
Determinants 386
Determinants by Cofactor Expansion 388
Nonsingular Matrices 389
Inversion-in-Place 390
Derivatives 393
References 395
Index 396