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Probability, Markov chains, queues and simulation.pdf

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    Cover
    Contents
    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
    7 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 Factorizations
    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/1 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
    11.3.1 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/E[sub(r)]/1 Queue
    12.2 The Erlang-r Arrival Model—The E[sub(r)]/M/1 Queue
    12.3 The M/H[sub(2)]/1 and H[sub(2)]/M/1 Queues
    12.4 Automating the Analysis of Single-Server Phase-Type Queues
    12.5 The H[sub(2)]/E[sub(3)]/1 Queue and General Ph/Ph/1 Queues
    12.6 Stability Results for Ph/Ph/1 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/E[sub(r)]/1 Queue Solved using z-Transforms
    13.3.5 The E[sub(r)]/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
    14.6.3 M/G/1: Preempt-Resume Priority Scheduling
    14.6.4 A Conservation Law and SPTF Scheduling
    14.7 The M/G/1/K Queue
    14.8 The G/M/1 Queue
    14.9 The G/M/1/K Queue
    14.10 Exercises
    15 Queueing Networks
    15.1 Introduction
    15.1.1 Basic Definitions
    15.1.2 The Departure Process—Burke's Theorem
    15.1.3 Two M/M/1 Queues in Tandem
    15.2 Open Queueing Networks
    15.2.1 Feedforward Networks
    15.2.2 Jackson Networks
    15.2.3 Performance Measures for Jackson Networks
    15.3 Closed Queueing Networks
    15.3.1 Definitions
    15.3.2 Computation of the Normalization Constant: Buzen's Algorithm
    15.3.3 Performance Measures
    15.4 Mean Value Analysis for Closed Queueing Networks
    15.5 The Flow-Equivalent Server Method
    15.6 Multiclass Queueing Networks and the BCMP Theorem
    15.6.1 Product-Form Queueing Networks
    15.6.2 The BCMP Theorem for Open, Closed, and Mixed Queueing Networks
    15.7 Java Code
    15.8 Exercises
    IV: SIMULATION
    16 Some Probabilistic and Deterministic Applications of Random Numbers
    16.1 Simulating Basic Probability Scenarios
    16.2 Simulating Conditional Probabilities, Means, and Variances
    16.3 The Computation of Definite Integrals
    16.4 Exercises
    17 Uniformly Distributed "Random" Numbers
    17.1 Linear Recurrence Methods
    17.2 Validating Sequences of Random Numbers
    17.2.1 The Chi-Square "Goodness-of-Fit" Test
    17.2.2 The Kolmogorov-Smirnov Test
    17.2.3 "Run" Tests
    17.2.4 The "Gap" Test
    17.2.5 The "Poker" Test
    17.2.6 Statistical Test Suites
    17.3 Exercises
    18 Nonuniformly Distributed "Random" Numbers
    18.1 The Inverse Transformation Method
    18.1.1 The Continuous Uniform Distribution
    18.1.2 "Wedge-Shaped" Density Functions
    18.1.3 "Triangular" Density Functions
    18.1.4 The Exponential Distribution
    18.1.5 The Bernoulli Distribution
    18.1.6 An Arbitrary Discrete Distribution
    18.2 Discrete Random Variates by Mimicry
    18.2.1 The Binomial Distribution
    18.2.2 The Geometric Distribution
    18.2.3 The Poisson Distribution
    18.3 The Accept-Reject Method
    18.3.1 The Lognormal Distribution
    18.4 The Composition Method
    18.4.1 The Erlang-r Distribution
    18.4.2 The Hyperexponential Distribution
    18.4.3 Partitioning of the Density Function
    18.5 Normally Distributed Random Numbers
    18.5.1 Normal Variates via the Central Limit Theorem
    18.5.2 Normal Variates via Accept-Reject and Exponential Bounding Function
    18.5.3 Normal Variates via Polar Coordinates
    18.5.4 Normal Variates via Partitioning of the Density Function
    18.6 The Ziggurat Method
    18.7 Exercises
    19 Implementing Discrete-Event Simulations
    19.1 The Structure of a Simulation Model
    19.2 Some Common Simulation Examples
    19.2.1 Simulating the M/M/1 Queue and Some Extensions
    19.2.2 Simulating Closed Networks of Queues
    19.2.3 The Machine Repairman Problem
    19.2.4 Simulating an Inventory Problem
    19.3 Programming Projects
    20 Simulation Measurements and Accuracy
    20.1 Sampling
    20.1.1 Point Estimators
    20.1.2 Interval Estimators/Confidence Intervals
    20.2 Simulation and the Independence Criteria
    20.3 Variance Reduction Methods
    20.3.1 Antithetic Variables
    20.3.2 Control Variables
    20.4 Exercises
    Appendix A: The Greek Alphabet
    Appendix B: Elements of Linear Algebra
    B.1 Vectors and Matrices
    B.2 Arithmetic on Matrices
    B.3 Vector and Matrix Norms
    B.4 Vector Spaces
    B.5 Determinants
    B.6 Systems of Linear Equations
    B.6.1 Gaussian Elimination and LU Decompositions
    B.7 Eigenvalues and Eigenvectors
    B.8 Eigenproperties of Decomposable, Nearly Decomposable, and Cyclic Stochastic Matrices
    B.8.1 Normal Form
    B.8.2 Eigenvalues of Decomposable Stochastic Matrices
    B.8.3 Eigenvectors of Decomposable Stochastic Matrices
    B.8.4 Nearly Decomposable Stochastic Matrices
    B.8.5 Cyclic Stochastic Matrices
    Bibliography
    Index
    A
    B
    C
    D
    E
    F
    G
    H
    I
    J
    K
    L
    M
    N
    O
    P
    Q
    R
    S
    T
    U
    V
    W
    Z
    PROBABILITY, MARKOV CHAINS, QUEUES, AND SIMULATION
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    Copyright c2009 by Princeton University Press Published by Princeton University Press, 41 William street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Stewart, William J., 1946– Probability, Markov chains, queues and simulation: the mathematical basis of performance modeling/William J. Stewart. – 1st ed. p. cm. ISBN 978-0-691-14062-9 (cloth : alk. paper) 1. Probability–Computer simulation. 2. Markov processes. 3. Queueing theory. I. Title. QA273.S7532 2009 519.201 2008041122 13–dc22 British Library Cataloging-in-Publication Data is available This book has been composed in Times Printed on acid-free paper. ∞ press.princeton.edu MATLAB is a registered trademark of The MathWorks, Inc. Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
    This book is dedicated to all those whom I love, especially My dear wife, Kathie, and my wonderful children Nicola, Stephanie, Kathryn, and William My father, William J. Stewart and the memory of my mother, Mary (Marshall) Stewart
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    Contents Preface and Acknowledgments I PROBABILITY 1 Probability 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Trials, Sample Spaces, and Events Probability Axioms and Probability Space Conditional Probability Independent Events Law of Total Probability Bayes’ Rule Exercises 2 Combinatorics—The Art of Counting 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Permutations Permutations with Replacements Permutations without Replacement Combinations without Replacement Combinations with Replacements Bernoulli (Independent) Trials Exercises 3 Random Variables and Distribution Functions Discrete and Continuous Random Variables The Probability Mass Function for a Discrete Random Variable The Cumulative Distribution Function The Probability Density Function for a Continuous Random Variable Functions of a Random Variable Conditioned Random Variables Exercises 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4 Joint and Conditional Distributions 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Joint Distributions Joint Cumulative Distribution Functions Joint Probability Mass Functions Joint Probability Density Functions Conditional Distributions Convolutions and the Sum of Two Random Variables Exercises 5 Expectations and More 5.1 5.2 5.3 Definitions Expectation of Functions and Joint Random Variables Probability Generating Functions for Discrete Random Variables xv 1 3 3 9 12 15 18 20 21 25 25 26 27 29 31 33 36 40 40 43 46 51 53 58 60 64 64 64 68 71 77 80 82 87 87 92 100
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