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.

Refining Square-Root Safety Staffing by Expanding Erlang C

“…Mandelbaum & Zeltyn (2007) report less favorable numerical results on conventional squareroot safety staffing in this setting; it therefore makes sense to include a correction term in this case. …”

“…Some studies focusing on obtaining limiting approximations for the steady-state distribution or for the timedependent process are Gamarnik Whitt (2005). Another body of work is concerned with optimization issues and developing asymptotically control policies, see for example Atar (2005), Borst et al (2004), Dai & Tezcan (2007) and Mandelbaum & Zeltyn (2007).…”

Service-Level Differentiation in Call Centers with Fully Flexible Servers

“…The most relevant references in this context are the papers by Borst et al (2004) and Mandelbaum and Zeltyn (2006), which cover in great generality staffing problems for the single-class M/M/N and M/M/N +G systems.…”

“…To generalize this approach, researchers have proposed using the Erlang-A model, which includes also customer abandonment (see, for example, Garnett et al 2002;Mandelbaum and Zeltyn 2006;Whitt 2004Whitt , 2006. This model has been increasingly adopted by developers of workforce management tools and is consequently becoming more prevalent in call centers.…”

When Promotions Meet Operations: Cross-Selling and Its Effect on Call Center Performance

“…In particular, we follow the asymptotic optimality framework approach first used by Borst et al [8], and adapted later to more complex settings ( [3], [4], [5], [6], [15] and [19]). …”