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Acad Emerg Med. 2013 Sep;20(9):939-46. doi: 10.1111/acem.12215.

An emergency department patient flow model based on queueing theory principles.

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Department of Emergency Medicine, University of Colorado School of Medicine, Aurora, CO; Division of Emergency Medicine, Washington University in St. Louis School of Medicine, St. Louis, MO.



The objective was to derive and validate a novel queuing theory-based model that predicts the effect of various patient crowding scenarios on patient left without being seen (LWBS) rates.


Retrospective data were collected from all patient presentations to triage at an urban, academic, adult-only emergency department (ED) with 87,705 visits in calendar year 2008. Data from specific time windows during the day were divided into derivation and validation sets based on odd or even days. Patient records with incomplete time data were excluded. With an established call center queueing model, input variables were modified to adapt this model to the ED setting, while satisfying the underlying assumptions of queueing theory. The primary aim was the derivation and validation of an ED flow model. Chi-square and Student's t-tests were used for model derivation and validation. The secondary aim was estimating the effect of varying ED patient arrival and boarding scenarios on LWBS rates using this model.


The assumption of stationarity of the model was validated for three time periods (peak arrival rate = 10:00 a.m. to 12:00 p.m.; a moderate arrival rate = 8:00 a.m. to 10:00 a.m.; and lowest arrival rate = 4:00 a.m. to 6:00 a.m.) and for different days of the week and month. Between 10:00 a.m. and 12:00 p.m., defined as the primary study period representing peak arrivals, 3.9% (n = 4,038) of patients LWBS. Using the derived model, the predicted LWBS rate was 4%. LWBS rates increased as the rate of ED patient arrivals, treatment times, and ED boarding times increased. A 10% increase in hourly ED patient arrivals from the observed average arrival rate increased the predicted LWBS rate to 10.8%; a 10% decrease in hourly ED patient arrivals from the observed average arrival rate predicted a 1.6% LWBS rate. A 30-minute decrease in treatment time from the observed average treatment time predicted a 1.4% LWBS. A 1% increase in patient arrivals has the same effect on LWBS rates as a 1% increase in treatment time. Reducing boarding times by 10% is expected to reduce LWBS rates by approximately 0.8%.


This novel queuing theory-based model predicts the effect of patient arrivals, treatment time, and ED boarding on the rate of patients who LWBS at one institution. More studies are needed to validate this model across other institutions.

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