A time-varying call center design via Lagrangian mechanics

We consider a multi-server delay queue with finite additional waiting spaces and time-varying arrival rates, where the customers waiting in the buffer may abandon. These are features that arise naturally from the study of service systems such as call centers. Moreover, we assume rewards for successful service completions and cost rates for service resources. Finally, we consider service level agreements that constrain the fractions of arriving customers that abandon as well as the ones that are blocked. Applying the theory of Lagrangian mechanics to the fluid limit of a related Marko-vian service network model, we obtain near profit-optimal staffing and provisioning schedules. The nature of this solution consists of three modes of operation. A key step in deriving this solution is combining the modified offered load approximation for loss systems with our fluid model to estimate effectively both our service level agreement metrics and the profit for the original queueing model. Second-order profit improve-ments are achieved through a modified offered load version of the conventional square root rule