Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication

ISBN-13: 9781461272601 / Angielski / Miękka / 2012 / 207 str.

N. U. Prabhu

Opis

A self-contained treatment of stochastic processes arising from models for queues, insurance risk, and dams and data communication, using their sample function properties. The approach is based on the fluctuation theory of random walks, L vy processes, and Markov-additive processes, in which Wiener-Hopf factorisation plays a central role. This second edition includes results for the virtual waiting time and queue length in single server queues, while the treatment of continuous time storage processes is thoroughly revised and simplified. With its prerequisite of a graduate-level course in probability and stochastic processes, this book can be used as a text for an advanced course on applied probability models.

Spis treści

1. A Class of Stochastic Models.- 1.1 The Single-Server Queueing System.- 1.2 Inventory Models.- 1.3 Storage Models.- 1.4 Insurance Risk.- 1.5 Continuous Time Inventory and Storage Models.- 2. Problems Arising from the Models.- 3. The Scope of This Book.- 4. Further Remarks.- 4.1 Queues.- 4.2 Inventory Models.- 4.3 Models for Dams.- 4.4 Insurance Risk.- 4.5 Data Communication Models.- Problems.- Selected Bibliography.- I. The Single Server Queue.- 1. The Queue GI/G/1.- 1. Orientation.- 2. Ladder Processes.- 3. Two Renewal Functions.- 4. Maximum and Minimum.- 5. Application to the Queue GI/G/1.- 6. The Principle of Duality.- 7. The Queue M/M/1.- 8. The Queues GI/M/1 and M/G/1.- 8.1 The Queue GI/M/1.- 8.2 The Queue M/G/1.- 9. Transforms.- 10. The Queues GI/Ek/1 and Ek/G/1.- 10.1 The Queue GI/Ek/1.- 10.2 The Queue Ek/G/1.- 11. Further Problems in the Queue GI/G/1.- 12. A Random Walk on the Half-Plane.- 13. Applications to the Queue GI/G/1.- 13.1 The Queue GI/M/1.- 13.2 The Queue M/G/1.- 14. The Busy and Idle Periods.- 15. Compound Renewal Processes.- 16. Further Remarks.- Problems.- 2. Further Results for the Queue GI/G/1.- 1. Orientation.- 2. The Workload and Idle Time.- 3. The Workload and Idle Time in GI/M/1.- 4. The Workload and Idle Time in M/G/1.- 5. The Queue Length in GI/M/1.- 6. The Queue Length in M/G/1.- 7. The Queue M/D/1.- 8. The Queue D/M/1.- 9. Bulk Queues.- 10. The Queue M(X)/G(Y)/1.- 11. The Queue G(X)/M(Y)/1.- 12. The Queue Length in Ek/G/1 and GI/Ek/1.- Problems.- 3. The Queue M/M/1.- 1. Introduction.- 2. The Net Input Process X (t).- 3. The Busy Period.- 4. The Distribution of Q(t).- 5. The Output.- Problems.- References.- II. Continuous Time Storage Models.- 4. The Basic Storage Model.- 1. Orientation.- 2. Lévy Processes.- 2.1 A Continuous Time Model for a Dam.- 3. A Generalized Storage Model.- 4. A First Passage Time.- 5. Applications to M/G/1 and Related Models.- 6. The Process {T(x), x ? 0} in M/G/1.- 7. The Process {T(x), x ? 0} in Dam Models.- 8. Limit Theorems for{T(x), x ? 0}.- 9. Insurance Risk: Two Special Cases.- 10. The Ladder Process {TR*Y(TR*}.- 11. Limit Theorems for the Storage Process.- 12. Dams of Infinite Depth.- 13. The Insurance Risk Problem: General Case.- 14. A Storage Model with Random Output.- 15. Queues with Static Priorities.- 16. Queues with Dynamic Priorities.- Problems.- References.- III. Markov-Modulated Storage Models.- 5. The Markov-Modulated Single Server Queue.- 1. Introduction.- 2. The Adjoint Queue.- 3. Wiener-Hopf Factorization; Fluctuation Theory.- 4. The Waiting Time and Idle Time.- 5. The Markov-Modulated M/M/1 Queue.- 6. Further Remarks.- Problems.- 6. A Fluid Model for Data Communication.- 1. Introductory Examples.- 2. The Storage Model.- 3. The Net Input Process.- 4. The MRW.- 5. The Busy Period.- 6. The Storage Process {Z(t), I(t), J(t)}.- 7. Further Remarks.- 7. A Data Communication Model with Packets.- 1. Introduction.- 2. The Model.- 3. The Busy Period Process.- 4. The Unsatisfied DemandI(t).- 5. The Storage Process {Z(t), I(t), J(t)}.- 6. Further Remarks.- Problems.- References.- Appendix A.- Appendix B.- Appendix C.