Staffing and Routing in a Two-Tier Call Center

This paper studies service systems with gatekeepers who diagnose a customer problem and then either refer the customer to an expert or attempt treatment. The probability of successful treatment by a gatekeeper decreases as the problem's complexity increases. We determine the optimal staffing levels and referral rates that minimize the system costs, where these costs include staffing, customer waiting, and mistreatment costs. We also compare the optimal gatekeeper system (a two-tier system) with a system staffed with only experts (a direct-access system). By drawing upon recent results showing the asymptotic optimality of square-root staffing rules for stand-alone queues, we show that the optimal design of a two-tier system can be reduced to determining an optimal referral rate. It is well known in the queueing literature that pooling resources can create economic benefits. However, in a two-tier system, it is not clear how to take advantage of pooling. We find that when waiting costs are high, having a direct-access system is preferred unless the gatekeepers have a high skill level. If the gatekeepers' skills are high enough, it is optimal to achieve pooling economies at the first tier for even very high values of the waiting costs. Finally, we use numerical experiments to demonstrate that an easily computed referral rate from a deterministic system is a reasonable approximation for the optimal referral rate whenever a two-tier system is preferred to a direct-access system