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

Part of: Special processes

Published online by Cambridge University Press:  09 April 2009

Wolfgang Stadje
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 = j ≧ l, 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

Secondary: 60K25: Queueing theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 89 - 100
Copyright
Copyright © Australian Mathematical Society 1990

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