*e.g.*, OR time) as well as the morale of staff. To confidently yet efficiently solve such a problem, one must be able to predict how much capacity to add to the congested unit in the workflow to reduce or eliminate delays. Alternatively, one must be able to predict the impact of increased OR volume in the face of constrained PACU capacity.

*i.e.*, the number of beds to have in the PACU) rather than ongoing staffing decisions. In particular, we sought to understand the following issues: (1) whether proposed modest increases of PACU capacity would have a meaningful impact on the waiting time for PACU beds at our hospital; (2) how long-term changes in the overall case volume might affect waiting for PACU beds and perioperative congestion in our hospital and other similar medical centers. In the longer term, we hope to use this tool in the workflow planning of the new OR building that is currently under construction.

^{1}Currently, the application areas include telecommunications, wireless communication, manufacturing, computer systems, networks, and a wide range of service areas such as call (contact) centers.

^{2}In a compact but insightful way, the theory captures fundamental trade-offs among the limited resources, implied waiting times, and inherent randomness of congested systems. However, effective application of queueing theory tools to capacity management in the healthcare industry has thus far been relatively limited. Relevant references in this area include work on inpatient bed occupancy,

^{3}access times for outpatient departments,

^{4}and access to critical care beds.

^{5,6}

^{7}Our results are mostly applicable in scenarios where case scheduling is, by design or by accident, mostly random, leading to a steady release of work into the PACU. In this paper, we will test whether this is the case in our hospital, but we were sensitive to the fact that our model may not be applicable in contexts with highly structured case scheduling. To get a rough sense of how commonly random (as opposed to well-structured) scheduling policies are encountered (and by extension, how broadly our model is applicable) we conducted an informal survey of scheduling policies in medium and large OR suites.

#### Materials and Methods

##### Discussion of Queueing Models

*a priori*unpredictable. Each of the jobs arriving to the system needs to be served by one of multiple servers with identical performance. However, the time it takes to serve a given job is, in general, also random and follows its own probability distribution. This creates another source of unpredictability in the system. Jobs are served according to some queueing discipline, in our case on a first-in-first-out basis. Upon arrival, a job may have to wait in the queue until there is an idle server, and all the jobs that arrived before that job are already served or in service. The arrival and service times are unpredictable; therefore, it is never

*a priori*certain exactly how long a given job will have to wait until it will be served. Many queueing models have been developed to characterize the unpredictable behavior of the system based on certain performance measures (

*e.g.*, the average waiting time of a job).

*i.e.*, patients) arrive to the servers and need to be served (

*i.e.*, recover from anesthesia). For each patient (job), the service time is simply the time they are in the PACU system. However, unlike most queuing models, the jobs in our model do not wait in the queue until they can be served. In particular, patients start their recovery from anesthesia after emergence, regardless of whether there is an available bed in the PACU. To capture this fact, we adapt the standard model and assume that there are infinitely many servers, out of which some are “real” servers (

*i.e.*, actual beds in the PACU) and some are “dummy” servers (

*i.e.*, waitlist slots for patients recovering in the ORs). This represents a nontrivial adaptation of standard queuing models because it accounts for a reality in medical care; the service times of jobs start upon their arrival into the queue and not upon the time they are delivered to the server (as would traditionally be the case in a classic queueing model). This is a fundamental difference between the jobs in our study (patients recovering from anesthesia) and the jobs in a traditional queueing system; therefore, we believe that the adaptation is novel and not a trivial departure from established modeling techniques. In the model we assume that patients arrive to the PACU with exponentially distributed interarrival times (between consecutive patients), but we impose no specific assumptions about the distribution of time in recovery and on the waitlist, and we allow maximum modeling flexibility of this aspect. The assumption of exponential interarrival times is justified based on known theory. We then want to analyze the fraction of time in which the dummy servers are used, which is exactly the time in which the PACU is completely full and patients have to recover in the ORs. The results from the theoretical model are then compared to and verified with actual data from our OR statistics database.

*M*arkovian (exponentially distributed interarrival times), G because the “processing” time (the time a patient spends in the PACU and on the waitlist) is assumed to have a

*g*eneral distribution (that is, we make no specific assumptions about this distribution), and ∞ because we place no limit on the number of patients in the joint PACU+waitlist system. Because there is no limit on the number of patients, our model actually does not have a queue

*per se*; all patients in the system are assumed to be in a state of processing (

*i.e.*, time spent on the waitlist “counts” towards total time required to recover from anesthesia). This reflects the fact that patients start the recovery phase immediately after surgery, regardless of whether there is an available bed in the PACU. The way in which we distinguish patients in waitlists

*versus*those in PACU is a novel aspect of our modeling, and it will be detailed below.

*e.g.*, PACU admissions per hour in our setting as λ), and the (arithmetic) mean length of stay in the joint PACU+waitlist system as 1/μ. It is a well-known result from queueing theory

^{8}that, under these assumptions, the steady-state distribution of the number of patients in the system follows a Poisson distribution, and the probability of having

*i*patients in the system can be expressed as follows:

*N*patients (

*i.e.*,

*N*beds), then the number of patients

*i*

_{PACU}in the PACU is:

*i*

_{Waitlist}is

*N*or fewer patients in the system, then all of them are in the PACU, but patients in excess of

*N*will be on the waitlist.

Equation 1 Image Tools |
Equation 3 Image Tools |
Equation (Uncited) Image Tools |

*i*

_{Waitlist}as follows:

##### Statistical Tests of Appropriateness of Underlying Model Assumptions

^{2}statistics to test the hypothesis that interarrival times of patients to the PACU follow an exponential distribution and whether, more generally, the queueing model gives good predictions about the system performance.

##### Simulation Experiments to Test Model Assumptions

^{9}as well as data from our own hospital. For the simulation, we sampled the distribution of actual procedure times. This distribution of procedure times can be described by their geometric mean (the antilog of the mean of the log-transformed data) and a measure of dispersion from the mean, the SD, here given as the antilogs of (mean of the log-transformed data minus SD of the log-transformed data) and (mean of the log-transformed data PLUS SD of the log-transformed data). Using this convention, the mean procedure time was 1.95 h (SD 0.97 h, 3.94 h), as expected in a right-skewed distribution. In the simulation, additional cases were added to each OR until the last case had a scheduled finish time later than 2:30 pm, at which point no more cases were added. Cases were added without consideration of their duration; in effect, random scheduling was simulated. The simulated OR days began at 8:00 am, and 30 min of scheduled turnover time (our hospital's average target) was added between each case. However, the actual time between surgeries using the convention above was described by a lognormal distribution with mean of 1.25 h (SD 0.68 h, 2.29 h). This longer and more variable time between cases corresponds to the actual time between cases measured at our hospital, and it is driven both by turnover time and longer scheduled and unscheduled gaps between cases. The simulation was repeated for 5,10, …, 50 ORs, although for some questions the number of ORs did not matter. For each number of ORs, the simulation was run for 1,000 days. The output of the simulation was the interarrival time between completed cases, and this was compared to the exponential distribution. We also used the simulations to generate the number of completed operations (

*i.e.*, PACU arrivals) per hour, again varying the number of hypothetical ORs feeding the PACU.

*i.e.*, PACU arrivals) for each half-hour of the day using the actual OR time data from each of the different suites.

##### Survey of Scheduling Policies at Other Hospitals

*i.e.*, a person in the anesthesia department with personal, direct knowledge of how the daily OR schedule was constructed. We then asked how many beds were in the unit considered to be the main PACU and how many ORs were in the largest contiguous group of ORs feeding that PACU. Next, we asked “Who decides how the cases on the OR schedule in the largest block of ORs feeding the main PACU are sequenced?” We then asked all respondents a follow-up question: “Are any strategies such as shortest-cases-first or ambulatory-cases-first pursued at your institution?” Finally, we asked directly: “Does your institution pursue any strategies to shape the flow of work entering the PACU such that it would be different from the patient stream occurring from surgeons simply adding cases to the schedule until their block was filled?” To analyze the results, we condensed the results of “Are any strategies such as shortest-cases-first or ambulatory-cases-first pursued at your institution?” and “Does your institution pursue any strategies to shape the flow of work entering the PACU such that it would be different from the patient stream occurring from surgeons simply adding cases to the schedule until their block was filled?” into a yes/no categorical response for whether case-sequencing was applied.

^{8}That is, one can use the average time 1/μ regardless of what the actual distribution is. In our case, we calculated the average processing time by taking the difference between the time a PACU bed was requested and the time of PACU departure. Sometimes patients in the PACU have to wait for beds to become available elsewhere in the hospital. This “waiting to leave PACU” congestion effect was not dynamically modeled; however, the average time spent waiting for floor beds was included in the processing time 1/μ. Even though equation (1) does not rely on any distributional assumptions of processing time, it does assume that the average time is the same throughout the day. However, our data suggested that the average time in PACU tended to increase somewhat over the day, primarily because departing patients more often had to wait for floor beds later in the day. We will return to this factor when we compare the predicted and actual patient distributions in the Results section.

*N*. The physical capacity of the PACU at our hospital is 28 beds. However, even when the system is busy it is rare that 28 patients are in the PACU simultaneously. Occasionally, the beds are not fully staffed. More importantly, when the system is busy, many patients are transitioning in and out of the recovery area; as a result, there are usually a few PACU beds that have just released one patient but not yet received the next waitlisted one from the ORs. We found that when a patient gets waitlisted during the steady-state period from 3:00 to 5:00 pm, there were 24.1 ± 2.1 (mean ± SD) patients in the PACU. We will use the (constant) value

*N*= 24 of “effective capacity” to represent the current situation.

*e.g.*, surgery volume, length of recovery,

*etc*.).

#### Results

##### Evaluation of Modeling Assumptions

Fig. 1 Image Tools |
Fig. 2 Image Tools |

*i.e.*, Mondays

*vs.*Tuesdays

*vs.*Wednesdays,

*etc*.), and we still found the same arrival rate during the steady-state period. We concluded that the work released into the PACU was constant during the study. The impact of λ that changes throughout the day and how to deal with such changes will be addressed in the Discussion section.

Fig. 3 Image Tools |
Fig. 4 Image Tools |
Table 1 Image Tools |

^{2}> 0.99). The results of the simulation have also been listed in table 1. Note that, as can be seen in figure 4, the temporal distribution of arrivals is independent of the number of ORs. That is, although changing the number of ORs will change the absolute number of PACU arrivals, figure 4 and our empirical data demonstrate that the average proportion of arrivals in different hours of the day only depends on the durations of operations and turnover times and the scheduling policy.

*i.e.*, PACU arrivals) for each 0.5 h of the day using the actual OR time data from each of the different suites. Figure 5A shows eight different grayscale lines plotting the completed operations from each of the 5-OR suites. The basic shape of each curve is identical to figure 2 (the actual rate of completed operations from the complete OR suite) and figure 4 (the numerical simulations). Specifically, the release of work from each 5-OR suite is constant from 10:00 am to 4:30 pm. Figure 5B highlights the arrival rate to the PACU from one of the mock 5-OR suites and shows the 95% confidence interval of an constant arrival rate model of 0.36 patients per hour. The actual curve compares quite well to this model.

##### Comparison of Actual and Modeled Patient Distributions

Fig. 6 Image Tools |
Fig. 7 Image Tools |

*i.e.*, emerged from anesthesia and waiting to enter the PACU) at a given time. This brings us to the next part of our verification effort: checking directly if the distribution of equation (1) fits with empirical data. A central question here is when and if the system, which starts the day empty, reaches the steady-state conditions assumed in equation (1). We must also consider the previously noted issue that the average processing time tended to increase somewhat over the day. Analyzing the distribution of patients at different times of the day revealed the following pattern. In the morning, the OR suite-PACU system was filling, and the number of patients was much smaller than the steady-state equation suggested. However, in the afternoon, the theoretical distribution

^{1}predicted the actual distribution of patients in the combined (PACU + waitlist) queue, assuming that we used a PACU processing time relevant for the corresponding time window (figs. 6 and 7). That is, we measured the average time spent in PACU around 1:00 pm and around 4:00 pm (the values were 1/μ

_{1PM}= 3.21 h/patient and 1/μ

_{4PM}= 3.64 h/patient), and we used these values in equation (1). Specifically, we measured the average time spent in PACU+waitlist for patients leaving between 12:00 pm and 2:00 pm and 3:00 pm and 5:00 pm, respectively. Stays longer than 5 h typically represented cases from the previous day, and were truncated at 5 h. We then compared the theoretical curve with the empirical distribution of patients around the same times, and we specifically performed a χ

^{2}hypothesis test on the 4: pm data, with the bins (≤12, 13–14, 15, 16, … 31, 32–33, ≥34). The hypothesis that the model could generate the distribution we observed was not rejected (

*P*= 0.217). In figures 6 and 7, we have also drawn the ranges within which the occupancy levels would be expected to be 95% of the time.

##### Survey of Case Sequencing Strategy Implementation

^{10}

#### Discussion

^{11,12}but it addresses a different problem. We assume that nurses are indeed available when needed and that the key system constraint being considered is physical bed availability. Our model might also be useful when staff is the scarce resource; however, we do assume that the scarce resource is constant. Thus, our work may not be applicable in situations when staffing resources are both variable and drive waitlist problems. At least in our hospital, nurse staffing does not seem to be a first-order bottleneck in the PACU environment. In terms of methodology, we depart from previous work by introducing a queueing model, rather than seeking answers only directly from data on past events. While our approach necessitates testable assumptions, it is conducive to managerial insight, and it enabled us to analyze future and hypothetical situations for which data-driven analyses based on past data are not applicable. The latter point is particularly important in the context of long-term planning.

*e.g.*, cancellations, delays, urgent and emergent cases, to name prominent examples).

^{13}This observation was predicted to be true in our numeric simulations (fig. 4) and verified for eight different 5-OR mock suites created by segmenting our actual OR process time data (fig. 5). In figure 5, each mock suite has a different average arrival rate; for each, however, the rate is constant from 10:00 am to 4:30 pm. This indicates that even for as few as five ORs, the superposition of arrival processes indeed converges to a Poisson process, with the rate equal to the sum of the rates of the individual processes. We observed this for eight mock OR suites created from our own actual data, so it is likely that the key assumption of steady-state arrivals is borne out in many OR suites.

*e.g.*, shortest case first). Indeed, such policies have been proposed as a means to reduce PACU congestion.

^{7}We think such benefits are difficult to realize in practice, especially at large hospitals, partly because of organizational resistance (patients and surgeons have other considerations than PACU congestion when scheduling cases) and partly because of substantial divergences between scheduled and actual events in real-world hospital environments. Finally, Marcon and Dexter found that the scheduling policies used in actual practice had little impact on peak PACU patient load.

^{10}

##### Additional Modeling Considerations

^{10}However, this generalization must be treated cautiously. In one instance, the observed arrival rates are best described by a triangular distribution—one in which the arrival rate is never constant.

^{14}Investigators wishing to use our model in a different hospital would have to check that these assumptions are applicable in that setting. Specifically, if a hospital employed case sequencing strategies that could make the release of work into the PACU nonrandom, then the simple queueing model might not work. In that instance, it would be necessary to compare the actual and predicted distributions of patients in the combined PACU + waitlist queues by a goodness of fit test.

*i.e.*, measure the arrival rate and recovery time for each scenario) for each day of the week and plan capacity accordingly. For example, in a hypothetical 5-OR suite in which one of the ORs releases one 8-h case into the PACU on Mondays and 15 short cases per day into the PACU on Tuesdays, it is likely that separate model parameters will be needed for the 2 days.

*et al.*

^{15}captures several aspects (such as transfer time and porter availability) not explicitly encompassed by our model. On the other hand, analytical formulae such as equation (1) also have benefits; they reveal key relationships in a compact and instructive way.

^{7}

^{16,17}rather than first going to (and potentially waiting for) other hospital beds. According to the model, if such efforts could have a meaningful impact on the average length of stay, this would translate to a reduced waitlist problem as well. However, our previous work in this setting suggests that, at least for one patient population, the practical impact of this strategy was quite limited.

^{16}On the other hand, we also had data (not shown) on what portion of time in PACU was spent waiting for beds elsewhere. If this portion could be taken out of 1/μ, then calculating waitlists from equations (1) to (3) suggests that the OR-to-PACU waitlist would be all but eliminated. In this sense, the PACU is not the “true” capacity constraint at our hospital.

*vs.*free access) is important because it affects the impact of interventions to reduce 1/μ by speeding ‘wakeup' in the PACU. If much of 1/μ, is attributable to waiting for hospital beds then it is not beneficial to expend resources to speed recovery.

*N*and λ. As we have seen, because of staff availability and especially turnover times for PACU beds, the effective PACU capacity is less than the actual physical capacity at our hospital; it is the former that should be used as

*N*in our model. If one could bring the effective capacity closer to physical capacity, for example, by using improved nurse scheduling procedures

^{7}and/or by reducing PACU turnover times, then the model would predict the resulting reduction of waitlists. We primarily wanted to understand the impact on waitlists of changing demand or changing PACU capacity; for a small increase it seems reasonable to assume that the effective number of beds will increase by approximately the same amount (

*e.g.*, that if the physical capacity goes from 28 to 31, then the effective capacity will go from 24 to 27).

^{18}). It is also in good agreement with the result of the data-driven analysis outlined at the end of the Results section (

*i.e.*, lowering the curve in fig. 8). It should also be acknowledged, with reference to figure 9, that the marked sensitivity to small changes in the input parameters makes the model results sensitive to estimation errors.

^{19–24}Preliminary analysis of this strategy indicates that there would be sufficient time saved and OR capacity realized to close a small OR and convert it to PACU space,

^{25,26}with the impact of adding a net three PACU beds to the perioperative system. Thus, the empirical initiative to expand parallel processing in our ORs and the queueing model work described in this paper provide convenient opportunities to test the validity of both approaches and provide methods by which other congested academic medical centers may increase their functional capacity for growth in OR patient volume.