Contents Preface The Exponential Distribution and the Poisson Process Introduction to Renewal Theory

Transcription

1 Contents Preface... v 1 The Exponential Distribution and the Poisson Process Introduction The Density, the Distribution, the Tail, and the Hazard Functions The Hazard Function and the Memoryless Property (Version 1) The Memoryless Property (Version 2) The Memoryless Property (Version 3) The Least Among Exponential Random Variables The Erlang Distribution The Hyperexponential Distribution A Mixture of Erlang Distributions The Poisson Process When Have They Actually Arrived? Thinning and Superpositioning of Poisson Processes Transforms The z-transform The Laplace-Stieltjes Transform Exercises Introduction to Renewal Theory Introduction Main Renewal Results The Length Bias Distribution and the Inspection Paradox The Age and the Residual Distributions The Memoryless Property (Versions 4 and 5) An Alternative Approach A Note on the Discrete Version Exercises ix

2 x Contents 3 Introduction to Markov Chains Introduction Some Properties of Markov Chains Time Homogeneity State Classification Transient and Recurrent Classes Periodicity Limit Probabilities and the Ergodic Theory Computing the Limit Probabilities The Time-Reversed Process and Reversible Processes Discrete Renewal Processes Revisited Transient Matrices Short-Circuiting States Exercises From Single Server Queues to M/G/ Introduction Why Do Queues Exist at All? Why Queues Are Long? Queueing Disciplines Basics in Single Server Queues The Utilization Level Little s Law Residual Service Times The Virtual Waiting Time Arrival and Departure Instants ASTA and the Khintchine Pollaczek Formula The M/G/1 Model Examples The Busy Period of an M/G/1 Queue Stand-By Customers and Externalities M/G/1 Queues with Vacations The G/G/1 Queue Lindley s Equation Exercises Priorities and Scheduling in M/G/ An M/G/1 Queue with Priorities Conservation Laws The Optimality of the C Rule Waiting Times in Priority Queues Shortest Job First (SJF) Preemptive Priority Exercises... 78

3 Contents xi 6 M/G/1 Queues Using Markov Chains and LSTs Introduction The Markov Chain Underlying the Departure Process The Limit Probabilities The Distribution of Time in the System Arrival, Departure, and Random Instants Observable Queues Busy Period in an M/G/1 Queue Revisited A Final Word Exercises The G/M/1 Queueing System Introduction and Modeling The Stationary Distribution at Arrival Instants The Balance Equations and Their Solution Exponential Waiting Times The Queue Length at Random Times Exercises Continuous-Time Markov Chains and Memoryless Queues The Model Examples The Limit Probabilities The Limit Probabilities and the Balance Equations The Embedded Process Uniformization The Cut Balancing Theorem The Time-Reversed Process A Condition on the Limit Probabilities The Time-Reversed Process Is Markovian Time-Reversible Processes Poisson Processes Stemming from Markov Processes Exercises Open Networks of Exponential Queues Open Networks of Exponential Queues: Model and Limit Probabilities Partial Balancedness Processes in Open Networks of Memoryless Queues Sojourn Times in Open Network of Queues The Unconditional Mean Waiting Time The Arrival Theorem Generalizations Exercises

4 xii Contents 10 Closed Networks of Exponential Queues The Model and the Limit Probabilities Partial Balancedness The Convolution Algorithm Short-Circuiting Stations The Arrival Theorem Mean Value Analysis (MVA) for Closed Networks of Queues The Mean Value Analysis (MVA) Algorithm Generalizations Exercises Insensitivity and Product-Form Queueing Models Introduction Symmetric Queues Examples Product Form and Insensitivity One-Chance Queues Examples Product Form and Insensitivity Proof of Theorem BCMP Network of Queues Exercises Two-Dimensional Markov Processes and Their Applications to Memoryless Queues Model Description Examples Example 1: The M=Er=1 Model Revisited Example 2: The Two Shortest Truncated Symmetric Queues Example 3: Two Servers: When Customers Join the First When Its Queue Is Not Too Long Example 4: Game with Tokens The Balance Equations Example 1 (Cont.) Example 2 (Cont.) Example 3 (Cont.) Example 4 (Cont.) Solving for the Limit Probabilities Example 1 (Cont.) Example 3 (Cont.) Example 4 (Cont.) The General Case Exercises References Index

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