A Queuing System with Service-Time Distribution of Mixed Chi-Squared Type

David M. G. Wishart
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David M. G. Wishart: Department of Pure Mathematics, The University of Birmingham, Birmingham, England

Operations Research, 1959, vol. 7, issue 2, 174-179

Abstract: In this paper Kendall's technique of the embedded Markov chain (Kendall, D. G. 1953. Stochastic processes in the theory of queues. Ann. Math. Stat. 24 338--354.) is applied to a queuing system with general independent input and a wide class of service-time distributions. The matrix of transition probabilities is found to be formally identical with that discussed in our earlier study (Wishart, D. M. G. 1956. A queuing system with (chi) 2 service-time distribution. Ann. Math. Stat. 27 768--779.), which will be taken as read in the present paper. Using the results of reference 9 we can write down the equilibrium distribution of waiting-times for customers in the more general system in terms of the roots of a transcendental equation. An example is considered that arose in Bailey's study of hospital systems (Bailey, N. T. J. 1952. A study of queues and appointment systems in hospital out-patient departments. J. Roy. Stat. Soc. B14 185--199.).

Date: 1959
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