J Austral Math Soc Ser A 48 pp89--100, 1990.

A New Approach to the Distribution of the Duration of the Busy Period for a G/G/1 Queueing System

Wolfgang Stadje

(Received 1 October 1992; revised 21 December 1992)

Abstract

For a G/G/1 queueing system let X_t be the number of customers present at time t and Y_t(Z_t) be the time elapsed since the last arrival of a customer (the last completion of a service) at time t. Let t be the time until the number of customers in the system is reduced from j to j - l, given that X₀ = j ³ l, Y₀ = y, Z₀ = z. For the joint distribution of t₁ and Y_t₁ and the Laplace transforms of the t_l intefral 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 fluctuation theoretic approach.

1980 AMS Subject Classification (1985 Revision): 60K25

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Authors

Wolfgang Stadje: Fachbereich Mathematik/Informatik, Universität Osnabrück, Postfach 4469, Albechtstrasse 28, 45 Osnabrück, West Germany.

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Last Modified: Wed Feb 19 10:27:52 2003