*ACP Hospitalist*content is available exclusively to ACP Members. This article is free to the public.

# Little's Law and you

To understand what's really happening in your hospital, you need to consider Little's Law.

It's been a really long day, and it's only 10 a.m. You look at the number of patients on your service, and it just keeps climbing. You look at your colleagues and they all have that beleaguered, the-census-is-too-high, don't-mess-with-me look on their faces. The emergency department wait time is escalating, and they are admitting patients who could be taken care of somewhere else. Anywhere else. The time frame for imaging studies is glacial; the speed of the consult services would make a snail proud.

Yet when the case manager comes to you and asks about discharging one of your patients,
you groan. You would rather have them stay an extra day or 2 than deal with a discharge.
What difference will it make if the hospital is packed anyway? (See “Top 10 reasons not to discharge your patient” in the October 2011 *ACP Hospitalist*.) To understand what's really happening in your hospital, you need to consider Little's
Law.

John Dutton Conant Little was born in 1928 and eventually became a professor at MIT. In studying inventory and flow, he developed what became known as Little's Law: The average number of customers in a system (over some interval) is equal to their average arrival rate, multiplied by their average time in the system. Little's Law can be expressed by the formula I=RT, where I is the inventory, R is the arrival rate, and T is the average processing time.

Goodness, you think, what could that possibly have to do with me and my service? We are dealing with patients, not widgets or cars. But the rule still applies. Instead of arrival rate, think admissions. Instead of processing time, think length of stay (LOS), and of course inventory is your hospital census! Understanding this basic formula can impact how you deal with and understand your overfilled facility.

Let's start in the ED, where capacity, scheduling, staffing, and flow issues are universal and throughput times are a core quality metric. An ED director who has data on the average patient arrival rate and average census metrics can determine the average time patients spend in the ED, potentially identifying targets for improved efficiencies. Little's Law can also be applied to predict demand for ancillary services when standardized times are available and the arrival rate is known. This is particularly useful for imaging, which often becomes a bottleneck in patient flow, especially with capacity limitations or overutilization.

But still, you are a hospitalist—do you really care about ED quality metrics and imaging utilization? The answer is yes, as these affect your own input and throughput of patients. Suppose you have an average LOS of 6 days and run an average census of 15 patients. How many admissions are you likely to get a day? I=RT, 15 patients=6 days × R. Therefore you average 2.5 admissions a day.

Now let's get back to that case manager trying to get you to discharge a patient. Suppose that if all the hospitalists in your group comply, you can drop the average LOS to 5 days. You are still getting 2.5 admissions per day, but now your average census will be 2.5 × 5=12.5 patients. Your service size just dropped from 15 to 12.5! And if it's applied to the whole hospital, perhaps the average census dropped from 750 to 625. Resources become more available, tests get done more quickly, and suddenly the inpatient exodus can begin. However, this is not a closed system. This kind of math only works and has this effect if the speed of care doesn't overwhelm the nurses and other resources and if staffing ratio metrics are adjusted to account for this shorter length of stay. There's also the potential risk that the drop in LOS causes the readmission rate to climb.

A useful offshoot of Little's Law is the takt time. “Takt” is the German word for beat and represents the pace of the process. It is the reciprocal of the throughput (admissions) and the maximum time the team can devote to a patient to keep up with demand. Suppose there are 20 admissions between 4 and 6 p.m. The takt time is 2/20 or 6 minutes (1 patient admitted every 6 minutes). If the average admission takes 60 minutes, you would need T/(1/R) or (60/6)=10 clinicians to admit these patients. Of course you know that patients generally don't agree to be admitted on a schedule of every 6 minutes—usually it's 10 admissions in 5 minutes and then nothing for 2 hours!

Understanding the flow of patients through the hospital and using that knowledge to identify bottlenecks, predict staffing needs, and make sense of the census is an important skill. And next time the LOS drops and the census falls, you'll know it wasn't a random effect, but Little's Law.