Algorithmic Analysis of Queueing Models

Queueing models have their origin in the study of design problems of automatic telephone exchanges and were first analysed by the queueing pioneer A.K. Erlang in the early 1900s. In planning telephone systems to meet given performance criteria, questions were asked such as: How many lines are required in order to give a certain grade of service? What is the probability that a delayed customer has to wait more than a certain time before getting a connection? Similar questions arise in the design of many other systems: How many terminals are needed in a computer system so that 80% of the users get access to a terminal within 20 seconds? What will be the effect on the average waiting time of customers when changing the size of a maintenance staff to service leased equipment? How much storage space is needed in buffers at workstations in an assembly line in order to keep the probability of blocking below a specified acceptable level? These design problems and many others concern, in fact, facilities serving a community of users, where both the times at which the users ask for service and the durations that the requests for service will occupy facilities are stochastic, so that inevitably congestion occurs and queues may build up. In the first stage of design the system engineer usually needs quick answers to a variety of questions like those posed above. Queueing theory constitutes a basic tool for making firstapproximation estimates of queue sizes and probabilities of delays. Such a simple tool should in general be preferred to simulation, especially when it is possible to have a large number of different configurations in the design problem. In this chapter we discuss a number of basic queueing models that have proved to be useful in analysing a wide variety of stochastic service systems. The emphasis will be on algorithms and approximations rather than on mathematical aspects. We feel that there is a need for such a treatment in view of the increased use of queueing models in modern technology. Actually, the application of queueing theory in the performance analysis of computer and communication systems has