Queueing Theory 2

Vladimir Anisimov is Full Professor in Applied Statistics. He works in the Center for Design & Analysis at Amgen Inc. in London, UK. His research interests include probability models and stochastic processes, clinical trials modeling, applied statistics, queueing models and asymptotic techniques. Nikolaos Limnios is Full Professor in Applied Mathematics at the University of Technology of Compiègne, part of the Sorbonne University Group, in France. His research interests include stochastic processes and statistics, Markov and semi-Markov processes, random evolutions with applications in reliability, queueing systems, earthquakes and biology.

• Preface xiChapter 1. Stability Analysis of Queueing Systems based on Synchronization of the Input and Majorizing Output Flows 1Larisa AFANASEVA1.1. Introduction 11.2. Model description 41.3. Auxiliary service process 61.4. Instability result for the case ρ ≥ 1 91.5. Stochastic boundedness for the case ρ < 1 101.6. Queueing system with unreliable servers and preemptive resume service discipline 111.7. Discrete-time queueing system with interruptions and preemptive repeat different service discipline 151.8. Queueing system with a preemptive priority discipline 181.9. Queueing system with simultaneous service of a customer by a random number of servers 201.10. Applications to transport systems analysis 231.11. Conclusion 281.12. Acknowledgment 291.13. References 29Chapter 2. Queueing Models in Services – Analytical and Simulation Approach 33Srinivas R. CHAKRAVARTHY2.1. Introduction 332.2. Phase-type distributions and the batch Markovian arrival process 342.2.1. Phase-type distributions 352.2.2. Some useful results related to continuous PH distributions 362.2.3. The batch Markovian arrival process 402.3. Generation of MAP processes for numerical purposes 422.4. Analysis of selected queueing models of BMAP/G/c type 442.4.1. MAP/PH/1 queueing model 442.4.2. The system performance measures 482.4.3. Illustrative numerical examples for MAP/PH/1 492.4.4. MAP/M/c queueing model 552.4.5. The system performance measures 572.4.6. Illustrative numerical examples for MAP/M/c 572.5. Simulated models of BMAP/G/c type queues 582.5.1. Simulated model validation using MAP/M/c type queues 592.5.2. Simulated model validation using MAP/PH/1 type queues 592.5.3. Selected simulated models of BMAP/G/c type queues 592.6. Analysis of selected queueing models of BMAP/G/c type with a vacation 662.6.1. MAP/PH/1 queueing model with a vacation 662.6.2. The system performance measures 692.6.3. Illustrative numerical examples for MAP/PH/1 with a vacation 692.6.4. Validation of the simulated model for vacation type queues 742.6.5. Selected simulated models of BMAP/G/c type queues with a vacation 752.7. Acknowledgment 782.8. References 78Chapter 3. Distributions and Random Processes Related to Queueing and Reliability Models 81Boyan DIMITROV3.1. Some useful notations, relationships and interpretations 813.2. Unreliable service model and reliability maintenance 853.3. Characterizations of exponential and geometric distributions via properties of service times 883.3.1. Instant repairs: characterization of geometric distribution 893.3.2. Instant repairs: characterizations of the exponential distribution 943.3.3. Various simplifying conditions 1013.3.4. Unreliable service, repair times included 1113.4. Probability distributions almost having lack of memory property 1153.4.1. Service time on an unreliable server: instantaneous repairs 1163.4.2. Properties of ALM distributions, and equivalent presentations 1193.4.3. Periodicity in natural phenomena 1263.5. Random processes with a periodic nature 1263.5.1. Counting processes 1273.5.2. Characterization of an NPP 1283.5.3. Applications in risk modeling 1313.6. Conclusions 1323.7. References 133Chapter 4. The Impact of Information Structure on Strategic Behavior in Queueing Systems 137Antonis ECONOMOU4.1. Introduction 1384.2. Game-theoretical framework in queueing 1394.3. The unobservable model 1424.4. The observable model 1464.5. Comparison of the unobservable and the observable models 1514.6. Partially observable models 1534.7. Heterogeneously observable models 1584.8. Observable-with-delay models 1624.9. Conclusions and literature review for further study 1674.10. Acknowledgments 1674.11. References 168Chapter 5. Non-extensive Maximum Entropy Formalisms and Inductive Inference of a Stable M/G/1 Queue with Heavy Tails 171Demetres D. KOUVATSOS and Ismail A. MAGEED5.1. Introduction 1725.2. General systems and inductive ME formalisms 1755.2.1. “Classical” Shannon’s EME formalism with short-range interactions 1755.2.2. Rényi’s and Tsallis’s NME formalisms with long-range interactions 1765.3. NME formalisms and EME consistency axioms 1775.4. A stable M/G/1 queue with long-range interactions 1795.4.1. Background: Shannon’s EME state probability of a stable M/G/1 queue 1795.4.2. Tsallis’ and Rényi’s NME state probabilities of a stable M/G/1 queue 1805.4.3. Exact Rényi’s and Tsallis’ NME state probabilities with distinct GEq-type service time distributions 1835.5. Numerical experiments and interpretations 1885.6. Conclusions 1955.7. Acknowledgments 1965.8. Appendix: Rényi’s NME formalisms versus EME consistency axioms 1965.8.1. Uniqueness 1965.8.2. Invariance 1975.8.3. System independence 1975.8.4. Subset independence 1985.9. References 199Chapter 6. Inventory with Positive Service Time: a Survey 201Achyutha KRISHNAMOORTHY, Dhanya SHAJIN and Viswanath C. NARAYANAN6.1. Introduction 2016.2. Queueing inventory models 2036.2.1. Single-commodity queueing-inventory systems 2066.2.2. Production inventory systems 2156.2.3. Multicommodity queueing-inventory system 2176.2.4. Retrial queues with inventory 2196.2.5. Queues requiring additional items for service 2226.2.6. Queueing-inventory: some work in progress and suggestions for future studies 2256.3. Acknowledgment 2276.4. References 227Chapter 7. A Stability Analysis Method of Regenerative Queueing Systems 239Evsey MOROZOV and Bart STEYAERT7.1. Introduction 2397.2. Preliminaries 2417.3. The single-server system 2447.4. The zero-delayed multiserver system 2487.5. The delayed multiserver system: finiteness of the first regeneration period 2527.6. Instability 2567.6.1. Some comments on the method 2607.7. Related research 2627.8. Acknowledgments 2667.9. References 266Chapter 8. Transient Analysis of Markovian Queueing Systems: a Survey with Focus on Closed-forms and Uniformization 269Gerardo RUBINO8.1. Introduction 2708.2. Basics on Markovian queues 2728.2.1. Markov models 2728.2.2. Uniformization 2738.3. First examples 2758.3.1. The Ehrenfest model in continuous-time 2758.3.2. The M/M/∞ model 2768.3.3. A queue with no server and catastrophes 2778.3.4. The fundamental M/M/1 model 2788.3.5. M/M/1 with bounded waiting room: the M/M/1/H model 2828.3.6. Comments 2848.4. An uniformization-based path for the M/M/1 with matrix generating functions 2848.4.1. General case 2868.4.2. Mean number of customers at time t in the M/M/1 2878.5. An uniformization-based path using duality 2908.5.1. Duality 2908.5.2. The path toward the transient state distributions using duality 2938.5.3. Application to the M/M/1 queueing system 2948.5.4. Application to the M/M/1/H queueing system 2958.5.5. Application to an M/M/1/H model with catastrophes 2978.6. Other transient results 2998.6.1. Busy period of the M/M/1 2998.6.2. Max backlog of the M/M/1 over a finite time interval 2998.6.3. M/E/1 3008.7. Conclusions 3028.8. References 302List of Authors 307Index 309