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.

I. Introduction.- 1. Some examples.- 2. The aims of an investigation of congestion.- 3. The arrival pattern.- 4. The service mechanism.- 5. The queue-discipline.- 6. The measurement of congestion.- 7. The investigation of queueing systems.- 8. The modification of queueing systems.- II. Some simple queues with random arrivals.- 1. Introduction.- 2. Statistical equilibrium.- 3. Single-server queue with random arrivals and exponential service-times.- 4. Arrival rates and service-times dependent on queue size.- 5. Further remarks on equilibrium conditions.- 6. Single-server queue with random arrivals and general distribution of service-times.- 7. The busy-period distribution.- III. More about simple queues.- 1. Non-equilibrium theory.- 2. Queues with many servers.- 3. Queues with priorities.- IV. Machine Interference.- 1. Statement of problem.- 2. The simplest probability model.- 3. Some complications.- V. More specialized topics.- 1. Introduction.- 2. The method of stages.- 3. The integral equation of a single-server queue.- 4. Simulation and Monte Carlo methods.- 5. Series of queues.- 6. The busy-period distribution.- Appendixes.- I. Bibliographical notes.- II. Exercises and further results.- III. Notes on the theory of probability and Laplace-Stieltjes transforms.- Author Index.