A little number crunching can show hospitals how many beds and staff members they really need.
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| David Ollier Weber |
Editor’s note: This is the first part in a two-part series on handling patient flow. Today the author discusses how mathematical formulas can help manage patient peaks and valleys. Next week, he will show how some variation can be smoothed out simply through better time management.
N = λT. That’s a good one, huh? Basic. It’s called Little’s law, and it enables hospital or clinic administrators to figure out how many patients (N) can be served when λ is their average arrival rate and T is the average total time each one spends waiting and being seen by caregivers.
Then there’s the Poisson distribution that describes the probability of unscheduled patient arrivals in any interval of time--the next 10 minutes, an hour, three hours, whatever:
p = (λ)n e-λ /n!
Umm . . . you seem to be nodding off. Those formulas are just so much gibberish, you say. You don’t need any stinking formulas. Like many health care leaders, you’re a people-person, intuitive, a brainstormer, a benchmarker.
Too bad. Really too bad. Those and similar equations are the theoretical backbone of many businesses like yours, in which random customer demand must be balanced against fixed capacity. These businesses include airlines, banks, telephone companies . . . indeed, the Internet could not exist without the application of such formulas. They’re the fundaments of what’s known as queuing theory.
But, as usual, your sector is only belatedly catching on to the one tool that can guide you through the problem that lies at the heart of all your problems--how to juggle staffing, beds and financial resources to match constantly fluctuating patient flows.
Quantifying Traffic Flow
Queuing (also spelled queueing) theory traces its origins to the Copenhagen Telephone Exchange at the turn of the 20th century. A brilliant young mathematician named Agner Erlang was employed to calculate how many circuits and operators would be needed to handle a given volume of telephone calls at an acceptable connection rate. (At the time, operators had to manually plug a jack into a switchboard.)
Erlang’s 1909 paper “The Theory of Probabilities and Telephone Conversations” laid the groundwork for the modern telecommunications and computer industries. He was honored for his pioneering insights into how to deal with queues--waiting lines that can form unpredictably, whether made up of packets of electrons, restaurant diners, bank depositors, travelers or emergency room patients--by having a statistical unit named after him: the erlang. The erlang, a measure of traffic, can be used to gauge the adequacy of resources allocated to a system.
Eugene Litvak, an amiable professor of health care and operations management at Boston University, director of its Program for the Management of Variability in Health Care Delivery and an adjunct professor at the Harvard School of Public Health, remembers some years ago asking the head of a large emergency department why the hospital didn’t apply queuing theory to figure out the level of resources needed to handle unpredictable patient arrivals.
“Thank you very much, Dr. Litvak,” the physician responded.
Litvak was startled. “For what?” he asked in the accent, rich as a plump piroshki, that attests to his origins and his doctorate in operations research from the Moscow Institute of Physics and Technology.
“Thanks for assuming,” the executive smiled, “that I’d know what you’re talking about.” He hadn’t a clue.
A Powerful Tool
Ten years on, not much has changed. At least, laments Litvak, not in the upper echelons of health care. But in conjunction with the Institute for Healthcare Improvement, he is now “trying to the best of my ability to create a certain community [of people conversant with the principles of queuing theory] to be educators at their own organizations.”
On a recent weekday morning, 36 health system representatives from 20 states, Canada and New Zealand wrapped up what they’d learned from Litvak in an IHI-sponsored, three-session, Web-cum-telephone seminar titled Queuing for Clinicians. They took turns describing a local patient flow issue to which they’d applied the queuing formulas he’d taught them.
Litvak patiently analyzed their scenarios and offered challenges such as, “Why don’t we think about whether queuing theory can apply at all here? You have to have a random demand. So . . . is your discharge activity random? I’d suggest not. Random demand is not a good thing, and discharges should not be random. Many hospitals schedule them. Couldn’t you?”
Litvak has trepidations about the damage that might be wrought by introducing so complex and heavily mathematical a discipline as queuing theory to an audience of health care managers--desperate, fad-prone and likely to swallow it half-chewed.
“There are probably 40 models” based on different queue management goals and service conditions, he explains, “so it’s better to be ignorant than to apply the wrong model.”
Still, for decision-makers struggling to pinpoint the exact number of beds, technology and personnel they need to cope with the vicissitudes of birth, contagion, human decrepitude and violence, “queuing models,” exclaims Litvak, “are the only solution! There are no other ways one can figure out the capacity needed given random demand. So to people who say they don’t like formulas, I say, ‘Then you don’t like figuring out your problem.’
“That doesn’t mean,” he adds quickly, “that the hospital CEO needs to sit and learn lambda [how to use the formulas]. But he should know queuing theory is available and hire the right people. This is a powerful tool. It really performs miracles.”
Next week: Math averse? Don’t despair. Before you’ll ever need to apply queuing theory, Litvak promises you can eliminate most of your patient flow problems by attacking the predictable peaks and valleys.
David Ollier Weber is principal of The Kila Springs Group in Mendocino, Calif. He is also a regular contributor to H&HN OnLine.
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This article 1st appeared on May 10, 2006 in HHN Magazine online site.
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