People perform tasks at different rates. For example, surgeons differ in how quickly they perform the same procedure.^{1–3} The coefficient of variation among surgeons of mean case durations for the same procedure was 30%.^{1} ^{,} ^{4} Similarly, assembly line workers differ in how quickly they inspect radios.^{5} The coefficient of variation of mean radios inspected per hour was 25% (see Table 2 of Ref. ^{5} ). Therefore, it is not surprising that practitioners at the authors’ hospitals differ in times for preanesthesia evaluation (each hospital P < 10^{−12} ).^{6} ,^{a} The coefficients of variation of mean evaluation times among practitioners at the 2 hospitals were 22% and 25%. Whether these differences in mean evaluation times among practitioners are large enough to be operationally important and whether there is something to be done with the information is unknown.

Many preanesthesia clinics are constrained as to the number of patients that can be seen during a given workday because of space and/or labor limitations. Thus, it is possible that the efficiency of the clinic could be enhanced by better management of the queue of patients in the waiting room. Understanding operations of preanesthesia clinics is important in part based on the prevalence of these clinics “staffed” by the “department of anesthesiology.” This was the model at the institutions of most attendees (69%) at the 2005 American Society of Anesthesiologists’ Meeting.^{7} All 8 University hospitals in The Netherlands included at least 1 anesthesiologist and anesthesia resident.^{8}

Example #1: There are 2 certified registered nurse practitioners (“practitioners”) working at a preanesthesia clinic. The slower practitioner takes, on average, twice as long to evaluate a patient as the faster practitioner. While the faster practitioner is evaluating a patient, another patient arrives. There are no other patients in the queue. The slower practitioner is reviewing charts. Should the slower practitioner evaluate the new patient or continue reviewing charts? Whether the slower practitioner should evaluate the new (waiting) patient is unclear because the faster practitioner could finish evaluating her current patient and then see the new patient in less time than the slower practitioner would take to evaluate the new patient. Although the new patient would wait longer to be seen, the patient would leave the clinic sooner.

Medical directors of preanesthesia evaluation clinics need to decide how to handle the decision of who sees the next patient. The strategy would be different if the primary goal of the clinic was to maximize patient throughput, or if the primary goal was to keep the practitioners happy by balancing their patient workloads. Suppose the faster practitioner’s response to being asked to see the next patient, even though her colleague is available, may be to take a coffee break. Then perhaps the clinic should have a financial incentive program that would encourage the faster practitioner to see the next patient instead of doing chart reviews.

Example #2: The slower practitioner is reviewing charts. Another patient arrives for evaluation, leaving 2 patients waiting to be seen. Should the slower practitioner now start to evaluate the first of the 2 waiting patients? Should the threshold for the slower practitioner to start evaluating waiting patients instead be when 3 patients are waiting?

Example #3: Both practitioners evaluate patients and the queue is reduced to no patients. The faster practitioner finishes evaluating her current patient. The slower practitioner will likely be available in a few minutes. A new patient arrives. The patient’s electronic medical record medication list shows no (0) medications (i.e., likely brief evaluation time).^{6} Should the faster practitioner see this patient because the practitioner is available and the patient is waiting? Should the faster practitioner wait for the next patient because the currently waiting patient will likely have a brief evaluation and thus is a better pair with the slower practitioner?

We studied how to manage the queue of patients waiting in the clinic for evaluation to reduce the total time patients spend in the clinic each day.^{9–11} ,^{b} We performed our analysis by first reviewing operations research studies that have identified conditions for which such management of the queue can be beneficial. Then, we used data from a preanesthesia evaluation clinic to test whether those conditions are typical for preanesthesia evaluation.

METHODS
Background (Review) to Identify Hypotheses Tested
Analytical models that evaluate the formation and depletion of waiting have been studied in the queuing system literature.^{12} ,^{c} We first considered the practice in a preanesthesia evaluation clinic such as those of Examples #1 to #3: a queuing system with 2 heterogeneous practitioners.^{13} The numbers of patients arriving each hour follows a Poisson distribution (i.e., the times between arrivals of successive patients follow an exponential distribution).^{13} The service times of patients also follow exponential distributions, but with different mean evaluation times depending on the assigned practitioner.^{13} The value to be minimized is the total time in the clinic, defined as the sum of the time patients wait in the queue and the times spent undergoing evaluation.^{13} Then, the optimal policy is for the faster practitioner to be evaluating patients whenever a patient is waiting.^{13} The slower practitioner is assigned a patient only when the number of patients waiting for evaluation exceeds a threshold value.^{13} For example, suppose that the threshold was 1 patient. Then, if there was 1 patient in the queue and the faster practitioner was busy, the patient would wait until the faster practitioner is available or until a second patient joins the queue. If there were 2 or more patients in the queue, both practitioners would evaluate patients. Such a threshold policy is also optimal (with additional assumptions) when there are more than 2 practitioners^{14} ^{,} ^{15} or to minimize the percentage of patients whose total time in the clinic exceeds some undesirable duration.^{15}

Consider instead a clinic with 3 practitioners, instead of 2 as in the preceding examples. The mean interval between arrivals of patients is 15 minutes. Suppose that the second slowest practitioner takes a mean of 30 minutes to evaluate patients. Then, subject to the preceding assumptions, the specific threshold is 1 patient or more if the fastest practitioner evaluates patients in ≤ 1/3 the time of the second slowest practitioner.^{16}

Hypothesis #1A: Consider sets of 3 practitioners. The fastest practitioner will not be more than 3 times as fast as the second fastest practitioner (i.e., such differences in mean evaluation times are unrealistically large for people).

If Hypothesis #1A holds, then the threshold policy would not be superior to the simpler policy that (i) no patient waits when there is an available practitioner, and (ii) when multiple practitioners are available (e.g., reviewing future charts) and a patient is ready for evaluation, the fastest of the available practitioners sees the patient. The latter policy is referred to as “Fastest Practitioner First,” and is equivalent to setting the threshold equal to zero patients.

Consider a different clinic with 3 practitioners, and the same statistical assumptions.^{14} The second slowest practitioner takes a mean of 30 minutes to evaluate patients. The fastest practitioner’s mean evaluation time is <1/2 the mean evaluation time of the slowest practitioner. When the patients routinely have long mean waiting times (43 minutes), as soon as a practitioner finishes with one patient, there will be another patient waiting to be seen. Consequently, there will be a negligible difference in mean waiting between the threshold policy and Fastest Practitioner First.^{14} Furthermore, the mean total clinic time will be only 1.5% less (1.0 minute) with Fastest Practitioner First than random selection of the practitioner to evaluate the next patient when more than 1 practitioner is available.^{14}

Hypothesis #1B: Consider sets of 3 practitioners. The fastest practitioner will not be more than 2 times as fast as the slowest practitioner, based on mean evaluation times.

For example, suppose that the fastest practitioner’s mean evaluation time was 6.7% less than the mean evaluation time of the slowest practitioner. Then, the mean total clinic time would be only 0.1% less (0.1 minute) with the fastest of the available practitioners evaluating the patient as compared with random.^{14}

Hypothesis #1C: Consider sets of 3 practitioners. The fastest practitioner will be at least 1.072 times as fast as the slowest practitioner, based on mean evaluation times, where 1.072 = 1/(1 − 0.067).

If the Hypotheses #1A, #1B, and #1C hold, then differences in mean evaluation times among practitioners will not be sufficiently large to warrant use of the threshold policy or Fastest Practitioner First as compared with whatever policy happens to be used at a preanesthesia clinic.

Suppose that the fastest of the available practitioners were to see the next ready patient, whenever the fastest practitioner is available. The fastest practitioner would then spend more time working directly with patients than the slower practitioners (i.e., have a higher percentage utilization measured from patient data).^{17} As for the Examples #1 to #3 and the first paragraph of the Methods section, consider a single patient queue and 2 practitioners, but let the practitioners adjust their evaluation times.^{17} The practitioners are salaried. Continuing, suppose that each chooses her evaluation time to maximize her personal (and/or professional) satisfaction. Each increase in her evaluation time results in either no change or a decrease in her satisfaction. Each increase in her time evaluating charts (or taking a break) rather than seeing new patients results in an increase in her satisfaction. Finally, suppose that whenever both practitioners are available to evaluate patients, the next patient arriving is seen by the fastest practitioner. Then, each practitioner would progressively evaluate patients at a slower pace, until the queue begins to lengthen.^{17} The reason for this is that the longer a practitioner takes to see the current patient, the greater the chance that the practitioner will get to review charts rather than see a new patient. In contrast, random assignment would result in what is referred to as a unique symmetric Nash equilibrium.^{17} Without knowing whether it will be the fastest or the slower practitioner who will evaluate the next patient, neither increases his or her satisfaction by working slower.^{17} Thus, random assignment results not only in fairness (equality) of clinical work, but briefer mean evaluation times. In practice, such behavior would be observed by the slower practitioners having percentage utilizations no less than those of the faster practitioners:

Hypothesis #2: Slower practitioners will have percentage utilizations no less than those of the faster practitioners, based on patient evaluation times, consistent with random assignment and inconsistent with Fastest Practitioner First assignment.

Preanesthesia Evaluations Studied
The study was approved by the Jefferson Medical College IRB without requirement for written patient consent. The overall mean preanesthesia evaluation time was 26 minutes (n = 69,654), very close to the mean of 28 minutes from one of the authors’ institution more than a decade ago.^{6} ^{,} ^{9} Details of the data were published previously.^{6} The times studied were from when the practitioner opened each patient’s record until the practitioner saved the evaluation.^{6} Nursing assistants did not evaluate patients before their being seen by the practitioner, unlike in the clinic studied previously by Zonderland et al.^{11} The evaluations were performed January 2, 2006 through May 27, 2011 (n = 1378 days). Every evaluation was used starting from when data were collected, so there was no statistical power analysis performed a priori.^{6} Times for follow-up review of consultations, imaging studies, etc., were not included, because these tasks are performed principally by the anesthesiologists assigned daily to the clinic.

Hypotheses #1A to C were tested by considering all combinations of days and triplets of practitioners (Table 1 ). All combinations were created using For-Next loops by using Excel Visual Basic for Applications (Microsoft, Redmond, WA). If a practitioner had fewer than 7 evaluations for the day, the next combination was considered. For each combination of practitioner and day with at least 7 evaluations, the mean of the evaluation times was calculated. For each day, the ratio of the briefest to second briefest of the means was calculated. With 1 ratio for each day, the median of the ratios was calculated, as was the conservative Clopper-Pearson confidence interval for the median. The analysis was repeated using the ratio of the briefest to the longest of the means. The analysis was repeated with at least 9 evaluations as a sensitivity analysis. The analysis was repeated with 7 evaluations and pairwise combinations of practitioners.^{d}

Table 1: Tests of Hypotheses #1A, #1B, and #1C, Ratios of Mean Evaluation Times

Hypothesis #2 was tested using Kendall τ_{b} correlation between each practitioner’s mean evaluation time and the practitioner’s percentage utilization evaluating patients. The sample size was n = 24 practitioners, this being the number of practitioners with at least 100 evaluations. A 1-sided P value was used to match the hypothesis. The percentage utilization was measured as the ratio of total time spent evaluating divided by the sum among workdays of the total time present for evaluations, the latter being measured using the time of start of the first evaluation to the time of end of the last evaluation.

RESULTS
Multiple practitioners evaluated patients on the same day. The fastest practitioner was typically 1.23 times faster than the second fastest practitioner and 1.61 times faster than the slowest of 3 practitioners (Table 1 ). These findings are consistent with Hypotheses #1A to C.

Anesthesia practitioners with longer mean evaluation times^{e} had larger percentage utilizations of working time (Kendall τ_{b} = 0.56, P = 0.0001, 95% lower confidence limit 0.39). This finding is inconsistent with preferential assignment of patients to the fastest practitioner(s) available (i.e., supports Hypothesis #2).

DISCUSSION
As summarized in the Introductory and Background sections, when a patient arrives for preanesthesia evaluation and more than 1 practitioner is not currently evaluating a patient, it is not obvious that patients waiting should be assigned to the available practitioners at random. Nor is it obvious that the practitioner who has been idle (from evaluating patients) the longest should evaluate the patient. If practitioners were machines, varying markedly in processing speed, then it would be the fastest practitioner who should evaluate the new patient. There would be use of so-called state-dependent assignment policy. However, practitioners are not like computers with markedly different “clock rates.” Table 1 shows that differences in speed are too small to warrant consideration when choosing who evaluates the next patient. Clinics aiming to reduce mean patient waiting should focus on reducing the overall mean evaluation time (e.g., by chart review ahead).^{9}

From the sensitivity analyses in the columns and legends of Table 1 , and the common coefficients of variations of mean service times among people of different professions (Introductory paragraph), our results are very unlikely to depend on the specific types of practitioners studied. Our results can be generalized to any preanesthesia evaluation clinic with multiple practitioners and a single queue of patients. Such facilities are the ones for which simulation can help make decisions.^{9–11}

The principal limitation of our results is that they will not apply to some small clinics. For example, somewhere there is an anesthesia evaluation clinic with 2 anesthesiologists daily, one of whom tends to acquire and record hardly any information (e.g., medications and airway) and the other who effectively repeats the entire medical history, often including a detailed social history (see footnote e in Results). Our results do not apply to this hypothetical clinic, not because the clinic has anesthesiologists, but because the hypothetical scenario has only 2 practitioners and one being so much slower than the other is uncommon.

In conclusion, practitioners’ speeds in evaluating patients do not differ enough to warrant use of information systems to choose who evaluates the next patient (i.e., state-dependent assignment policy). Clinics aiming to reduce mean patient waiting (e.g., to increase patient satisfaction)^{21} should focus on reducing the overall mean evaluation time (e.g., by chart review ahead), appropriately scheduling patients, and having the right numbers of nursing assistants and practitioners.^{9–11}

RECUSE NOTE
Dr. Franklin Dexter is the Statistical Editor and Section Editor for Economics, Education, and Policy for the Journal. This manuscript was handled by Dr. Steven L. Shafer, Editor-in-Chief, and Dr. Dexter was not involved in any way with the editorial process or decision.

DISCLOSURES
Name : Franklin Dexter, MD, PhD.

Contribution : This author helped design the study, conduct the study, analyze the data, and write the manuscript.

Attestation : Franklin Dexter has approved the final manuscript.

Name : Hyun-Soo Ahn, PhD.

Contribution : This author helped design the study, analyze the data, and write the manuscript.

Attestation : Hyun-Soo Ahn approved the final manuscript.

Name : Richard H. Epstein, MD, CPHIMS.

Contribution : This author helped design the study, conduct the study, analyze the data, and write the manuscript.This author is also the archival author.

Attestation : Richard Epstein approved the final manuscript.

a Data used are those described below. At the primary hospital analyzed there were 69,654 evaluations performed by 30 providers. At the secondary hospital with data in Table 1 , there were 1735 evaluations performed by 35 providers. Analysis of variance was used to compare the logarithms of evaluation times.^{6} At the primary hospital, 28 of the 30 providers performed at least 25 evaluations, the minimum being 28. The coefficient of variation of the means was 25%. At the secondary hospital, 25 of the 35 providers performed at least 25 evaluations. The coefficient of variation was 22%. Cited Here

b Previous operations research studies of preanesthesia clinics are different, having evaluated patient scheduling and the choice of the numbers of nursing assistants and providers present.^{9–11} For example, Zonderland et al.^{11} simulated and then implemented the rescheduling of appointments to achieve a more homogeneous patient arrival pattern throughout the day. Cited Here

c Research in queuing systems models the system as stochastic processes and study the key performance indexes such as the mean waiting time and the mean number of patients in the system. For classical work and extension, readers can refer to Chen and Yao (2001).^{12} Cited Here

d Combinations of 3 providers (and 2 in sensitivity analysis) were used instead of 4 providers for 3 reasons. One was practical, that being the number of combinations in Table 1 of Ref. 14 was for 3 providers. Second, the examples in the Introduction were for 2 providers. Third, in our experience, when a clinic has 2 providers seeing patients, routinely both providers are idle (i.e., the examples in the Introduction are typical). However, when there are 4 providers seeing patients, hardly ever are they all idle. Our experience is a logical consequence of queues. Consider the observed^{6} mean evaluation time of 26 minutes and a low (1.50) ratio for medical clinics of mean patient waiting time to evaluation time.^{6} Follow the assumptions above.^{13–16} With 2, 3, and 4 providers, all would be idle for 12.7%, 4.3%, and 1.5% of time, respectively. Cited Here

e The number of different medications is a better predictor of mean evaluation time than other variables including age, weight, ASA physical status, and physiological complexity of surgery^{18–20} (all P < 10^{−5} ).^{6} However, there was no relationship between mean evaluation time of each provider and mean numbers of medications used by patients (i.e., no heterogeneity in patient conditions among providers)^{6} (Kendall τ_{b} = 0.16, P = 0.28). In contrast, mean numbers of characters including spaces in evaluations were predictive (Kendall τ_{b} = 0.40, P = 0.007). Thus, if an organization wants to reduce evaluation times of slower providers, consider first reviewing lengths of notes and whether the information in the notes is used. These findings are preliminary and highly limited because there are limited numbers of providers (i.e., 95% confidence intervals for τ_{b} are very wide, −0.15 to 0.48 and 0.09 to 0.70, respectively). Cited Here

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