An Introduction to Stochastic Modeling

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  • Edition: 4th
  • Format: Paperback
  • Copyright: 2010-12-10
  • Publisher: Elsevier Science Ltd
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Serving as the foundation for a one-semester course in stochastic processes for students familiar with elementary probability theory and calculus, Introduction to Stochastic Modeling, Third Edition, bridges the gap between basic probability and an intermediate level course in stochastic processes. The objectives of the text are to introduce students to the standard concepts and methods of stochastic modeling, to illustrate the rich diversity of applications of stochastic processes in the applied sciences, and to provide exercises in the application of simple stochastic analysis to realistic problems.

Table of Contents

Preface to the Fourth Editionp. xi
Preface to the Third Editionp. xiii
Preface to the First Editionp. xv
To the Instructorp. xvii
Acknowledgmentsp. xix
Introductionp. 1
Stochastic Modelingp. 1
Stochastic Processesp. 4
Probability Reviewp. 4
Events and Probabilitiesp. 4
Random Variablesp. 5
Moments and Expected Valuesp. 7
Joint Distribution Functionsp. 8
Sums and Convolutionsp. 10
Change of Variablep. 10
Conditional Probabilityp. 11
Review of Axiomatic Probability Theoryp. 12
The Major Discrete Distributionsp. 19
Bernoulli Distributionp. 20
Binomial Distributionp. 20
Geometric and Negative Binominal Distributionsp. 21
The Poisson Distributionp. 22
The Multinomial Distributionp. 24
Important Continuous Distributionsp. 27
The Normal Distributionp. 27
The Exponential Distributionp. 28
The Uniform Distributionp. 30
The Gamma Distributionp. 30
The Beta Distributionp. 31
The Joint Normal Distributionp. 31
Some Elementary Exercisesp. 34
Tail Probabilitiesp. 34
The Exponential Distributionp. 37
Useful Functions, Integrals, and Sumsp. 42
Conditional Probability and Conditional Expectationp. 47
The Discrete Casep. 47
The Dice Game Crapsp. 52
Random Sumsp. 57
Conditional Distributions: The Mixed Casep. 58
The Moments of a Random Sump. 59
The Distribution of a Random Sump. 61
Conditioning on a Continuous Random Variablep. 65
Martingalesp. 71
The Definitionp. 72
The Markov Inequalityp. 73
The Maximal Inequality for Nonnegative Martingalesp. 73
Markov Chains: Introductionp. 79
Definitionsp. 79
Transition Probability Matrices of a Markov Chainp. 83
Some Markov Chain Modelsp. 87
An Inventory Modelp. 87
The Ehrenfest Urn Modelp. 89
Markov Chains in Geneticsp. 90
A Discrete Queueing Markov Chainp. 92
First Step Analysisp. 95
Simple First Step Analysesp. 95
The General Absorbing Markov Chainp. 102
Some Special Markov Chainsp. 111
The Two-State Markov Chainp. 112
Markov Chains Defined by Independent Random Variablesp. 114
One-Dimensional Random Walksp. 116
Success Runsp. 120
Functionals of Random Walks and Success Runsp. 124
The General Random Walkp. 128
Cash Managementp. 132
The Success Runs Markov Chainp. 134
Another Look at First Step Analysisp. 139
Branching Processesp. 146
Examples of Branching Processesp. 147
The Mean and Variance of a Branching Processp. 148
Extinction Probabilitiesp. 149
Branching Processes and Generating Functionsp. 152
Generating Functions and Extinction Probabilitiesp. 154
Probability Generating Functions and Sums of Independent Random Variablesp. 157
Multiple Branching Processesp. 159
The Long Run Behavior of Markov Chainsp. 165
Regular Transition Probability Matricesp. 165
Doubly Stochastic Matricesp. 170
Interpretation of the Limiting Distributionp. 171
Examplesp. 178
Including History in the State Descriptionp. 178
Reliability and Redundancyp. 179
A Continuous Sampling Planp. 181
Age Replacement Policiesp. 183
Optimal Replacement Rulesp. 185
The Classification of Statesp. 194
Irreducible Markov Chainsp. 195
Periodicity of a Markov Chainp. 196
Recurrent and Transient Statesp. 198
The Basic Limit Theorem of Markov Chainsp. 203
Reducible Markov Chainsp. 215
Poisson Processesp. 223
The Poisson Distribution and the Poisson Processp. 223
The Poisson Distributionp. 223
The Poisson Processp. 225
Nonhomogeneous Processesp. 226
Cox Processesp. 227
The Law of Rare Eventsp. 232
The Law of Rare Events and the Poisson Processp. 234
Proof of Theorem 5.3p. 237
Distributions Associated with the Poisson Processp. 241
The Uniform Distribution and Poisson Processesp. 247
Shot Noisep. 253
Sum Quota Samplingp. 255
Spatial Poisson Processesp. 259
Compound and Marked Poisson Processesp. 264
Compound Poisson Processesp. 264
Marked Poisson Processesp. 267
Continuous Time Markov Chainsp. 277
Pure Birth Processesp. 277
Postulates for the Poisson Processp. 277
Pure Birth Processp. 278
The Yule Processp. 282
Pure Death Processesp. 286
The Linear Death Processp. 287
Cable Failure Under Static Fatiguep. 290
Birth and Death Processesp. 295
Postulatesp. 295
Sojourn Timesp. 296
Differential Equations of Birth and Death Processesp. 299
The Limiting Behavior of Birth and Death Processesp. 304
Birth and Death Processes with Absorbing Statesp. 316
Probability of Absorption into State 0p. 316
Mean Time Until Absorptionp. 318
Finite-State Continuous Time Markov Chainsp. 327
A Poisson Process with a Markov Intensityp. 338
Renewal Phenomenap. 347
Definition of a Renewal Process and Related Conceptsp. 347
Some Examples of Renewal Processesp. 353
Brief Sketches of Renewal Situationsp. 353
Block Replacementp. 354
The Poisson Process Viewed as a Renewal Processp. 358
The Asymptotic Behavior of Renewal Processesp. 362
The Elementary Renewal Theoremp. 363
The Renewal Theorem for Continuous Lifetimesp. 365
The Asymptotic Distribution of N(t)p. 367
The Limiting Distribution of Age and Excess Lifep. 368
Generalizations and Variations on Renewal Processesp. 371
Delayed Renewal Processesp. 371
Stationary Renewal Processesp. 372
Cumulative and Related Processesp. 372
Discrete Renewal Theoryp. 379
The Discrete Renewal Theoremp. 383
Deterministic Population Growth with Age Distributionp. 384
Brownian Motion and Related Processesp. 391
Brownian Motion and Gaussian Processesp. 391
A Little Historyp. 391
The Brownian Motion Stochastic Processp. 392
The Central Limit Theorem and the Invariance Principlep. 396
Gaussian Processesp. 398
The Maximum Variable and the Reflection Principlep. 405
The Reflection Principlep. 406
The Time to First Reach a Levelp. 407
The Zeros of Brownian Motionp. 408
Variations and Extensionsp. 411
Reflected Brownian Motionp. 411
Absorbed Brownian Motionp. 412
The Brownian Bridgep. 414
Brownian Meanderp. 416
Brownian Motion with Driftp. 419
The Gambler's Ruin Problemp. 420
Geometric Brownian Motionp. 424
The Ornstein-Uhlenbeck Processp. 432
A Second Approach to Physical Brownian Motionp. 434
The Position Processp. 437
The Long Run Behaviorp. 439
Brownian Measure and Integrationp. 441
Queueing Systemsp. 447
Queueing Processesp. 447
The Queueing Formula L = X Wp. 448
A Sampling of Queueing Modelsp. 449
Poisson Arrivals, Exponential Service Timesp. 451
The M/M/1 Systemp. 452
The M/M/$ Systemp. 456
The M/M/s Systemp. 457
General Service Time Distributionsp. 460
The M/G/1 Systemp. 460
The M/G/$ Systemp. 465
Variations and Extensionsp. 468
Systems with Balkingp. 468
Variable Service Ratesp. 469
A System with Feedbackp. 470
A Two-Server Overflow Queuep. 470
Preemptive Priority Queuesp. 472
Open Acyclic Queueing Networksp. 480
The Basic Theoremp. 480
Two Queues in Tandemp. 481
Open Acyclic Networksp. 482
Appendix: Time Reversibilityp. 485
Proof of Theorem 9.1p. 487
General Open Networksp. 488
The General Open Networkp. 492
Random Evolutionsp. 495
Two-State Velocity Modelp. 495
Two-State Random Evolutionp. 498
The Telegraph Equationp. 500
Distribution Functions and Densities in the Two-State Modelp. 501
Passage Time Distributionsp. 505
JV-State Random Evolutionp. 507
Finite Markov Chains and Random Velocity Modelsp. 507
Constructive Approach of Random Velocity Modelsp. 507
Random Evolution Processesp. 508
Existence-Uniqueness of the First-Order System (10.26)p. 509
Single Hyperbolic Equationp. 510
Spectral Properties of the Transition Matrixp. 512
Recurrence Properties of Random Evolutionp. 515
Weak Law and Central Limit Theoremp. 516
Isotropic Transport in Higher Dimensionsp. 521
The Rayleigh Problem of Random Flightsp. 521
Three-Dimensional Rayleigh Modelp. 523
Characteristic Functions and Their Applicationsp. 525
Definition of the Characteristic Functionp. 525
Two Basic Properties of the Characteristic Functionp. 526
Inversion Formulas for Characteristic Functionsp. 527
Fourier Reciprocity/Local Non-Uniquenessp. 530
Fourier Inversion and Parseval's Identity Inversionp. 531
Formula for General Random Variablesp. 532
The Continuity Theoremp. 533
Proof of the Continuity Theoremp. 534
Proof of the Central Limit Theoremp. 535
Stirling's Formula and Applicationsp. 536
Poisson Representation of n!p. 537
Proof of Stirling's Formulap. 538
Local deMoivre-Laplace Theoremp. 539
Further Readingp. 541
Answers to Exercisesp. 543
Indexp. 557
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