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A new Rx for crowded hospitals: Math

Operations management expert brings queuing theory to health care

From the December ACP Hospitalist, copyright © 2007 by the American College of Physicians

By Stacey Butterfield

Eugene Litvak, PhD, wants to make U.S. hospitals more efficient. As director of the program for management of variability in health care delivery at the Boston University Health Policy Institute, Dr. Litvak conducts research and advises hospital leaders on the use of operations management techniques to control health care costs and improve quality of care.

Queuing theory, a mathematical formula used by operations management experts in many fields, is one of the most critical tools available for improving hospital efficiency, Dr. Litvak said. He recently taught an Institute for Healthcare Improvement seminar on how hospitals can use queuing theory to more closely align their fixed capacity (from the number of beds and CT scanners to the size of hospitalist staffs) with patient demand.

Dr. Litvak spoke with ACP Hospitalist about what queuing theory is and why everyone in the health care industry needs to know more about it.

Q: What is queuing theory and how is it used?

A: Queuing theory is a methodology that helps match random demand to fixed capacity. Queuing theory was first used in telecommunications and then was adopted by all major industries, like airlines, the Internet and most service-delivery organizations. However, in the health care industry, when we had unlimited cash flow, nobody cared very much about how to match demand and capacity. We used to throw around as much money as we needed to meet demand and not necessarily worry about being efficient or frugal. I would say it's probably the last industry where queuing theory has not been used, until very recently.

Q: What kinds of problems can queuing theory solve for hospitals?

A: A lot. For example, how many nurses do we need to triage in the emergency department for different parts of the day or different days of the week? How many beds do we need? To build one bed in the hospital, the capital investment exceeds $1 million, not to mention the operating cost that usually exceeds a quarter-million annually. That's the type of money we are talking about. When you don't have extra funds, as we don't have now in health care, you better be wise in how you are using them.

Q: Could you give an example?

A: Let's say you have one ICU with five beds, another with 10. The patient acuity, and therefore the average length of stay, is the same for both units—let's say 2.5 days. For the unit with five beds, the average demand is one patient per day. In the larger unit, the average number of patients who need admission per day is two. What would be the waiting time for these units?

Our first reaction is to say the waiting time would be the same, because the capacity of the larger ICU is twice is big and the demand is twice as big. In fact the difference is tenfold—the waiting time for the larger unit is ten times shorter.

[This] means that when we are talking about baby boomers coming and we say we are going to have 20% more patients in five years, it doesn't mean that you need 20% more beds or nurses, you need much less. It also means that you cannot benchmark small and large hospitals, even if everything is the same. The large hospital has an advantage just because of its size. That's very counterintuitive, and in health care we are used to relying on intuition.

Q: How much interest have you gotten from hospitals in applying the theory?

A: I started this journey trying to convince hospitals to do it about 10 years ago. Every other industry was using [it]. If you compare us with a "normal" industry, we are probably 50 years behind in using operations management and queuing theory as a part of operations management.

When I started presenting queuing theory to the health care industry, I was saying that nothing could be worse than health care not using [it]. Since then it's become a fashionable subject. When I present now, I say that nothing could be worse than health care using queuing theory. It is frequently misused due to a lack of understanding.

Q: How is it misused?

A: Most of the models that people use in health care queuing theory assume random demand. When the patient comes to the ED because he breaks his leg, that's random appearance. Scheduled admissions are not random. In our study of over 100 hospitals in the U.S., Europe, Australia and other countries, we found that arrivals for scheduled admission like surgery are less predictable than when a patient breaks a leg and comes to the ED. Our scheduled admissions fluctuate more than admissions in the ED. Does that make them random? No. Those fluctuations are determined by individual priorities rather than patient needs. It's unpredictable, it's highly variable, but it's not random, so the models do not work. If you enter the wrong data, you will get the wrong outcome.

In health care, queuing theory should come together with variability methodology. If you look at hospital admissions, you'll find [that they are] extremely variable. If you apply queuing theory to the overall demand, you will be making a mistake. You have to first eliminate your artificial ups and downs. Once you eliminate those artificial peaks, and you are left only with natural patient-driven variability, then you can apply queuing theory. You cannot apply queuing theory to your overall patient load because you will get meaningless results. You can apply it only after you have smoothed your elective admissions. After your elective admissions are more or less stable from day to day, then is the time to apply a queuing model to unscheduled demand.

Q: How does a hospital get started on this process?

A: At this point, they need help. Many hospitals have operations improvement teams. It's very important that those teams have analysts. But when I was [beginning to teach] queuing and we were thinking about who should be invited, we encouraged hospital executives to come with their analysts. Why are analysts alone not enough? If you just have data analysts doing calculations, even very accurate ones, it is very difficult for [them] to convince a hospital CEO that she should make decisions that could result in multi-million-dollar changes. The executives don't know the formulas or do the calculations, and they do not have to, but they have to go through basic education to understand the kinds of problems that can be resolved. If you exclude [CEOs] from the process, the analysts would do their calculations but they will not be used for decision-making.

Q: How can physicians use queuing theory?

A: If you say that we have to hire more physicians or nurses for this unit, prove it to me. Tell me exactly how many. Everybody wants more. The difference that queuing theory makes is that it allows you to justify your demand. This is not limited to the hospital environment. Queuing theory has as much application in the PCP office. What is the waiting time? How many exam rooms do you have to have in your office so that waiting time does not exceed one hour, two hours?

Q: What other health care issues could be improved by the theory?

A: EDs all over the country are overcrowded. You talk to any single director of an ED and ask, Why is your ED overcrowded? Their answer would be 'Because we don't have enough ICU or floor beds,' right? So the question is 'How many do you need?' Hundreds, thousands, three, 15? If you open an extra 10 beds because you don't know how many you need, then you are talking about an extra $10 million right away plus $2 to $8 million annually. Believe me when I say that a mistake on 10 beds is not realistic. The mistakes that hospitals are making are much, much in excess of 10 beds.

We have a shortage of nurses—another national problem in health care. How do you decide how many nurses you need objectively based on patient demand? If you want to figure out how many nurses do you need in a unit, how many beds do you need in a unit, how many physicians do you need, there is absolutely no way to do that without queuing theory. Everything else would be guessing.

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