An adaptive multistage queueing system

With the ultimate aim of controlling the queue size, the conventional M/M/1 queueing model can be modified, in a number of ways, to allow for dependence of arrival and service processes. In this article one such modification is introduced and the resulting model is analyzed in detail. The service time of each customer is assumed to be the sum of a random number of independent exponential variables, each of which represents one stage of service. The number of stages has a presumed distribution, with the added adaptive mechanism of curtailing the nth service time upon the completion of the stage during which the first customer, whose arrival time is after the commencement of the nth service, joins the system. Initially, an expression is derived for the moment-generating function of the effective service times. By differentiating this expression, the effective traffic intensity is given which, as expected, depends on the mean number of stages. Then, by using the transition probabilities of an imbedded Markov chain, the stationary queue size distribution at departure times is obtained. An approximate expression is also given for the mean waiting time. The paper is concluded with examples of specific distributions (geometric and Poisson) for the number of stages, along with the degenerate case---namely, one stage with probability one which corresponds to M/M/1.