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On a single-server queue with group arrivals

Published online by Cambridge University Press:  14 July 2016

J. W. Cohen
J. W. Cohen*
Affiliation:
Mathematical Institute, University of Utrecht
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Abstract

The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.

Keywords

GI/G/1 QUEUEGROUP ARRIVALSWAITING TIMESTATIONARY DISTRIBUTIONSREGENERATIVE PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 619 - 622
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Le Gall, P. (1962) Les Systèmes avec ou sans Attente et les Processus Stochastiques. Dunod, Paris.Google Scholar
[2] Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[3] Burke, P. J. (1975) Delays in single server queues with batch input. Operat. Res. 23, 830–833.Google Scholar
[4] Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Operations Research and Mathematical Systems. Springer Verlag, Heidelberg.Google Scholar
[5] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
[6] Stidham, S. (1972) Regenerative processes in the theory of queues, with applications to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.CrossRefGoogle Scholar