Queues : A Course in Queueing Theory

Online-Ressource (XIV, 221 p. 32 illus, digital)

International Series in Operations Research & Management Science ; 191

Description based upon print version of record

Preface; Contents; List of Figures; 1 The Exponential Distribution and the Poisson Process; 1.1 Introduction; 1.2 The Density, the Distribution, the Tail,and the Hazard Functions; 1.2.1 The Hazard Function and the Memoryless Property (Version 1); 1.2.2 The Memoryless Property (Version 2); 1.2.3 The Memoryless Property (Version 3); 1.2.4 The Least Among Exponential Random Variables; 1.2.5 The Erlang Distribution; 1.2.6 The Hyperexponential Distribution; 1.2.7 A Mixture of Erlang Distributions; 1.3 The Poisson Process; 1.3.1 When Have They Actually Arrived?

1.3.2 Thinning and Superpositioning of Poisson Processes1.4 Transforms; 1.4.1 The z-Transform; 1.4.2 The Laplace-Stieltjes Transform; 1.5 Exercises; 2 Introduction to Renewal Theory; 2.1 Introduction; 2.2 Main Renewal Results; 2.2.1 The Length Bias Distribution and the Inspection Paradox; 2.2.2 The Age and the Residual Distributions; 2.2.3 The Memoryless Property (Versions 4 and 5); 2.3 An Alternative Approach; 2.4 A Note on the Discrete Version; 2.5 Exercises; 3 Introduction to Markov Chains; 3.1 Introduction; 3.2 Some Properties of Markov Chains; 3.3 Time Homogeneity

3.4 State Classification3.5 Transient and Recurrent Classes; 3.6 Periodicity; 3.7 Limit Probabilities and the Ergodic Theory; 3.7.1 Computing the Limit Probabilities; 3.8 The Time-Reversed Process and Reversible Processes; 3.9 Discrete Renewal Processes Revisited; 3.10 Transient Matrices; 3.11 Short-Circuiting States; 3.12 Exercises; 4 From Single Server Queues to M/G/1; 4.1 Introduction; 4.2 Why Do Queues Exist at All?; 4.3 Why Queues Are Long?; 4.4 Queueing Disciplines; 4.5 Basics in Single Server Queues; 4.5.1 The Utilization Level; 4.5.2 Little's Law; 4.5.3 Residual Service Times

4.5.4 The Virtual Waiting Time4.5.5 Arrival and Departure Instants; 4.6 ASTA and the Khintchine-Pollaczek Formula; 4.7 The M/G/1 Model; 4.7.1 Examples; 4.7.2 The Busy Period of an M/G/1 Queue; 4.7.3 Stand-By Customers and Externalities; 4.7.4 M/G/1 Queues with Vacations; 4.8 The G/G/1 Queue; 4.8.1 Lindley's Equation; 4.9 Exercises; 5 Priorities and Scheduling in M/G/1; 5.1 An M/G/1 Queue with Priorities; 5.1.1 Conservation Laws; 5.1.2 The Optimality of the Cμ Rule; 5.1.3 Waiting Times in Priority Queues; 5.1.4 Shortest Job First (SJF); 5.1.5 Preemptive Priority; 5.2 Exercises

6 M/G/1 Queues Using Markov Chains and LSTs6.1 Introduction; 6.2 The Markov Chain Underlying the Departure Process; 6.2.1 The Limit Probabilities; 6.3 The Distribution of Time in the System; 6.3.1 Arrival, Departure, and Random Instants; 6.3.2 Observable Queues; 6.4 Busy Period in an M/G/1 Queue Revisited; 6.5 A Final Word; 6.6 Exercises; 7 The G/M/1 Queueing System; 7.1 Introduction and Modeling; 7.2 The Stationary Distribution at Arrival Instants; 7.2.1 The Balance Equations and Their Solution; 7.2.2 Exponential Waiting Times; 7.2.3 The Queue Length at Random Times; 7.3 Exercises

8 Continuous-Time Markov Chains and Memoryless Queues

10.1007/978-1-4614-6765-6 [DOI]

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