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The suprema of the actual and virtual waiting times during a busy cycle of the Km/Kn/1 queueing system

Published online by Cambridge University Press:  01 July 2016

J. W. Cohen
J. W. Cohen*
Affiliation:
Technological University, Delft
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Abstract

For the single server queueing system, whose distributions of service and inter-arrival times have rational Laplace-Stieltjes transforms, limit theorems are derived for the supremum of the virtual waiting time during k successive busy cycles for k→∞. Similarly, for the supremum of the actual waiting times of all customers arriving in k successive busy cycles. Only the cases with the load of the system less than one and equal to one are considered. The limit distributions are extreme value distributions. The results are obtained by first deriving a number of asymptotic expressions for the quantities which govern the analytic description of the system Km/Kn/1. Using these asymptotic relations limit theorems for entrance times can also be obtained, a few examples are given.

Keywords

WAITING TIMELIMIT THEOREMS FOR QUEUESASYMPTOTIC BEHAVIOURENTRANCE TIMESFINITE, INFINITE CAPACITYWIENER PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 339 - 356
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Cohen, J. W. (1969) The Single Server Queue. North-Holland Publ. Co., Amsterdam.Google Scholar
[2] van Vugt, E. M. (1969) Limit theorems for the distribution of the supremum of the actual waiting time in a K m /G/1 and a G/K n /1 queueing system. . Technological University, Delft. (In Dutch.) Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol.11. Wiley, New York.Google Scholar
[4] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic I, II, III. Adv. Appl. Prob. 2, 150–177, 355–369, 370–375.CrossRefGoogle Scholar
[5] Iglehart, D. L. (1971) Extreme values in the GI/G/1 queue. Ann. Math. Statist. To appear.CrossRefGoogle Scholar
[6] Cohen, J. W. (1969) Single server queues with restricted accessibility. J. Engineering Mathematics 3, 265–284.Google Scholar