Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T15:59:55.813Z Has data issue: false hasContentIssue false
  • English
  • Français

On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Sergei Grishechkin
Sergei Grishechkin*
Affiliation:
Gubkin Institute of Oil and Gas, Moscow
*
∗Postal address: Warshavskoje shosse, d.88, kv. 43, 113 556 Moscow, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The M/G/1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/1 queues with shortest residual processing time discipline.

Keywords

SHORTEST RESIDUAL PROCESSING TIME DISCIPLINEHEAVY TRAFFICOVERLOADED QUEUES

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 3 , September 1992 , pp. 653 - 698
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Van Den Berg, J. L. and Boxma, O. J. (1987) Sojourn times in feedback queues. Report OS-R 8710, Centre for Mathematics and Computer Science.Google Scholar
Van Den Berg, J. L. and Boxma, O. J. (1989) Sojourn times in feedback and processor sharing queues. Teletraffic Science for new Cost-Effective Systems, Networks and Services , ITC 12.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Cohn, H. (1985) A martingale approch to supercritical (CMJ) branching processes. Ann. Prob. 13, 1179–1191.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. 2. Wiley, New York.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Grishechkin, S. A. (1988a) On study of queueing systems with random selection discipline by branching process methods. Dokl. Akad. Nauk SSSR 300, 23–26 (in Russian). English translation: Sov. Math. Dokl. 37, 610–613.Google Scholar
Grishechkin, S. A. (1988b) One-channel systems with round robin or processor sharing and branching processes. Matem. Zametki 44, 433–448 (in Russian). English translation: Math. Notes. Google Scholar
Grishechkin, S. A. (1989) Branching processes close to explosive with applications to priority queues. In Probl. Ustojchiv. Stoch. Model. , 34–42. Moscow. (In Russian).Google Scholar
Grishechkin, S. A. (1990a) Branching processes and queueing systems with repeated orders or with the random discipline. Teor. Verojatn. i Primen. 35, 35–50 (in Russian). English translation: Theory Prob. Appl. Google Scholar
Grishechkin, S. A. (1990b) On connection between the theory of branching processes and queueing theory. In Probability Theory and Mathematical Statistics , Vol. 1, 455–462. Proceedings of the Fifth Vilnius Conference 25 June–1 July 1989. VSP, Utrecht/Mokslas, Vilnius.Google Scholar
Grishechkin, S. A. (1991a) Crump–Mode–Jagers branching processes as a method for investigation the M/G/1 system with processor sharing. Teor. Verojatn. i Primen. 36, 16–33 (in Russian). English translation: Theory Prob. Appl. Google Scholar
Grishechkin, S. A. (1991b) Interoutput times in M/G/1/PS queue with instantaneous feedback. Submitted to Teor. Verojat. i Primenen. Google Scholar
Grishechkin, S. A. (1991c) Multiclass batch arrival retrial queues analyzed as branching processes with immigration. QUESTA. Submitted.CrossRefGoogle Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, Chichester.Google Scholar
Jagers, P. and Nerman, O. (1984) The growth and composition of branching populations. Adv. Appl. Prob. 16, 221–259.CrossRefGoogle Scholar
Kallenberg, O. (1976) Random Measures. Academic-Verlag, Berlin.Google Scholar
Kawazu, K. and Watanabe, S. (1971) Branching processes with immigration and related limit theorems. Teor. Verojatn. i Primen. 16, 34–51.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kendall, D. G. (1951) Some problems in the theory of queues. J Roy. Statist. Soc. B 13, 151–185.Google Scholar
Neuts, M. F. (1969) The queue with Poisson input and general service times, treated as a branching process. Duke Math. J. 36, 215–232.CrossRefGoogle Scholar
Neuts, M. F. (1970) Two servers in series, studied in terms of a Markov renewal branching process. Adv. Appl. Prob. 2, 110–149.CrossRefGoogle Scholar
Neuts, M. F. (1974) The Markov renewal branching process. In Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 98, 1–21. Springer-Verlag, Berlin.Google Scholar
Ott, T. J. (1984) The sojourn-time distribution in the M/G/1 queue with processor sharing. J. Appl. Prob. 21, 360–378.CrossRefGoogle Scholar
Resing, J., Hooghiemstra, G. and Keane, M. (1990) The M/G/1 processor sharing queue as the almost sure limit of feedback queues. J. Appl. Prob. 27, 913–918.CrossRefGoogle Scholar
Rudin, W. (1976) Principles of Mathematical Analysis. McGraw-Hill, New York.Google Scholar
Sagitov, S. (1992) General branching processes: convergence towards Jirina processes. In Probl. Ustojchiv. Stoch. Model. Moscow (in Russian). To appear.Google Scholar
Schassberger, R. (1984) A new approach to the M/G/1 processor-sharing queue. Adv. Appl. Prob. 16, 202–213.CrossRefGoogle Scholar
Schassberger, R. (1988) The steady-state distribution of spent service times present in the M/G/1 foreground-background processor-sharing queue. J. Appl. Prob. 25, 194–203.CrossRefGoogle Scholar
Schassberger, R. (1990) The steady-state appearance of the M/G/1 queue under the discipline of shortest remaining processing time. Adv. Appl. Prob. 22, 456–479.CrossRefGoogle Scholar
Schrage, E. and Miller, L. (1966) The queue M/G/1 with the shortest remaining processing time discipline. Operat. Res. 14, 670–684.CrossRefGoogle Scholar
Seneta, E. (1976) Regularly Varying Functions. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Sevast'Yanov, B. A. (1971) Vetvyasciesya Processy. Mir, Moscow (in Russian). German translation: Verzweigungsprozesse, Academic-Verlag, Berlin (1974).Google Scholar
Shalmon, M. (1988) Analysis of the GI/GI/1 queue and its variations via the LCFS preemptive resume discipline and its random walk interpretation. Prob. Engin. and Inform. Sciences 2, 215–230.CrossRefGoogle Scholar
Teugels, J. L. and De Meyer, A. (1980) On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1. J. Appl. Prob. 17, 802–813.Google Scholar
Yashkov, S. F. (1983) A derivation of response time distribution for a M/G/1 processor-sharing queue. Problems of Control and Information Theory 12, 133–148.Google Scholar
Yashkov, S. F. (1987) Processor-sharing queues: some progress in analysis. QUESTA 2, 1–17.Google Scholar