Fluid approximation of a call center model with redials and reconnects

In many call centers, callers may call multiple times. Some of the calls are reattempts after abandonments (redials), and some are re-attempts after connected calls (reconnects). The combination of redials and reconnects has not been considered when making staffing decisions, while ignoring them will inevitably lead to underor overestimation of call volumes, which results in improper and hence costly staffing decisions. Motivated by this, in this paper we study call centers where customers can abandon, and abandoned customers may redial, and when a customer finishes his conversation with an agent, he may reconnect. We use a fluid model to derive first order approximations for the number of customers in the redial and reconnect orbits in the heavy traffic. We show that the fluid limit of such a model is the unique solution to a system of three differential equations. Furthermore, we use the fluid limit to calculate the expected total arrival rate, which is then given as an input to the Erlang A model for the purpose of calculating service levels and abandonment rates. The performance of such a procedure is validated in the case of single intervals as well as multiple intervals with changing parameters.