On performance comparison of MR/GI/1 queues

In this paper the authors are interested in the effect that dependencies in the arrival process to a queue have on queueing properties such as mean queue length and mean waiting time. They start with a review of the well known relations used to compare random variables and random vectors, e.g., stochastic orderings, stochastic increasing convexity, and strong stochastic increasing concavity. These relations and others are used to compare interarrival times in Markov renewal processes first in the case where the interarrival time distributions depend only on the current state in the underlying Markov chain and then in the general case where these interarrival times depend on both the current state and the next state in that chain. These results are used to study a problem previously considered by Patuwo et al. Then, in order to keep the marginal distributions of the interarrival times constant, the authors build a particular transition matrix for the underlying Markov chain depending on a single parameter, p. This Markov renewal process is used in the Patuwo et al. problem so as to investigate the behavior of the mean queue length and mean waiting time on a correlation measure depending only on p. As constructed, the interarrival time distributions do not depend on p so that the effects the authors find depend only on correlation in the arrival process. As a result of this latter construction, they find that the mean queue length is always larger in the case where correlations are non-zero than they are in the more usual case of renewal arrivals (i.e., where the correlations are zero). The implications of the present results are clear.