Research Methods and Statistics Arena

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• Price: $99.95 $89.96
• Hardback: 563 pages
• Also available in e-Book
• Published: December 2009
• ISBN: 978-1-4398087-5-7
• Publisher: Chapman & Hall

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Based on the author’s more than 25 years of teaching experience, Modeling and Analysis of Stochastic Systems, Second Edition covers the most important classes of stochastic processes used in the modeling of diverse systems, from supply chains and inventory systems to genetics and biological systems. For each class of stochastic process, the text includes its definition, characterization, applications, transient and limiting behavior, first passage times, and cost/reward models. Along with reorganizing the material, this edition revises and adds new exercises and examples.

New to the Second Edition

• A new chapter on diffusion processes that gives an accessible and non-measure-theoretic treatment with applications to finance
• A more streamlined, application-oriented approach to renewal, regenerative, and Markov regenerative processes
• Two appendices that collect relevant results from analysis and differential and difference equations

Rather than offer special tricks that work in specific problems, this book provides thorough coverage of general tools that enable the solution and analysis of stochastic models. After mastering the material in the text, students will be well-equipped to build and analyze useful stochastic models for various situations.

A collection of MATLAB^®-based programs can be downloaded from the author’s website and a solutions manual is available for qualifying instructors.

Table of Contents

Introduction

What in the World Is a Stochastic Process?

How to Characterize a Stochastic Process

What Do We Do with a Stochastic Process?

Discrete-Time Markov Chains: Transient Behavior

Definition and Characterization

Examples

DTMCs in Other Fields

Marginal Distributions

Occupancy Times

Computation of Matrix Powers

DTMCs: First Passage Times

Definitions

Cumulative Distribution Function of T

Absorption Probabilities

Expectation of T

Generating Function and Higher Moments of T

DTMCs: Limiting Behavior

Exploring the Limiting Behavior by Examples

Irreducibility and Periodicity

Recurrence and Transience

Determining Recurrence and Transience: Infinite DTMCs

Limiting Behavior of Irreducible DTMCs

Examples: Limiting Behavior of Infinite State-Space Irreducible DTMCs

Limiting Behavior of Reducible DTMCs

DTMCs with Costs and Rewards

Reversibility

Poisson Processes

Exponential Distributions

Poisson Process: Definitions

Event Times in a Poisson Process

Superposition and Splitting of Poisson Processes

Non-Homogenous Poisson Process

Compound Poisson Process

Continuous-Time Markov Chains

Definitions and Sample Path Properties

Examples

Transient Behavior: Marginal Distribution

Transient Behavior: Occupancy Times

Computation of P(t): Finite State-Space

Computation of P(t): Infinite State-Space

First-Passage Times

Exploring the Limiting Behavior by Examples

Classification of States

Limiting Behavior of Irreducible CTMCs

Limiting Behavior of Reducible CTMCs

CTMCs with Costs and Rewards

Phase-Type Distributions

Reversibility

Queueing Models

Introduction

Properties of General Queueing Systems

Birth and Death Queues

Open Queueing Networks

Closed Queueing Networks

Single Server Queues

Retrial Queue

Infinite Server Queue

Renewal Processes

Introduction

Properties of N(t)

The Renewal Function

Renewal-Type Equation

Key Renewal Theorem

Recurrence Times

Delayed Renewal Processes

Alternating Renewal Processes

Semi-Markov Processes

Renewal Processes with Costs/Rewards

Regenerative Processes

Markov Regenerative Processes

Definitions and Examples

Markov Renewal Process and Markov Renewal Function

Key Renewal Theorem for MRPs

Extended Key Renewal Theorem

Semi-Markov Processes: Further Results

Markov Regenerative Processes

Applications to Queues

Diffusion Processes

Brownian Motion

Sample Path Properties of BM

Kolmogorov Equations for Standard Brownian Motion

First Passage Times

Reflected SBM

Reflected BM and Limiting Distributions

BM and Martingales

Cost/Reward Models

Stochastic Integration

Stochastic Differential Equations

Applications to Finance

Epilogue

Appendix A: Probability of Events

Appendix B: Univariate Random Variables

Appendix C: Multivariate Random Variables

Appendix D: Generating Functions

Appendix E: Laplace–Stieltjes Transforms

Appendix F: Laplace Transforms

Appendix G: Modes of Convergence

Appendix H: Results from Analysis

Appendix I: Difference and Differential Equations

Answers to Selected Problems

References

Index

Exercises appear at the end of each chapter.

Author Biography

Vidyadhar G. Kulkarni is a Norman Johnson Professor in the Department of Statistics and Operations Research at the University of North Carolina at Chapel Hill.