An Introduction to Stochastic Modeling, Fourth Edition [4th ed.] 0123814162, 9780123814166

Table of contents :
Contents......Page f005.djvu
Preface to the Fourth Edition......Page f011.djvu
Preface to the Third Edition......Page f013.djvu
Preface to the First Edition......Page f015.djvu
To the Instructor......Page f017.djvu
Acknowledgments......Page f019.djvu
1.1 Stochastic Modeling......Page p001.djvu
1.2.1 Events and Probabilities......Page p004.djvu
1.2.2 Random Variables......Page p005.djvu
1.2.3 Moments and Expected Values......Page p007.djvu
1.2.4 Joint Distribution Functions......Page p008.djvu
1.2.6 Change of Variable......Page p010.djvu
1.2.7 Conditional Probability......Page p011.djvu
1.2.8 Review of Axiomatic Probability Theory......Page p012.djvu
1.3 The Major Discrete Distributions......Page p019.djvu
1.3.2 Binomial Distribution......Page p020.djvu
1.3.3 Geometric and Negative Binominal Distributions......Page p021.djvu
1.3.4 The Poisson Distribution......Page p022.djvu
1.3.5 The Multinomial Distribution......Page p024.djvu
1.4.1 The Normal Distribution......Page p027.djvu
1.4.2 The Exponential Distribution......Page p028.djvu
1.4.4 The Gamma Distribution......Page p030.djvu
1.4.6 The Joint Normal Distribution......Page p031.djvu
1.5.1 Tail Probabilities......Page p034.djvu
1.5.2 The Exponential Distribution......Page p037.djvu
1.6 Useful Functions, Integrals, and Sums......Page p042.djvu
2.1 The Discrete Case......Page p047.djvu
2.2 The Dice Game Craps......Page p052.djvu
2.3 Random Sums......Page p057.djvu
2.3.1 Conditional Distributions: The Mixed Case......Page p058.djvu
2.3.2 The Moments of a Random Sum......Page p059.djvu
2.3.3 The Distribution of a Random Sum......Page p061.djvu
2.4 Conditioning on a Continuous Random Variable......Page p065.djvu
2.5 Martingales......Page p071.djvu
2.5.1 The Definition......Page p072.djvu
2.5.3 The Maximal Inequality for Nonnegative Martingales......Page p073.djvu
3.1 Definitions......Page p079.djvu
3.2 Transition Probability Matrices of a Markov Chain......Page p083.djvu
3.3.1 An Inventory Model......Page p087.djvu
3.3.2 The Ehrenfest Urn Model......Page p089.djvu
3.3.3 Markov Chains in Genetics......Page p090.djvu
3.3.4 A Discrete Queueing Markov Chain......Page p092.djvu
3.4.1 Simple First Step Analyses......Page p095.djvu
3.4.2 The General Absorbing Markov Chain......Page p102.djvu
3.5 Some Special Markov Chains......Page p111.djvu
3.5.1 The Two-State Markov Chain......Page p112.djvu
3.5.2 Markov Chains Defined by Independent Random Variables......Page p114.djvu
3.5.3 One-Dimensional Random Walks......Page p116.djvu
3.5.4 Success Runs......Page p120.djvu
3.6 Functional of Random Walks and Success Runs......Page p124.djvu
3.6.1 The General Random Walk......Page p128.djvu
3.6.2 Cash Management......Page p132.djvu
3.6.3 The Success Runs Markov Chain......Page p134.djvu
3.7 Another Look at First Step Analysis......Page p139.djvu
3.8 Branching Processes......Page p146.djvu
3.8.1 Examples of Branching Processes......Page p147.djvu
3.8.2 The Mean and Variance of a Branching Process......Page p148.djvu
3.8.3 Extinction Probabilities......Page p149.djvu
3.9 Branching Processes and Generating Functions......Page p152.djvu
3.9.1 Generating Functions and Extinction Probabilities......Page p154.djvu
3.9.2 Probability Generating Functions and Sums of Independent Random Variables......Page p157.djvu
3.9.3 Multiple Branching Processes......Page p159.djvu
4.1 Regular Transition Probability Matrices......Page p165.djvu
4.1.1 Doubly Stochastic Matrices......Page p170.djvu
4.1.2 Interpretation of the Limiting Distribution......Page p171.djvu
4.2.1 Including History in the State Description......Page p178.djvu
4.2.2 Reliability and Redundancy......Page p179.djvu
4.2.3 A Continuous Sampling Plan......Page p181.djvu
4.2.4 Age Replacement Policies......Page p183.djvu
4.2.5 Optimal Replacement Rules......Page p185.djvu
4.3 The Classification of States......Page p194.djvu
4.3.1 Irreducible Markov Chains......Page p195.djvu
4.3.2 Periodicity of a Markov Chain......Page p196.djvu
4.3.3 Recurrent and Transient States......Page p198.djvu
4.4 The Basic Limit Theorem of Markov Chains......Page p203.djvu
4.5 Reducible Markov Chains......Page p215.djvu
5.1.1 The Poisson Distribution......Page p223.djvu
5.1.2 The Poisson Process......Page p225.djvu
5.1.3 Nonhomogeneous Processes......Page p226.djvu
5.1.4 Cox Processes......Page p227.djvu
5.2 The Law of Rare Events......Page p232.djvu
5.2.1 The Law of Rare Events and the Poisson Process......Page p234.djvu
5.2.2 Proof of Theorem 5.3......Page p237.djvu
5.3 Distributions Associated with the Poisson Process......Page p241.djvu
5.4 The Uniform Distribution and Poisson Processes......Page p247.djvu
5.4.1 Shot Noise......Page p253.djvu
5.4.2 Sum Quota Sampling......Page p255.djvu
5.5 Spatial Poisson Processes......Page p259.djvu
5.6.1 Compound Poisson Processes......Page p264.djvu
5.6.2 Marked Poisson Processes......Page p267.djvu
6.1.1 Postulates for the Poisson Process......Page p277.djvu
6.1.2 Pure Birth Process......Page p278.djvu
6.1.3 The Yule Process......Page p282.djvu
6.2 Pure Death Processes......Page p286.djvu
6.2.1 The Linear Death Process......Page p287.djvu
6.2.2 Cable Failure Under Static Fatigue......Page p290.djvu
6.3.1 Postulates......Page p295.djvu
6.3.2 Sojourn Times......Page p296.djvu
6.3.3 Differential Equations of Birth and Death Processes......Page p299.djvu
6.4 The Limiting Behavior of Birth and Death Processes......Page p304.djvu
6.5.1 Probability of Absorption into State 0......Page p316.djvu
6.5.2 Mean Time Until Absorption......Page p318.djvu
6.6 Finite-State Continuous Time Markov Chains......Page p327.djvu
6.7 A Poisson Process with a Markov Intensity......Page p338.djvu
7.1 Definition of a Renewal Process and Related Concepts......Page p347.djvu
7.2.1 Brief Sketches of Renewal Situations......Page p353.djvu
7.2.2 Block Replacement......Page p354.djvu
7.3 The Poisson Process Viewed as a Renewal Process......Page p358.djvu
7.4 The Asymptotic Behavior of Renewal Processes......Page p362.djvu
7.4.1 The Elementary Renewal Theorem......Page p363.djvu
7.4.2 The Renewal Theorem for Continuous Lifetimes......Page p365.djvu
7.4.3 The Asymptotic Distribution of N(t)......Page p367.djvu
7.4.4 The Limiting Distribution of Age and Excess Life......Page p368.djvu
7.5.1 Delayed Renewal Processes......Page p371.djvu
7.5.3 Cumulative and Related Processes......Page p372.djvu
7.6 Discrete Renewal Theory......Page p379.djvu
7.6.1 The Discrete Renewal Theorem......Page p383.djvu
7.6.2 Deterministic Population Growth with Age Distribution......Page p384.djvu
8.1.1 A Little History......Page p391.djvu
8.1.2 The Brownian Motion Stochastic Process......Page p392.djvu
8.1.3 The Central Limit Theorem and the Invariance Principle......Page p396.djvu
8.1.4 Gaussian Processes......Page p398.djvu
8.2 The Maximum Variable and the Reflection Principle......Page p405.djvu
8.2.1 The Reflection Principle......Page p406.djvu
8.2.2 The Time to First Reach a Level......Page p407.djvu
8.2.3 The Zeros of Brownian Motion......Page p408.djvu
8.3.1 Reflected Brownian Motion......Page p411.djvu
8.3.2 Absorbed Brownian Motion......Page p412.djvu
8.3.3 The Brownian Bridge......Page p414.djvu
8.3.4 Brownian Meander......Page p416.djvu
8.4 Brownian Motion with Drift......Page p419.djvu
8.4.1 The Gambler's Ruin Problem......Page p420.djvu
8.4.2 Geometric Brownian Motion......Page p424.djvu
8.5 The Ornstein-Uhlenbeck Process......Page p432.djvu
8.5.1 A Second Approach to Physical Brownian Motion......Page p434.djvu
8.5.2 The Position Process......Page p437.djvu
8.5.3 The Long Run Behavior......Page p439.djvu
8.5.4 Brownian Measure and Integration......Page p441.djvu
9.1 Queueing Processes......Page p447.djvu
9.1.1 The Queueing Formula L = lambda W......Page p448.djvu
9.1.2 A Sampling of Queueing Models......Page p449.djvu
9.2 Poisson Arrivals, Exponential Service Times......Page p451.djvu
9.2.1 The M/M/1 System......Page p452.djvu
9.2.2 The M/M/infty System......Page p456.djvu
9.2.3 The M/M/s System......Page p457.djvu
9.3.1 The M/G/1 System......Page p460.djvu
9.3.2 The M/G/infty System......Page p465.djvu
9.4.1 Systems with Balking......Page p468.djvu
9.4.2 Variable Service Rates......Page p469.djvu
9.4.4 A Two-Server Overflow Queue......Page p470.djvu
9.4.5 Preemptive Priority Queues......Page p472.djvu
9.5.1 The Basic Theorem......Page p480.djvu
9.5.2 Two Queues in Tandem......Page p481.djvu
9.5.3 Open Acyclic Networks......Page p482.djvu
9.5.4 Appendix: Time Reversibility......Page p485.djvu
9.5.5 Proof of Theorem 9.1......Page p487.djvu
9.6 General Open Networks......Page p488.djvu
9.6.1 The General Open Network......Page p492.djvu
10.1 Two-State Velocity Model......Page p495.djvu
10.1.1 Two-State Random Evolution......Page p498.djvu
10.1.2 The Telegraph Equation......Page p500.djvu
10.1.3 Distribution Functions and Densities in the Two-State Model......Page p501.djvu
10.1.4 Passage Time Distributions......Page p505.djvu
10.2.2 Constructive Approach of Random Velocity Models......Page p507.djvu
10.2.3 Random Evolution Processes......Page p508.djvu
10.2.4 Existence-Uniqueness of the First-Order System (10.26)......Page p509.djvu
10.2.5 Single Hyperbolic Equation......Page p510.djvu
10.2.6 Spectral Properties of the Transition Matrix......Page p512.djvu
10.2.7 Recurrence Properties of Random Evolution......Page p515.djvu
10.3 Weak Law and Central Limit Theorem......Page p516.djvu
10.4.1 The Rayleigh Problem of Random Flights......Page p521.djvu
10.4.2 Three-Dimensional Rayleigh Model......Page p523.djvu
11.1 Definition of the Characteristic Function......Page p525.djvu
11.1.1 Two Basic Properties of the Characteristic Function......Page p526.djvu
11.2 Inversion Formulas for Characteristic Functions......Page p527.djvu
11.2.1 Fourier Reciprocity/Local Non-Uniqueness......Page p530.djvu
11.2.2 Fourier Inversion and Parseval's Identity......Page p531.djvu
11.3 Inversion Formula for General Random Variables......Page p532.djvu
11.4 The Continuity Theorem......Page p533.djvu
11.4.1 Proof of the Continuity Theorem......Page p534.djvu
11.5 Proof of the Central Limit Theorem......Page p535.djvu
11.6 Stirling's Formula and Applications......Page p536.djvu
11.6.1 Poisson Representation of n!......Page p537.djvu
11.6.2 Proof of Stirling's Formula......Page p538.djvu
11.7 Local deMoivre-Laplace Theorem......Page p539.djvu
Further Reading......Page p541.djvu
Answers to Exercises......Page p543.djvu
Index......Page p557.djvu