Introduction to Probability Models

Preface1 Introduction to Probability Theory 1.1 Introduction 1.2 Sample Space and Events 1.3 Probabilities Defined on Events 1.4 Conditional Probabilities 1.5 Independent Events 1.6 Bayes’ Formula Exercises References2 Random Variables 2.1 Random Variables 2.2 Discrete Random Variables 2.2.1 The Bernoulli Random Variable 2.2.2 The Binomial Random Variable 2.2.3 The Geometric Random Variable 2.2.4 The Poisson Random Variable 2.3 Continuous Random Variables 2.3.1 The Uniform Random Variable 2.3.2 Exponential Random Variables 2.3.3 Gamma Random Variables 2.3.4 Normal Random Variables 2.4 Expectation of a Random Variable 2.4.1 The Discrete Case 2.4.2 The Continuous Case 2.4.3 Expectation of a Function of a Random Variable 2.5 Jointly Distributed Random Variables 2.5.1 Joint Distribution Functions 2.5.2 Independent Random Variables 2.5.3 Covariance and Variance of Sums of Random Variables 2.5.4 Joint Probability Distribution of Functions of Random Variables 2.6 Moment Generating Functions 2.6.1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population 2.7 The Distribution of the Number of Events that Occur 2.8 Limit Theorems 2.9 Stochastic Processes Exercises References3 Conditional Probability and Conditional Expectation 3.1 Introduction 3.2 The Discrete Case 3.3 The Continuous Case 3.4 Computing Expectations by Conditioning 3.4.1 Computing Variances by Conditioning 3.5 Computing Probabilities by Conditioning 3.6 Some Applications 3.6.1 A List Model 3.6.2 A Random Graph 3.6.3 Uniform Priors, Polyas Urn Model, and Bose–Einstein Statistics 3.6.4 Mean Time for Patterns 3.6.5 The k-Record Values of Discrete Random Variables 3.6.6 Left Skip Free Random Walks 3.7 An Identity for Compound Random Variables 3.7.1 Poisson Compounding Distribution 3.7.2 Binomial Compounding Distribution 3.7.3 A Compounding Distribution Related to the Negative Binomial Exercises4 Markov Chains 4.1 Introduction 4.2 Chapman–Kolmogorov Equations 4.3 Classification of States 4.4 Limiting Probabilities 4.5 Some Applications 4.5.1 The Gamblers Ruin Problem 4.5.2 A Model for Algorithmic Efficiency 4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem 4.6 Mean Time Spent in Transient States 4.7 Branching Processes 4.8 Time Reversible Markov Chains 4.9 Markov Chain Monte Carlo Methods 4.10 Markov Decision Processes 4.11 Hidden Markov Chains 4.11.1 Predicting the States Exercises References5 The Exponential Distribution and the Poisson Process 5.1 Introduction 5.2 The Exponential Distribution 5.2.1 Definition 5.2.2 Properties of the Exponential Distribution 5.2.3 Further Properties of the Exponential Distribution 5.2.4 Convolutions of Exponential Random Variables 5.3 The Poisson Process 5.3.1 Counting Processes 5.3.2 Definition of the Poisson Process 5.3.3 Interarrival and Waiting Time Distributions 5.3.4 Further Properties of Poisson Processes 5.3.5 Conditional Distribution of the Arrival Times 5.3.6 Estimating Software Reliability 5.4 Generalizations of the Poisson Process 5.4.1 Nonhomogeneous Poisson Process 5.4.2 Compound Poisson Process 5.4.3 Conditional or Mixed Poisson Processes Exercises References6 Continuous-Time Markov Chains 6.1 Introduction 6.2 Continuous-Time Markov Chains 6.3 Birth and Death Processes 6.4 The Transition Probability Function Pij(t) 6.5 Limiting Probabilities 6.6 Time Reversibility 6.7 Uniformization 6.8 Computing the Transition Probabilities Exercises References7 Renewal Theory and Its Applications 7.1 Introduction 7.2 Distribution of N(t) 7.3 Limit Theorems and Their Applications 7.4 Renewal Reward Processes 7.5 Regenerative Processes 7.5.1 Alternating Renewal Processes 7.6 Semi-Markov Processes 7.7 The Inspection Paradox 7.8 Computing the Renewal Function 7.9 Applications to Patterns 7.9.1 Patterns of Discrete Random Variables 7.9.2 The Expected Time to a Maximal Run of Distinct Values 7.9.3 Increasing Runs of Continuous Random Variables 7.10 The Insurance Ruin Problem Exercises References8 Queueing Theory 8.1 Introduction 8.2 Preliminaries 8.2.1 Cost Equations 8.2.2 Steady-State Probabilities 8.3 Exponential Models 8.3.1 A Single-Server Exponential Queueing System 8.3.2 A Single-Server Exponential Queueing System Having Finite Capacity 8.3.3 Birth and Death Queueing Models 8.3.4 A Shoe Shine Shop 8.3.5 A Queueing System with Bulk Service 8.4 Network of Queues 8.4.1 Open Systems 8.4.2 Closed Systems 8.5 The System M/G/1 8.5.1 Preliminaries: Work and Another Cost Identity 8.5.2 Application of Work to M/G/1 8.5.3 Busy Periods 8.6 Variations on the M/G/1 8.6.1 The M/G/1 with Random-Sized Batch Arrivals 8.6.2 Priority Queues 8.6.3 An M/G/1 Optimization Example 8.6.4 The M/G/1 Queue with Server Breakdown 8.7 The Model G/M/1 8.7.1 The G/M/1 Busy and Idle Periods 8.8 A Finite Source Model 8.9 Multiserver Queues 8.9.1 Erlangs Loss System 8.9.2 The M/M/k Queue 8.9.3 The G/M/k Queue 8.9.4 The M/G/k Queue Exercises References9 Reliability Theory 9.1 Introduction 9.2 Structure Functions 9.2.1 Minimal Path and Minimal Cut Sets 9.3 Reliability of Systems of Independent Components 9.4 Bounds on the Reliability Function 9.4.1 Method of Inclusion and Exclusion 9.4.2 Second Method for Obtaining Bounds on r(p) 9.5 System Life as a Function of Component Lives 9.6 Expected System Lifetime 9.6.1 An Upper Bound on the Expected Life of a Parallel System 9.7 Systems with Repair 9.7.1 A Series Model with Suspended Animation Exercises References10 Brownian Motion and Stationary Processes 10.1 Brownian Motion 10.2 Hitting Times, Maximum Variable, and the Gambler's Ruin Problem 10.3 Variations on Brownian Motion 10.3.1 Brownian Motion with Drift 10.3.2 Geometric Brownian Motion 10.4 Pricing Stock Options 10.4.1 An Example in Options Pricing 10.4.2 The Arbitrage Theorem 10.4.3 The Black-Scholes Option Pricing Formula 10.5 White Noise 10.6 Gaussian Processes 10.7 Stationary and Weakly Stationary Processes 10.8 Harmonic Analysis of Weakly Stationary Processes Exercises References11 Simulation 11.1 Introduction 11.2 General Techniques for Simulating Continuous Random Variables 11.2.1 The Inverse Transformation Method 11.2.2 The Rejection Method 11.2.3 The Hazard Rate Method 11.3 Special Techniques for Simulating Continuous Random Variables 11.3.1 The Normal Distribution 11.3.2 The Gamma Distribution 11.3.3 The Chi-Squared Distribution 11.3.4 The Beta (n,m) Distribution 11.3.5 The Exponential Distribution—The Von Neumann Algorithm 11.4 Simulating from Discrete Distributions 11.4.1 The Alias Method 11.5 Stochastic Processes 11.5.1 Simulating a Nonhomogeneous Poisson Process 11.5.2 Simulating a Two-Dimensional Poisson Process 11.6 Variance Reduction Techniques 11.6.1 Use of Antithetic Variables 11.6.2 Variance Reduction by Conditioning 11.6.3 Control Variates 11.6.4 Importance Sampling 11.7 Determining the Number of Runs 11.8 Generating from the Stationary Distribution of a Markov Chain 11.8.1 Coupling from the Past 11.8.2 Another Approach Exercises ReferencesAppendix: Solutions to Starred ExercisesIndex