DDHA 8203 Week 3 Queue Analysis Example

Running at Seventy Percent and Still Lined Up at Nine: A Queue Analysis of One Imaging Registration Desk Through Arrivals, Service Time and Variation

Student Name

Walden University

DDHA 8203: Advanced Healthcare Operations Management

Instructor Name

Month Day, Year

Running at Seventy Percent and Still Lined Up at Nine: A Queue Analysis of One Imaging Registration Desk Through Arrivals, Service Time and Variation

The Queue and the Question

Patients arriving at the outpatient imaging department of a composite regional hospital I will call Tamarack Valley Medical Center pass through one registration desk with two positions before they are called for a scan. Across the working day the desk is not overloaded. Yet patients and technologists complain about the same thing: on most weekdays the line at registration reaches into the corridor between roughly 8:30 and 10:00, scans start late, and the delay carries into late morning. Managers have proposed a third registration position. This paper explains the line through three quantities, the arrival pattern, the service time and the variation in both, and argues that the line is produced less by the average load than by variation in arrivals and service during the morning peak.

The figures below come from the case data supplied for the assignment: arrival timestamps for 20 weekdays and registration times for a sample of 400 patients. Any number the case did not supply is labeled as an assumption in the sentence that uses it.

Arrivals by Hour

Over the 20 weekdays, an average of 175 patients registered between 7:00 and 17:00, or 17.5 per hour. That daily figure hides the shape that matters. Arrivals climb steeply after 8:00, stay high until 11:00, fall over the lunch hour and rise again after 13:00, as Table 1 shows.

The morning hours are not only busier on average; they are also less predictable. Scheduled patients are asked to arrive fifteen minutes early, and many do, while others arrive late from parking or from another appointment. The case data show that the number of arrivals in the 8:00 hour ranged from 17 to 31 across the 20 days.

Table 1

Mean Arrivals and Desk Utilization by Hour, 20 Weekdays

Capacity is two positions at a mean service time of 4.68 minutes, about 12.8 patients per position per hour. Utilization is mean arrivals divided by capacity.

Service Time by Case Type

Registration is not one task. For most patients it consists of confirming identity, the order and the insurance already on file, which took a median of 3 minutes in the sample. For 28% of patients, registration also required verifying insurance that was new, changed or missing a prior authorization number, and these cases took a median of 9 minutes. Treating the two as one average, 4.68 minutes, describes neither of them.

The verification cases matter more than their share suggests. A single twenty-minute verification occupies one of the two positions for the time it would take to register six routine patients, and while it lasts the desk has half its capacity.

Table 2

Registration Time by Case Type, Sample of 400 Patients

The mean for all patients is computed from the shares and medians in the first two rows; the case did not supply a mean for each type, so using medians there is an assumption.

Variation on Both Sides

If patients arrived evenly and every registration took exactly 4.68 minutes, a desk with capacity for 25.6 patients an hour could absorb 24 arrivals in the 8:00 hour with almost no line. The line forms because neither side is even. Kingman (1961) showed that the waiting time in a queue near full use grows in proportion to the variability of arrivals and service, not only with their averages, and that it rises very steeply as utilization approaches one. Green (2006), applying queueing models to health care settings, makes the practical version of the point: delays increase sharply well before a service is fully used, so a unit that looks only moderately busy on average can still produce long waits.

To show the scale of the effect at this desk, I used a standard approximation for a two-server queue. I assumed that arrivals within an hour are close to random, which is consistent with the spread in the 8:00 counts, and that service time variation is the spread between the two case types, which gives a coefficient of variation of roughly 0.6. Both are assumptions. Under them, the expected wait in line at 11:00, at 70% utilization, is about 3 minutes. At 9:00, at 90% utilization, it is about 14 minutes, and in the 8:00 hour it is longer still. The desk's load rises by about a third between those hours, while the wait rises roughly fivefold. That nonlinear relationship between utilization and waiting is the central finding of this analysis.

One Application of Little's Law

Little (1961) proved that, in a stable queue, the average count of items present equals how fast items arrive times how long each one stays. For the 9:00 hour, an arrival rate of 23 patients per hour and an expected wait of about 14 minutes imply that roughly five to six patients are standing in line on average. The case data include a lobby headcount taken at 9:30 on each of the 20 days, which averaged six. The two figures agree closely enough to suggest that the approximation describes this desk reasonably well. The agreement matters because it ties the model to something staff can count.

Why a Third Position Is the Wrong First Answer

A third registration position would lower utilization in the 8:00 hour from 94% to about 62% and would shorten the line. But it would add a full-time salary to cover roughly three busy hours, leaving the position idle for much of the afternoon. The analysis points instead to the source of the variation. Moving insurance checks for booked patients to the previous afternoon, by phone or through the patient portal, would remove most of the nine-minute cases from the morning desk. If even two-thirds of verification cases were completed in advance, which is an assumption to be tested, the mean service time would fall to about 3.6 minutes and the spread between case types would narrow. Both changes lower the wait, and the second lowers it more than the averages alone would suggest.

A smaller change addresses the arrival side. Staggering the early-arrival instruction, so that patients with 8:00 and 8:15 appointments are not asked to arrive at the same moment, would flatten the first peak without changing the schedule.

Limits of the Analysis

The approximation treats each hour as if it were in steady state, but a line that builds at 8:30 carries into the next hour, so the 9:00 wait partly reflects the 8:00 overload. The case supplies medians rather than means for each case type, and the coefficient of variation is estimated from two groups rather than measured from individual times. Finally, the analysis covers registration only. If scans themselves start late for other reasons, a shorter registration line will move some of the waiting to the scanner waiting area rather than remove it.

Conclusion

The registration desk at Tamarack Valley runs at about 70% utilization across the day and still forms a line every morning. Arrivals by hour, service time by case type and the variation in both explain why: in the peak hours the desk runs close to full use, and near full use waiting rises steeply with variability. The best first response is to reduce that variability, by moving insurance verification out of the morning and smoothing early arrivals, before paying for capacity the desk would use for three hours a day.

References

Green, L. (2006). Queueing analysis in healthcare. In R. W. Hall (Ed.), Patient flow: Reducing delay in healthcare delivery (pp. 281-307). Springer. https://doi.org/10.1007/978-0-387-33636-7_10

Kingman, J. F. C. (1961). The single server queue in heavy traffic. Mathematical Proceedings of the Cambridge Philosophical Society, 57(4), 902-904. https://doi.org/10.1017/S0305004100036094

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