When queueing theory breaks down

In an appointment-based healthcare system congestion can easily be caused by patients arriving late. There is also the universally accepted idea that people don’t like waiting. With these factors in mind, we came up with a simple sounding idea. If patients have to sit in a waiting room to see a service provider because the system has fallen behind schedule, and they have an appointment to follow up on that visit, they will be more likely to arrive later for the next appointment. If I show up on time and have to wait 20 minutes, I might learn something and show up 10 minutes later the next time, and so on. There are some settings where this is particularly relevant. For example, in a Radiation Oncology clinic that we studied patients are required to arrive for daily treatments for up to 30 days in a row. If a patient has to wait day after day, there is a good chance that they will eventually compensate by coming in late. At least this was our expectation.

Surprise, surprise, surprise!!!

However, when we looked at arrival time data in this and other settings we made a surprising discovery. Waiting on one visit was not correlated with a later arrival time on the next visit. Even when we looked at the pattern across 2, 3, or 4 visits the expected behavior was not in evidence. This was baffling. It seemed to suggest that making patients wait for service had no longer term impact on the arrival process. This is a big deal for a variety for reasons. One of which is that queueing models of service systems assume that arrival processes and service processes are stationary. This means that once we estimate the average times between arrivals and the average service times we have a complete picture of the system at hand. Our initial observations seemed to be completely consistent with this idea. On the other hand this makes no sense. If I consistently have to wait to see you in a clinic, surely I will adjust my behavior based on this information.

Thinking that there was something wrong with the way we were looking at the data, we eventually decided to look at the cycle times that the patients experienced. In this context, cycle time means the span between arrival to and exit from a clinic. If we keep track of this we can ask a more subtle question. We can ask whether there was a relationship between a clinic falling behind schedule (which causes patients to wait) and Cycle times for the patients involved. If the system speeds up for patients that are forced to wait, this may compensate for the extra waiting time and mitigate the damage done to the reputation of the providers and the system.

We gathered data on patient arrival times, appointment times, whether the patient visit started on time, and how long the patient spent in the system. This lead to a surprising discovery. When the clinic fell behind schedule, the service providers compensated by going faster. In hindsight this was common sense. If I am behind schedule, I work harder to catch up. However, this phenomenon was not well documented in outpatient clinics. In addition, all of the prior work that used simulation models or queueing models to look at clinic operations assumed that the distributions of activity times were independent of clinic status. If service providers actually change their behavior based on system status, that would suggest that a lot of earlier research on the behavior of such systems is simply wrong. Ignoring the strategic behavior of agents in a system leads to erroneous conclusions.

In our work presented in Williams, et.al. 2014, and Williams, et.al. 2016 we document a number of key findings. When an Attending physician in a small private practice with low patient volumes falls behind she can go faster to catch up, but this has its limits. Observed physicians cut about 6 minutes from face time in these instances. In other clinics with higher volumes and a shared Resident, the Cycle time could be cut by up to 12 minutes. This was done by using the Resident strategically and eliminating the Resident from some visits. We note that the Attendings rarely if ever report doing this. It is simply an instinctive response to a time crunch and will not be noted in the records or information systems for the clinic. In extreme cases with larger, high volume clinics, the reduction could be even greater by eliminating the Resident entirely. We note that we found this behavior in multiple settings, including both private practice and academic medical centers (AMCs), involving different specialties, different doctors, and different geographical locations. This doesn’t prove that the behavior is universal, but it does imply that this was not an isolated event or an anomaly.

Take Aways

The first lesson is that service providers adjust their behavior based on system status. Another important lesson is that the educational mission of the AMC provides options that are absent from other settings (such as omitting the resident from the process flow) and this has to be accounted for in process measurement and design. However, the most significant finding may be that healthcare research that does not involve extensive on-site observations has to be taken with a rather large “grain of salt.” Nuances of human behavior like we saw here dramatically alter system performance and are quickly forgotten about by the agents involved. This flies in the face of roughly a hundred years of efforts to apply queueing models to healthcare settings based on the assumption that activity times come from a stationary distribution. (This means that the shape of the activity time distribution does not change over a clinic session.) Since this basic assumption is not correct, standard results and cookie-cutter solutions are likely to produce unexpected results.

One additional lesson should be listed here as well. Many experts who work to improve these systems rely on analysis of datasets drawn from existing systems without spending time in the clinic under consideration. This naturally leads to acceptance of “standard” assumptions about how they work. If one really wants to improve these systems, direct observation is a necessary step because it is often the only way to figure out what assumptions hold and which do not.

As always, our efforts are documented in a number of published works that are available on our Documents page. Full discussions are available in out text available for purchase at Amazon.com.

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