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A mathematical investigation has been made of a simple queueing process in which customers arrive at random, form a single queue in order of arrival, and are served in batches, the size of each batch having a fixed maximum. In most of the results obtained the time intervals between successive occasions of service are assumed to be independently distributed in χ2 distributions with an even number of degrees of freedom (including the special cases of a random distribution and a constant service interval). The equilibrium distribution of queue length has been studied by the imbedded Markov chain method, and ergodicity established when the average demand is less than the average supply available. Expressions for the mean and variance of queue length, and the mean waiting-time are given. A useful inequality for the latter is also available in the special case of a constant service interval. An immediate application can be made to hospital out-patient departments, with possible extensions to lifts, buses and so on.
Journal of the Royal Statistical Society. Series B (Methodological) © 1954 Royal Statistical Society