Introduction to Stochastic Modeling

ISBN-10: 0123814162

ISBN-13: 9780123814166

Edition: 4th 2011

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Description: Serving as the foundation for a one-semester course in stochastic processes for students familiar with elementary probability theory and calculus, Introduction to Stochastic Modeling, Third Edition, bridges the gap between basic probability and an intermediate level course in stochastic processes. The objectives of the text are to introduce students to the standard concepts and methods of stochastic modeling, to illustrate the rich diversity of applications of stochastic processes in the applied sciences, and to provide exercises in the application of simple stochastic analysis to realistic problems.

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Book details

List price: $60.99
Edition: 4th
Copyright year: 2011
Publisher: Elsevier Science & Technology
Publication date: 1/13/2011
Binding: Paperback
Pages: 584
Size: 6.13" wide x 9.21" long x 1.25" tall
Weight: 2.288
Language: English

Preface to the Fourth Edition
Preface to the Third Edition
Preface to the First Edition
To the Instructor
Acknowledgments
Introduction
Stochastic Modeling
Stochastic Processes
Probability Review
Events and Probabilities
Random Variables
Moments and Expected Values
Joint Distribution Functions
Sums and Convolutions
Change of Variable
Conditional Probability
Review of Axiomatic Probability Theory
The Major Discrete Distributions
Bernoulli Distribution
Binomial Distribution
Geometric and Negative Binominal Distributions
The Poisson Distribution
The Multinomial Distribution
Important Continuous Distributions
The Normal Distribution
The Exponential Distribution
The Uniform Distribution
The Gamma Distribution
The Beta Distribution
The Joint Normal Distribution
Some Elementary Exercises
Tail Probabilities
The Exponential Distribution
Useful Functions, Integrals, and Sums
Conditional Probability and Conditional Expectation
The Discrete Case
The Dice Game Craps
Random Sums
Conditional Distributions: The Mixed Case
The Moments of a Random Sum
The Distribution of a Random Sum
Conditioning on a Continuous Random Variable
Martingales
The Definition
The Markov Inequality
The Maximal Inequality for Nonnegative Martingales
Markov Chains: Introduction
Definitions
Transition Probability Matrices of a Markov Chain
Some Markov Chain Models
An Inventory Model
The Ehrenfest Urn Model
Markov Chains in Genetics
A Discrete Queueing Markov Chain
First Step Analysis
Simple First Step Analyses
The General Absorbing Markov Chain
Some Special Markov Chains
The Two-State Markov Chain
Markov Chains Defined by Independent Random Variables
One-Dimensional Random Walks
Success Runs
Functionals of Random Walks and Success Runs
The General Random Walk
Cash Management
The Success Runs Markov Chain
Another Look at First Step Analysis
Branching Processes
Examples of Branching Processes
The Mean and Variance of a Branching Process
Extinction Probabilities
Branching Processes and Generating Functions
Generating Functions and Extinction Probabilities
Probability Generating Functions and Sums of Independent Random Variables
Multiple Branching Processes
The Long Run Behavior of Markov Chains
Regular Transition Probability Matrices
Doubly Stochastic Matrices
Interpretation of the Limiting Distribution
Examples
Including History in the State Description
Reliability and Redundancy
A Continuous Sampling Plan
Age Replacement Policies
Optimal Replacement Rules
The Classification of States
Irreducible Markov Chains
Periodicity of a Markov Chain
Recurrent and Transient States
The Basic Limit Theorem of Markov Chains
Reducible Markov Chains
Poisson Processes
The Poisson Distribution and the Poisson Process
The Poisson Distribution
The Poisson Process
Nonhomogeneous Processes
Cox Processes
The Law of Rare Events
The Law of Rare Events and the Poisson Process
Proof of Theorem 5.3
Distributions Associated with the Poisson Process
The Uniform Distribution and Poisson Processes
Shot Noise
Sum Quota Sampling
Spatial Poisson Processes
Compound and Marked Poisson Processes
Compound Poisson Processes
Marked Poisson Processes
Continuous Time Markov Chains
Pure Birth Processes
Postulates for the Poisson Process
Pure Birth Process
The Yule Process
Pure Death Processes
The Linear Death Process
Cable Failure Under Static Fatigue
Birth and Death Processes
Postulates
Sojourn Times
Differential Equations of Birth and Death Processes
The Limiting Behavior of Birth and Death Processes
Birth and Death Processes with Absorbing States
Probability of Absorption into State 0
Mean Time Until Absorption
Finite-State Continuous Time Markov Chains
A Poisson Process with a Markov Intensity
Renewal Phenomena
Definition of a Renewal Process and Related Concepts
Some Examples of Renewal Processes
Brief Sketches of Renewal Situations
Block Replacement
The Poisson Process Viewed as a Renewal Process
The Asymptotic Behavior of Renewal Processes
The Elementary Renewal Theorem
The Renewal Theorem for Continuous Lifetimes
The Asymptotic Distribution of N(t)
The Limiting Distribution of Age and Excess Life
Generalizations and Variations on Renewal Processes
Delayed Renewal Processes
Stationary Renewal Processes
Cumulative and Related Processes
Discrete Renewal Theory
The Discrete Renewal Theorem
Deterministic Population Growth with Age Distribution
Brownian Motion and Related Processes
Brownian Motion and Gaussian Processes
A Little History
The Brownian Motion Stochastic Process
The Central Limit Theorem and the Invariance Principle
Gaussian Processes
The Maximum Variable and the Reflection Principle
The Reflection Principle
The Time to First Reach a Level
The Zeros of Brownian Motion
Variations and Extensions
Reflected Brownian Motion
Absorbed Brownian Motion
The Brownian Bridge
Brownian Meander
Brownian Motion with Drift
The Gambler's Ruin Problem
Geometric Brownian Motion
The Ornstein-Uhlenbeck Process
A Second Approach to Physical Brownian Motion
The Position Process
The Long Run Behavior
Brownian Measure and Integration
Queueing Systems
Queueing Processes
The Queueing Formula L = X W
A Sampling of Queueing Models
Poisson Arrivals, Exponential Service Times
The M/M/1 System
The M/M/$ System
The M/M/s System
General Service Time Distributions
The M/G/1 System
The M/G/$ System
Variations and Extensions
Systems with Balking
Variable Service Rates
A System with Feedback
A Two-Server Overflow Queue
Preemptive Priority Queues
Open Acyclic Queueing Networks
The Basic Theorem
Two Queues in Tandem
Open Acyclic Networks
Appendix: Time Reversibility
Proof of Theorem 9.1
General Open Networks
The General Open Network
Random Evolutions
Two-State Velocity Model
Two-State Random Evolution
The Telegraph Equation
Distribution Functions and Densities in the Two-State Model
Passage Time Distributions
JV-State Random Evolution
Finite Markov Chains and Random Velocity Models
Constructive Approach of Random Velocity Models
Random Evolution Processes
Existence-Uniqueness of the First-Order System (10.26)
Single Hyperbolic Equation
Spectral Properties of the Transition Matrix
Recurrence Properties of Random Evolution
Weak Law and Central Limit Theorem
Isotropic Transport in Higher Dimensions
The Rayleigh Problem of Random Flights
Three-Dimensional Rayleigh Model
Characteristic Functions and Their Applications
Definition of the Characteristic Function
Two Basic Properties of the Characteristic Function
Inversion Formulas for Characteristic Functions
Fourier Reciprocity/Local Non-Uniqueness
Fourier Inversion and Parseval's Identity Inversion
Formula for General Random Variables
The Continuity Theorem
Proof of the Continuity Theorem
Proof of the Central Limit Theorem
Stirling's Formula and Applications
Poisson Representation of n!
Proof of Stirling's Formula
Local deMoivre-Laplace Theorem
Further Reading
Answers to Exercises
Index
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