| 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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