# Doc APProvED: Queuing Theory Can Solve Flow and Wait Problems

Mohseni, Alex MD

doi: 10.1097/01.EEM.0000459002.81450.02
Doc APProvED

Dr. Mohseniis an emergency physician in the Washington, D.C., metropolitan area and the counsel to the president for innovation and technology of Emergency Medicine Associates. He is the editor of his own blog,http://CreativeHealthLabs.com. Follow him @amohseni, and read his past columns athttp://bit.ly/MohseniDocAPProvED.

As much talk as there is of hospitals reengineering themselves using Lean, TQM, and Six Sigma methodologies, there is a surprising dearth of mathematics in these discussions. Hospitals and EDs that determine staffing levels, calculate necessary bed capacities, and design workflow based on averages without considering variability are doomed to long waits, inefficient flow, and patient dissatisfaction.

Queuing theory tells us that variability has a significant impact on flow and wait times in the ED. Variability in the ED presents itself in two main aspects: patient arrivals and service times. Patients don't arrive every six minutes, nicely spaced apart, of course; they arrive in randomly with occasional, if not frequent, boluses. Service times also vary greatly: Some patients require just reassurance and are discharged within 15 minutes, while others occupy a bed for 14 hours waiting for a consultant.

Queuing theory gives us a way to measure and predict waiting room time, length of stay, and bed capacity while also accounting for variability. The results can be shocking. Let's take a simple example: Your ED averages 14 patient arrivals per hour during the evening rush. Your triage process takes eight minutes per patient, and you have two triage nurses. Two triage nurses average 15 triages per hour combined, which is more than the 14 arriving per hour. So, everything should be fine, right?

Wrong. Queuing theory tells us that the variability in arrivals and service times (some patients will require a three-minute triage, others 12 minutes) will result in an average of 62 minutes in the waiting room (door-to-bed). It also tells us that there will be an average of 12 patients in the waiting room waiting to be triaged. These are unacceptable numbers, of course.

Using a queuing theory calculator, you could calculate the impact of reducing the average triage service time from eight minutes (7.5 patients/hour) to six minutes (10 patients/hour) by forgoing medication lists during triage, for example. This simple change is predicted to result in a dramatic reduction in door-to-bed time from 62 minutes to 12 minutes, and a reduction in the number of patients waiting for triage from 12 to 1.3.

Many queuing theory calculators and tutorials are on the web and available for Excel, which can help calculate and predict the impact of different staffing and process interventions. I am particularly fond of supositorio.com (http://bit.ly/1scZufV) because of its integrated charts. Let's start using math to help make better decisions.

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