Call center staffing: Service-level constraints and index priorities

Motivated by the theory and practice of staffing call centers with multiple customer classes we study a question of formulation choice. Staffing problems are concerned with minimizing capacity-related costs while meeting pre-specified Quality-of-Service (QoS) targets. Taken "at face value", the mathematical solutions to some prevalent staffing problems contradict implicit intuitions about service-level differentiation (SLD). We make the SLD requirement mathematically explicit and study it in the context of the two-customer-class V model. We derive insights on the role of SLD and, particularly, on the impact of the SLD degree on capacity costs and optimal prioritization rules. We prove that a decrease in the SLD (sub-perfect SLD) can lead to comparably significant decreases in costs. To materialize this reduction, however, one must accept prioritization policies that are non-monotone (customers arriving at periods of high congestion wait less) and discontinuous (consecutive customers - from the same class - can have starkly different service experiences). Perfect SLD, on the other hand, is equivalent to restricting the family of prioritization rules to the appealing family of monotone-index rules. In other words, we find the prevalent use of monotone index rules in call centers is consistent from a mathematical optimality standpoint with a perfect SLD requirement.