Counterexamples for comparisons of queues with finite waiting rooms

Abstract

We constructG/G/1/k queueing models that fail to have anticipated monotonicity properties with respect to the capacityk. In one model the long-run average number of customers in the system is arbitrarily close to the capacityk, but it decreases to an arbitrarily small value when the capacity is increased. In another model the throughput is arbitrarily close to the arrival rate when the capacity isk, but the throughput decreases to an arbitrarily small value when the capacity is increased. These examples involving non i.i.d. service times, which are associated with external arrivals instead of being assigned when service begins, show that stochastic assumptions and arguments involving more than direct sample-path comparisons are essential for obtaining useful bounds and positive comparison results.