Foundations of Queueing Theory, 1997 International Series in Operations Research & Management Science Series, Vol. 7

3. 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.

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.