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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