Erlang-R: A Time-Varying Queue with Reentrant Customers, in Support of Healthcare Staffing

We analyze a queueing model that we call Erlang-R, where the R stands for reentrant customers. Erlang-R accommodates customers who return to service several times during their sojourn within the system, and its modeling power is most pronounced in time-varying environments. Indeed, it was motivated by healthcare systems, in which offered-loads vary over time and patients often go through a repetitive service process. Erlang-R helps answer questions such as how many servers (physicians/nurses) are required to achieve predetermined service levels. Formally, it is merely a two-station open queueing network, which, in a steady state, evolves like an Erlang-C (M/M/s) model. In time-varying environments, on the other hand, the situation differs: here one must account for the reentrant nature of service to avoid excessive staffing costs or undesirable service levels. We validate Erlang-R against an emergency ward (EW) operating under normal conditions as well as during a mass casualty event (MCE). In both scenarios, we apply time-varying fluid and diffusion approximations: the EW is critically loaded and the MCE is overloaded. In particular, for the EW we propose a time-varying square-root staffing policy, based on the modified offered-load, which is proved to perform well over small-to-large systems.