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.

I n a call center, staffing decisions must be made before the call arrival rate is known with certainty. Once the arrival rate becomes known, the call center may be over-staffed, in which case staff are being paid to be idle, or understaffed, in which case many callers hang-up in the face of long wait times. Firms that have chosen to keep their call center operations in-house can mitigate this problem by co-sourcing; that is, by sometimes outsourcing calls. Then, the required staffing N depends on how the firm chooses which calls to outsource in real time, after the arrival rate realizes and the call center operates as a M/M/N + M queue with an outsourcing option. Our objective is to find a joint policy for staffing and call outsourcing that minimizes the long-run average cost of this two-stage stochastic program when there is a linear staffing cost per unit time and linear costs associated with abandonments and outsourcing. We propose a policy that uses a square-root safety staffing rule, and outsources calls in accordance with a threshold rule that characterizes when the system is "too crowded." Analytically, we establish that our proposed policy is asymptotically optimal, as the mean arrival rate becomes large, when the level of uncertainty in the arrival rate is of the same order as the inherent system fluctuations in the number of waiting customers for a known arrival rate. Through an extensive numerical study, we establish that our policy is extremely robust. In particular, our policy performs remarkably well over a wide range of parameters, and far beyond where it is proved to be asymptotically optimal. by Michael Pinedo, after 2 revisions. which include service quality costs that are hard to

We study an inventory system that consists of two demand classes. The orders in the first class need to be satisfied immediately, whereas the orders in the second class are to be filled in a given demand lead time. The two classes are also of different criticality. For this system, we propose a policy that rations the non-critical orders. Under a one-for-one replenishment policy with backordering and for Poisson demand arrivals for both classes, we first derive expressions for the service levels of both classes. The service level for the critical class is an approximation, whereas the service level for the non-critical class is exact. We then conduct a computational study to show that our approximation works reasonably, the benefits of rationing can be substantial, and the incorporation of demand lead time provides more value when the demand class with demand lead time is the critical class. The research is motivated by the spare parts service system of a major capital equipment manufacturer that faces two types of demand. For this company, the critical down orders need to be satisfied immediately, while the less critical maintenance orders can be satisfied after a fixed demand lead time. We conduct a case study with 64 representative parts and show that significant savings (as much as 14% on inventory on hand) are possible through incorporation of demand lead times and rationing.

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