A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

Aligning service mechanism and demand is achieved essentially in two ways: either service and/or arrival parameters are managed to vary with system state, or consecutive inter-arrival intervals and service times are not assumed to be independent. The former is by now well studied. In the latter, a bivariate distribution with negative exponential marginals, which can be constructed in many ways, constitutes a first attempt. With a particular construction involving a modified Bes-el function of order zero, the waiting time density (as well as its stationary counterpart) of such a partially correlated generalization of M

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As a sequel to the author's and Conolly and Choo's articles, this paper further investigates the effects of positive correlation between the underlying processes in queueing models. Having established that the waiting time distribution is hyperexponential for a certain service interarrival joint density, we examined the sensitivity of this distribution to the value of the correlation coefficient r(0 < r < 1). Since the extreme cases of r = 0 (conventional models) and r = 1 have long been known, we are able to illustrate the convergence of the waiting time density and respective moments to their limiting values.

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.