Routing and Staffing in Large-Scale Service Systems: The Case of Homogeneous Impatient Customers and Heterogeneous Servers

Motivated by call centers, we study large-scale service systems with homogeneous impatient customers and heterogeneous servers; the servers differ with respect to their speed of service. For this model, we propose staffing and routing rules that are jointly asymptotically optimal in the heavy-traffic many-server QED, ED, and ED + QED regimes, respectively. For the QED regime, our proposed routing rule is FSF, that assigns customers to the fastest server available first. In the ED and ED + QED regimes, all work-conserving policies perform (asymptotically) equally well. In all these regimes, the form of the asymptotically optimal staffing is consistent with the asymptotically optimal staffing in the same regimes in the single-pool case, respectively. In particular, the total service capacity is (asymptotically) equal to a term that is proportional to the arrival rate plus, possibly, a term that is proportional to the square-root of the arrival rate, with both terms being regime dependent. Our specific proposed approximation for the optimal staffing vector is obtained via a straightforward solution to a deterministic optimization problem subject to a linear feasible region.