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E-grāmata: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling

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  • Formāts: 776 pages
  • Izdošanas datums: 06-Jul-2009
  • Izdevniecība: Princeton University Press
  • Valoda: eng
  • ISBN-13: 9781400832811
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Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance ModelingPalielināt
  • Formāts: 776 pages
  • Izdošanas datums: 06-Jul-2009
  • Izdevniecība: Princeton University Press
  • Valoda: eng
  • ISBN-13: 9781400832811
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Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics. The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolomogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation. Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available.Numerous examples illuminate the mathematical theories Carefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approach Each chapter concludes with an extensive set of exercises An instructor's solution manual, in which all exercises are completely worked out, is available

Recenzijas

"The book represents a valuable text for courses in statistics and stochastic processes, so it is strongly recommended to libraries."--Hassan S. Bakouch, Journal of Applied Statistics

Papildus informācija

This is an excellent book on the topics of probability, Markov chains, and queuing theory. Extremely well-written, the contents range from elementary topics to quite advanced material and include plenty of well-chosen examples. -- Adarsh Sethi, University of Delaware Clear and pleasant to read, this book distinguishes itself from comparable textbooks by its inclusion of the computational aspects of the material. -- Richard R. Muntz, University of California, Los Angeles
Preface and Acknowledgments xv
I PROBABILITY
1(190)
Probability
3(22)
Trials, Sample Spaces, and Events
3(6)
Probability Axioms and Probability Space
9(3)
Conditional Probability
12(3)
Independent Events
15(3)
Law of Total Probability
18(2)
Bayes' Rule
20(1)
Exercises
21(4)
Combinatorics---The Art of Counting
25(15)
Permutations
25(1)
Permutations with Replacements
26(1)
Permutations without Replacements
27(2)
Combinations without Replacement
29(2)
Combinations with Replacements
31(2)
Bernoulli (Independent) Trials
33(3)
Exercises
36(4)
Random Variables and Distribution Functions
40(24)
Discrete and Continuous Random Variables
40(3)
The Probability Mass Function for a Discrete Random Variable
43(3)
The Cumulative Distribution Function
46(5)
The Probability Density Function for a Continuous Random Variable
51(2)
Functions of a Random Variable
53(5)
Conditioned Random Variables
58(2)
Exercises
60(4)
Joint and Conditional Distributions
64(23)
Joint Distributions
64(1)
Joint Cumulative Distribution Functions
64(4)
Joint Probability Mass Functions
68(3)
Joint Probability Density Functions
71(6)
Conditional Distributions
77(3)
Convolutions and the Sum of Two Random Variables
80(2)
Exercises
82(5)
Expectations and More
87(28)
Definitions
87(5)
Expectation of Functions and Joint Random Variables
92(8)
Probability Generating Functions for Discrete Random Variables
100(3)
Moment Generating Functions
103(5)
Maxima and Minima of Independent Random Variables
108(2)
Exercises
110(5)
Discrete Distribution Functions
115(19)
The Discrete Uniform Distribution
115(1)
The Bernoulli Distribution
116(1)
The Binomial Distribution
117(3)
Geometric and Negative Binomial Distributions
120(4)
The Poisson Distribution
124(3)
The Hypergeometric Distribution
127(1)
The Multinomial Distribution
128(2)
Exercises
130(4)
Continuous Distribution Functions
134(46)
The Uniform Distribution
134(2)
The Exponential Distribution
136(5)
The Normal or Gaussian Distribution
141(4)
The Gamma Distribution
145(4)
Reliability Modeling and the Weibull Distribution
149(6)
Phase-Type Distributions
155(21)
The Erlang-2 Distribution
155(3)
The Erlang-r Distribution
158(4)
The Hypoexponential Distribution
162(2)
The Hyperexponential Distribution
164(2)
The Coxian Distribution
166(2)
General Phase-Type Distributions
168(3)
Fitting Phase-Type Distributions to Means and Variances
171(5)
Exercises
176(4)
Bounds and Limit Theorems
180(11)
The Markov Inequality
180(1)
The Chebychev Inequality
181(1)
The Chernoff Bound
182(1)
The Laws of Large Numbers
182(2)
The Central Limit Theorem
184(3)
Exercises
187(4)
II MARKOV CHAINS
191(192)
Discrete- and Continuous- Time Markov Chains
193(92)
Stochastic Processes and Markov Chains
193(2)
Discrete- Time Markov Chains: Definitions
195(7)
The Chapman-Kolmogorov Equations
202(4)
Classification of States
206(8)
Irreducibility
214(4)
The Potential, Fundamental, and Reachability Matrices
218(10)
Potential and Fundamental Matrices and Mean Time to Absorption
219(4)
The Reachability Matrix and Absorption Probabilities
223(5)
Random Walk Problems
228(7)
Probability Distributions
235(13)
Reversibility
248(5)
Continuous-Time Markov Chains
253(12)
Transition Probabilities and Transition Rates
254(3)
The Chapman-Kolmogorov Equations
257(2)
The Embedded Markov Chain and State Properties
259(3)
Probability Distributions
262(3)
Reversibility
265(1)
Semi-Markov Processes
265(2)
Renewal Processes
267(8)
Exercises
275(10)
Numerical Solution of Markov Chains
285(98)
Introduction
285(5)
Setting the Stage
285(2)
Stochastic Matrices
287(2)
The Effect of Discretization
289(1)
Direct Methods for Stationary Distributions
290(11)
Iterative versus Direct Solution Methods
290(1)
Gaussian Elimination and LU Factorizations
291(10)
Basic Iterative Methods for Stationary Distributions
301(18)
The Power Method
301(4)
The Iterative Methods of Jacobi and Gauss-Seidel
305(6)
The Method of Successive Overrelaxation
311(2)
Data Structures for Large Sparse Matrices
313(3)
Initial Approximations, Normalization, and Convergence
316(3)
Block Iterative Methods
319(5)
Decomposition and Aggregation Methods
324(8)
The Matrix Geometric/Analytic Methods for Structured Markov Chains
332(22)
The Quasi-Birth-Death Case
333(7)
Block Lower Hessenberg Markov Chains
340(5)
Block Upper Hessenberg Markov Chains
345(9)
Transient Distributions
354(21)
Matrix Scaling and Powering Methods for Small State Spaces
357(4)
The Uniformization Method for Large State Spaces
361(4)
Ordinary Differential Equation Solvers
365(10)
Exercises
375(8)
III QUEUEING MODELS
383(228)
Elementary Queueing Theory
385(59)
Introduction and Basic Definitions
385(17)
Arrivals and Service
386(9)
Scheduling Disciplines
395(1)
Kendall's Notation
396(1)
Graphical Representations of Queues
397(1)
Performance Measures---Measures of Effectiveness
398(2)
Little's Law
400(2)
Birth-Death Processes: The M/M/1 Queue
402(11)
Description and Steady-State Solution
402(4)
Performance Measures
406(6)
Transient Behavior
412(1)
General Birth-Death Processes
413(6)
Derivation of the State Equations
413(2)
Steady-State Solution
415(4)
Multiserver Systems
419(6)
The M/M/c Queue
419(6)
The M/M/∞ Queue
425(1)
Finite-Capacity Systems---The M/M/1/K Queue
425(7)
Multiserver, Finite-Capacity Systems---The M/M/c/K Queue
432(2)
Finite-Source Systems---The M/M/c//M Queue
434(3)
State-Dependent Service
437(1)
Exercises
438(6)
Queues with Phase-Type Laws: Neuts' Matrix-Geometric Method
444(31)
The Erlang-r Service Model---The M/Er/1 Queue
444(6)
The Erlang-r Arrival Model---The Er/M/1 Queue
450(4)
The M/H2/1 and H2/M/1 Queues
454(4)
Automating the Analysis of Single-Server Phase-Type Queues
458(2)
The H2/E3/1 Queue and General Ph/Ph/1 Queues
460(6)
Stability Results for Ph/Ph/1 Queues
466(2)
Performance Measures for Ph/Ph/1 Queues
468(1)
Matlab code for Ph/Ph/1 Queues
469(2)
Exercises
471(4)
The z-Transform Approach to Solving Markovian Queues
475(34)
The z-Transform
475(3)
The Inversion Process
478(6)
Solving Markovian Queues using z-Transforms
484(22)
The z-Transform Procedure
484(1)
The M/M/1 Queue Solved using z-Transforms
484(2)
The M/M/1 Queue with Arrivals in Pairs
486(2)
The M/Er/1 Queue Solved using z-Transforms
488(8)
The Er/M/1 Queue Solved using z-Transforms
496(7)
Bulk Queueing Systems
503(3)
Exercises
506(3)
The M/G/1 and G/M/1 Queues
509(50)
Introduction to the M/G/1 Queue
509(1)
Solution via an Embedded Markov Chain
510(5)
Performance Measures for the M/G/1 Queue
515(8)
The Pollaczek-Khintchine Mean Value Formula
515(3)
The Pollaczek-Khintchine Transform Equations
518(5)
The M/G/1 Residual Time: Remaining Service Time
523(3)
The M/G/1 Busy Period
526(5)
Priority Scheduling
531(11)
M/M/1: Priority Queue with Two Customer Classes
531(2)
M/G/1: Nonpreemptive Priority Scheduling
533(3)
M/G/1: Preempt-Resume Priority Scheduling
536(2)
A Conservation Law and SPTF Scheduling
538(4)
The M/G/1/K Queue
542(4)
The G/M/1 Queue
546(5)
The G/M/1/K Queue
551(2)
Exercises
553(6)
Queueing Networks
559(52)
Introduction
559(4)
Basic Definitions
559(1)
The Departure Process---Burke's Theorem
560(2)
Two M/M/1 Queues in Tandem
562(1)
Open Queueing Networks
563(5)
Feedforward Networks
563(1)
Jackson Networks
563(4)
Performance Measures for Jackson Networks
567(1)
Closed Queueing Networks
568(14)
Definitions
568(2)
Computation of the Normalization Constant: Buzen's Algorithm
570(7)
Performance Measures
577(5)
Mean Value Analysis for Closed Queueing Networks
582(9)
The Flow-Equivalent Server Method
591(3)
Multiclass Queueing Networks and the BCMP Theorem
594(8)
Product-Form Queueing Networks
595(3)
The BCMP Theorem for Open, Closed, and Mixed Queueing Networks
598(4)
Java Code
602(5)
Exercises
607(4)
IV SIMULATION
611(108)
Some Probabilistic and Deterministic Applications of Random Numbers
613(12)
Simulating Basic Probability Scenarios
613(5)
Simulating Conditional Probabilities, Means, and Variances
618(2)
The Computation of Definite Integrals
620(3)
Exercises
623(2)
Uniformly Distributed ``Random'' Numbers
625(22)
Linear Recurrence Methods
626(4)
Validating Sequences of Random Numbers
630(14)
The Chi-Square ``Goodness-of-Fit'' Test
630(3)
The Kolmogorov-Smirnov Test
633(1)
``Run'' Tests
634(6)
The ``Gap'' Test
640(1)
The ``Poker'' Test
641(3)
Statistical Test Suites
644(1)
Exercises
644(3)
Nonuniformly Distributed ``Random'' Numbers
647(33)
The Inverse Transformation Method
647(7)
The Continuous Uniform Distribution
649(1)
``Wedge-Shaped'' Density Functions
649(1)
``Triangular'' Density Functions
650(2)
The Exponential Distribution
652(1)
The Bernoulli Distribution
653(1)
An Arbitrary Discrete Distribution
653(1)
Discrete Random Variates by Mimicry
654(3)
The Binomial Distribution
654(1)
The Geometric Distribution
655(1)
The Poisson Distribution
656(1)
The Accept-Reject Method
657(5)
The Lognormal Distribution
660(2)
The Composition Method
662(8)
The Erlang-r Distribution
662(1)
The Hyperexponential Distribution
663(1)
Partitioning of the Density Function
664(6)
Normally Distributed Random Numbers
670(3)
Normal Variates via the Central Limit Theorem
670(1)
Normal Variates via Accept-Reject and Exponential Bounding Function
670(2)
Normal Variates via Polar Coordinates
672(1)
Normal Variates via Partitioning of the Density Function
673(1)
The Ziggurat Method
673(3)
Exercises
676(4)
Implementing Discrete-Event Simulations
680(17)
The Structure of a Simulation Model
680(2)
Some Common Simulation Examples
682(13)
Simulating the M/M/1 Queue and Some Extensions
682(4)
Simulating Closed Networks of Queues
686(3)
The Machine Repairman Problem
689(3)
Simulating an Inventory Problem
692(3)
Programming Projects
695(2)
Simulation Measurements and Accuracy
697(22)
Sampling
697(10)
Point Estimators
698(6)
Interval Estimators/Confidence Intervals
704(3)
Simulation and the Independence Criteria
707(4)
Variance Reduction Methods
711(5)
Antithetic Variables
711(2)
Control Variables
713(3)
Exercises
716(3)
APPENDIX A: THE GREEK ALPHABET
719(2)
APPENDIX B: ELEMENTS OF LINEAR ALGEBRA
721(24)
Vectors and Matrices
721(1)
Arithmetic on Matrices
721(2)
Vector and Matrix Norms
723(1)
Vector Spaces
724(2)
Determinants
726(2)
Systems of Linear Equations
728(6)
Gaussian Elimination and LU Decompositions
730(4)
Eigenvalues and Eigenvectors
734(4)
Eigenproperties of Decomposable, Nearly Decomposable, and Cyclic Stochastic Matrices
738(7)
Normal Form
738(1)
Eigenvalues of Decomposable Stochastic Matrices
739(2)
Eigenvators of Decomposable Stochastic Matrices
741(2)
Nearly Decomposable Stochastic Matrices
743(1)
Cyclic Stochastic Matrices
744(1)
Bibliography 745(4)
Index 749
William J. Stewart is professor of computer science at North Carolina State University. He is the author of "An Introduction to the Numerical Solution of Markov Chains" (Princeton).
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