Fluid approximations and control of queues in emergency departments

Long queues in emergency departments (EDs) lead to overcrowding, a phenomenon that can potentially compromise patient care when medical interventions are delayed. There are several causes of this problem, one of which is inadequate resource allocation. In this paper, we propose using a modified version of the square root staffing (SRS) rule to satisfy the probability of delay target. We use the concepts of kinetics and biological modeling to approximate the fluid behavior of the queueing process. We are then able to estimate the offered load and the appropriate service grade necessary to construct a staffing policy that meets the target. Additionally, we show how to utilize Pontryagin's maximum principle to find the optimal number of providers that minimizes delay and staffing costs. Finally, we demonstrate the implementation of our model using data from a hospital in upstate New York.