Asymptotic Optimal Control of Multi-Class G/G/1 Queues with Feedback

Abstract

In this chapter, we extend the well-known Klimov problem, i.e. the multiclass M/G/1 queueing control problem, to a general G/G/1 setting. Specifically, we consider a single server multi-class queueing system with Bernoulli feedback, where the arrival stream is a renewal process and the service times per class are i.i.d. random variables of general distribution, independent of the arrival process and other classes. A linear holding cost is incurred by each job in the system each unit time. The problem is to determine the optimal control so as to minimize the long-run average holding cost. We show that as the traffic intensity is close to 1, the control policy specified in Klimov [10] is strongly asymptotically optimal in the sense that its absolute difference from the optimal value is bounded from above while the optimal value is approaching infinity. To establish such strong asymptotic optimality, we present a method of measuring the tightness of control policies to optimality by developing close bounds for the corresponding system performance. We further give an example to show that typical Brownian approximations may not be sufficient to lead to solutions that are strongly asymptotically optimal, due to the loss of non-heavy traffic information in the Brownian limit.