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A new approach to the distribution of the duration of the busy period for a G/G/l queueing system

Published online by Cambridge University Press:  09 April 2009

Wolfgang Stadje
Affiliation:
Fachbereich Mathematik/Informatik, Universität Osnabr¨ckPostfach 4469 Albrechtstrasse 28 45 Osnabrück, West Germany
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Abstract

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For a G/G/l queueing system let Xt be the number of customers present at time t and Yt(Zt) be the time elapsed since the last arrival of a customer (the last completion of a service) at time t. Let τl be the time until the number of customers in the sustem is reduced from j to j – l, given that X0 = jl, Y0 = y, Z0 = z. For the joint distribution of τl and Yτl and the Laplace transforms of the τl intergral equations are derived. Under slight conditions these integral equations have unique solutions which can be determined by standard methods. Our results offer a method for calculating the busy period distribution which is completely different from the usual fluctuatuion theoretic approach.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1990

References

Cohen, J. W. (1982), The single server queue (North-Holland, Amsterdam-New York-Oxford).Google Scholar
Feller, W. (1971), An introduction to probability theory and its applications, volume II (Wiley, New York).Google Scholar
Finch, P. (1961), ‘On the busy period in the queueing sustem GI/G/1’, J. Austral Math. Soc. 2, 217227.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980), Table of integrals, series, and products (Academic Press, New York).Google Scholar
Keilson, J. and Kooharian, A. (1960), ‘On time-dependent queueing processes’, Ann. Math. Statist. 31, 104112.CrossRefGoogle Scholar
Keilson, J. and Kooharian, A. (1962), ‘On the general time-dependent queue with a single server’, Ann. MAth. Statist. 33, 767791.CrossRefGoogle Scholar
Kingman, J. F. C. (1962), ‘The use of Spitzer's indentity in the investigation of the busy period and other quantities in the queue GI/G/l’, J. Austral Math. Soc. 2, 345356.CrossRefGoogle Scholar
Prabhu, N. U. (1980), Stochastic storage processes (Springer, New York-Berlin-Heidelberg).CrossRefGoogle Scholar