Monotonicity properties of single-server retrial queues

Abstract

A single server retrial queue is a queueing system consisting of a primary queue with finite capacity, an orbit and a server. Customers can arrive at the primary queue either from outside the system or from the orbit. If the primary queue is full, an arriving customer joins the orbit and conducts a retrial later. Otherwise, he enters the primary queue, waits for service and then leaves the system after service completion. We investigate the effect of retrial times on the behavior of the system. In particular, we assume that the retrial time distributions are phase type and introduce a new relation, which we call K-dominance (short for Kalmykov), on these distributions. Longer retrial times with respect to this K-dominance are shown to result in a more congested system in the stochastic sense. From these results, we derive monotonicity properties of several system performance measures of interest.