Expecting the Unexpected: Towards Better Planning in Healthcare Systems (Part Two)

When is a hospital (or hospital unit) “undersized” to meet ongoing (or future!) patient demand?

In Part One of this series we looked at variability in unscheduled demand for systems like an Emergency Department from a statistical point of view (using standard deviations and the Poisson distribution).? Two key lessons learned were:

1. Systems with unscheduled demand likely have significantly more variability in demand than one might intuitively expect.
2. Systems with low demand will have more (relative) variability than ones with larger demand.

That article (which we recommend you read first) focused solely on unscheduled patient visits (and their expected variability).? It did not consider length of stay (and its expected variability, particularly with long stay outliers).?

So, if we’re planning for a new hospital, how many acute care beds are needed to “fit” most days give the ups and downs in patient demand??

In this article we’ll attempt to provide some thoughts on this question using queueing theory.? Those that stick around to the end will also find a link to a simple simulation application they can use to see the theory in action.?

Note that though this article focused on inpatient hospital units, the same thinking can be applied to any clinic or service.

A beginners guide to target occupancy rates ????

Hospitals and their units often describe their current or historical busyness in terms of occupancy, which is just their average census (e.g., the number of patients present at midnight, or some other census time) as a ratio of their planned capacity.? A 20-bed unit that is running at 75% occupancy means on an average day that 75% of its capacity is occupied (i.e., 15 patients on average, with 5 beds “open”).?

Target occupancy rates are often set to help evaluate current hospital or unit performance in terms of its level of busyness.? If the target occupancy rate of the above unit was 90%, then one may conclude that the unit is not as “full” as we’d like it to ideally be.

Target occupancy rates are also often used in planning exercises to help determine future capacity needs. They can set the intended amount of “flex” the unit is allowed to meet variability in patient demand. A lower target occupancy implies more "flex" capacity.

Consider a unit with a planned average census of 13.7 patients at midnight.

• At 90% occupancy, 13.7 census ÷ 90% occupancy = 15.2 beds are indicated
• At 70% occupancy, 13.7 census ÷ 70% occupancy = 19.6 beds are indicated

Clearly the choice of target occupancy is extremely consequential; leaving aside operational factors, this is an over 4 bed difference in calculated need for a relatively small unit.?

So, what is the right target occupancy rate? And does it change depending on the kind of unit (e.g., an adult medical bed versus a pediatric medical bed)?

Traditional thinking about target unit occupancy rates ??

When it comes to target occupancy rates, many organizations and governments have a set of numbers in mind that probably look similar to this list:

• Medicine units should run at 90% to 95% occupancy.
• Surgery units should run at 85% to 90% occupancy.
• Intensive Care Units and Pediatric units should run at 70% occupancy.

In our experience in Canadian healthcare planning, these target occupancy rates are frequently used and non-controversial.

The lived experience of many hospitals is of course completely removed from even a 95% occupancy rate.? As an example, this data release by British Columbia’s Ministry of Health estimated the average bed in a BC hospital operated at 103% occupancy in 2020/21-Q3 (i.e., for every 100 beds there were 103 patients).?

For many managers “planning to run at only 95% occupancy” may sound like a dream.? One may also looks at the limited public funds and/or nursing resources and concludes a 95% target occupancy may be “the best we can do anyways”.

The supply of hospital beds (and the dollars and staff to go with them) is, of course, a reality; yet, failure to account for natural and expected variation in patient volumes carries their own significant costs, including longer lengths of stay and adverse patient and staff outcomes.

So, leaving supply considerations aside, what variation in demand should a hospital or unit(s) expect?

Operationalized (“staffed”) beds versus physical beds ???

One nuance that is sometimes overlooked in planning is distinguishing between operationalized bed need (or “staffed” beds) versus physical bed need.

In planning a new hospital, we may choose to build more physical beds than we typically plan to operationalize on a daily basis (sometimes called “flex” or “overcapacity” beds).? This provides the ability to provide access to care during extremely busy periods (such as the winter flu season) while providing some measure of fiscal prudence (i.e., not staffing beds that would be typically empty).

Going forward in this article, we assume that any estimates of total bed capacity need should consider:

1. An operationalized bed total to meet demand at least 75% of calendar days.
2. A physical bed total to meet demand at least 95% of calendar days (beds above the operationalized bed total would be staffed by redeployed nurses and/or additional temporary nursing, and likely paid at overtime rates).
3. The remaining 5% of days would require some patients be cared for in a non-ideal area, such as the Emergency Room.

To be clear: this is not suggesting target unit occupancy rates of 75%/95%, rather a service target to indicate sufficient beds to meet demand 75%/95% of days. Different projects can of course make different choices in their service targets.? The theory that follows though remains the same.

Queueing Theory to the rescue! ??????

Agner Krarup Erlang was a statistician who worked for the Danish telephone company in the early 1900s.? Telephone exchanges at that time required human operators to manually complete a circuit to connect callers together.? If an operator was unavailable, a customer would be unable to place a call.

Erlang’s work with this telephone company essentially involved estimating the number of operators that would be required to meet call demand throughout the day.? His work pioneered the field of Queueing Theory, which is the study of queues and wait times.? It provides us a mathematical framework to understand capacity needs (such as the number of inpatient beds) given desired service levels (e.g. “placing 75% of patients within 4 hours of arrival in an inpatient bed).

There are various models that we can use to estimate system capacity needs.? The one we will look at here is often referred to as an “M/M/c queue” (try not to worry about the technical sounding name) which has the following requirements:

1. New customers (patients) arrive at a rate λ according to a Poisson process (as described in "Part One").
2. Service times (length of stay) s have an exponential distribution (which has long "right tails" to recognize some 'customers' will have extreme 'outlier' service times -- one can see this in unscheduled length of stay histograms).
3. Customers are seen in a “first-in/first-out” (FIFO) order by c servers (beds). If there are fewer than c customers, then some servers (beds) will be idle (empty). If there are more than c customers (patients), then new arrivals will queue in a buffer.?
4. The buffer is of unlimited size.? Patients will not “drop out” of the queue for any reason.

There are formulas that describe the behaviours of an M/M/c queue which allow us to reverse engineer what bed capacity would be needed to meet a given service target, such as:

• Place 75% of patients within 4 hours of arrival (i.e., operational bed need).
• Place 95% of patients within 4 hours of arrival (i.e., physical bed need).

Using these formulas, the below chart illustrates how a unit’s target occupancy rate changes based on the average census of the unit (holding length of stay constant), allowing us to evaluate how many beds are needed to meet demand 75% and/or 95% of days.

(Note: this specific example assumes an average length of stay of 7.2 days, aiming to place 75% and 95% of new patient arrivals in a bed within 4 hours – but this general shape holds for different parameters).?

?

We can see in the above example to meet demand 75% of days that given a planned average census of…

• … 5 patients that a target occupancy of 63% is needed (i.e., 8 total beds).
• … 10 patients that a target occupancy of 71% is needed (i.e., 14 total beds).
• … 20 patients that a target occupancy of 80% is needed (i.e., 25 total beds).
• … 40 patients that a target occupancy of 87% is needed (i.e., 46 total beds).

… and so forth. As average census increases, target occupancy increases as well.

(To be clear: for larger hospitals we would treat similar or interchangeable units as if it was one large unit, even if physically they may be sub-divided).

These results mimic what we saw in our previous article with the Poisson distribution: smaller units require lower target occupancy rates (i.e., more ‘flex space’) than larger ones.? These target occupancy rates are often quite a bit lower than conventional wisdom may suggest.

To return then to our original question about target unit occupancy rates, Queueing Theory provides us a model to determine a unit’s needed capacity size based solely on its planned average census.

Note that we did not consider the kind of patient population in the Queueing Theory model.? Medicine, Mental Health, Pediatric, and Critical Care, etc.? populations are all treated the same (at least in terms of estimating bed need – staffing requirements will, of course, be much different).? As Critical Care and Pediatric units tend to have smaller census, this naturally results in lower target occupancy rates.

A model is nice, but what about the ‘real world’!? ??

Of course, real life is more complicated. ?For example:

1. Patients do get tired of waiting and will drop out if they wait too long.
2. Arrivals and length of stay at a hospital are very context-dependent on provider availability, day of week, and overall unit or hospital pressures at the moment.
3. When hospitals get overwhelmed that may divert patients (e.g., ambulances going on ‘bypass’) and/or transfer patients to other sites with capacity.

One could argue though that these context-specific situations, if anything, make a queuing theory model underestimate true need, particularly given unmet demand (due to long wait times), day of week variability, and the cost of hospital to hospital transfers.?

Can I see these concepts in my hospital data? ??

Absolutely!? Here’s what you need to do:

1. Gather daily census data (one year or longer) for a given patient population. Make sure that you include all patients in the hospital, including off-service patients on other units or patients that are present in the Emergency Department. Include days with 0 patients too as a data point.
2. Calculate the average census A.
3. Calculate the 75th percentile census B and 95th percentile census C.?
4. Divide the average census into the 75th percentile census (i.e., A÷B).? This is the target unit occupancy required to meet operational demand 75% of days.
5. Divide the average census into the 95th percentile census (i.e., A÷C).? This is the target unit occupancy required to meet physical demand 95% of days.

Note that we would be happy to work with an organization to demonstrate these trends in their data for free if they are open to sharing the results in a follow-up to this article.

I don’t have time for that! ??

Fair enough!? We’ve developed a simple application that, given an average number of admissions and length of stay, will simulate a 3-year period of a hospital census.?

In addition to providing a simulation result of capacity need to fit 75% and 95% of days, it also performs the above mentioned M/M/c queue calculations to provide a theoretical result as well.

The application can be accessed at https://johnlarusic.github.io/hospital-census/.? Source code is freely available on GitHub under a MIT license.

What are the implications for healthcare system planning?

1. We should not confuse planning for operationalized/staffed beds with planning for physical beds. Sites need beds they can temporarily open/flex into so as to meet expected variation in patient demand.
2. The amount of planned "flex" for a unit should be based on its size, not its function (acknowledging operational concerns, such as staffing ratios, play a role as well).
3. Smaller hospitals and units require "more flex" (i.e., lower target occupancy rates) than larger ones.
4. The more you silo or segregate patient populations, the more beds (and likely staff) will be needed.

What’s next? ??

In our final article, we’ll see about trying to understand the impact of day of week variations in admissions and discharges, as well as better understanding the bed needs of a scheduled surgical population.

About the author ????

John LaRusic is the Principal of Compass Healthcare, a Canadian management consulting firm that specializes in analytics and modeling to support clinical service planning.

Comments are corrections are very much appreciated: [email protected].