《随机模型概论(英文版第4版)》(作者品斯基、卡尔林)适合作为一学期随机过程课程的教材，需要读者有初等概率论和微积分基础。本书的目标是介绍随机建模的基本概念和方法，阐明随机过程在科学领域中的各种应用，其主要内容包括条件概率与条件期望、马尔可夫链、泊松过程、连续时间马尔可夫链、布朗运动及相关过程、排队系统、随机发展和特征函数及其应用等。此外，每节后都配有大量与实际问题相关的练习，便于读者巩固、提高。

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

2.313 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 Functionals 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 = ~.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/oo System

9.2.3 The M/M/s System

9.3 General Service Time Distributions

9.3.1 TheM~G~1 System

9.3.2 The M/G/oo 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