Applied Probability Models with Optimization Applications: (Dover Books on Mathema 1.4tics)

Concise advanced-level introduction to stochastic processes that arise in applied probability. Poisson process, renewal theory, Markov chains, Brownian motion, much more. "A clarity of style and a conciseness of treatment which students will find most welcome. The material is valuable and well organised... an excellent introduction to applied probability." - Journal of the American Statistical Association. This book offers a concise introduction to some of the stochastic processes that frequently arise in applied probability. Emphasis is on optimization models and methods, particularly in the area of decision processes. After reviewing some basic notions of probability theory and stochastic processes, the author presents a useful treatment of the Poisson process, including compound and nonhomogeneous Poisson processes. Subsequent chapters deal with such topics as renewal theory and Markov chains; semi-Markov, Markov renewal, and regenerative processes; inventory theory; and Brownian motion and continuous time optimisation models. Each chapter is followed by a section of useful problems that illustrate and complement the text. There is also a short list of relevant references at the end of every chapter. Students will find this a largely self-contained text that requires little previous knowledge of the subject. It is especially suited for a one-year course in applied probability at the advanced undergraduate or beginning postgraduate level.

Table of Contents:
1. INTRODUCTION TO STOCHASTIC PROCESSES 1.1. Random Variables and Probability Theory 1.2. Conditional Expectation 1.3. Stochatic Processes Problems 2. THE POISSON RPCESS 2.1 Introduction and Definitions 2.2 Interarrival and Waiting Time Distributions 2.3 Conditional Distribution of the Arrival Times 2.4 Compound and Nonhomogenous Poisson Processes 2.5 Stationary Point Processes Problems References 3. RENEWAL THEORY 3.1 Introduction and Preliminaries 3.2 Renewal Equation and Generalizations 3.3 Limit Theorems 3.4 Wald's Equation 3.5 Back to Renewal Theory 3.6 Excess Life and Age Distribution 3.7 Delayed Renewal Processes 3.8 Counter Models 3.9 Renewal Reward Process 3.10 Nonterminating versus Terminating Renewal Processes 3.11 Age Dependent Branching Processes Problems References 4. MARKOV CHAINS 4.1 Preliminaries and Examples 4.2 Classification of States 4.3 Limit Theorems 4.4 Transitions Among Classes 4.5 Branching Processes 4.6 Transient States Problems References 5. "SEMI-MARKOV, MARKOV RENEWAL AND REGERNERATIVE PROCESSES" 5.1 Introduction and Preliminaries 5.2 Classification of States 5.3 Some Simple Relationships 5.4 Regenerative Processes 5.5 A Queueing Application 5.6 Back to Markov Renewal Processes-Limiting Probabilities 5.7 Limiting Distributions of the Markov Renewal Process 5.8 Continuous Time Markov Chains 5.9 Birth and Death Processes Problems References 6. MARKOV DECISION PROCESSES 6.1 Introduction 6.2 Expected Discounted Cost 6.3 Some Examples 6.4 "Positive Costs, No Discounting" 6.5 Applications: Optimal Stopping and Sequential Analysis 6.6 Expected Average Cost Criterion-Introduction and Counter examples 6.7 Expected Average Cost Criterion 6.8 Finite State Space-Computational Approaches Problems References 7. SEMI-MARKOV DECISION PROCESSES 7.1 Introduction 7.2 Discounted Cost Criterion 7.3 Average Cost-Preliminaries and Equality of Criteria 7.4 Average Cost-Results 7.5 Some Examples Problems References 8. INVENTORY THEORY 8.1 Introduction 8.2 A Single Period Model 8.3 Multi-Period Models 8.4 A Multi-Period Stationary Optimal Policy 8.5 Inventory Issuing Policies Problems References 9. BROWNIAN MOTION AND CONTINUOUS TIME OPTIMIZATION MODELS 9.1 Introduction and Preliminaries 9.2 Maximum of the Wiener Process 9.3 The Wiener Process and Optimization 9.4 The Maximum Variable-A Renewal Application 9.5 Optimal Dispatching of a Poisson Process 9.6 Infinitesimal Look-Ahead Stopping Rules Problems Reference APPENDICES INDEX