Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T00:26:16.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytical solution of finite capacity M/D/1 queues

Part of: Special processes Computer 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.
Get access Rights & Permissions [Opens in a new window]

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
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 

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

[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