Queues for service of one kind or another arise in many different fields of activity. In recent years a considerable amount of research has been conducted into the properties of simplified mathematical models of such queueing systems. Our objects in the present mono­ graph are threefold. First we have tried, especially in Chapter I, to give an account of the general ideas that are useful in describing and thinking about queueing systems. Secondly, we have illustrated by examples some of the mathematical techniques that are useful for the study of these systems. Finally, we have given some explicit mathematical results which may be useful in practical investigations. A recent bibliography gave some 600 papers on queueing and allied subjects. Clearly we cannot, in the modest limits of the present work, cover more than a small proportion of the huge amount of material available. However, some results that we have not had space to discuss in detail have been given in outline in the form of exercises.