概率论、马尔科夫链、排队和模拟

书籍目录

Preface and Acknowledgments

Ⅰ PROBABILITY

1 Probability

1.1 Trials, Sample Spaces, and Events

1.2 Probability Axioms and Probability Space

1.3 Conditional Probability

1.4 Independent Events

1.5 Law of Total Probability

1.6 Bayes' Rule

1.7 Exercises

2 Combinatorics-The Art of Counting

2.1 Permutations

2.2 Permutations with Replacements

2.3 Permutations without Replacement

2.4 Combinations without Replacement

2.5 Combinations with Replacements

2.6 Bernoulli (Independent) Trials

2.7 Exercises

3 Random Variables and Distribution Functions

3.1 Discrete and Continuous Random Variables

3.2 The Probability Mass Function for a Discrete Random Variable

3.3 The Cumulative Distribution Function

3.4 The Probability Density Function for a Continuous Random Variable

3.5 Functions of a Random Variable

3.6 Conditioned Random Variables

3.7 Exercises

4 Joint and Conditional Distributions

4.1 Joint Distributions

4.2 Joint Cumulative Distribution Functions

4.3 Joint Probability Mass Functions

4.4 Joint Probability Density Functions

4.5 Conditional Distributions

4.6 Convolutions and the Sum of Two Random Variables

4.7 Exercises

5 Expectations and More

5.1 Definitions

5.2 Expectation of Functions and Joint Random Variables

5.3 Probability Generating Functions for Discrete Random Variables

5.4 Moment Generating Functions

5.5 Maxima and Minima of Independent Random Variables

5.6 Exercises

6 Discrete Distribution Functions

6.1 The Discrete Uniform Distribution

6.2 The Bernoulli Distribution

6.3 The Binomial Distribution

6.4 Geometric and Negative Binomial Distributions

6.5 The Poisson Distribution

6.6 The Hypergeometric Distribution

6.7 The Multinomial Distribution

6.8 Exercises

Continuous Distribution Functions

7.1 The Uniform Distribution

7.2 The Exponential Distribution

7.3 The Normal or Gaussian Distribution

7.4 The Gamma Distribution

7.5 Reliability Modeling and the Weibull Distribution

7.6 Phase-Type Distributions

7.6.1 The Erlang-2 Distribution

7.6.2 The Erlang-r Distribution

7.6.3 The Hypoexponential Distribution

7.6.4 The Hyperexponential Distribution

7.6.5 The Coxian Distribution

7.6.6 General Phase-Type Distributions

7.6.7 Fitting Phase-Type Distributions to Means and Variances

7.7 Exercises

8 Bounds and Limit Theorems

8.1 The Markov Inequality

8.2 The Chebychev Inequality

8.3 The Chernoff Bound

8.4 The Laws of Large Numbers

8.5 The Central Limit Theorem

8.6 Exercises

Ⅱ MARKOV CHAINS

9 Discrete- and Continuous-Time Markov Chains

9.1 Stochastic Processes and Markov Chains

9.2 Discrete-Time Markov Chains: Definitions

9.3 The Chapman-Kolmogorov Equations

9.4 Classification of States

9.5 Irreducibility

9.6 The Potential, Fundamental, and Reachability Matrices

9.6.1 Potential and Fundamental Matrices and Mean Time to Absorption

9.6.2 The Reachability Matrix and Absorption Probabilities

9.7 Random Walk Problems

9.8 Probability Distributions

9.9 Reversibility

9.10 Continuous-Time Markov Chains

9.10.1 Transition Probabilities and Transition Rates

9.10.2 The Chapman-Kolmogorov Equations

9.10.3 The Embedded Markov Chain and State Properties

9.10.4 Probability Distributions

9.10.5 Reversibility

9.11 Semi-Markov Processes

9.12 Renewal Processes

9.13 Exercises

10 Numerical Solution of Markov Chains

10.1 Introduction

10.1.1 Setting the Stage

10.1.2 Stochastic Matrices

10.1.3 The Effect of Discretization

10.2 Direct Methods for Stationary Distributions

10.2.1 Iterative versus Direct Solution Methods

10.2.2 Gaussian Elimination and LU Factorizattons

10.3 Basic Iterative Methods for Stationary Distributions

10.3.1 The Power Method

10.3.2 The Iterative Methods of Jacobi and Gauss-Seidel

10.3.3 The Method of Successive Overrelaxation

10.3.4 Data Structures for Large Sparse Matrices

10.3.5 Initial Approximations, Normalization, and Convergence

10.4 Block Iterative Methods

10.5 Decomposition and Aggregation Methods

10.6 The Matrix Geometric/Analytic Methods for Structured Markov Chains

10.6.1 The Quasi-Birth-Death Case

10.6.2 Block Lower Hessenberg Markov Chains

10.6.3 Block Upper Hessenberg Markov Chains

10.7 Transient Distributions

10.7.1 Matrix Scaling and Powering Methods for Small State Spaces

10.7.2 The Uniformization Method for Large State Spaces

10.7.3 Ordinary Differential Equation Solvers

10.8 Exercises

Ⅲ QUEUEING MODELS

11 Elementary Queueing Theory

11.1 Introduction and Basic Definitions

11.1.1 Arrivals and Service

11.1.2 Scheduling Disciplines

11.1.3 Kendall's Notation

11.1.4 Graphical Representations of Queues

11.1.5 Performance Measures--Measures of Effectiveness

11.1.6 Little's Law

11.2 Birth-Death Processes: The M/M/I Queue

11.2.1 Description and Steady-State Solution

11.2.2 Performance Measures

11,2.3 Transient Behavior

11.3 General Birth-Death Processes

11,3. I Derivation of the State Equations

11.3.2 Steady-State Solution

11.4 Multiserver Systems

11.4.1 The M/M/c Queue

11.4.2 The M/M/∞ Queue

11.5 Finite-Capacity Systems--The M/M/1/K Queue

11.6 Multiserver, Finite-Capacity Systems--The M/M/c/K Queue

11.7 Finite-Source Systems-The M/M/c//M Queue

11.8 State-Dependent Service

11.9 Exercises

12 Queues with Phase-Type Laws: Neuts' Matrix-Geometric Method

12.1 The Erlang-r Service Model--The M/Er/l Queue

12.2 The Erlang-r Arrival Model-The Er/M/] Queue

12.3 The M/H2/1 and H2/M/1 Queues

12.4 Automating the Analysis of Single-Server Phase-Type Queues

12.5 The H2/E3/1 Queue and General Ph/Ph/1 Queues

12.6 Stability Results for Ph/Ph/l Queues

12.7 Performance Measures for Ph/Ph/1 Queues

12.8 Matlab code for Ph/Ph/1 Queues

12.9 Exercises

13 The z-Transform Approach to Solving Markovian Queues

13.1 The z-Transform

13.2 The Inversion Process

13.3 Solving Markovian Queues using z-Transforms

13.3.1 The z-Transform Procedure

13.3.2 The M/M/1 Queue Solved using z-Transforms

13.3.3 The M/M/1 Queue with Arrivals in Pairs

13.3.4 The M/Er/1 Queue Solved using z-Transforms

13.3.5 The Er/M/1 Queue Solved using z-Transforms

13.3.6 Bulk Queueing Systems

13.4 Exercises

14 The M/G/1 and G/M/1 Queues

14.1 Introduction to the M/G/1 Queue

14.2 Solution via an Embedded Markov Chain

14.3 Performance Measures for the M/G/1 Queue

14.3.1 The Pollaczek-Khintchine Mean Value Formula

14.3.2 The Pollaczek-Khintchine Transform Equations

14.4 The M/G/1 Residual Time: Remaining Service Time

14.5 The M/G/1 Busy Period

14.6 Priority Scheduling

14.6.1 M/M/1: Priority Queue with Two Customer Classes

14.6.2 M/G/1: Nonpreemptive Priority Scheduling

……

Ⅳ SIMULATION

Appendix A: The Greek Alphabet

Appendix B: Elements of Linear Algebra

Bibliography

Index

作者简介

斯图尔特著的《概率论马尔科夫链排队和模拟》是一部讲述如何挖掘蕴藏在模型表现形式背后的数学过程的权威作品。详细的数学推导和大量图例使得更加易于研究生和高年级本科生作为学习随机过程的教材和参考资料。本书应用广泛，也适用于计算科学、工程、运筹学、统计和数学等学科。

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