Admission control for a multi-server queue with abandonment

Abstract

In a M/M/N + M queue, when there are many customers waiting, it may be preferable to reject a new arrival rather than risk that arrival later abandoning without receiving service. On the other hand, rejecting new arrivals increases the percentage of time servers are idle, which also may not be desirable. We address these trade-offs by considering an admission control problem for a M/M/N + M queue when there are costs associated with customer abandonment, server idleness, and turning away customers. First, we formulate the relevant Markov decision process (MDP), show that the optimal policy is of threshold form, and provide a simple and efficient iterative algorithm that does not presuppose a bounded state space to compute the minimum infinite horizon expected average cost and associated threshold level. Under certain conditions we can guarantee that the algorithm provides an exact optimal solution when it stops; otherwise, the algorithm stops when a provided bound on the optimality gap is reached. Next, we solve the approximating diffusion control problem (DCP) that arises in the Halfin-Whitt many-server limit regime. This allows us to establish that the parameter space has a sharp division. Specifically, there is an optimal solution with a finite threshold level when the cost of an abandonment exceeds the cost of rejecting a customer; otherwise, there is an optimal solution that exercises no control. This analysis also yields a convenient analytic expression for the infinite horizon expected average cost as a function of the threshold level. Finally, we propose a policy for the original system that is based on the DCP solution, and show that this policy is asymptotically optimal. Our extensive numerical study shows that the control that arises from solving the DCP achieves a very similar cost to the control that arises from solving the MDP, even when the number of servers is small.

Dynamic control of a single-server system with abandonments

“…While in our work the only possibilities after arrival are that a customer is either served or abandons, there is a line of work that attempts to compensate for potential abandonments in other manners. In [15] Koçaga and Ward study an admission control problem for a multiserver queue with a single class of customers who may abandon. In Armony et al [5] customers are provided with delay estimates to influence their behavior, while in Armony and Maglaras [3,4] a call-back option is proposed to allow potential abandonments to be contacted at a future point in time (when presumably servers are less busy).…”

Optimal Admission Control for Many-Server Systems with QED-Driven Revenues

“…315-341, © 2017 Wallace (2005), Whitt (2005), and Whitt (2004). General admission control, however, has only received limited attention in the QED regime, see for instance Koçağa and Ward (2010), Weerasinghe and Mandelbaum (2013). These studies specifically account for abandonments, which create a trade-off between the rejection of a new arrival and the risk of that arrival later abandoning without receiving service, with the associated costly increase of server idleness.…”

An SMDP model for a multiclass multi-server queueing control problem considering conversion times

“…They also prove several conditional stochastic decomposition properties. Koçagǎ and Ward [11] formulate a M/M/N+M queue with abandonment as a Markov decision process (MDP), show that the optimal policy is of threshold form, and provide a simple and efficient iterative algorithm. They solve the approximating diffusion control problem (DCP) and obtain a convenient analytic expression for the infinite horizon expected average cost as a function of the threshold level.…”