# Staffing many-server queues with impatient customers: Constraint satisfaction in call centers

## Abstract

Motivated by call center practice, we study asymptotically optimal staffing of many-server queues with abandonment. A call center is modelled as an M/M/n+G queue, which is characterized by Poisson arrivals, exponential service times, n servers, and generally distributed patience times of customers. Our asymptotic analysis is performed as the arrival rate, and hence the number of servers n, increases indefinitely. We consider a constraint satisfaction problem, where one chooses the minimal staffing level n that adheres to a given cost constraint. The cost can incorporate the fraction abandoning, average wait, and tail probabilities of wait. Depending on the cost, several operational regimes arise as asymptotically optimal: Efficiency-Driven (ED), Quality and Efficiency-Driven (QED), and also a new ED+QED operational regime that enables QED tuning of the ED regime. Numerical experiments demonstrate that, over a wide range of system parameters, our approximations provide useful insight as well as excellent fit to exact optimal solutions. It turns out that the QED regime is preferable either for small-to-moderate call centers or for large call centers with relatively tight performance constraints. The other two regimes are more appropriate for large call centers with loose constraints. We consider two versions of the constraint satisfaction problem. The first one is constraint satisfaction on a single time interval, say one hour, which is common in practice. Of special interest is a constraint on the tail probability, in which case our new ED+QED staffing turns out asymptotically optimal. We also address a global constraint problem, say over a full day. Here several time intervals, say 24 hours, are considered, with interval-dependent staffing levels allowed; one seeks to minimize staffing levels, or more generally costs, given the overall performance constraint. In this case, there is the added flexibility of trading service levels among time intervals, but we demonstrate that only little gain is associated with this flexibility if one is concerned with the fraction abandoning. © 2009 INFORMS.