urban and rural clinics.

Finally, Column 5 includes additional covariates at the provider level, showing that doctors and other para-professionals have slightly higher caseloads compared to nurses. Because we assume that caseloads are evenly distributed within facilities, this implies that doctors are, on average, located in busier facilities. A final result is that at the provider level, we could not explain more than 25.6% of the variation in caseloads, of which only 5.6% is explained by facility location and provider characteristics (the remaining 20% is explained by country). This results from the fact that facilities with any given combination of characteristics and staffing had a very wide range of per-provider caseloads in each country.

A first implication of this wide unexplained variation in caseloads is that the experience of the average patient differs from that of the average provider. To see this in our data, we weighted each provider by the number of patients they saw, and we report the “crowd” faced by the median patient in **Table 1** , Column 6. The median patient was seen by a provider who sees 25.7 patents each day, with country-level medians ranging from 6.6 patients a day in Nigeria to 64.6 a day in Malawi.

Using queuing theory simulations, we find that substantial queues can develop at these crowd sizes if patients arrivals and consultation lengths vary randomly (as opposed to uniformly). **Figure 2** demonstrates the method, showing demonstrative queuing and service patterns over a single simulated working day for a single provider facing various average caseloads, using randomly distributed patient arrivals and departures and evaluating the queue flow minute-by-minute over an assumed six hour outpatient shift (see the **Supplemental Appendix** for details). In these examples, the provider works 2.3 hours out of the 6-hour day when there are an average of 15 patients, and patients wait under a minute each; there is nearly never a queue. But when the provider faces 40 patients, they are serving patients the entire day, and patient wait times and queue lengths increase dramatically.

**Figure 3** generalizes this approach to show the wide variation in utilization and queue lengths that can occur given any level of “average” utilization. With 20 or fewer patients per day, providers are frequently underutilized over the course of a six-hour workday. With 30 or more patients per day, average wait times become long on average, and excess capacity quickly declines. At 40 patients per day, provider working time ranged from 60% to 100%, and patient wait times ranged from five minutes to over one hour. Any “clumping” of arrival times onto specific days of the week or times of the day substantially lengthened the queues, driving wait times to “infinity” in some cases. This means that not all patients who showed up would be seen by closing time, since a facility working at or above capacity would have a constantly-growing queue. Thus, the caseloads we estimated in many countries would be consistent with long waiting times for the typical patient under conditions of even moderate clumping.

Under these conditions of variation, caseloads were observed to be substantial at the busiest facilities in each country, but only the very busiest providers appeared to have caseloads that would commit them to consistently serving patients throughout a full workday. Combining these observations and simulation results with the facility survey

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