Motivated by call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called Queue-and- Idleness-Ratio (QIR) rules. 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 state-dependent proportion of the total queue length. An arriving customer is routed to the agent pool whose idleness most exceeds a specified state-dependent proportion of the total idleness. We identify regularity conditions on the network structure and system parameters under which QIR produces an important state-space collapse (SSC) result in the Quality-and-Efficiency-Driven (QED) many-server heavy-traffic limiting regime. The SSC result is applied here to prove stochastic-process limits and in subsequent papers to solve important staffing and control problems for large-scale service systems.

In a recent paper we introduced the fixed-queue-ratio (FQR) family of routing rules for many-server service systems with multiple customer classes and server pools. A newly available server 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. Under fairly general conditions, FQR produces an important state-space collapse as the total arrival rate and the numbers of servers increase in a coordinated way. That state-space collapse was previously used to delicately balance service levels for the different customer classes. In this sequel, we show that a special version of FQR stochastically minimizes convex holding costs in a finite-horizon setting when the service rates are restricted to be pool-dependent. Under additional regularity conditions, the special version of FQR reduces to a simple policy: Linear costs produce a priority-type rule, in which the least-cost customers are given low priority. Strictly convex costs (plus other regularity conditions) produce a many-server analogue of the generalized-c¹ (Gc¹) rule, under which a newly available server selects a customer from the class experiencing the greatest marginal cost at that time.