Dynamic control of a single-server system with abandonments

In this paper, we discuss the dynamic server control in a two-class service system with abandonments. Two models are considered. In the first case, rewards are received upon service completion, and there are no abandonment costs (other than the lost opportunity to gain rewards). In the second, holding costs per customer per unit time are accrued, and each abandonment involves a fixed cost. Both cases are considered under the discounted or average reward/cost criterion. These are extensions of the classic scheduling question (without abandonments) where it is well known that simple priority rules hold. The contributions in this paper are twofold. First, we show that the classic c-μ rule does not hold in general. An added condition on the ordering of the abandonment rates is sufficient to recover the priority rule. Counterexamples show that this condition is not necessary, but when it is violated, significant loss can occur. In the reward case, we show that the decision involves an intuitive tradeoff between getting more rewards and avoiding idling. Secondly, we note that traditional solution techniques are not directly applicable. Since customers may leave in between services, an interchange argument cannot be applied. Since the abandonment rates are unbounded we cannot apply uniformization-and thus cannot use the usual discrete-time Markov decision process techniques. After formulating the problem as a continuous-time Markov decision process (CTMDP), we use sample path arguments in the reward case and a savvy use of truncation in the holding cost case to yield the results. As far as we know, this is the first time that either have been used in conjunction with the CTMDP to show structure in a queueing control problem. The insights made in each model are supported by a detailed numerical study. © 2010 Springer Science+Business Media, LLC

Dynamic routing of customers with general delay costs in a multiserver queuing system

We consider a network of parallel service stations each modeled as a single-server queue. Each station serves its own dedicated customers as well as generic customers who are routed from a central controller. We suppose that the cost incurred by a customer is an increasing function of her time spent in the system. In a significant advance on most previous work, we do not require waiting costs to be convex, still less linear. With the objective of minimizing the long-run average waiting cost, we develop two heuristic routing policies, one of which is based on dynamic programming policy improvement and the other on Lagrangian relaxation. In developing the latter policy, we show that each station is “indexable” under mild conditions for customers’ waiting costs and also prove some structural results on the admission control problem that naturally arises as a result of the Lagrangian relaxation. We then test the performance of our heuristics in an extensive numerical study and show that the Lagrangian heuristic demonstrates a strong level of performance in a range of traffic conditions. In particular, it clearly outperforms both a greedy heuristic, which is a standard proposal in complex routing problems, and a recent proposal from the heavy traffic literature