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Preface to the Fourth Edition | p. xi |
Preface to the Third Edition | p. xiii |
Preface to the First Edition | p. xv |
To the Instructor | p. xvii |
Acknowledgments | p. xix |
Introduction | p. 1 |
Stochastic Modeling | p. 1 |
Stochastic Processes | p. 4 |
Probability Review | p. 4 |
Events and Probabilities | p. 4 |
Random Variables | p. 5 |
Moments and Expected Values | p. 7 |
Joint Distribution Functions | p. 8 |
Sums and Convolutions | p. 10 |
Change of Variable | p. 10 |
Conditional Probability | p. 11 |
Review of Axiomatic Probability Theory | p. 12 |
The Major Discrete Distributions | p. 19 |
Bernoulli Distribution | p. 20 |
Binomial Distribution | p. 20 |
Geometric and Negative Binominal Distributions | p. 21 |
The Poisson Distribution | p. 22 |
The Multinomial Distribution | p. 24 |
Important Continuous Distributions | p. 27 |
The Normal Distribution | p. 27 |
The Exponential Distribution | p. 28 |
The Uniform Distribution | p. 30 |
The Gamma Distribution | p. 30 |
The Beta Distribution | p. 31 |
The Joint Normal Distribution | p. 31 |
Some Elementary Exercises | p. 34 |
Tail Probabilities | p. 34 |
The Exponential Distribution | p. 37 |
Useful Functions, Integrals, and Sums | p. 42 |
Conditional Probability and Conditional Expectation | p. 47 |
The Discrete Case | p. 47 |
The Dice Game Craps | p. 52 |
Random Sums | p. 57 |
Conditional Distributions: The Mixed Case | p. 58 |
The Moments of a Random Sum | p. 59 |
The Distribution of a Random Sum | p. 61 |
Conditioning on a Continuous Random Variable | p. 65 |
Martingales | p. 71 |
The Definition | p. 72 |
The Markov Inequality | p. 73 |
The Maximal Inequality for Nonnegative Martingales | p. 73 |
Markov Chains: Introduction | p. 79 |
Definitions | p. 79 |
Transition Probability Matrices of a Markov Chain | p. 83 |
Some Markov Chain Models | p. 87 |
An Inventory Model | p. 87 |
The Ehrenfest Urn Model | p. 89 |
Markov Chains in Genetics | p. 90 |
A Discrete Queueing Markov Chain | p. 92 |
First Step Analysis | p. 95 |
Simple First Step Analyses | p. 95 |
The General Absorbing Markov Chain | p. 102 |
Some Special Markov Chains | p. 111 |
The Two-State Markov Chain | p. 112 |
Markov Chains Defined by Independent Random Variables | p. 114 |
One-Dimensional Random Walks | p. 116 |
Success Runs | p. 120 |
Functionals of Random Walks and Success Runs | p. 124 |
The General Random Walk | p. 128 |
Cash Management | p. 132 |
The Success Runs Markov Chain | p. 134 |
Another Look at First Step Analysis | p. 139 |
Branching Processes | p. 146 |
Examples of Branching Processes | p. 147 |
The Mean and Variance of a Branching Process | p. 148 |
Extinction Probabilities | p. 149 |
Branching Processes and Generating Functions | p. 152 |
Generating Functions and Extinction Probabilities | p. 154 |
Probability Generating Functions and Sums of Independent Random Variables | p. 157 |
Multiple Branching Processes | p. 159 |
The Long Run Behavior of Markov Chains | p. 165 |
Regular Transition Probability Matrices | p. 165 |
Doubly Stochastic Matrices | p. 170 |
Interpretation of the Limiting Distribution | p. 171 |
Examples | p. 178 |
Including History in the State Description | p. 178 |
Reliability and Redundancy | p. 179 |
A Continuous Sampling Plan | p. 181 |
Age Replacement Policies | p. 183 |
Optimal Replacement Rules | p. 185 |
The Classification of States | p. 194 |
Irreducible Markov Chains | p. 195 |
Periodicity of a Markov Chain | p. 196 |
Recurrent and Transient States | p. 198 |
The Basic Limit Theorem of Markov Chains | p. 203 |
Reducible Markov Chains | p. 215 |
Poisson Processes | p. 223 |
The Poisson Distribution and the Poisson Process | p. 223 |
The Poisson Distribution | p. 223 |
The Poisson Process | p. 225 |
Nonhomogeneous Processes | p. 226 |
Cox Processes | p. 227 |
The Law of Rare Events | p. 232 |
The Law of Rare Events and the Poisson Process | p. 234 |
Proof of Theorem 5.3 | p. 237 |
Distributions Associated with the Poisson Process | p. 241 |
The Uniform Distribution and Poisson Processes | p. 247 |
Shot Noise | p. 253 |
Sum Quota Sampling | p. 255 |
Spatial Poisson Processes | p. 259 |
Compound and Marked Poisson Processes | p. 264 |
Compound Poisson Processes | p. 264 |
Marked Poisson Processes | p. 267 |
Continuous Time Markov Chains | p. 277 |
Pure Birth Processes | p. 277 |
Postulates for the Poisson Process | p. 277 |
Pure Birth Process | p. 278 |
The Yule Process | p. 282 |
Pure Death Processes | p. 286 |
The Linear Death Process | p. 287 |
Cable Failure Under Static Fatigue | p. 290 |
Birth and Death Processes | p. 295 |
Postulates | p. 295 |
Sojourn Times | p. 296 |
Differential Equations of Birth and Death Processes | p. 299 |
The Limiting Behavior of Birth and Death Processes | p. 304 |
Birth and Death Processes with Absorbing States | p. 316 |
Probability of Absorption into State 0 | p. 316 |
Mean Time Until Absorption | p. 318 |
Finite-State Continuous Time Markov Chains | p. 327 |
A Poisson Process with a Markov Intensity | p. 338 |
Renewal Phenomena | p. 347 |
Definition of a Renewal Process and Related Concepts | p. 347 |
Some Examples of Renewal Processes | p. 353 |
Brief Sketches of Renewal Situations | p. 353 |
Block Replacement | p. 354 |
The Poisson Process Viewed as a Renewal Process | p. 358 |
The Asymptotic Behavior of Renewal Processes | p. 362 |
The Elementary Renewal Theorem | p. 363 |
The Renewal Theorem for Continuous Lifetimes | p. 365 |
The Asymptotic Distribution of N(t) | p. 367 |
The Limiting Distribution of Age and Excess Life | p. 368 |
Generalizations and Variations on Renewal Processes | p. 371 |
Delayed Renewal Processes | p. 371 |
Stationary Renewal Processes | p. 372 |
Cumulative and Related Processes | p. 372 |
Discrete Renewal Theory | p. 379 |
The Discrete Renewal Theorem | p. 383 |
Deterministic Population Growth with Age Distribution | p. 384 |
Brownian Motion and Related Processes | p. 391 |
Brownian Motion and Gaussian Processes | p. 391 |
A Little History | p. 391 |
The Brownian Motion Stochastic Process | p. 392 |
The Central Limit Theorem and the Invariance Principle | p. 396 |
Gaussian Processes | p. 398 |
The Maximum Variable and the Reflection Principle | p. 405 |
The Reflection Principle | p. 406 |
The Time to First Reach a Level | p. 407 |
The Zeros of Brownian Motion | p. 408 |
Variations and Extensions | p. 411 |
Reflected Brownian Motion | p. 411 |
Absorbed Brownian Motion | p. 412 |
The Brownian Bridge | p. 414 |
Brownian Meander | p. 416 |
Brownian Motion with Drift | p. 419 |
The Gambler's Ruin Problem | p. 420 |
Geometric Brownian Motion | p. 424 |
The Ornstein-Uhlenbeck Process | p. 432 |
A Second Approach to Physical Brownian Motion | p. 434 |
The Position Process | p. 437 |
The Long Run Behavior | p. 439 |
Brownian Measure and Integration | p. 441 |
Queueing Systems | p. 447 |
Queueing Processes | p. 447 |
The Queueing Formula L = X W | p. 448 |
A Sampling of Queueing Models | p. 449 |
Poisson Arrivals, Exponential Service Times | p. 451 |
The M/M/1 System | p. 452 |
The M/M/$ System | p. 456 |
The M/M/s System | p. 457 |
General Service Time Distributions | p. 460 |
The M/G/1 System | p. 460 |
The M/G/$ System | p. 465 |
Variations and Extensions | p. 468 |
Systems with Balking | p. 468 |
Variable Service Rates | p. 469 |
A System with Feedback | p. 470 |
A Two-Server Overflow Queue | p. 470 |
Preemptive Priority Queues | p. 472 |
Open Acyclic Queueing Networks | p. 480 |
The Basic Theorem | p. 480 |
Two Queues in Tandem | p. 481 |
Open Acyclic Networks | p. 482 |
Appendix: Time Reversibility | p. 485 |
Proof of Theorem 9.1 | p. 487 |
General Open Networks | p. 488 |
The General Open Network | p. 492 |
Random Evolutions | p. 495 |
Two-State Velocity Model | p. 495 |
Two-State Random Evolution | p. 498 |
The Telegraph Equation | p. 500 |
Distribution Functions and Densities in the Two-State Model | p. 501 |
Passage Time Distributions | p. 505 |
JV-State Random Evolution | p. 507 |
Finite Markov Chains and Random Velocity Models | p. 507 |
Constructive Approach of Random Velocity Models | p. 507 |
Random Evolution Processes | p. 508 |
Existence-Uniqueness of the First-Order System (10.26) | p. 509 |
Single Hyperbolic Equation | p. 510 |
Spectral Properties of the Transition Matrix | p. 512 |
Recurrence Properties of Random Evolution | p. 515 |
Weak Law and Central Limit Theorem | p. 516 |
Isotropic Transport in Higher Dimensions | p. 521 |
The Rayleigh Problem of Random Flights | p. 521 |
Three-Dimensional Rayleigh Model | p. 523 |
Characteristic Functions and Their Applications | p. 525 |
Definition of the Characteristic Function | p. 525 |
Two Basic Properties of the Characteristic Function | p. 526 |
Inversion Formulas for Characteristic Functions | p. 527 |
Fourier Reciprocity/Local Non-Uniqueness | p. 530 |
Fourier Inversion and Parseval's Identity Inversion | p. 531 |
Formula for General Random Variables | p. 532 |
The Continuity Theorem | p. 533 |
Proof of the Continuity Theorem | p. 534 |
Proof of the Central Limit Theorem | p. 535 |
Stirling's Formula and Applications | p. 536 |
Poisson Representation of n! | p. 537 |
Proof of Stirling's Formula | p. 538 |
Local deMoivre-Laplace Theorem | p. 539 |
Further Reading | p. 541 |
Answers to Exercises | p. 543 |
Index | p. 557 |
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