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An Introduction to Stochastic Modeling.pdf
Contents
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the First Edition
To the Instructor
Acknowledgments
1 Introduction
1.1 Stochastic Modeling
1.1.1 Stochastic Processes
1.2 Probability Review
1.2.1 Events and Probabilities
1.2.2 Random Variables
1.2.3 Moments and Expected Values
1.2.4 Joint Distribution Functions
1.2.5 Sums and Convolutions
1.2.6 Change of Variable
1.2.7 Conditional Probability
1.2.8 Review of Axiomatic Probability Theory
1.3 The Major Discrete Distributions
1.3.1 Bernoulli Distribution
1.3.2 Binomial Distribution
1.3.3 Geometric and Negative Binominal Distributions
1.3.4 The Poisson Distribution
1.3.5 The Multinomial Distribution
1.4 Important Continuous Distributions
1.4.1 The Normal Distribution
1.4.2 The Exponential Distribution
1.4.3 The Uniform Distribution
1.4.4 The Gamma Distribution
1.4.5 The Beta Distribution
1.4.6 The Joint Normal Distribution
1.5 Some Elementary Exercises
1.5.1 Tail Probabilities
1.5.2 The Exponential Distribution
1.6 Useful Functions, Integrals, and Sums
2 Conditional Probability and Conditional Expectation
2.1 The Discrete Case
2.2 The Dice Game Craps
2.3 Random Sums
2.3.1 Conditional Distributions: The Mixed Case
2.3.2 The Moments of a Random Sum
2.3.3 The Distribution of a Random Sum
2.4 Conditioning on a Continuous Random Variable
2.5 Martingales
2.5.1 The Definition
2.5.2 The Markov Inequality
2.5.3 The Maximal Inequality for Nonnegative Martingales
3 Markov Chains: Introduction
3.1 Definitions
3.2 Transition Probability Matrices of a Markov Chain
3.3 Some Markov Chain Models
3.3.1 An Inventory Model
3.3.2 The Ehrenfest Urn Model
3.3.3 Markov Chains in Genetics
3.3.4 A Discrete Queueing Markov Chain
3.4 First Step Analysis
3.4.1 Simple First Step Analyses
3.4.2 The General Absorbing Markov Chain
3.5 Some Special Markov Chains
3.5.1 The Two-State Markov Chain
3.5.2 Markov Chains Defined by Independent Random Variables
3.5.3 One-Dimensional Random Walks
3.5.4 Success Runs
3.6 Functional of Random Walks and Success Runs
3.6.1 The General Random Walk
3.6.2 Cash Management
3.6.3 The Success Runs Markov Chain
3.7 Another Look at First Step Analysis
3.8 Branching Processes
3.8.1 Examples of Branching Processes
3.8.2 The Mean and Variance of a Branching Process
3.8.3 Extinction Probabilities
3.9 Branching Processes and Generating Functions
3.9.1 Generating Functions and Extinction Probabilities
3.9.2 Probability Generating Functions and Sums of Independent Random Variables
3.9.3 Multiple Branching Processes
4 The Long Run Behavior of Markov Chains
4.1 Regular Transition Probability Matrices
4.1.1 Doubly Stochastic Matrices
4.1.2 Interpretation of the Limiting Distribution
4.2 Examples
4.2.1 Including History in the State Description
4.2.2 Reliability and Redundancy
4.2.3 A Continuous Sampling Plan
4.2.4 Age Replacement Policies
4.2.5 Optimal Replacement Rules
4.3 The Classification of States
4.3.1 Irreducible Markov Chains
4.3.2 Periodicity of a Markov Chain
4.3.3 Recurrent and Transient States
4.4 The Basic Limit Theorem of Markov Chains
4.5 Reducible Markov Chains
5 Poisson Processes
5.1 The Poisson Distribution and the Poisson Process
5.1.1 The Poisson Distribution
5.1.2 The Poisson Process
5.1.3 Nonhomogeneous Processes
5.1.4 Cox Processes
5.2 The Law of Rare Events
5.2.1 The Law of Rare Events and the Poisson Process
5.2.2 Proof of Theorem 5.3
5.3 Distributions Associated with the Poisson Process
5.4 The Uniform Distribution and Poisson Processes
5.4.1 Shot Noise
5.4.2 Sum Quota Sampling
5.5 Spatial Poisson Processes
5.6 Compound and Marked Poisson Processes
5.6.1 Compound Poisson Processes
5.6.2 Marked Poisson Processes
6 Continuous Time Markov Chains
6.1 Pure Birth Processes
6.1.1 Postulates for the Poisson Process
6.1.2 Pure Birth Process
6.1.3 The Yule Process
6.2 Pure Death Processes
6.2.1 The Linear Death Process
6.2.2 Cable Failure Under Static Fatigue
6.3 Birth and Death Processes
6.3.1 Postulates
6.3.2 Sojourn Times
6.3.3 Differential Equations of Birth and Death Processes
6.4 The Limiting Behavior of Birth and Death Processes
6.5 Birth and Death Processes with Absorbing States
6.5.1 Probability of Absorption into State 0
6.5.2 Mean Time Until Absorption
6.6 Finite-State Continuous Time Markov Chains
6.7 A Poisson Process with a Markov Intensity
7 Renewal Phenomena
7.1 Definition of a Renewal Process and Related Concepts
7.2 Some Examples of Renewal Processes
7.2.1 Brief Sketches of Renewal Situations
7.2.2 Block Replacement
7.3 The Poisson Process Viewed as a Renewal Process
7.4 The Asymptotic Behavior of Renewal Processes
7.4.1 The Elementary Renewal Theorem
7.4.2 The Renewal Theorem for Continuous Lifetimes
7.4.3 The Asymptotic Distribution of N(t)
7.4.4 The Limiting Distribution of Age and Excess Life
7.5 Generalizations and Variations on Renewal Processes
7.5.1 Delayed Renewal Processes
7.5.2 Stationary Renewal Processes
7.5.3 Cumulative and Related Processes
7.6 Discrete Renewal Theory
7.6.1 The Discrete Renewal Theorem
7.6.2 Deterministic Population Growth with Age Distribution
8 Brownian Motion and Related Processes
8.1 Brownian Motion and Gaussian Processes
8.1.1 A Little History
8.1.2 The Brownian Motion Stochastic Process
8.1.3 The Central Limit Theorem and the Invariance Principle
8.1.4 Gaussian Processes
8.2 The Maximum Variable and the Reflection Principle
8.2.1 The Reflection Principle
8.2.2 The Time to First Reach a Level
8.2.3 The Zeros of Brownian Motion
8.3 Variations and Extensions
8.3.1 Reflected Brownian Motion
8.3.2 Absorbed Brownian Motion
8.3.3 The Brownian Bridge
8.3.4 Brownian Meander
8.4 Brownian Motion with Drift
8.4.1 The Gambler's Ruin Problem
8.4.2 Geometric Brownian Motion
8.5 The Ornstein-Uhlenbeck Process
8.5.1 A Second Approach to Physical Brownian Motion
8.5.2 The Position Process
8.5.3 The Long Run Behavior
8.5.4 Brownian Measure and Integration
9 Queueing Systems
9.1 Queueing Processes
9.1.1 The Queueing Formula L = lambda W
9.1.2 A Sampling of Queueing Models
9.2 Poisson Arrivals, Exponential Service Times
9.2.1 The M/M/1 System
9.2.2 The M/M/infty System
9.2.3 The M/M/s System
9.3 General Service Time Distributions
9.3.1 The M/G/1 System
9.3.2 The M/G/infty System
9.4 Variations and Extensions
9.4.1 Systems with Balking
9.4.2 Variable Service Rates
9.4.3 A System with Feedback
9.4.4 A Two-Server Overflow Queue
9.4.5 Preemptive Priority Queues
9.5 Open Acyclic Queueing Networks
9.5.1 The Basic Theorem
9.5.2 Two Queues in Tandem
9.5.3 Open Acyclic Networks
9.5.4 Appendix: Time Reversibility
9.5.5 Proof of Theorem 9.1
9.6 General Open Networks
9.6.1 The General Open Network
10 Random Evolutions
10.1 Two-State Velocity Model
10.1.1 Two-State Random Evolution
10.1.2 The Telegraph Equation
10.1.3 Distribution Functions and Densities in the Two-State Model
10.1.4 Passage Time Distributions
10.2 N-State Random Evolution
10.2.1 Finite Markov Chains and Random Velocity Models
10.2.2 Constructive Approach of Random Velocity Models
10.2.3 Random Evolution Processes
10.2.4 Existence-Uniqueness of the First-Order System (10.26)
10.2.5 Single Hyperbolic Equation
10.2.6 Spectral Properties of the Transition Matrix
10.2.7 Recurrence Properties of Random Evolution
10.3 Weak Law and Central Limit Theorem
10.4 Isotropic Transport in Higher Dimensions
10.4.1 The Rayleigh Problem of Random Flights
10.4.2 Three-Dimensional Rayleigh Model
11 Characteristic Functions and Their Applications
11.1 Definition of the Characteristic Function
11.1.1 Two Basic Properties of the Characteristic Function
11.2 Inversion Formulas for Characteristic Functions
11.2.1 Fourier Reciprocity/Local Non-Uniqueness
11.2.2 Fourier Inversion and Parseval's Identity
11.3 Inversion Formula for General Random Variables
11.4 The Continuity Theorem
11.4.1 Proof of the Continuity Theorem
11.5 Proof of the Central Limit Theorem
11.6 Stirling's Formula and Applications
11.6.1 Poisson Representation of n!
11.6.2 Proof of Stirling's Formula
11.7 Local deMoivre-Laplace Theorem
Further Reading
Answers to Exercises
Index
n to
An Introductio
Stochastic
Fourth Edition
Modeling
Mark A. Pinsky
of Mathematics
Department
Northwestern
University
Illinois
Evanston,
Samuel Karlin
Department
of Mathematics
University
Stanford
Stanford,
California
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Contents
to the Fourth Edition
to the Third Edition
to the First Edition
Preface
Preface
Preface
To the Instructor
Acknowledgments
1 Introduction
1 . 1 Stochastic
1 . 1 . 1
Modeling
Stochastic
Processes
and Probabilities
1 .2 Probability Review
1 .2.1 Events
1 .2.2 Random Variables
1 .2.3 Moments and Expected Values
1 .2.4 Joint
1 .2.5 Sums and Convolutions
1 .2.6 Change
1 .2.7 Conditional
1 .2.8 Review of Axiomatic
Distribution
of Variable
1 .3 The Major Discrete Distrib
utions
xi
xiii
XV
xvii
xix
1
Theory
Probability
Probability
Functions
1
4
4
4
5
7
8
10
10
1 1
1 2
19
20
20
and Negative Binominal Distributions 2 1
22
24
27
27
28
30
30
3 1
3 1
34
34
37
42
Distribution
Normal Distribution
1 .3.1 Bernoulli Distribution
1 .3.2 Binomial Distribution
1.3.3 Geometric
1 .3.4 The Poisson Distribution
1 .3.5 The Multinomial
Distribution
1 .4 Important Continuous Distributions
Distribution
1 .4.1 The Normal
1 .4.2 The Exponential
1 .4.3 The Uniform Distribution
1 .4.4 The Gamma
1 .4.5 The Beta Distribution
1 .4.6 The Joint
Distribution
1 .5 Some Elementary Exercises
1 .5.1 Tail Probabilities
1 .5.2 The Exponential
Distribution
1 .6 Useful Functions, Integrals, and Sums
Content�
47
Probability and Conditional
Expectation
Conditional
2.1 The Discrete Case
2.2 The Dice Game Craps
2.3 Random Sums
Distributions
: The Mixed Case
2.3.1 Conditional
2.3.2 The Moments
2.3.3 The Distributio
47
52
57
58
59
61
65
71
2.5.1 The Definition
72
2.5.2 The Markov Inequality
73
2.5.3 The Maximal Inequality for Nonnegative Martingales 73
of a Random Sum
n of a Random Sum
on a Continuous Random Variable
2.4 Conditioning
2.5 Martingales
vi
2
3
Markov Chains: Introduction
3.1 Definitions
3.2 Transition Probability Matrices of a Markov Chain
3.3 Some Markov Chain Models
3.3.1 An Inventory Model
3.3.2 The Ehrenfest Urn Model
3.3.3 Markov Chains in Genetics
3.3.4 A Discrete
3.4 First Step Analysis
Queueing Markov Chain
3.4.1 Simple First Step Analyses
3.4.2 The General Absorbing Markov
Chain
3.5 Some Special Markov Chains
Markov Chain
3.5.1 The Two-State
3.5.2 Markov Chains Defined by Independent
79
79
83
87
87
89
90
92
95
95
102
Ill
112
3.7 Another
3.8 Branching
3.6 Functionals of Random Walks and Success
Runs
Runs
Random Variables
3.5.3 One-Dimensional Random Walks
3.5.4 Success
3.6.1 The General Random Walk
3.6.2 Cash Management
3.6.3 The Success
114
116
120
124
128
132
134
139
146
3.8. 1 Examples of Branching Processes
147
3.8.2 The Mean and Variance of a Branching Process 148
3.8.3 Extinction
149
152
Look at First Step Analysis
Runs Markov Chain
Processes
Processes
Probabi
3.9.1 Generating Functions
3.9.2 Probability Generating Functions
and Generating Functions
and Extinction
and Sums of
Probabilities 154
lities
Independent Random Variables
3.9.3 Multiple Branching
Processes
157
159
3.9 Branching
Contents
4 The Long Run Behavior of Markov Chains
4.1 Regular Transition
Probability Matrices
4.1 . 1 Doubly Stochastic
4.1 .2 Interpretation
Matrices
of the Limiting
Distribution
4.2 Examples
4.2.1 Including History in the State Description
4.2.2 Reliability and Redundancy
4.2.3 A Continuous Sampling Plan
4.2.4 Age Replacement Policies
4.2.5 Optimal Replacement Rules
4.3 The Classifica
tion of States
4.3.1 Irreducible Markov Chains
4.3.2 Periodicity
of a Markov Chain
4.3.3 Recurrent and Transient
States
4.4 The Basic Limit Theorem
4.5 Reducible Markov Chains
of Markov Chains
vii
165
1 65
1 70
1 7 1
1 78
1 78
1 79
1 8 1
1 83
1 85
194
1 95
196
1 98
203
2 1 5
5 Poisson
Processes
5.1 The Poisson Distribution
and the Poisson Process
5.1 . 1 The Poisson Distribution
5.1 .2 The Poisson Process
5.1 .3 Nonhomogeneous Processes
5.1 .4 Cox Processes
223
223
223
225
226
227
232
and the Poisson Process 234
237
241
247
253
255
259
264
264
267
Processes
Events
5.2 The Law of Rare
Rare Events
5.2.1 The Law of
5.2.2 Proof of Theorem 5.3
5.3 Distributions
5.4 The Uniform Distribution
5.4.1 Shot Noise
5.4.2 Sum Quota Sampling
5.5 Spatial Poisson Processes
5.6 Compound and Marked Poisson
Associated with the Poisson Process
and Poisson Processes
5.6.1 Compound Poisson Processes
5.6.2 Marked Poisson Processes
6 Continuous Time Markov Chains
6.1 Pure Birth Processes
6.1 . 1 Postulates for the Poisson Process
6.1 .2 Pure Birth Process
6.1 .3 The Yule Process
6.2 Pure Death Processes
6.2.1 The Linear Death Process
6.2.2 Cable Failure Under Static Fatigue
277
277
277
278
282
286
287
290
viii
Content�
7
8
6.3 Birth and Death Processes
6.3.1 Postulates
6.3.2 Sojourn Times
6.3.3 Differential
Equations
295
295
296
of Birth and Death Processes 299
of Birth and Death Processes 304
316
316
318
327
338
Absorption
Behavior
6.4 The Limiting
6.5 Birth and Death Processes with Absorbing States
6.5 .1 Probability of Absorption into State 0
6.5.2 Mean Time Until
6.6 Finite-State Continuous Time Markov Chains
6. 7 A Poisson Process with a Markov Intensity
Processes
Renewal
of Renewal Situations
Process and Related Concepts
Renewal Phenomena
7.1 Definition of a
7.2 Some Examples of Renewal
7 .2.1 Brief Sketches
7 .2.2 Block Replacement
7.3 The Poisson Process Viewed as
7.4 The Asymptotic
34 7
353
353
354
358
362
363
ous Lifetimes 365
367
368
7.5 Generalizations and Variations on Renewal Processes 3 71
7 .4.1 The Elementary Renewal Theorem
7 .4.2 The Renewal
7 .4.3 The Asymptotic
7 .4.4 The Limiting
Behavior of Renewal
Distribution
Distribution
of Age and Excess
Theorem for Continu
a Renewal Process
Processes
of N(t)
Life
347
7 .5.1 Delayed Renewal
Processes
7 .5.2 Stationary Renewal Processes
7 .5.3 Cumulative and Related Processes
7.6 Discrete Renewal
Theory
371
372
372
379
383
384
391
7.6.1 The Discrete Renewal
Theorem
7 .6.2 Deterministic Population Growth
with Age Distribution
Brownian Motion and Related Processes
8.1 Brownian Motion
8.1 . 1 A Little
8.1 .2 The Brownian Motion
8.1 .3 The Central Limit
8.1 .4 Gaussian Processes
and Gaussian
History
Theorem
Processes
391
391
392
and the Invariance Principle 396
Process
Stochastic
398
8.3
8.2 The Maxin1um Variable and the Reflection
Principle
8.2.1 The Reflection
8.2.2 The Time to First Reach a Level
8.2.3 The Zeros of Brownian Motion
Variations and Extensions
8.3.1 Reflected Brownian Motion
8.3.2 Absorbed Brownian Motion
8.3.3 The Brownian Bridge
8.3.4 Brownian Meander
Principle 405
406
407
408
411
411
412
4 1 4
4 1 6
Contents
ix
8.4 Brownian Motion
with Drift
8.4.1 The Gambler's Ruin Problem
8.4.2 Geometric
Brownian Motion
8.5 The Orn
stein-Uhlenbeck Process
4 19
420
424
432
ysical Brownian Motion 434
437
439
Approach to Ph
8.5.1 A Second
8.5.2 The Position Process
8.5.3 The Long Run Behavior
8.5.4 Brownian Measure and Integration 441
9 Queueing
Systems
9.1 Queueing Processes
9.1 . 1 The Queueing Formula L = A. W
9.1 .2 A Sampling of Queueing
Models
9.2 Poisson Arrivals, Exponential Service Times
I System
9.2.1 TheM I M I
9.2.2 The MIMioo System
9.2.3 The MIMis System
9.3 General Service
Time Distributions
9.3.1 TheMIG il System
9.3.2 The MIGioo System
9.4 Variations and Extensions
9.4.1 Systems with Balking
9.4.2 Variable Service
9.4.3 A System with Feedback
9.4.4 A Two-Server Overflow Queue
9.4.5 Preemptive Priority Queues
Rates
9.5 Open Acyclic Queueing Networks
9.5.1 The Basic Theorem
9.5.2 Two Queues in Tandem
9.5.3 Open Acyclic Networks
9.5.4 Appendix: Time Reve
9.5.5 Proof of
Theorem 9.1
rsibility
9.6 General Open Networks
9.6.1 The General Open Network
447
447
448
449
451
452
456
457
460
460
465
468
468
469
470
470
472
480
480
481
482
485
487
488
492
495
495
498
500
10 Random Evolutions
1 0. 1 Two-State Velocity Model
I 0.1 . 1 Two-State Random Evolution
I 0.1 .2 The Telegraph Equation
I 0.1 .3 Distribution
Two-State Model
Functions and Densities
1 0.1 .4 Passage Time Distributions
501
505
507
1 0.2.1 Finite Markov Chains and Random Velocity Models 507
1 0.2.2 Constructive Approach
of Random Velocity Models 507
in the
1 0.2 N-State Random Evolution
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