As pressure on the health system grows, intensive care units (ICUs) are increasingly operating close to their capacity. This has led a number of authors to describe a link between admission and discharge behaviours, labelled variously as: ‘bumping’, ‘demand-driven discharge’, ‘premature discharge’ etc. These labels all describe the situation that arises when a patient is discharged to make room for the more acute arriving patient. This link between the admission and discharge behaviours, and other potential occupancy-management behaviours, can create a correlation between the arrival process and LOS distribution. In this paper, we demonstrate the considerable problems that this correlation structure can cause capacity models built on queueing theory, including discrete event simulation (DES) models; and provide a simple and robust solution to this modelling problem. This paper provides an indication of the scope of this problem, by showing that this correlation structure is present in most of the 37 ICUs in Australia. An indication of the size of the problem is provided using one ICU in Australia. By incorrectly assuming that the arrival process and LOS distribution are independent (i.e. that the correlation structure does not exist) for an occupancy DES model, we show that the crucial turn-away rates are markedly inaccurate, whilst the mean occupancy remains unaffected. For the scenarios tested, the turn-away rates were over-estimated by up to 46 days per year. Finally, we present simple and robust methods to: test for this correlation, and account for this correlation structure when simulating the occupancy of an ICU.
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Kendall’s notation is used to characterise queueing systems. It is given by A/S/c/K/N/D, where A describes the inter-arrival process, S indicates the distribution of service time, c indicates the number of servers, K indicates the system capacity, N indicates the size of the customer population, and D indicates the queue’s discipline.
Also called critical care units in the literature.
APAHCE stands for Acute Physiology and Chronic Health Evaluation. It is a severity-of-disease classification system, and this is the third iteration of this system
A&E: accident and emergency
OT: operating theatre
Note the shift pattern in the study ICU is: Day shift - 7:00a.m. to 3:00p.m., Afternoon shift - 3:00p.m. to 11:00p.m., Night shift - 11:00p.m. to 7:00a.m..
Bai et al.  report patients grouped into 2, 3, 4, 8 or m classes, where no value for m is given. This is because a number of papers do not provide the exact number of patient classes.
We used both the graphical method and the confidence interval method described by Robinson  to determine that n = 100 was sufficient for our simulation. The confidence interval for the mean at n = 100 is 1%, sufficiently less than the 5% interval we were aiming for.
This was arbitrarily chosen as sufficiently large to say a correlation existed, and was not spurious. Other thresholds could have been considered.
Night time discharges were discharges from 10pm to 6:59am.
Day light hour discharges are all those which are not night time discharges.
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The authors would like to recognise the engagement of the ICU staff at the RAH. In particular, for providing practical insights into results, and for making the data accessible which was used to carry out this research. This publication is based on a project funded by the Premier’s International Research Fund. The international partners for the research grant are the international partners Cumberland Initiative and The AnyLogic Company. The views expressed in the paper are those of the authors and do not necessarily the views of the any other parties involved in this research.
Appendix A: ANZICs data
A summary of the ANZICS data is shown in Table 5.
Appendix B: Arrival rate and LOS analysis
Based on work in [13, 25, 33], discussions with clinicians, and our own observations we chose three likely classes. These classes are: 1. Emergency or elective patient, 2. Arrival day and shift, and 3. A patients’ APACHE III code.
B.1 Emergency and elective arrivals
Emergency and elective patients are commonly considered different classes in ICU modelling [25, 33], due to their differing arrival rates and differing LOSs. In this study ICU elective patients predominantly arrive on weekdays, during day and afternoon shifts (see Figs. 8 and 9). This is not surprising, since elective patients are scheduled in the normal business hours of the hospital. In comparison, emergency arrivals have a more even arrival pattern across the week and across shifts (see Figs. 8 and 9). This is expected, as emergency arrivals are unplanned/random arrivals.
Emergency and elective patients also differ in their LOSs. As Fig. 8 shows, emergency patients tend to have longer LOSs than elective patients. Both the emergency and elective patient LOS distributions are long-tailed, as expected. However, the emergency patient LOS distribution has a much heavier tail than the elective patient LOS distribution. Note that Fig. 8 is cut-off at 16 days LOS, so the full length of the tails are not shown.
Based on this data it is clear that emergency and elective arrivals need to be considered as different classes of patients.
B.2 Time of arrival
In the previous section, we have seen graphically that there might be different LOS distributions depending on a patient’s day-of-week-of-arrival (see Fig. 8). Hence, we conducted one way ANOVA tests on LOS based on day-of-week-of-arrival. This analysis showed that the mean LOS for some day-of-week-of-arrival was different to other day-of-week-of-arrivals, with a p-value of 0.02. We then further subdivided the data into emergency and elective patients, and repeated the above ANOVA test on day-of-week-of-arrival, both of these ANOVA tests passed (with p-values of 0.87 and 0.36 respectively), which indicates that the LOS means based on day-of-week-of-arrival are not statistically different. Despite this last analysis, we still chose ‘day-of-week-of-arrival’ as a separate class for our data set. This is because, a count of the number of discharges, by day of week, shows a structure in the results (see Fig. 10). There are fewer discharges on weekends for emergency patients, and fewer discharges of elective patients on Sundays and Mondays. This structure indicates that there could be a correlation between the arrival process and LOSs based on day-of-week-of-arrival, so it is more conservative to choose each day-of-week-of-arrival as a class. Further, we would like this method to apply across hospitals, and so it is more robust to include each day-of-week-of-arrival as a class.
Additionally, we have also just shown in Fig. 9, that there are significant differences in arrival and discharge counts based on the hour-of-day-of-arrival and hour-of-day-of-discharge for the study ICU. To investigate this idea, we constructed histograms of LOS, where each bin was set at 1/3 of a day. This bin-width was chosen because there are three shifts per day and because the arrival and discharge counts appear to be grouped into those three shifts. This plot, further divided into emergency and elective patients, with LOS truncated at 10 days, is shown in Fig. 2. It shows a significant stratification in the LOS profiles, indicating that LOS is indeed correlated with arrival time, specifically the arrival shift. The stratification can be explained as follows: if you arrive on day shift, you are likely to stay between (1 to 4/3) + n days (where n is a non-negative integer), because most discharges occur late on day shift and early afternoon shift. Similarly, if you arrive on afternoon shift your LOS is likely to be (2/3 to 4/3) + n days. Due to this correlation, we further class our data into arrival-shift: 1. day shift 2. afternoon shift, and 3. night shift.
B.3 APACHE III code
Another class suggested by clinicians is the APACHE III code. The ICU provides care for a wide range of patients. We cannot create classes for each APACHE diagnosis in the ANZICS data, because this would make the size of the groups to be permuted too small to be meaningful. Therefore we have simplified, based on clinical advice and operational considerations, the APAHCE III codes to the following four diagnostic codes (DCs): 1. Cardiovascular related codes (Cardio), 2. Respiratory and gastrointestinal related codes (RespGastro), 3. Neurological and trauma related codes (NT), and 4. all other codes (other).
Note that these diagnostics codes were chosen for the study-ICU and may not be appropriate for all ICUs, because the patient cohort can vary significantly between ICUs. For example, the study-ICU treats emergency and elective patients, whilst other ICUs may only treat emergency patients, or only treat elective patients. Further, because the study-ICU is the main hospital in the area, it treats all presenting diagnostic codes, excluding paediatrics and obstetrics. In comparison, other ICUs may only, for example, treat paediatric patients or elective cardiac patients.
Boxplots for the LOS separated into 56 classes of: day-of-week-of-arrival (7 classes), diagnostic code (4 classes) and emergency/elective patients (2 classes) are shown in Fig. 11. Based on this figure we conducted a number of one way ANOVA tests. The first of these was on LOSs with diagnostic code, which showed that not all diagnostic codes have the same mean LOS. Then we subdivided each diagnostic code class into emergency and elective classes. We then conducted two one way ANOVA tests, one for elective patients and one for emergency patients, and in each ANOVA test we compared diagnostic code and LOS. The results from both ANOVA tests showed that not all diagnostic codes (for emergency or elective patients) have the same mean LOS.
Appendix C: Fitting arrival rate and LOS distributions
Based on our analysis of the data, arrival and LOS data was divided into the following classes:
When patients die in ICU, shortly after admission. Hence their LOS profile is quite different to other classes of patients. Additionally, our research sponsor was interested in scenarios involving increasing LOS to all patients who depart alive. Hence, constructing this class of patient was necessary for this modelling purpose.
Patients were divided according to whether they were emergency or elective patients.
Patients were divided according to the shift they arrived on, that is day shift, afternoon shift, or night shift. This class is in place due to the stratification in LOS based on arrival shift shown in Fig. 2. Note that there were very few elective arrivals on night-shift, hence not large enough for modelling purposes, so this group was set to zero arrivals.
This class is in place due to elective patients who have significantly different arrival patterns on weekdays in comparison to the weekends. Hence, elective arrivals were grouped as Monday to Friday and Saturday to Sunday. However, the elective Saturday to Sunday group was almost empty, hence not large enough for modelling purposes, so this group was set to zero arrivals. Emergency arrivals were not grouped according to day-of-week, as we found no significant change in arrival rate or LOS based on the day-of-week.
C.2 Fitting arrival rates
The key parameters for arrival rates were estimated using maximum-likelihood fitting. As expected, the arrival distributions were all found to be Poisson distributions (see Table 6).
C.2 Fitting LOS distributions
The LOS data was transformed into discrete time intervals, representing the number of ‘shifts’ each patient stayed in the ICU. For example, if the patient arrived and left during the same shift, their LOS was 0 shifts. If a patient arrived on long-day shift and left on the subsequent long-night shift, then their LOS was 1 shift.
Calculating LOS in this manner means that all the LOS distributions (apart from deaths) show significant stratification (see Fig. 12), creating a significant modelling challenge. Clearly, a distribution like this cannot be modelled simply - using a single theoretical distribution. We successfully (and simply) dealt with this stratification by noticing that the proportion of discharges on day-shift, afternoon-shift and night-shift for a single day, was similar for one whole distribution, and further, across many distributions. This meant that we could fit a a typical discrete negative binomial distribution to daily LOS data, then use these percentages to transform the LOS (by day) distribution, into a LOS (by shift) distributions. The procedure we used is outlined below (Fig. 13).
Determine the early shifts which are best modelled as point densities. Record the point densities and remove that data. The point density data is shown in Table 7.
Transform the LOS (by shift) distributions into LOS (by day) distributions by aggregating the count for day-shift, afternoon-shift and night-shift for each day.
Determine the best fit for these distributions. These are given in Table 7.
Using the best fit theoretical distributions determined in Step 3, the LOS (by day) distributions, were transformed back into LOS (by shift) distributions as follows: the density for each day was spread across day-shift, afternoon-shift and night-shift (for that day) according the percentages shown in Table 8.
Note that, the deaths distributions did not need this procedure as they do not show any significant stratification. This is because the stratification in LOS is due to patients rarely leaving ICU on night-shift, less often on afternoon-shift etc. However, this logic does not affect deaths. Hence the death distributions were simply modelled using a discrete binomial distribution.
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Varney, J., Bean, N. & Mackay, M. The self-regulating nature of occupancy in ICUs: stochastic homoeostasis. Health Care Manag Sci 22, 615–634 (2019). https://doi.org/10.1007/s10729-018-9448-4
- Decision analysis
- Health services
- Bed capacity