The Mathematical Basis of Performance Modeling
Buch, Englisch, 776 Seiten, Format (B × H): 186 mm x 260 mm, Gewicht: 1517 g
ISBN: 978-0-691-14062-9
Verlag: Princeton University Press
Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics. The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolmogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation. Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available (to professors only).Numerous examples illuminate the mathematical theories Carefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approach Each chapter concludes with an extensive set of exercises Professors: A supplementary Solutions Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://press.princeton.edu/class_use/solutions.html
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Preface and Acknowledgments xv
PART I PROBABILITY 1
Chapter 1: Probability 3
1.1 Trials, Sample Spaces, and Events 3
1.2 Probability Axioms and Probability Space 9
1.3 Conditional Probability 12
1.4 Independent Events 15
1.5 Law of Total Probability 18
1.6 Bayes? Rule 20
1.7 Exercises 21
Chapter 2: Combinatorics--The Art of Counting 25
2.1 Permutations 25
2.2 Permutations with Replacements 26
2.3 Permutations without Replacement 27
2.4 Combinations without Replacement 29
2.5 Combinations with Replacements 31
2.6 Bernoulli (Independent) Trials 33
2.7 Exercises 36
Chapter 3: Random Variables and Distribution Functions 40
3.1 Discrete and Continuous Random Variables 40
3.2 The Probability Mass Function for a Discrete Random Variable 43
3.3 The Cumulative Distribution Function 46
3.4 The Probability Density Function for a Continuous Random Variable 51
3.5 Functions of a Random Variable 53
3.6 Conditioned Random Variables 58
3.7 Exercises 60
Chapter 4: Joint and Conditional Distributions 64
4.1 Joint Distributions 64
4.2 Joint Cumulative Distribution Functions 64
4.3 Joint Probability Mass Functions 68
4.4 Joint Probability Density Functions 71
4.5 Conditional Distributions 77
4.6 Convolutions and the Sum of Two Random Variables 80
4.7 Exercises 82
Chapter 5: Expectations and More 87
5.1 Definitions 87
5.2 Expectation of Functions and Joint Random Variables 92
5.3 Probability Generating Functions for Discrete Random Variables 100
5.4 Moment Generating Functions 103
5.5 Maxima and Minima of Independent Random Variables 108
5.6 Exercises 110
Chapter 6: Discrete Distribution Functions 115
6.1 The Discrete Uniform Distribution 115
6.2 The Bernoulli Distribution 116
6.3 The Binomial Distribution 117
6.4 Geometric and Negative Binomial Distributions 120
6.5 The Poisson Distribution 124
6.6 The Hypergeometric Distribution 127
6.7 The Multinomial Distribution 128
6.8 Exercises 130
Chapter 7: Continuous Distribution Functions 134
7.1 The Uniform Distribution 134
7.2 The Exponential Distribution 136
7.3 The Normal or Gaussian Distribution 141
7.4 The Gamma Distribution 145
7.5 Reliability Modeling and the Weibull Distribution 149
7.6 Phase-Type Distributions 155
7.6.1 The Erlang-2 Distribution 155
7.6.2 The Erlang-r Distribution 158
7.6.3 The Hypoexponential Distribution 162
7.6.4 The Hyperexponential Distribution 164
7.6.5 The Coxian Distribution 166
7.6.6 General Phase-Type Distributions 168
7.6.7 Fitting Phase-Type Distributions to Means and Variances 171
7.7 Exercises 176
Chapter 8: Bounds and Limit Theorems 180
8.1 The Markov Inequality 180
8.2 The Chebychev Inequality 181
8.3 The Chernoff Bound 182
8.4 The Laws of Large Numbers 182
8.5 The Central Limit Theorem 184
8.6 Exercises 187
PART II MARKOV CHAINS 191
Chapter 9: Discrete- and Continuous-Time Markov Chains 193
9.1 Stochastic Processes and Markov Chains 193
9.2 Discrete-Time Markov Chains: Definitions 195
9.3 The Chapman-Kolmogorov Equations 202
9.4 Classification of States 206
9.5 Irreducibility 214
9.6 The Potential, Fundamental, and Reachability Matrices 218
9.6.1 Potential and Fundamental Matrices and Mean Time to Absorption 219
9.6.2 The Reachability Matrix and Absorption Probabilities 223
9.7 Random Walk Problems 228
9.8 Probability Distributions 235
9.9 Reversibility 248
9.10 Continuous-Time Markov Chains 253
9.10.1 Transition Probabilities and Transition Rates 254
9.10.2 The Chapman-Kolmogorov Equations 257
9.10.3 The Embedded Markov Chain and State Properties 259
9.10.4 Probability Distributions 262
9.10.5 Reversibility 265
9.11 Semi-Markov Processes 265
9.12 Renewal Processes 267
9.13 Exercises 275
Chapter 10: Numerical Solution of Markov Chains 285
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