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Some Time-Dependent Properties of Symmetric M/G/1 Queues

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella ,
Bert Zwart  and
Onno Boxma
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Bert Zwart*
Affiliation:
Eindhoven University of Technology and CWI
Onno Boxma*
Affiliation:
EURANDOM, Eindhoven University of Technology and CWI
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: mskella@mscc.huji.ac.il
∗∗Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
∗∗Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

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We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.

Keywords

Symmetric queuetime-dependent analysisinsensitivityorder statisticrandom permutationtail behavior

MSC classification

Secondary: 60K25: Queueing theory 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 223 - 234
Copyright
© Applied Probability Trust 2005 

Footnotes

∗

Supported in part by grant 819/03 from the Israel Science Foundation.

∗∗∗

Supported by an NWO VENI grant.

∗∗∗∗

Work carried out within the Euro-NGI project

References

Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975). Open, closed, and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
Bertoin, J. (1995). Lévy Processes. Cambridge University Press.Google Scholar
Bonald, T. and Proutière, A. (2002). Insensitivity in processor-sharing networks. Performance Evaluation 49, 193–209.CrossRefGoogle Scholar
Burke, P. J. (1956). The output of a queueing system. Operat. Res. 4, 699–704.CrossRefGoogle Scholar
Cohen, J. W. (1979). The multiple phase service network with generalized processor sharing. Acta Informatica 12, 245–284.CrossRefGoogle Scholar
Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi + 147 pp.Google Scholar
Jackson, J. R. (1963). Jobshop-like queueing systems. Manag. Sci. 10, 131–142.CrossRefGoogle Scholar
Kalashnikov, V. and Tsitsiashvili, G. (1999). Tails of waiting times and their bounds. Queueing Systems 32, 257–283.CrossRefGoogle Scholar
Kelly, F. P. (1976). Networks of queues. Adv. Appl. Prob. 8, 416–432.CrossRefGoogle Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.Google Scholar
Kitaev, M. Yu. (1993). The M/G/1 processor-sharing model: transient behavior. Queueing Systems 14, 239–273.CrossRefGoogle Scholar
Limic, V. (2001). A LIFO queue in heavy traffic. Ann. Appl. Prob. 11, 301–331.CrossRefGoogle Scholar
Maulik, K. and Zwart, B. (2004). Tail asymptotics for exponential functionals of Lévy processes. EURANDOM Report 2004-036.Google Scholar
O'Connell, N. and Yor, M. (2001). Brownian analogues of Burke's theorem. Stoch. Process. Appl. 96, 285–304.CrossRefGoogle Scholar
Rosenkrantz, W. (1983). Calculation of the Laplace transform of the length of the busy period for the M/G/1 queue via martingales. Ann. Prob. 11, 817–818.CrossRefGoogle Scholar
Shalmon, M. (1988). Analysis of the GI/G/1 queue and its variations via the LCFS preemptive resume discipline and its random walk interpretation. Prob. Eng. Inf. Sci. 2, 215–230.CrossRefGoogle Scholar
Sigman, K. (1996). Queues under preemptive LIFO and ladder height distributions for risk processes: a duality. Stoch. Models 12, 725–735.CrossRefGoogle Scholar