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Analytical solution of finite capacity M/D/1 queues

Part of:Special processesComputer system organization

Published online by Cambridge University Press:  14 July 2016

Olivier Brun and
Jean-Marie Garcia
Olivier Brun*
Affiliation:
LAAS-CNRS
Jean-Marie Garcia*
Affiliation:
LAAS-CNRS
*
∗ Postal address: Laboratoire d'Analyse et d'Architecture des Systèmes du CNRS, Toulouse, France.
∗ Postal address: Laboratoire d'Analyse et d'Architecture des Systèmes du CNRS, Toulouse, France.
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Abstract

Although the M/D/1/N queueing model is well solved from a computational point of view, there is no known analytical expression of the queue length distribution. In this paper, we derive closed-form formulae for the distribution of the number of customers in the system in the finite-capacity M/D/1 queue. We also give an explicit solution for the mean queue length and the average waiting time.

Keywords

Finite capacity M/D/1 queuesembedded Markov chainqueue length distributionmean number of customersaverage waiting time

MSC classification

Primary:60K25: Queueing theory
Secondary:68M20: Performance evaluation; queueing; scheduling

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000, pp. 1092 - 1098
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

[1] Brun, O., and Garcia, J. M. (1999). Analytical solution of finite capacity M/D/1 queues. Res. Rept 99372, LAAS, Toulouse.Google Scholar
[2] Cohen, J. W. (1969). The Single Server Queue. North Holland, Amsterdam.Google Scholar
[3] Crommelin, C. D. (1933). Delay probability formulae. Post Office Elec. Eng. J. 26.Google Scholar
[4] Doyon, G. (1989). Systèmes et Réseaux de Télécommunications en Régime Stochastique (Collection Technique et Scientifique des Télécommunications). Masson, Paris.Google Scholar
[5] Gravey, A., Louvion, J. R., and Boyer, P. (1990). On the Geo/D/1 and Geo/D/1/N Queues. Perf. Eval. 11, 117–125.Google Scholar
[6] Kleinrock, L. (1975). Queueing Systems. Volume I: Theory. John Wiley, New York.Google Scholar
[7] Kleinrock, L. (1976). Queueing Systems. Volume II: Applications. John Wiley, New York.Google Scholar
[8] Roberts, J., Mocci, U., and Virtamo, J. (eds) (1996). Broadband Network Teletraffic. Final Report of Action Cost 242. Springer, Berlin.Google Scholar
[9] Takacs, L. (1962). Introduction to the Theory of Queues. Oxford University Press.Google Scholar
[10] Tijms, H. C. (1986). Stochastic Modelling and Analysis: A Computational Approach. John Wiley, New York.Google Scholar
[11] Tijms, H. C. (1994). Stochastic Models: An Algorithmic Approach. John Wiley, New York.Google Scholar
[12] Vicari, N., and Tran-Gia, P. (1996). A numerical analysis of the Geo/D/N queueing system. Tech. Rept 151, Institute of Computer Science, University of Würzburg.Google Scholar