An Optimal Callback Policy for General Arrival Processes: A Pathwise Analysis

Abstract

This paper studies the callback option as an instrument to effectively mitigate congestion due to temporary surges in arrivals to a call center. The call arrival process can be an arbitrary point process, allowing uncertainty and temporary surges in the arrival rate, provided that the system is stable. When a customer arrives, the call center manager examines the system state and decides whether to offer the incoming customer the callback option or not. When the callback option is offered, the customer decides whether to accept the offer. The customer is routed to the offline queue (to be called back later) only if he is offered the callback option and accepts it. Otherwise, he is routed to the online queue. For each customer in the online queue, the call center manager incurs a waiting cost of h per time unit. Similarly, whenever she routes a customer to the offline queue (for a callback later), she incurs a one-time penalty of p. Initially, we allow complete foresight policies that look into the entire future. We show that a simple lookahead policy that looks into the future arrivals and service completion times for the next p/h time units and uses the current number of customers in the system who previously rejected a callback offer (but does not look into the accept/reject decisions of future customers) is pathwise optimal. Building on the insights gleaned from the optimal lookahead policies, we also propose a non-anticipating (and implementable) policy by interpreting the lookahead policy in the fluid model and show its optimality in the fluid model. In particular, we show that this policy reduces to the so called line policy, if the arrival rate process follows a Cox-Ingersoll-Ross process. Lastly, we conduct a simulation study which shows that the proposed policies perform well.