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Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals

Published online by Cambridge University Press:  01 July 2016

J. C. W. Van Ommeren
J. C. W. Van Ommeren*
Affiliation:
Vrije Universiteit, Amsterdam
*
∗Postal address: Operations Research Group, Dept. of Econometrics, Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam, The Netherlands.
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Abstract

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.

Keywords

WAITING-TIME DISTRIBUTIONCOMPLEX INVERSION FORMULAARITHMETIC BATCH-SIZE DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 880 - 895
Copyright
Copyright © Applied Probability Trust 1988 

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References

Burke, P. J. (1975) Delays in single server queues with batch inputs. Operat. Res. 23, 830–833.CrossRefGoogle Scholar
Chaudhry, M. L. and Templeton, J. G. C. (1983) A First Course in Bulk Queues. Wiley, New York.Google Scholar
Cohen, J. W. (1976) On a single server queue with group arrivals. J. Appl. Prob. 13, 619–622.CrossRefGoogle Scholar
Cohen, J. W. (1982) The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
Eikeboom, A. M. and Tijms, H. C. (1987) Waiting time percentiles in the multiserver Mx/G/c queue with batch arrivals. Prob. Eng. Inf. Sci. 1, 75–96.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Iglehart, D. L. (1972) Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.CrossRefGoogle Scholar
Lukacs, E. (1970) Characteristic Functions, 2nd edn. Griffin, London.Google Scholar
Takahashi, Y. (1981) Asymptotic exponentiality of the tail of the waiting time distribution in a Ph/Ph/c queue. Adv. Appl. Prob. 13, 619–630.CrossRefGoogle Scholar
Titchmarsh, E. C. (1939) The Theory of Functions, 2nd edn. Clarendon Press, Oxford. University Press.Google Scholar
Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton, NJ.Google Scholar