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

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 χ² 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.).