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Statistical Analysis by Monte-Carlo Simulation of the Impact of Administrative and Medical Delays in Discharge from the Postanesthesia Care Unit on Total Patient Care Hours

Dexter, Franklin MD, PhD*,; Penning, Donald H. MS, MD†,; Traub, Rodney D. PhD

doi: 10.1097/00000539-200105000-00026
Case Report: Case Report

*Division of Management Consulting, Department of Anesthesia, University of Iowa, Iowa City, Iowa, †Sunnybrook and Women’s Health Sciences Centre and the Department of Anesthesiology, University of Toronto, Toronto, Ontario, Canada, and ‡College of Business Administration, North Dakota State University, Fargo, North Dakota

This project was funded by Sunnybrook and Women’s Health Sciences Centre.

January 23, 2001.

Address correspondence to Franklin Dexter, Division of Management Consulting, Department of Anesthesia, University of Iowa, Iowa City, IA 52242. Address e-mail to

IMPLICATIONS: We review in this case study how we used computer simulation to estimate for a postanesthesia care unit (PACU) the impact on total length of stay of administrative delays from hospital wards not being available to receive patients in a timely manner from the PACU. The methodology can be applied to other delays in discharge, be they administrative or medical.

Nursing salaries account for the majority of postanesthesia care unit (PACU) costs (1). PACU management efforts often focus on decreasing PACU length of stay (LOS) to decrease nursing costs. Discharges from a PACU can be delayed for many reasons, including administrative delays such as a lack of open hospital beds. If these delays could be reduced or eliminated, PACU staffing costs may be less.

To predict the financial impact of eliminating delays in PACU discharge, a manager could add together the delays for many patients and divide by the total time required to care for them. However, generally this would be inaccurate, because it would not be known precisely when each delay began. For example, although it may be clear that a patient had a prolonged PACU LOS because of vomiting, typically there is no one precise discharge time if the patient had not been vomiting. The same dilemma arises for administrative delays. If, as soon as a patient arrives from the operating room (OR), the PACU nurses know that a hospital room will not be available for several hours, then that knowledge will often affect the patients’ care (e.g., when central venous and intraarterial catheters are removed). Thus, knowing that a patient is likely to have an administrative delay can affect when the patient is ready for discharge from the PACU, making efforts to precisely time the start of the delay invalid.

Although estimates of the durations of delays can be inaccurate, it is usually possible to determine whether a patient’s discharge from the PACU has been delayed. Thus, to predict the impact of delays on total LOS among all patients, the LOS of patients who had and did not have delays can be compared. Such analyses have been performed in research studies quantifying the effects of nausea and vomiting (1,2), pain (2), and all adverse medical events (2) on total PACU LOS among all patients. The same approach can be applied to total PACU nursing workload (3). These studies measured the percentage decreases in total PACU LOS that could be achieved by eliminating these causes of delayed PACU exit.

Although this methodology has been used for research purposes, it is important for routine PACU management. Hopefully, before a manager focuses administrative efforts on eliminating a cause of delays in PACU discharge, the manager would evaluate whether total PACU LOS would be reduced by a financially or medically important amount if the delays were eliminated successfully. In this economic case study, we show how it is possible to ask ahead of time: “If patients never again had to wait for admission to a ward from the PACU, what effect would this have on total PACU LOS?”

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Case Study—Impact of an Administrative Delay on Discharge

In this case study, we review the use of computer simulation to estimate the impact of an administrative delay on total PACU LOS. In particular, the administrative delay of concern was the delay resulting when hospital wards were not available to receive patients in a timely manner from the PACU.

The analysis was performed using the times of admission and discharge of all patients cared for in the PACU between April 1, 1999 and March 30, 2000. The other datum used for each patient was whether his or her discharge from the PACU was delayed because his or her hospital ward was not ready to receive patients.

The phase I PACU under consideration was in a tertiary, academic surgical suite with a mixture of outpatient, inpatient, and same day admit cases. Admission times, discharge times, and patient delays and the cause of such delays were recorded routinely at the PACU. Of the 14 surgical departments working in the surgical suite, the 4 with a more than average rate of administrative delays accounted for 37% of the surgical cases and 53% of the delays.

We performed the “what if” analysis using a spreadsheet program (Excel 2000; Microsoft, Redmond, WA), its built in macro language (Excel 2000 Visual Basic, Microsoft), and an add-in package for fitting probability distributions to data (BestFit 4.0; Palisade Corporation, Newfield, NY). We performed the analysis in the following sequence (4).

First, the probability distribution for the chance that a patient will be delayed was identified. At the PACU studied, 8.4% or 635 of the 7,578 consecutive patients had a delay in discharge due to the hospital wards being unable to accept the patients. Whether or not each patient was delayed was therefore a Bernoulli random variable with an 8.4% probability that each patient’s discharge would be delayed. Generating such a Bernoulli random variable is analogous to flipping a coin that is biased so that it lands “heads” 8.4% of the time (patient is delayed) and “tails” 91.6% of the time (patient is not delayed).

Second, PACU length of stays were fitted to three- parameter log normal distributions (1). The statistical distribution has three parameters: a distribution shift, a mean, and a standard deviation (5). Three-parameter log normal distributions were established separately for those patients who were not delayed and those who were delayed (Fig. 1). For patients who were not delayed, after shifting the LOS distributions to the right by 0.04 hours the natural logarithms had mean ± sd of 0.29 ± 0.65 (n = 6,943). Among delayed patients, after a shift of 0.33 hours, the mean ± sd were 0.82 ± 0.58 (n = 635).

Figure 1

Figure 1

Third, two simulated LOS were generated. One PACU LOS was generated from the log normal distribution with parameters estimated from patients who did not incur a delay. The second PACU LOS was generated from the log normal distribution with parameters estimated from patients who incurred a delay. If the LOS generated using parameters from patients without a delay was not shorter than the LOS generated using parameters from patients with a delay, then additional simulated LOS were generated from the log normal distribution with parameters appropriate for patients who incurred a delay. We stopped generating such additional LOS when one was longer than the simulated LOS for a patient without a delay. The difference between the final two LOS represents the additional time that a patient would spend in the PACU if the patient were to incur a delay.

Fourth, we generated a patient arrival, and used the two LOS from the third step. A Bernoulli random number was generated indicating whether the patient was delayed. If delayed, the potential percentage decrease in total PACU time from the elimination of delays equaled 100 multiplied by the difference between the delayed LOS and the nondelayed LOS, divided by the delayed LOS. If there was no delay, the potential percentage decrease for that patient equaled 0%.

Fifth, the third and fourth steps were repeated n−1 times, where n refers to the number of simulated patient arrivals. The mean of the n percentages was calculated, and taken as the result.

The number of simulations, n, to be performed was chosen based on the desired accuracy of the mean of the n percentages, the result. The mathematics of choosing n is the routine method of finding a confidence interval for the mean of one group from a clinical trial. If the mean and standard deviation of the percentage changes in total time are represented by and s, respectively, then the confidence interval for the true percentage difference in time MATH. For management decision making, a result will usually be sufficiently accurate if there is a 95% certainty that the mean percentage change in total time () is within 5% of the true change in time (μ). For large values of n, the Z value from tables of normal probability distributions for a two-sided 95% confidence interval is 1.96.

We ran n = 100,000 simulations. Execution time was 3 s on a 650 MHz Pentium III microprocessor with 256 MB RAM running Windows 2000. Our result was a mean of 4.54% and standard deviation of 16.5%. This implies that if administrative delays from hospital wards not being available to receive patients in a timely manner were eliminated, total PACU LOS would be 4.54% less.

We chose the value of n for the width of the confidence interval to be <0.2%, where 0.2% is 5% of 4.54%. With the n of 100,000, the width of the 95% confidence interval was 0.2%, whereMATH

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We performed 100,000 simulations. This is a typical “n” for such analyses (4). The fact that 100,000 patients would have been needed shows one of the two reasons why addressing management “what if” questions is typically done by simulation, not experimentation. The other reason is that the goal of the analysis is to predict what will happen before the manager expends the political capital within the organization to affect change. As this case study shows, if the manager had tried to change hospital admission practices, and had succeeded, the manager would still have had little direct financial benefit on the end-point of interest: PACU staffing. We included the execution time of 3 s to point out that performing 100,000 computer simulations for this problem is practical; some simulations for other fields take weeks of computer time.

The reason for focusing on the percentage change in total PACU patient care hours is because the peak number of patients is, approximately, proportional to the overall (total) LOS (1). The decrease in the number of nurses needed to care for the patients is, approximately, proportional to the percentage decrease in the total PACU patient care hours. The exact relationship between changing LOS and nursing is likely to be complicated, and to depend on lunch shifts, sick leave, nurses being used in other hospital units, etc. The achievable percentage decrease in PACU staffing will typically be less than the percentage decrease in total PACU patient care hours. However, focusing on percentage changes in total patient care hours is useful for research purposes (1–3). It is the experience of the authors, from consulting work, that it is adequate for assisting managers in deciding whether to focus efforts on eliminating the delays. For the PACU described in the case study, even if all administrative delays were eliminated, total PACU time would have been decreased by <5%. It is very unlikely that doing so could result in decreased PACU staffing costs. Management efforts were therefore focused elsewhere.

This relationship between total PACU patient care hours and the peak number of patients applies provided the delays occur throughout the day. In the first and second steps, we estimated parameters for the probability distributions using data from throughout the day. If the delays of interest to the PACU manager were to occur at only some times of the day, then performing one analysis using data from the period of time with the delay and a second using data from times of the day without the delay may be worthwhile. Before performing such an analysis, however, the PACU manager may benefit from evaluating whether such results could be combined with his or her quantitative staffing plans (6,7) to achieve reduced staffing costs. The latter can be quite challenging statistically, in part because the differences throughout the day in PACU staffing requirements may depend less on variation in the incidence or severity of the delays and more on differences in admission rates from the OR (1).

The statistical analysis described in this article is predicated on the assumption that the care of one patient does not affect the care of another. This assumption is not reasonable in other settings, such as the presurgical holding area. There, a sick patient being prepared for urgent surgery may require the attention of most of the staff, resulting in delays for the other patients. In the PACU, the LOS of one patient is generally not affected by the LOS of another patient. Adjusting PACU nurses’ staffing on an hour-to-hour basis (e.g., for breaks) based on the patients’ acuity may improve patient outcome. However, such adjustments in staffing are unlikely to have important effects on how long patients remain in the PACU past the time when they can be discharged. Furthermore, because staffing decisions with full-time nurses are made months ahead of time, there will generally be no or little benefit financially in such hour-to-hour adjustment in staffing.

Because one patient’s LOS will generally not affect another’s, delays experienced by each group of patients can be analyzed independently. If, for example, one surgical group has a disproportionate percentage of its patients with administrative delays, and if eliminating delays in PACU discharge is important financially, then focusing attention on that surgical group’s patients may be worthwhile (10).

PACU LOS are approximately log normally distributed (1,2). However, this will not always be the case. Application of administrative rules (9) such as discharge criteria with a minimum number of minutes that patients must remain in the PACU can affect this value (e.g., many patients may be discharged at precisely 1 hour after admission). Then, an empirical probability distribution or truncated probability distribution may need to be used (4,9). This means that the actual measured values from patients are sampled with interpolation to generate future values.

The methodology that we used to calculate log normal distributions applies when there are a relatively large number (> 20) data for the undesired delays (10). However, we have not found this to be a problem in practice. The problem of delayed discharge must be sufficiently common at a PACU for the analysis to be worthwhile.

In summary, a PACU manager can focus attention on uncommon but prolonged delays or common but brief delays. Neither may be important financially. In this paper, we have reviewed how computer simulation can be used to estimate the impact of delays in PACU discharge on total PACU hours. Before making organizational changes, managers can estimate whether total PACU LOS could be reduced by a financially important amount if the delays were successfully eliminated.

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    © 2001 International Anesthesia Research Society