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.

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CWI
CWI. Probability, Networks and Algorithms [PNA]
Stochastics

Borst, S.C, Mandelbaum, A, & Reiman, M.I. (2000). Dimensioning large call centers. CWI. Probability, Networks and Algorithms [PNA]. CWI.