Foundations of Queueing Theory

2 The Busy Period 43 3. 3 The M 1M IS System with Last Come, First Served 50 3. 4 Comparison of FCFS and LCFS 51 3. 5 Time-Reversibility of Markov Processes 52 The Output Process 54 3. 6 3. 7 The Multi-Server System in a Series 55 Problems for Solution 3. 8 56 4 ERLANGIAN QUEUEING SYSTEMS 59 4. 1 Introduction 59 4. 2 The System M I E/c/1 60 4. 3 The System E/cl Mil 67 4. 4 The System MIDI1 72 4. 5 Problems for Solution 74 PRIORITY SYSTEMS 79 5 5. 1 Description of a System with Priorities 79 Two Priority Classes with Pre-emptive Resume Discipline 5. 2 82 5. 3 Two Priority Classes with Head-of-Line Discipline 87 5. 4 Summary of Results 91 5. 5 Optimal Assignment of Priorities 91 5. 6 Problems for Solution 93 6 QUEUEING NETWORKS 97 6. 1 Introduction 97 6. 2 A Markovian Network of Queues 98 6. 3 Closed Networks 103 Open Networks: The Product Formula 104 6. 4 6. 5 Jackson Networks 111 6. 6 Examples of Closed Networks; Cyclic Queues 112 6. 7 Examples of Open Networks 114 6. 8 Problems for Solution 118 7 THE SYSTEM M/G/I; PRIORITY SYSTEMS 123 7. 1 Introduction 123 Contents ix 7. 2 The Waiting Time in MIGI1 124 7. 3 The Sojourn Time and the Queue Length 129 7. 4 The Service Interval 132 7.

Klappentext

The order and presentation of Foundations of Queueing Theory is drawn from Professor Prabhu's extensive experience as a researcher, teacher, expositor, and editor. The book deals with the foundations of queueing theory and is intended as an advanced text for courses on queueing theory, and as a professional reference to queueing research. The results of queueing theory are established within the context of the book's instructional framework. The central results of queueing theory are stated in the form of theorems for accessibility and reading ease. Current research has shown the need to pay more attention to the basic concepts and techniques of queueing theory. These include the busy period, imbedded chains, regeneration points, Wiener-Hopf technique, time-reversibility, output, vector Markov processes, remaining workload and completion times. The use of these concepts and techniques considerably simplifies the analysis of models involving last come, first served queue discipline, priorities, networks, set-up times and server vacations. It is these conceptional techniques that form the foundations of queueing theory.

Zusammenfassung
` ... presented in a clear and attractive fashion. The book can be warmly recommended to graduate students and teachers of applied mathematics and operations research.'
OR News, 7 (1999)

Inhalt
1 Introduction.- 1.1 Description of a Queueing System.- 1.2 The Basic Model GI/G/S.- 1.3 Processes of Interest.- 1.4 The Nature of Congestion.- 1.5 Little's Formula L = ?W.- 1.6 Control of Queueing Systems.- 1.7 Historical Remarks.- 2 Markovian Queueing Systems.- 2.1 Introduction.- 2.2 The System M/M/1.- 2.3 The System M/M/s.- 2.4 A Design Problem.- 2.5 M/M/s System with Finite Source.- 2.6 The Machine Interference Problem.- 2.7 The System M/M/s with Finite Capacity.- 2.8 Loss Systems.- 2.9 Social Versus Self-Optimization.- 2.10 The System M/M/s with Balking.- 2.11 The System M/M/s with Reneging.- 2.12 Problems for Solution.- 3 The Busy Period, Output and Queues in Series.- 3.1 Introduction.- 3.2 The Busy Period.- 3.3 The M/M/S System with Last Come, First Served.- 3.4 Comparison of FCFS and LCFS.- 3.5 Time-Reversibility of Markov Processes.- 3.6 The Output Process.- 3.7 The Multi-Server System in a Series.- 3.8 Problems for Solution.- 4 Erlangian Queueing Systems.- 4.1 Introduction.- 4.2 The System M/Ek/1.- 4.3 The System Ek/M/1.- 4.4 The System M/D/1.- 4.5 Problems for Solution.- 5 Priority Systems.- 5.1 Description of a System with Priorities.- 5.2 Two Priority Classes with Pre-emptive Resume Discipline.- 5.3 Two Priority Classes with Head-of-Line Discipline.- 5.4 Summary of Results.- 5.5 Optimal Assignment of Priorities.- 5.6 Problems for Solution.- 6 Queueing Networks.- 6.1 Introduction.- 6.2 A Markovian Network of Queues.- 6.3 Closed Networks.- 6.4 Open Networks: The Product Formula.- 6.5 Jackson Networks.- 6.6 Examples of Closed Networks; Cyclic Queues.- 6.7 Examples of Open Networks.- 6.8 Problems for Solution.- 7 The System M/G/1; Priority Systems.- 7.1 Introduction.- 7.2 The Waiting Time in M/G/1.- 7.3 The Sojourn Time and the Queue Length.- 7.4 The ServiceInterval.- 7.5 The M/G/1 System with Exceptional Service.- 7.6 The Busy Period in M/G/1.- 7.7 Completion Times in Priority Systems.- 7.8 Low Priority Waiting Time.- 7.9 Problems for Solution.- 8 The System GI/G/1; Imbedded Markov Chains.- 8.1 Imbedded Markov Chains.- 8.2 The System GI/G/1.- 8.3 The Wiener-Hopf Technique; Examples.- 8.4 Set-up Times; Server Vacations.- 8.5 The Queue Length and Waiting Time in GI/M/1.- 8.6 The Queue Length in M/G/1.- 8.7 Time Sharing Systems.- 8.8 The M/M/1 System with RR Discipline.- 8.9 Problems for Solution.- A Appendix.- A.1 The Poisson Process.- A.2 Renewal Theory.- A.3 The Birth-And-Death Process.- A.4 Markov Processes with a Countable State Space.- A.5 Markov Chains.- A.6 Two Theorems on Functional Equations.- A.7 Review Problems in Probability and Stochastic Processes.- B Bibliography.