Dimensioning large call centers

We develop a framework for asymptotic optimization of a queueing system. The motivation is the staffing problem of call centers with 100's of agents (or more). Such a call center is modeled as an M/M/N queue, where the number of agents~$N$ is large. Within our framework, we determine the asymptotically optimal staffing level~$N^*$ that trades off agents' costs with service quality: the higher the latter, the more expensive is the former. As an alternative to this optimization, we also develop a constraint satisfaction approach where one chooses the least~$N^*$ that adheres to a given constraint on waiting cost. Either way, the analysis gives rise to three regimes of operation: quality-driven, where the focus is on service quality; efficiency-driven, which emphasizes agents' costs; and a rationalized regime that balances, and in fact unifies, the other two. Numerical experiments reveal remarkable accuracy of our asymptotic approximations: over a wide range of parameters, from the very small to the extremely large, $N^*$ is {em exactly/ optimal, or it is accurate to within a single agent. We demonstrate the utility of our approach by revisiting the square-root safety staffing principle, which is a long-existing rule-of-thumb for staffing the M/M/N queue. In its simplest form, our rule is as follows: if $c$ is the hourly cost of an agent, and $a$ is the hourly cost of customers' delay, then $N^* = R + y^*({a over c) sqrt R$, where $R$ is the offered load, and $y^*(cdot)$ is a function that is easily computable.