An Introduction to Stochastic Modeling

Preface1 Introduction 1.1 Stochastic Modeling 1.2 Probability Review 1.3 The Major Discrete Distributions 1.4 Important Continuous Distributions 1.5 Some Elementary Exercises 1.6 Useful Functions, Integrals, and Sums2 Conditional Probability and Conditional Expectation 2.1 The Discrete Case 2.2 The Dice Game Craps 2.3 Random Sums 2.4 Conditioning on a Continuous Random Variable3 Markov Chains: Introduction 3.1 Definitions 3.2 Transition Probability Matrices of a Markov Chain 3.3 Some Markov Chain Models 3.4 First Step Analysis 3.5 Some Special Markov Chains 3.6 Functionals of Random Walks and Success Runs 3.7 Another Look at First Step Analysis4 The Long Run Behavior of Markov Chains 4.1 Regular Transition Probability Matrices 4.2 Examples 4.3 The Classification of States 4.4 The Basic Limit Theorem of Markov Chains 4.5 Reducible Markov Chains 4.6 Sequential Decisions and Markov Chains5 Poisson Processes 5.1 The Poisson Distribution and the Poisson Process 5.2 The Law of Rare Events 5.3 Distributions Associated with the Poisson Process 5.4 The Uniform Distribution and Poisson Processes 5.5 Spatial Poisson Processes 5.6 Compound and Marked Poisson Processes6 Continuous Time Markov Chains 6.1 Pure Birth Processes 6.2 Pure Death Processes 6.3 Birth and Death Processes 6.4 The Limiting Behavior of Birth and Death Processes 6.5 Birth and Death Processes with Absorbing States 6.6 Finite State Continuous Time Markov Chains 6.7 Set Valued Processes7 Renewal Phenomena 7.1 Definition of a Renewal Process and Related Concepts 7.2 Some Examples of Renewal Processes 7.3 The Poisson Process Viewed as a Renewal Process 7.4 The Asymptotic Behavior of Renewal Processes 7.5 Generalizations and Variations on Renewal Processes 7.6 Discrete Renewal Theory8 Branching Processes and Population Growth 8.1 Branching Processes 8.2 Branching Processes and Generating Functions 8.3 Geometrically Distributed Offspring 8.4 Variations on Branching Processes 8.5 Some Stochastic Models of Plasmid Reproduction and Plasmid Copy Number Partition 8.6 Population Growth Processes with Interacting Types 8.7 Deterministic Population Growth with Age Distribution9 Queueing Systems 9.1 Queueing Processes 9.2 Poisson Arrivals and Exponentially Distributed Service Times 9.3 The /W/G/1 and M/G/∞ Systems 9.4 Variations and Extensions 9.5 Open Acyclic Queueing Networks 9.6 General Open NetworksFurther ReadingsSolutions to Selected ExercisesIndex