Service-level differentiation in many-server service systems via queue-ratio routing

Motivated by telephone call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. To minimize staffing costs subject to service-level constraints, where we delicately balance the service levels (SLs) of the different classes, we propose a family of routing rules called fixed-queue-ratio (FQR) rules. With FQR, a newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified proportion of the total queue length. The proportions can be set to achieve desired SL targets. The FQR rule achieves an important state-space collapse (SSC) as the total arrival rate increases, in which the individual queue lengths evolve as fixed proportions of the total queue length. In the current paper we consider a variety of service-level types and exploit SSC to construct asymptotically optimal solutions for the staffing-and-routing problem. The key assumption in the current paper is that the service rates depend only on the agent pool.