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 exploit SSC to construct asymptotically feasible solutions for staffing-and-routing problem. In a sequel, we show that these solutions are asymptotically optimal when the service rates are pool-dependent. In another sequel, we show that a simulation-based search finds a good solution based on FQR when an asymptotically optimal solution is not yet available.