A Discrete Queueing Problem with Variable Service Times

Methods from the theory of Markov chains are used to analyze a simple single-server queueing model. The model is of the sort that arises naturally in information-handling contexts, in that a discrete time basis is used, which matches the cyclic character of processors. Considerable generality is attained, in that no appeal is made to the exponential or other conventional forms for the probability distributions governing the number of arrivals per cycle and the service times. The principal object of study is the queue length; the stationary distribution governing this quantity is calculated, along with various associated averages. The relation between the present method and the more usual continuous-variable method is illustrated by the derivation of some of the classical equations from a limiting case of our model.