Fluid Approximations and Control of Queues in Emergency Departments Jerome Niyirora a,* , Jun Zhuang b a SUNY Polytechnic Institute, Utica, New York, USA b University at Buffalo, SUNY, Buffalo, New York, USA Abstract 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. Keywords: OR in Health Services, Emergency Department, Queues, Square Root Staffing, Optimal Control. 1. Introduction Long queues in emergency departments (EDs) lead to overcrowding, a phenomenon that can potentially lead to adverse patient care when medical interventions are delayed (Warden et al., 2006). Additionally, increased mortality has been linked to overcrowding (Pines et al., 2011; Hoyle, 2013). Recent surveys in the United States (US) have indicated that, due to overcrowded EDs, about 500,000 ambulances are diverted each year (McCaig et al., 2009; Pitts et al., 2008). Moreover, an estimated 2 million patients leave EDs, annually, without receiving medical care (Warden et al., 2006). Some of the proposed solutions to this problem include: alleviating delays in processing lab and radiology tests (American College of Emergency Physicians - ACEP, 2004), reducing delays in patients transportation (Au et al., 2009; Green & Hall, 2006), and minimizing delays in the schedules of operating rooms for ED patients (Litvak et al., 2001). It is also believed that solving the problem of boarding admitted patients in the ED (Shi et al., 2015) and transferring mental health patients in a timely manner could lessen long delays (Government Accountability Office - GAO, 2009). * Corresponding author. Tel.:+0113157927430 Email address: jerome.niyirora@sunyit.edu (Jerome Niyirora) Preprint submitted to Europian Journal of Operational Research March 1, 2017