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

This is a revised and expanded version of the earlier edition. The new material is on Markov-modulated storage processes arising from queueing and data commu­ nication models. The analysis of these models is based on the fluctuation theory of Markov-additive processes and their discrete time analogues, Markov random walks. The workload and queue length processes, omitted from the earlier edition, are also presented. In addition, many sections have been rewritten, with new re­ sults and proofs, as well as further examples. The mathematical level and style of presentation, however, remain the same. Chapter I contains a comprefensive treatment of the waiting time and related quantities in a single server queue, combining Chapters 1 and 2 of the earlier edition. In Chapter 2 we treat the (continuous time) workload and queue length processes using their semiregenerative properties. Also included are bulk queues omitted from the earlier edition, but included in its Russian translation. The queue MIMIl is presented in Chapter 3. This is the so-called simple queue, but its treat­ ment in most of the literature is far from simple. Our analysis of the queue length process is elementary and yields explicit results for various distributions of interest. are treated in Chapter 4, combining Chapters 3 Continuous time storage models and 4 of the earlier edition. We present extensive new material, omitting much of the old Chapter 4. This has resulted in a streamlined account of this important class of models.

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{TnSnJn}.- 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.