Introduction

Throughout the health system, both within Canada and elsewhere, excessive waiting times are severely hampering our ability to provide adequate health care. While there can be little argument that the primary reasons for excessive waiting times are limited resources and increasing demands, there are secondary factors that impede the efficient use of existing resources. Among these secondary factors, perhaps none is more important than the variability in the demand. One day a resource is stretched beyond its capacity, the next day it stands idle for large portions of the day. We will address the challenge of excessive waiting times by developing an approach to improve resource utilization and then demonstrate via simulation the resultant effect on the growth rate of outpatient (OP) waiting time. We will concentrate on the growth rate in OP waiting time as the most relevant statistic as the issue of waiting time is of most import in those instances where demand exceeds capacity and therefore where queues are long and growing.

The problem

At the most general level, this paper addresses a situation where there is a fluctuating demand for a limited resource with multiple priority levels. Such a scenario occurs for instance in scheduling patients for operating room time (eg Gerchak et al, 1996; Strum et al, 1999; Kim and Horowitz, 2002), or else booking magnetic resonance imaging (MRI) or computed tomography (CT) scans. The key challenge is that the low-priority demand must be booked before knowing the high-priority demand. Therefore, a significant portion of the total capacity must be reserved for this unknown high-priority demand—leading inevitably to unused capacity on those days when the high-priority demand is lower than expected.

We develop a policy that reserves a certain amount of resource time to each priority level. Gerchak et al (1996) demonstrated that such a cutoff policy (in the case of booking elective and emergency surgeries) is potentially sub-optimal but they do so under the assumption that there are no fixed limits on overtime. In reality, most hospitals do work with a limited overtime budget. Gerchak and Gupta also acknowledge that the relative difference between the optimal policy and the cutoff policy is minor. Their main contention is that optimal cutoff policies are computationally difficult to determine. However, in the case of limited overtime there is, as we will show, a very simple means of determining an optimal cutoff policy. The potential for improving the ability to achieve reasonable waiting times for OPs by following Gerchak and Gupta's approach of allowing a flexible overtime limit that increases in proportion to the length of the queue is definitely worth pursuing.

The motivation for this paper grew out of a detailed analysis of CT operations at the Vancouver General Hospital (VGH) where there is a large and fluctuating inpatient (IP) demand and three OP priority classes. OP priority levels are broken down according to the maximum recommended waiting time. What makes this a scarce resource problem is that waiting times for OP are, at present, significantly longer than recommended. One long-term goal of the project is to provide a reservation policy that will cover all priority levels. For the purposes of this paper, however, we will assume only one OP category. All IPs are currently viewed as high priority—meaning that the demand must be met the day it arrives. As OP demand is booked weeks in advance, it is clear that, for any given day, the low-priority demand (OP) is booked prior to knowing the high-priority demand (IP) for that day. In order not to overtax the system on days when IP demand is high, the department is forced to reserve a significant portion of scanning time for this unknown IP demand, leaving little room for OP. This results in unused capacity on days when IP demand is low and thus longer waiting times for OP than might be the case if this unused capacity could be utilized.

This trade-off between unused capacity (resulting in longer waiting times) and overtime has received significant attention in the literature. Gerchak and Gupta model this trade-off in the case of elective surgery scheduling. This situation is analogous to our own as the challenge is to schedule elective surgeries while leaving room for possible emergency surgeries. Strum et al (1999) develop a minimal cost analysis model based on over-utilization and under-utilization applied to OR blocked time for subspecialties. Liu and Liu (1998) analyse an OP clinic with multiple doctors (with the complication that doctor arrival times are stochastic) and Ridge et al (1998) model the effect of balancing the cost of unused beds in the intensive care unit against the cost of insufficient capacity. Finally, Rohleder and Klassen (2002) present a simulation-based model that compares various potential methods for reducing waiting lists in the presence of uncertain demand with a fixed number of slots reserved for emergencies. Our contribution is to make use of some flexibility within the highest priority class in order to reduce the variability in IP demand. Whether this flexibility is present in all the scenarios described above is a point for investigation. However, it is clear that this flexibility is present in the case of CT scheduling in a hospital that has both IP and OP demand.

The proposed scheduling approach

Our attempt to make use of the unused capacity in the system hinges on the fact that, contrary to current practice, not all IP need to be scanned the day the request for a scan is placed. Our proposal is to:

• Divide IP into two categories—high-priority inpatients (HIPs) who must be scanned the day the request is placed and low-priority inpatients (LIPs) whose scans can be delayed 1 day.
• Identify a group of OP who will be on-call for scans with 1-day notice.
• Reserve a specified number of slots for HIP.
• Reserve a specified number of slots for possible carry over of LIP to the next day.
• Limit overtime (OT) usage.

How to make this operational will be described in the remainder of this paper.

A division of IPs as suggested above already exists at VGH. The prioritization scheme recently put in place contains a category for IP that can wait up to 24 h. This group contains pre-operative investigation for head and neck, routine follow-up for intracranial disease, pre-operative oncology staging, numerous fractures and some chest infections. However, this distinction is currently being ignored since all IP are scanned the day the request arrives regardless of the priority classification.

The advantage of such a division is that there is a pool of patients who are available to be scanned on the current day but do not necessarily need to be. One might liken LIP to stand-by passengers on flights. Just as stand-by passengers provide the airlines with the ability to fill last minute unused capacity, so LIP would provide the hospital with the same flexibility. The major difference is that LIP must be scanned within the next 24 h. There is potentially some useful crossover here as substantial research has gone into developing reservation policies for numerous priority levels within the airline industry (Van Ryzin and Vulcano, 2003). However, the airline industry is generally dealing with a small number of flights at a time over a finite horizon whereas a CT department is booking over an infinite rolling horizon where each day could be viewed as a separate flight.

We assume that the resource units are 15 min scanning slots since the vast majority of scans are scheduled for 15, 30, 45 or 60 min. Thus, a 60-min long scan would equate to four units of resource time. The capacity of the system is the number of 15 min scanning slots that can be scheduled for any given day. Initially, we will assume that all scans are scheduled for 15 min and later investigate, through simulation, the effect of introducing multiple exam lengths. Of course, actual exam lengths are stochastic and vary quite substantially from the scheduled length (see Strum et al (2000) for a comparison of normal versus log normal exam lengths). However, from a booking point of view, it is the scheduled length of the exam rather than the actual length that is relevant. We will discuss the effect of introducing stochastic exam lengths later in the paper.

Assume there are a total of C slots available. Each day, H slots are reserved for IP and an additional L slots for any LIP carried over from the previous day. The remaining C-H-L slots are available for booking OP demand. Note that the trade-off is clear—the more slots reserved for IP (H+L), the less scanning time is available for OP resulting in longer OP waiting times. Our proposed system operates as follows. On any given day perform all HIP scans and as many LIP scans as possible in the H reserved slots. At the end of the day, any additional LIP scans not performed are booked into the L slots reserved for tomorrow. If any of the L slots are not filled, then on-call OP from the waiting list are called to fill the remaining slots—reducing the total number of reserved slots for tomorrow's IP to H. In other words, looking forward days in advance, the policy reserves a total of H+L slots for IP but on the day before, it reduces this to H reserved slots by booking any remaining LIP scans and/or additional OP scans into tomorrow's L slots. (We impose the further restriction that no more than L LIP scans are carried over to the following day in order to avoid any snowball effect. If more LIP scans arrive on any given day than can be dealt with through the H slots reserved for IP today and the L slots reserved for carry-over to tomorrow, then the excess is dealt with through overtime.)

The following diagram (Figure 1) describes a typical day. At the beginning of the day, there are H slots reserved for incoming IP demand. The rest of the capacity is a mixture of OP demand and some LIP carried over from the day before. Tomorrow still has H+L reserved slots for IP with the rest of the capacity booked with OP.

Figure 1.

Diagrammatic representation of a typical day.

Full figure and legend (25K)

A short numerical example further illustrates the proposed reservation policy. Table 1 presents hypothetical daily IP demand for a given week. (Note that although the demand for the whole week is given, each day's demand is not known until the day it arrives.)

Table 1 - Illustrative IP demand data for 1 week.

Full table

Assume that the number of IP reserved slots, H, is set at 65 and the number of carryover reserved slots, L, is set at 10. On day 1, all 60 HIP and five LIP are scanned while the other five LIP are carried over to day 2. No overtime is required and there are no empty slots. Since only five of the 10 carryover slots are used by LIP, an additional five on-call OP can be scheduled for day 2. This reduces the total number of reserved slots on day 2 to H. On day 2, demand is low so all HIP and LIP patients can be accommodated (45+12<H) without requiring overtime or holding any LIP over to day 3. However, there are eight (65–57) unused slots. Since no LIP are held over, an additional 10 on-call OP can be booked for day 3.

On day 3, total IP demand spikes and 10 overtime scans are required just to meet the HIP demand. In addition, since no more than 10 LIP can be held over from 1 day to the next, five of the third day's LIP must be scanned using overtime with the other 10 being carried over until day 4. No additional OP can be booked for day 4 as all L slots are filled with LIP. For the week, the number of unused slots, the overtime required, the number of LIP carried over and the number of additional OP booked are given in Table 2. The units again are the number of 15 min scanning slots.

Table 2 - Summary of performance of the proposed reservation policy (H=65,L=10) for the data in Table 1.

Full table

Contrasting this with the policy that simply reserves 75 slots for IP (Table 3) gives an indication (despite the fictitious nature of the data) of the potential increase in resource utilization gained by the introduction of LIP. In particular, we have been able to fill 30 of the 38 unused slots (under the current policy) with additional OP under the proposed policy. Note also that this is done without increasing overtime.

Table 3 - Performance for the reservation policy (H=75,L=0) for the data in Table 1.

Full table

Is the proposed system practical?

It is worth, at this point, pausing to consider the practicality of implementing the proposed policy. First, while it is known that several patients fall into the LIP category, no data are currently available to determine the proportion of the total IP demand that would fit such a classification. What we will show, however, is that the number of patients in the LIP category need not be a large percentage of the total IP demand in order to have a significant impact on OP waiting time. Second, there is the practical issue of having a pool of on-call OP able to respond to a single day's notice. Again, we will show that the proportion of OP willing to be on-call need not be a large percentage of the total OP demand. It seems reasonable, at least for a city the size of Vancouver, to assume the existence of a pool of OP within easy distance of the hospital who would gladly be on-call in order to shorten their waiting time. Gerchak et al (1996) assume, not unreasonably, that all OP can be called in to receive their surgery earlier than initially scheduled. Third, there is a question of increased workload on the department staff. However, the only additional work placed on the staff is that of calling the on-call OP the night before. In our opinion, this added load is more than justified by the benefits of improved resource utilization. (In fact, one recommendation of the VGH study was to call OP the night before their scan in order to minimize no-shows so, in reality, there would be no additional work.) Fourth, there is a legitimate concern that such a system would increase costs due to increased IP hospitalization time. However, judicial use of the LIP category (ie not classifying as LIP those whose hospital stay might be shortened by a prompt scan) should alleviate this problem. Finally, we will show that such a system does not, as might be supposed, increase overtime.

The initial feedback that we have received both from hospital administration and the staff within the CT department has been favourable. Their major concern is the size of the LIP category. No one could give us much insight as to what numbers to expect, although they were cautiously optimistic that our proposed scheme could work—perhaps not at Vancouver General which is a tertiary care hospital, but at a community hospital where patients spend extended periods of time in hospital.

Choosing L and H

Of course, the obvious question concerns the choice of L and H. From a practical standpoint, the most likely scenario is that a hospital will have a set overtime budget that should not be exceeded. Thus, one possible approach is to choose L and H to minimize the expected number of empty slots subject to a constraint on OT. The papers referred to above do not put a limit on OT but rather assign a penalty to OT and then seek a policy to minimize cost. This is mathematically much more difficult and the benefit is dubious. Assigning a 'cost' to unused capacity is complex and thus one ends up adjusting the parameters to see what works regardless. Our approach is to impose an OT limit and then adjust to see the effect. From a practical standpoint, this is reasonable as hospitals generally do have an OT budget that they are trying not to exceed.

The above scenario generates a simple optimization problem that can be solved for any given breakdown of IP demand into HIP and LIP. For instance, if we assume the daily arrival rates for HIP and LIP follow independent, stationary Poisson distributions with means _H and _L, respectively, then the optimization problem is to minimize

subject to an OT constraint. The random variables D_HIP and D_LIP are the demand for HIP and LIP, respectively. Since they are independent, their sum is Poisson with parameter _H+_L. The above equation follows from the fact that there is only unused capacity if the sum of all IP demand is less than H. The OT constraint can take one of two forms—a limit on the probability of OT (or the probability OT reaches a certain limit) or else a limit on the expected number of units of OT. These lead to two optimization problems

and

The probability of overtime can be written as

Note that the H in the above equation refers to today's H reserved slots and the L refers to tomorrow's L slots for carryover LIP demand, as it is these two numbers that determine today's OT. This also holds true in the following equations.

Computing expected overtime is more complicated as it depends on whether the OT is due to HIP, LIP or both. Thus, the expected amount of OT is given by

Again, if we assume independent, stationary Poisson distributions for the arrival rates of both HIP and LIP we get the following equation for the expected amount of OT

In our opinion, the second OT constraint is more appropriate since it is easier to place a dollar figure on the expectation than on the probability. Our assumption of Poisson arrival rates is based on the data we collected for IP arrivals at VGH. We also tested the data both for a 'day of the week' effect and for any autocorrelation with negative results in both cases. There are, however, no figures on how much of the total IP demand could be classified as LIP. Thus, our assumption that the two IP categories form two independent Poisson distributions, though reasonable, needs to be verified.

The optimization problem with a constraint on expected overtime was solved by a simple directed search in MATLAB. The code searches for the smallest H for which there exists an L so that the pair (H, L) satisfies the OT constraint. This will minimize unused capacity (subject to the OT constraint) since unused capacity depends only on H. (We are assuming here that there is enough OP on call to fill any of the L slots not used by carryover LIP.) The algorithm starts with an H large enough so that, together with L=0, it satisfies the OT constraint. It proceeds by reducing H until the OT constraint is violated and then, holding H constant, increasing L until the constraint is once more satisfied. While keeping L constant, the program then reduces H by one and attempts to find an L so that the combination (H,L) once again satisfies the OT constraint. It terminates when no such combination can be found and returns the last viable combination. This combination minimizes expected unused capacity.

What improvement can we expect?

System performance can be measured by two metrics—the expected throughput rate (expected number of scans per day) and the expected growth rate in OP waiting time. The first can be calculated theoretically and the second will be analysed through a simulation of the scheduling process.

Once H and L have been specified, it is a straightforward calculation to determine the expected throughput of the system. In such a scenario, the expected number of scans per day in the case where all IP are treated as HIP is given by

where H is the number of slots reserved for IP and C is the total capacity. The first term represents the number of scheduled OP scans and the second the expected number of IP scans performed per day.

If a certain percentage of IP can be classified as LIP, the expected number of scans per day can be calculated as follows (provided we assume that we can fill all slots that are made available for on-call OP):

(Note that the H here may be different from the H generated for the initial case with zero LIP.) The three terms that constitute the coefficient multiplying the probabilities consist, respectively, of the number of HIP, the number of LIP scanned due to excess capacity (ie low number of HIP) and the number of LIP scanned due to having more than L LIP carried over.

The above throughput rates as well as the resulting OP waiting times depend heavily on certain parameters—namely, the limit placed on OT, the IP demand rate, the percentage of IP classified as LIP and the capacity of the system.

The rest of the paper will analyse system performance based on a simulation model of the scheduling process and study the sensitivity of results to variations in the parameters. Finally, we will consider the effect of introducing multiple exam lengths and stochastic service times as well as varying the percentage of OP willing to be on-call.

The simulation model

Our simulation model, created in ARENA, allows us to compare the effect of the model parameters on the growth rate in OP waiting time over any given period. We assume independent Poisson arrival rates for all priority levels. (Thus when we talk about LIP being 10% of total IP demand we mean that the mean arrival rate for HIP is nine times that of LIP.)

A more contentious assumption is that all incoming IP demand for any given day arrives at the beginning of the day. This is unrealistic since, though most IP demand is known at the beginning of the day; emergency demand occurs throughout the day and is therefore not known at the outset. However, our results are a comparison of three possible reservation policies. If anything, the intra-day variation would be more detrimental (increasing OT and unused capacity) in the case where all IP are assumed to be HIP than under our proposed policies since LIP patients would be identified early in the day (they would not be emergencies) and therefore would act as a pool of patients who could be scanned if and when space was available. As long as OPs are generally scheduled earlier in the day (as at VGH) the intra-day variation will have little impact. Our own observation of the CT department revealed that for the majority of the day, the department had a pool of IP waiting for a scan that they called down when space was available. Thus unused capacity is unlikely to be affected by intra-day variation in IP demand. As for OT, it is possible that a late arriving emergency could cause additional OT but this scenario is equally likely under all reservation policies and therefore does not affect the comparison.

We assume that OP demand arrives at the beginning of the day and is scheduled for the earliest day available. A certain percentage of the total OP demand is tentatively booked but also set aside as 'on call'. These are the OP who can be called the day before to fill any of the L carry-over slots not filled by LIP for the next day. After each day, the simulation cycles through these 'on-call' OP and places them into the available slots (on days when less than L LIP have been carried over to the next day) on a first in first out basis. As the simulation progresses, waiting times are recorded by priority level along with the arrival rates for the various categories, the amount of daily OT or unused capacity as well as the growth in OP waiting time from the previous day. We assume that an OP cannot be booked the day the request arrives.

Our initial testing of the simulation consisted of 100 replications of 1000 days each (with the first 500 days being a warm up period) for a system with a mean arrival rate for IP of 120 (15 min scans) per day and a mean arrival rate for OP of 50 per day. The regular-hour capacity is set at 165 which is five less than the average daily demand. We initially assume that all scans are 15 min and that an expected OT of five scans per day is acceptable. Thus, the total capacity including OT is equal to the average demand. Finally, we assume that 10% of OP are willing to be on-call and that 10% of IP can be classified as LIP. The results of the 100 replications were very similar so we present only the combined results with the half width for a 95% confidence interval.

We assess three reservation policies that we will call:

1. The constrained reservation policy
2. The unconstrained reservation policy
3. Current practice (ie treating all IP as HIP)

The constrained policy, (H=110, L=9), is the optimum of the above optimization problem under the additional constraint that the total H+L should be no more than would be reserved if all IP were HIP. This policy will perform at least as well as current practice since the same number of slots is reserved. This modified optimization problem can be solved in precisely the same manner as the previous one except that the viable combinations of L and H are restricted. The unconstrained policy, (H=107, L=15), is the optimum of the above optimization problem without the additional constraint. Finally, we compare both these to the current practice (H=119, L=0) of treating all IP as HIP.

Table 4 shows the average per day over all replications for the relevant statistics as well as the average weekly growth rate.

Table 4 - Simulation results (numbers are means with 95% confidence limits).

Full table

Both the constrained and unconstrained policies have significantly higher throughput rates and lower growth rates in OP waiting time than current practice. After 1000 days (starting from empty queues), the waiting time for OP under current practice grows to approximately 84 days while the waiting time for OP under the constrained optimal policy (column 1) only increases to 29 days. This significant reduction is achieved by identifying only 10% of total IP demand as LIP. It is accomplished with no additional expenditure (note that OT remained the same) and would seem to more than justify the procedural changes required. The fact that the more conservative constrained policy (column 1) outperforms (in reducing growth rate and increasing throughput) the optimal policy (column 2) can be attributed (as will be shown later) to the fact that a 10% on call rate for OP is simply not large enough to take full advantage of the flexibility provide by the (H=107, L=15) policy. Thus, the higher L value is detrimental since it means that there is less scanning time available for those OP not on call.

The effect of classifying 10% of IP as LIP is to increase the number of scans per day by almost 3. This may not seem like a huge increase but, as Praeter (2001) points out, it has been shown within queuing theory that 'small changes to a heavily congested systems bring disproportionate changes in queue size' and hence waiting times. Where demand outstrips throughput (under current practice) by just four scans per day, increasing the number of daily scans by almost 3 is a substantial step towards meeting total demand. Three additional scans per day over 1000 days results in 3000 more scans being performed than under the current practice. In that light it is perhaps understandable why there is such a significant difference in OP waiting time.

Clearly, none of the three policies satisfy demand in the above instance (average number of scans per day for all three is less than the combined arrival rate) thus waiting times are increasing in all cases. However, as the table indicates, it is the rate of increase in OP waiting time that is significantly reduced. Figure 2 shows the growth in OP waiting time from a typical simulation run for the constrained reservation policy and for current practice demonstrating the significantly lower growth rate in OP waiting time.

Figure 2.

Growth of OP waiting time for the constrained policy and the no LIP policy.

Full figure and legend (19K)

The rest of the paper seeks to demonstrate the robustness of these results to variations in the simulation parameters.

Varying the OT limit

Table 5 presents the expected unused capacity and expected number of scans per day based on Equations (5) and (6), as a function of the limit on expected OT for the three reservation policies. As one would expect, the more OT allowed, the less potential for unused capacity and therefore the less difference between the three cases. It is worth noting that we are assuming here that there is sufficient capacity to be able to reserve enough scanning time for IP so as to keep OT as low as required. The trade-off is simply that the more scanning slots reserved for IP, the less slots available for OP booking resulting in longer OP waiting times.

Table 5 - Effect of OT limit on capacity usage.

Full table

Of note is the fact that the overall number of reserved slots (H+L) is less under the current practice of treating all IPs as HIPs than it is under the unconstrained policy. This is precisely the reason for including the constrained policy in the analysis, restricting the number of reserved slots to be the same as under current practice. The row with a limit of (an average of) five OT scans per day, demonstrates that the simulation results given earlier are a very good fit to the theoretical results presented here. The unconstrained policy under-performs in the simulation but this will be shown to be due to the lack of OP on-call.

It is also of note that the current policy must increase the OT limit to eight before throughput matches demand. In contrast, both the constrained and unconstrained policies require an OT limit of six in order for throughput to match demand. At VGH, CT technicians are paid time and a half for OT giving an approximate cost of $60 per hour of OT. Thus, assuming two technicians per scan, an increase in OT by half an hour costs $60 in technician salaries per day yielding a $15 000 cost over the year. However, if OT exceeds 2 h then technicians are paid double leading to a cost of $80 in technician salaries for two scans yielding a $20 000 cost over the year. CT technician salaries are clearly only the most obvious cost associated with OT. Increased OT would also imply longer hours for CT support staff, radiologists and porters. Thus, the true cost of increased OT is probably significantly higher.

Varying IP demand

In order to determine the effect of the IP demand rate on the growth rate in OP waiting time, we ran the simulation with the IP arrival rate being 60 per day and 90 per day. Capacity was set at five under the total demand rate while all other parameters remained the same. Figure 3 compares the effect on the growth rate in OP waiting time for the three reservation policies, optimized for the different demand rates.

Figure 3.

Effect of IP arrival rate on growth rate of OP waiting time.

Full figure and legend (37K)

It is clear that reducing IP demand does not negate the benefit of the proposed reservation policy. Note also that the unconstrained policy is only outperformed by the constrained policy when the IP arrival rate reaches 120. This confirms the fact that the poorer results of this policy are due to having insufficient numbers of OP on-call in order to take advantage of the flexibility provided by a policy with a large L value.

Varying the percentage of LIP

As the percentage of LIP patients increases, the expectation is that the added flexibility would improve system throughput. To substantiate this, we ran the simulation assuming that 20% of IP could be classified as LIP and with 100% of OP on-call to determine the maximum benefit that might arise from an additional 10% LIP. Again, we ran the simulation for the optimal unconstrained policy (H=94,L=35) together with the optimal constrained policy (H=100,L=19) and compared them to the current practice. The growth rate in OP waiting time was dramatically reduced (from 0.432 days/week to 0.018 days/week) with throughput coming very close to matching demand. This was accomplished without increasing OT, thus the improvement is due entirely to a reduction in unused capacity.

Varying capacity

To determine the effect of increased capacity, we ran the simulation with the same initial conditions but increasing regular hour capacity. It is clear from Figure 4 that as capacity begins to meet demand the effect of introducing LIP becomes less pronounced though percentage wise the reduction in the weekly growth rate in OP waiting time in fact increases.

Figure 4.

Effect of increasing capacity.

Full figure and legend (44K)

It is also worth noting that as capacity approaches demand, some of the benefit derived from introducing an LIP category is transferred from increasing throughput to decreasing OT. With a capacity of 169, the average OT per day for the three policies was 3.39, 2.92 and 4.84. Thus, even in a system that is meeting demand through OT, there is a benefit to introducing LIP.

Effect of multiple examination lengths

The above results are based on the unrealistic assumption that all examinations are 15 min long. In reality, scan lengths can be scheduled for anywhere from 15 min to 90 min (although scan lengths greater than 60 min are rare). Thus, although it is possible to simply view a 60 min scan as four 15-min scans, the result is that arrivals now come in batches that would, of course, affect the theoretical calculations. To determine the effect of multiple examination lengths, we ran the simulation with examination lengths varying from 15 to 60 min. For both OP and IP, we assigned 50% of total demand to 15 min scans, 30% to 30 min scans, 10 to 45 min scans and the remaining 10% to 60 min scans. We adjusted the overall arrival rates so that the total demand on the resource was unchanged at 120 scanning slots (of 15 min each) per day for IP and 50 scanning slots per day for OP. The results are given in Table 6 and show that the effect is to increase the advantage of introducing LIP to the system. However, once multiple examination lengths are added, the amount of OT increases beyond the allowable limit for all three policies considered. Again, the units are the number of 15 min scanning slots.

Table 6 - Multiple examination lengths comparison.

Full table

Increasing on-call percentage

One of the challenges facing this proposed reservation policy is the need to have a pool of OP who are on 24-h call—that is, willing to be scanned if given a call the night before. In a city the size of Vancouver, this is unlikely to be a major problem. To test the range of results possible for the different reservation policies, we tested both the constrained policy (H=110, L=9) and the unconstrained policy (H=107, L=15) with the on-call percentage being 0, 10, 20 and 30% of the total OP demand. The resultant effect on the weekly growth rate in OP waiting times is given in Figure 5.

Figure 5.

Effect of percentage of OP on-call.

Full figure and legend (43K)

As we would expect, increasing the percentage of OP on-call increases the benefit of the proposed system up to a point, after which it makes little difference. (Note that the policy (H=110, L=9) with 0% on call gives the same growth rate in waiting time for OP as the current practice of treating all IP as HIP.)

Clearly, as the percentage of OP on-call is reduced to zero, the effect on OP waiting times of introducing LIP is lost, although there remains the advantage of reduced OT. In fact, for the policy (H=107, L=15), the effect is detrimental since the sum of H+L is greater than would be reserved under current practice. Conversely, if the percentage of OP on call is sufficient to take advantage of the flexibility provided by LIP, then the policy (H=107, L=15) is the most effective at reducing the growth rate in OP waiting time. However, the potential gain by using the unconstrained optimal policy is minimal, and thus the more conservative policy (H=110, L=9) that insures that we do no worse than under the current practice of treating all IP as HIP would seem to be the wiser option. The salient message, however, is that the percentage of OP on-call does not need to be a significant portion of the total OP demand in order for the introduction of LIP to have a significant impact on OP waiting times.

Stochastic service times

The above results assume that a 15-min scan takes 15 min which is, of course, highly unrealistic. For instance, at VGH, the time required for a scan is systematically overestimated. Thus, unused capacity would be much higher than reported in the simulation. However, our proposal looks at reducing the number of scheduling slots that stand idle in a given day. The efficiency within each scheduling slot is another issue entirely and largely depends on aligning the actual scanning time required for a given examination with the time scheduled. The overall impact on OT that would result from introducing stochastic scanning times is not, in our opinion, a significant factor—provided scheduled times are a fair reflection of the actual required scanning time and provided that regular hour capacity is sufficiently large. If these two criteria are met then the law of averages will mitigate the effect on OT of introducing stochastic scanning times. Moreover, whatever the effect produced, it would be the same for each policy and therefore would not affect the comparison.

To substantiate the above claim we ran the simulation with the same reservation policies but allowing for stochastic service times. We ran the simulation with the original parameters and allowed service times to vary from 10 to 20 min. The results are almost identical to the initial simulation results in Table 4 with only a slight increase in OT (Table 7).

Table 7 - Introducing stochastic service times.

Full table

Implementation challenges

There are, no doubt, a number of implementation challenges associated with any attempt to improve a system through process modification. We briefly discuss a few of the more obvious ones.

Firstly, our simulation assumes a 5-day working week whereas reality is somewhat different. VGH conducts OP scans Monday to Friday but performs IP scans 7 days a week. Since the weekends are only IP, the situation is actually better (favouring the proposed scheme over current practice) in that Monday starts off afresh with no carry over LIP, and all of Friday's carry over LIP can be finished at the weekend (capacity is currently underutilized at weekends). Our proposed scheduling scheme would therefore fit the current situation, with the only adjustment being that we need only reserve H slots on Monday (as we know in advance there will be no LIP) and hence can book more OP for Monday.

Secondly, although our own data collection suggests that a Poisson distribution is a reasonable fit to the data, this may not be true in other instances. It would be interesting to see the effect of running the simulation with different underlying arrival distributions. For instance, it could be argued that the Poisson does not represent enough variation. However, our simulation runs under the Poisson assumption resulted in a range of almost 90 slots per day for an average arrival rate of 120. This seems adequately to reflect the fluctuation in demand we observed.

Finally, it is worth noting that the introduction of multiple OP classes does not affect the results given in this paper. One can view the overall problem as two sub-problems. First, how much capacity does one need to reserve for IP in order to meet the required limits on OT? Second, given that one has set aside a certain amount of overall capacity for IP demand, how does one proceed to schedule the various OP classes into the remaining capacity? The optimal answer to the first question is entirely independent of the second.

There are other implementation challenges (such as how no shows would affect the results) but none, in our view, negate the fact that the introduction of the proposed reservation policy significantly increases throughput and reduces OP waiting times without increasing costs.

Conclusion

The introduction of a low-priority IP class does not represent a huge procedural change on the part of a CT department. It requires some careful thought on the part of the radiologists to ensure that the appropriate IPs are classified as LIPs and it requires some additional phone calling on the part of the staff. There are no additional financial costs involved. The results of the simulation show clearly that introducing a 1 day flexibility to only 10% of IP demand provides a significant reduction in the growth rate of OP waiting times—even if only 10% of OPs are willing to be on-call. Varying the parameters of the simulation, as well as introducing variability in the duration of each scan, did not affect the ability of the proposed policy to improve throughput and thereby reduce OP waiting times.

The proposal presented here applies readily to the CT operations that provided the impetus for this research. However, it could potentially be applied to any situation where there is a fluctuating demand and two or more priority levels (such as MRI-scheduling or operating room scheduling). The key is to determine whether there is any flexibility present in the high priority class and how to best exploit that flexibility. A potential application outside of health care might be the management and shipment of inventory. Inventory often involves multiple priority classes. The only question therefore is whether there is any flexibility in the scheduling of the higher priority class that is not currently being used efficiently. In the case of inventory management, Cattani and Souza (2002) compare a number of rationing policies in a scenario with two demand classes. However, all of the policies only attempt to manipulate the lower demand class. Lawson and Porteus (2000) have demonstrated the advantage of allowing for the possible expediting of a shipment. However, it might also be advantageous to have a priority class that can potentially be delayed in order to insure that shipments leave with near full capacity. This would be analogous to the LIP category.

The research in this paper could be viewed as clearing the ground for the more complex problem that includes multiple OP priority classes. Imposing an OT limit, as proposed above, allows for the separation of the total capacity between IP and emergency on the one hand and OP on the other. However, although a directed search was sufficient to determine the capacity required for IP demand, it does not answer the issue of when to schedule OP—particularly if there are numerous priority classes. Our future research will solve the scheduling problem for numerous OP priority classes through approximate dynamic programming. It will then be possible to vary the OT constraint (for IP scans) to determine the net effect on OP waiting times. It would also be interesting to know how to adjust H and L to adapt to shifting patterns in demand. If demand was shown to vary in a predictable fashion (ie weekly or annual patterns) then one might look at solving for H and L through dynamic programming. If, on the other hand, demand was shown to be increasing unpredictably one might instead look at adjusting H and L through adaptive control.

There is no doubt that limited resources are going to continue to challenge the provision of health care. It would seem reasonable therefore to insure that we are using the resources that we have as effectively as possible which is precisely what the proposal outlined in this paper attempts to do.

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