1. The study of queues has been of interest to mathematicians and engineers for the past forty years, and a very extensive literature on the subject exists. The applications of the theory are many and varied, from Erlang's original work on telephone engineering, to present-day studies in the design of airports. The number of papers which deal with the theoretical side of the subject is, however, small, and it therefore seems desirable to attempt to develop a general theory which will cover the diverse practical requirements, so that the unity in the applications will become apparent. This paper gives such a development in the case where there is a single queue and a single server attending to it; the theory of multiple queues or many servers seems, except under simplifying assumptions which do not always correspond to reality, to be a problem of considerable difficulty. Previous work on the subject has mainly been confined to a special case where the customers arrive at random, as, for example, in a recent paper by Kendall(5);* the present theory makes no such assumption and allows the customers to join the queue in other ways, though the theory simplifies when the more restrictive assumption is made. The centre of interest in the present theory is the waiting times of the customers.
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