HA425 Unit 4 ED Capacity and Little's Law Analysis Example

The Line Is at Triage, the Constraint Is in the Rooms: A Capacity Analysis of a Composite Community Hospital Emergency Department Using Station Utilization and Little's Law to Show Evening Demand for Twenty-Two Rooms Against Twenty Staffed

[Student Name]

Purdue University Global

HA425: Operational Analysis and Quality Improvement

Unit 4 Capacity Analysis

[Instructor Name]

[Date]

The department, its rates and every figure are composite and bracketed; the model is a simplified course analysis, not a staffing standard.

Demand

Figure 1 overlays arrivals per hour on staffed rooms per hour. Arrivals climb to about [6.2] an hour from 11:00 and hold near that level until 21:00. Staffed rooms stay at [24] until 19:00, when a day nurse's twelve-hour shift ends and four rooms close, leaving [20].

Station by Station

For each station, capacity per hour is the number of servers times the rate each can serve. Utilization is peak demand divided by capacity.

1. Registration: one clerk, about [3] minutes per patient, capacity [20] an hour; utilization about [31] percent.

2. Triage: one nurse, about [7] minutes per patient, capacity about [8.6] an hour; utilization about [72] percent.

3. Providers: three on the evening schedule, about [2.5] new patients an hour each, capacity [7.5] an hour; utilization about [83] percent.

4. Rooms: analyzed with Little's Law below.

Why Triage Looks Like the Problem

The line at triage is visible because patients stand there. But a station at [72] percent utilization has slack; the line forms when several patients arrive together and clears within minutes. Staff see the line, not the utilization. The most visible queue is often not the binding constraint; the constraint is wherever demand meets or exceeds capacity for hours at a time.

Rooms and Little's Law

A room is held for the whole time a patient is in treatment, not for a single service step, so room capacity is better analyzed with Little's Law. Little (1961) proved that, for a system in long-run balance, the average count inside it is the product of how fast items arrive and how long each stays. For the evening: Arrival rate needing a room: about [6.2] an hour. Average time in a room for these patients: about [3.5] hours. Average rooms occupied: [6.2] × [3.5] = about [21.7]. Staffed rooms after 19:00: [20]. Before 19:00, [24] staffed rooms give a room utilization near [90] percent, already high. After 19:00, demand for about [22] rooms exceeds the [20] available, so the queue for rooms grows every hour until arrivals fall after 21:00.

Why High Utilization Matters

Queueing analysis shows that waits rise sharply as utilization approaches 100 percent, especially when arrival and service times vary, as they do in an emergency department. A study applying queueing models to emergency department provider staffing found that adjusting staffing to match arrival patterns reduced the share of patients who left without being seen (Green et al., 2006). Rooms at more than 100 percent of capacity cannot reach a steady state at all. A systematic review of crowding likewise identified inadequate staffing and limited treatment space among the main throughput causes of long waits (Hoot & Aronsky, 2008).

Providers as a Second Constraint

Providers at [83] percent are close behind. If rooms were added without changing provider coverage, the constraint would shift to providers. Any fix should check both.

Checking the Numbers Against the Map

The process map found a median room wait of [58] minutes for evening patients. A queue that grows by about two patients an hour for two hours, with rooms turning over every three and a half hours, produces waits of that order. The model and the map agree.

Options

1. Keep [24] rooms staffed until 23:00 by moving one nurse's shift later.

2. Reduce time in room by moving patients awaiting results to a results-pending area, freeing rooms.

3. Add a fourth evening provider.

Option 1 addresses the room constraint directly; option 2 reduces the time term in Little's Law; option 3 addresses the next constraint.

Testing the Options in the Model

With [24] rooms: [21.7] ÷ [24] = about [90] percent, the same as the day. With time in room reduced to [3.0] hours: [6.2] × [3.0] = [18.6] rooms needed, within [20]. Combining both gives the most slack.

What Staff Will See if Rooms Stay Open

If [24] rooms stay staffed until 23:00, the charge nurse should see the waiting room empty by about 21:30 instead of midnight, and the triage line, which staff blame now, should look the same, because triage was never the constraint. Explaining this in advance will help staff judge the change by the right measure.

Limits

Little's Law assumes a stable system on average; evenings are not stable, so the model shows direction rather than exact waits. Time in room varies by patient type. Boarding of admitted patients would raise room time and is not modeled here.

Conclusion

The evening constraint is rooms, not triage. Demand for about [22] rooms meets [20] staffed, and the queue grows. Keeping rooms open later or shortening time in rooms are the two levers the model supports, and the root cause analysis should ask why the staffing template drops rooms at the peak.

References

Green, L. V., Soares, J., Giglio, J. F., & Green, R. A. (2006). Using queueing theory to increase the effectiveness of emergency department provider staffing. Academic Emergency Medicine, 13(1), 61-68. https://doi.org/10.1197/j.aem.2005.07.034

Hoot, N. R., & Aronsky, D. (2008). Systematic review of emergency department crowding: Causes, effects, and solutions. Annals of Emergency Medicine, 52(2), 126-136. https://doi.org/10.1016/j.annemergmed.2008.03.014

Little, J. D. C. (1961). A proof for the queuing formula: L = λW. Operations Research, 9(3), 383-387. https://doi.org/10.1287/opre.9.3.383