Optimally Scheduling Heterogeneous Impatient Customers

We study the question of scheduling impatient customers in parallel server queuing systems. At the time of arrival, customers can be identified as one of many classes, where the class represents the service time and patience time distributions, and cost characteristics. From the system's perspective, customers that are of the same type at time of arrival get further differentiated on their residual patience time as they wait in the system. This is due to the fact that at time of arrival, the system only knows the overall patience distribution from which the customer's patience value is drawn, and as time elapses, this estimate can be further updated for customers who are still in the system using the information that the customer has not yet abandoned. For non-exponential patience distributions, such an update indeed reveals additional information. In this paper, we use a fluid approach to characterize the cost-minimizing policy that schedules customers on two dimensions of heterogeneity: class and time-in-queue information. We propose a multi-class time-in-queue policy that prioritizes both across customer classes, and within each customer class using a simple rule, and further we show that most of the gains of such a policy can be achieved by deviating from within-class First Come First Serve (FCFS) for at most one customer class. Finally, for systems with exponential abandonment times, our policy reduces to a simple priority-based policy, which we prove to be asymptotically optimal with an optimality gap that does not grow with system scale