Serfozo Basics of Applied Stochastic Processes

1. Auflage 2009
ISBN: 978-3-540-89332-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 443 Seiten

Stochastic processes are mathematical models of random phenomena that evolve according to prescribed dynamics. Processes commonly used in applications are Markov chains in discrete and continuous time, renewal and regenerative processes, Poisson processes, and Brownian motion. This volume gives an in-depth description of the structure and basic properties of these stochastic processes. A main focus is on equilibrium distributions, strong laws of large numbers, and ordinary and functional central limit theorems for cost and performance parameters. Although these results differ for various processes, they have a common trait of being limit theorems for processes with regenerative increments. Extensive examples and exercises show how to formulate stochastic models of systems as functions of a system's data and dynamics, and how to represent and analyze cost and performance measures. Topics include stochastic networks, spatial and space-time Poisson processes, queueing, reversible processes, simulation, Brownian approximations, and varied Markovian models. The technical level of the volume is between that of introductory texts that focus on highlights of applied stochastic processes, and advanced texts that focus on theoretical aspects of processes.

Autoren/Hrsg.

Serfozo, Richard

Inhaltsverzeichnis

1;Preface;7
2;Contents;10
3;Markov Chains;14
3.1;1.1 Introduction;15
3.2;1.2 Probabilities of Sample Paths;18
3.3;1.3 Construction of Markov Chains;21
3.4;1.4 Examples;23
3.5;1.5 Stopping Times and Strong Markov Property;29
3.6;1.6 Classification of States;32
3.7;1.7 Hitting and Absorbtion Probabilities;39
3.8;1.8 Branching Processes;43
3.9;1.9 Stationary Distributions;46
3.10;1.10 Limiting Distributions;53
3.11;1.11 Regenerative Property and Cycle Costs;55
3.12;1.12 Strong Laws of Large Numbers;58
3.13;1.13 Examples of Limiting Averages;63
3.14;1.14 Optimal Design of Markovian Systems;66
3.15;1.15 Closed Network Model;68
3.16;1.16 Open Network Model;72
3.17;1.17 Reversible Markov Chains;74
3.18;1.18 Markov Chain Monte Carlo;81
3.19;1.19 Markov Chains on Subspaces;84
3.20;1.20 Limit Theorems via Coupling;86
3.21;1.21 Criteria for Positive Recurrence;89
3.22;1.22 Review of Conditional Probabilities;94
3.23;1.23 Exercises;97
4;Renewal and Regenerative Processes;112
4.1;2.1 Renewal Processes;112
4.2;2.2 Strong Laws of Large Numbers;117
4.3;2.3 The Renewal Function;120
4.4;2.4 Future Expectations;127
4.5;2.5 Renewal Equations;127
4.6;2.6 Blackwell’s Theorem;129
4.7;2.7 Key Renewal Theorem;131
4.8;2.8 Regenerative Processes;134
4.9;2.9 Limiting Distributions for Markov Chains;139
4.10;2.10 Processes with Regenerative Increments;139
4.11;2.11 Average Sojourn Times in Regenerative Processes;142
4.12;2.12 Batch-Service Queueing System;145
4.13;2.13 Central Limit Theorems;148
4.14;2.14 Terminating Renewal Processes;152
4.15;2.15 Stationary Renewal Processes;157
4.16;2.16 Refined Limit Laws;161
4.17;2.17 Proof of the Key Renewal Theorem*;164
4.18;2.18 Proof of Blackwell’s Theorem*;166
4.19;2.19 Stationary-Cycle Processes*;168
4.20;2.20 Exercises;169
5;Poisson Processes;182
5.1;3.1 Poisson Processes on R+;183
5.2;3.2 Characterizations of Classical Poisson Processes;186
5.3;3.3 Location of Points;189
5.4;3.4 Functions of Point Locations;192
5.5;3.5 Poisson Processes on General Spaces;194
5.6;3.6 Integrals and Laplace Functionals of Poisson Processes;196
5.7;3.7 Poisson Processes as Sample Processes;201
5.8;3.8 Deterministic Transformations of Poisson Processes;203
5.9;3.9 Marked and Space-Time Poisson Processes;207
5.10;3.10 Partitions and Translations of Poisson Processes;209
5.11;3.11 Markov/Poisson Processes;214
5.12;3.12 Poisson Input-Output Systems;216
5.13;3.13 Network of Mt/Gt/8 Stations;219
5.14;3.14 Cox Processes;224
5.15;3.15 Compound Poisson Processes;227
5.16;3.16 Poisson Law of Rare Events;229
5.17;3.17 Poisson Convergence Theorems*;231
5.18;3.18 Exercises;238
6;Continuous-Time Markov Chains;254
6.1;4.1 Introduction;255
6.2;4.2 Examples;258
6.3;4.3 Markov Properties;260
6.4;4.4 Transition Probabilities and Transition Rates;264
6.5;4.5 Existence of CTMCs;266
6.6;4.6 Uniformization, Travel Times and Transition Probabilities;268
6.7;4.7 Stationary and Limiting Distributions;271
6.8;4.8 Regenerative Property and Cycle Costs;276
6.9;4.9 Ergodic Theorems;277
6.10;4.10 Expectations of Cost and Utility Functions;282
6.11;4.11 Reversibility;285
6.12;4.12 Modeling of Reversible Phenomena;290
6.13;4.13 Jackson Network Processes;295
6.14;4.14 Multiclass Networks;300
6.15;4.15 Poisson Transition Times;304
6.16;4.16 Palm Probabilities;312
6.17;4.17 PASTA at Poisson Transitions;316
6.18;4.18 Relating Palm and Ordinary Probabilities;319
6.19;4.19 Stationarity Under Palm Probabilities;323
6.20;4.20 G/G/1, M/G/1 and G/M/1 Queues;327
6.21;4.21 Markov-Renewal Processes*;334
6.22;4.22 Exercises;336
7;Brownian Motion;354
7.1;5.1 Definition and Strong Markov Property;355
7.2;5.2 Brownian Motion as a Gaussian Process;358
7.3;5.3 Maximum Process and Hitting Times;362
7.4;5.4 Special Random Times;365
7.5;5.5 Martingales;367
7.6;5.6 Optional Stopping of Martingales;371
7.7;5.7 Hitting Times for Brownian Motion with Drift;374
7.8;5.8 Limiting Averages and Law of the Iterated Logarithm;377
7.9;5.9 Donsker’s Functional Central Limit Theorem;381
7.10;5.10 Regenerative and Markov FCLTs;386
7.11;5.11 Peculiarities of Brownian Sample Paths;390
7.12;5.12 Brownian Bridge Process;392
7.13;5.13 Geometric Brownian Motion;396
7.14;5.14 Multidimensional Brownian Motion;398
7.15;5.15 Brownian/Poisson Particle System;400
7.16;5.16 G/G/1 Queues in Heavy Traffic;402
7.17;5.17 Brownian Motion in a Random Environment;406
7.18;5.18 Exercises;407
8;Appendix;418
8.1;6.1 Probability Spaces and Random Variables;418
8.2;6.2 Table of Distributions;420
8.3;6.3 Random Elements and Stochastic Processes;422
8.4;6.4 Expectations as Integrals;423
8.5;6.5 Functions of Stochastic Processes;425
8.6;6.6 Independence;428
8.7;6.7 Conditional Probabilities and Expectations;430
8.8;6.8 Existence of Stochastic Processes;432
8.9;6.9 Convergence Concepts;434
9;Bibliographical Notes;440
10;References;442
11;Notation;448
12;Index;450