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Some general results for many server queues

Published online by Cambridge University Press:  01 July 2016

J. H. A. De Smit
J. H. A. De Smit*
Affiliation:
Center for Operations Research and Econometrics, University of Louvain
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Abstract

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

Keywords

MANY SERVER QUEUEPOLLACZEK'S METHODANALYTIC METHODACTUAL WAITING TIMEVIRTUAL WAITING TIMEBUSY CYCLEBUSY PERIODQUEUE LENGTHOUTPUT PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 1 , April 1973 , pp. 153 - 169
Copyright
Copyright © Applied Probability Trust 1973 

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References

Apostol, T. M. (1957) Mathematical Analysis. Addison-Wesley, Reading, Mass.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
De Smit, J. H.A. (1971) Many Server Queueing Systems. Ph. D. dissertation, University of Technology, Delft.Google Scholar
De Smit, J. H. A. (1973) On the many server queue with exponential service times. Adv. Appl. Prob. 5, 170–182.Google Scholar
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
Le Gall, P. (1962) Les Systèmes avec ou sans Attente et les Processus Stochastiques. Dunod, Paris.Google Scholar
Pollaczek, F. (1957) Problèmes Stochastiques Posés par le Phénomène de Formation d'une Queue d Attente à un Guichet et par des Phenomènes Apparentés. Gauthier-Villars, Paris.Google Scholar
Pollaczek, F. (1951) Théorie Analytique des Problèmes Stochastiques Relatifs à un Groupe de Lignes Téléphoniques avec Dispositif d'Attente. Gauthier-Villars, Paris.Google Scholar
Pollaczek, F. (1965) Concerning an analytic method for the treatment of queueing problems. Proc. Symp. on Congestion Theory. Eds. Smith, W. L. and Wilkinson, W. E.. University of North Carolina Press, Chapel Hill, N. C. Google Scholar
Roes, P. B. M. (1970) On the expected number of crossings of a level in certain stochastic processes. J. Appl. Prob. 7, 766–770.CrossRefGoogle Scholar
Syski, R. (1960) Introduction to Congestion Theory in Telephone Systems. Oliver and Boyd, London.Google Scholar
Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton, N.J.Google Scholar