Routing and staffing in emergency departments: A multiclass queueing model with workload dependent service times

Efficient patient flow through an emergency department is a critical factor that contributes to a hospital's performance, which influences overall patient health outcomes. In this work, we model a multiclass multiserver queueing system where patients of varying acuity receive care from one of several wards, each ward is attended by several nurses who work as a team. Supported by empirical evidence that a patient's time-in-ward is a function of the nurse-patient ratio in that ward, we incorporate state-dependent service times into our model. Our objective is to reduce patient time in system and to control nurse workload by jointly optimizing patient routing and nurse allocation decisions. Due to the computational challenges in formulating and solving the queueing model representation, we study a corresponding deterministic fluid model which serves as a first-order approximation of the multiclass queueing model. Next, we formulate and solve an optimization model using the first-order control equations and input the results into a discrete-event simulation to estimate performance measures, such as patient length-of-stay and ward workload. Finally, we present a case study using retrospective data from a real hospital which highlights the importance of accounting for nurse workload and service behavior in developing routing and staffing policies.