By Phillip M. Stephens, DHSc, PA-C
Understanding overcrowded emergency departments is an elusive exercise, but an important one. Emergency department managers are coming to understand it is largely a math problem, with concepts like “capacity over demand” appearing in current literature. But the constructs still are not widely implemented. Higher math applied with dynamic rather than static applications is the best utilization of these mathematical models, and we must educate managers about these new paradigms. What are these new ways of analyzing flow?
A decade-old conceptual model describes analyzing the problem by input, throughput and output methodologies. But even these models lack the arithmetical power to convey the necessary meaning.
The unique formulas of game theory proven by mathematician John von Neumann and applied by professor John Nash , which are used to understand scenarios of conflict and cooperation, are more appropriate, as human behavior, however difficult to measure, must be factored into any equation. Personal interactions, with their substantial effects on ED patient flow, are simply more complex than the literature suggests.
Consider our local emergency department designed for around 200 patients in a 24-hour period. Typically, this number is often exceeded. But even when it is not, wait times are unpredictable. Why?
Understanding Unpredictable Wait Times
With a 40-bed capacity, a 200-patient per day demand requires each bed to be turned over 5 times during a 24-hour period – or every 4.8 hours. That seems like a reasonably achievable goal, but rarely is the department able to function at exactly 100% capacity, and there the problem begins.
Typically, the department is functioning at between 40 and 60 percent of capacity. To simplify the math, let’s assume an average of 50% capacity, meaning half the beds are being used for boarding. Boarding is a national problem best understood as a facility-wide issue, but typically treated as solely an ED problem. It’s a huge mistake to approach boarding departmentally rather than system-wide–but that’s another discussion. Today we are just doing math, not psychology.
At 50% capacity, the department is down to 20 open beds that now must be turned 10 times in a 24-hour period or once every 2.4 hours. The impact of the boarding problem on efficiency is then clear, substantial and pervasive. But again, this is overly simplistic. Managers tend to understand the construct to this point but ignore the even more pervasive influence of variation. Understanding variation requires higher math.
The higher math is not too painful. Where k is the constant of variation and y varies inversely to x by the same factor, we represent this inverse variation by the formula xy=k. Now rarely do emergency departments experience this simplistic algebraic form, as volume cycles are not proportional. But the three points are:
- Volume is dynamic
- Volume occurs in waves
- Volume’s impact is complex.
Volume rises at one period during the 24-hour cycle while asymmetrically falling during another portion. But rather than apply algebraic principles to staffing and planning in dynamic fashion by matching resources to flow, many managers simply plan for the mean. Planning a department based on the mean results in appropriate staffing only half the time. Managers envision a bell curve when planning for the mean when a calculus sine wave provides a more accurate depiction of the reality. If boarding is the pervasive problem, variation is the elusive problem. It’s like hitting a moving target and is a formulaic fact we must teach administrators. Managers see numbers and pretend the numbers mean something in a vacuum. But external influences and cycles must also be factored.
Consider our example emergency department once again. In 2011, the average arrival rate per hour was 8.54 patients, equaling around 204 patients each 24-hour period. However, the variation was quite broad. The lowest hourly arrival rate was over a narrow period of time between 3 a.m. and 6 a.m. with an average of 3 patients per hour. The highest arrival rate was longer and sustained. Between 9 a.m. and 6 p.m. the arrival rate was 11 patients per hour. That’s broad variation. Staffing to address a volume of 8 patients an hour is insufficient when capacity over demand modeling indicates that expecting no fewer than 12 patients an hour is required for optimum system performance.
Imagine an ED that sees an average of 8 patients an hour and staffs for 8.3. There already is little wiggle room when 0.3 more arrive than staffing permits, and 0.3 adds up. And when 11 patients arrive in an hour and the situation is compounded by a problem such as boarding that has already diminished capacity, patient volume cascades into the lobby.
But this discussion has focused on only two long-standing variables – boarding and variation. There are many newer concepts we must teach administrators who help manage departments and incorporate them into our thinking and analysis of department efficiency. The new paradigm is a comprehensive view of the multitude of variables that disturb flow.
For example, despite the broad variation of patient arrival, departments rarely feel the broad variation; even when arrival times reach a low of only 3 an hour; patients are still backed up from the previous peak. Why isn’t the variation felt as it seems busy all the time?
Why do things back up when efficiency studies and supporting departments say our departments are at top efficiency, funding and staffing? The answer is found in multidisciplinary research.
Automobile Traffic Research Sheds Some Light
The National Science Foundation funded traffic flow pattern studies. Traffic engineers published the results in New Scientist Magazine . Even in deterministic models, which predicted uniform traffic flow, phantom traffic jams still occurred when a critical threshold was reached, seemingly for no reason.
Automobile drivers making small, preventive-driving corrections (human behavior) resulted in waves of traffic jams not unlike waves seen in water due to small perturbations that amplify themselves. Natural human behavior had been difficult to mathematically factor. The amplification of small behaviors results in a mathematical constant, whether in water, traffic or emergency departments. Perturbation theory has become a mathematical concept all its own. Anything that has flow is affected by small perturbations that amplify through systems. It is why one vehicle suddenly slowing from 55 mph to 50 mph by human reaction causes traffic to stand still 20 miles behind it if the viscosity of vehicles is at a critical threshold.
This also is why game theory models are proving more accurate in predicting ED flow, as they account for human behavior. Perturbations occur in the actual flow, and their causes are complex and human:
- Incentive structures adjust between hourly versus productivity pay,
- Individual provider efficiency varies,
- Staffing motivation fluctuates, and
- Patient acuity levels change.
Even so, a recent study at our facility revealed that despite the multitude of factors that affect length of stay, volume alone explained nearly one-third of the variation (r2 = 0.28). This does of course mean that ED flow remains largely dependent on the number of patients who arrive seeking care. Nevertheless, we have the potential–and on behalf of our patients and our staff, the duty– to influence the remaining two-thirds of what we can actually control. We all see the negative impact that small factors can have on patient flow; we must remain determined to detect and correct those that can have a positive impact.
The ironic downside to these new paradigms of analyzing flow is that if we solved all of these analytical problems, there is yet another variable of human nature known as the Pigou-Knight-Downs paradox, which takes effect. Traffic engineers discovered that traffic flow did not in fact improve when they fixed the apparent mathematical problem by building new bridges designed to have the capacity to eliminate the backups that were occurring. In short, they demonstrated that increasing the capacity of a bridge to any value less than twice the traffic flow has no ultimate effect on travel time. Why? “If you build it, they will come.” This means the ED that effectively solves many of its flow problems will realize more flow to its department – just as people adjust driving patterns to a newer, bigger bridge to get to work–and throughput times may paradoxically be no better than before.
The overarching point is that we don’t currently measure emergency department efficiency with the needed sophistication.
We also, despite sharing similar problems, do not approach flow with the sophistication of the traffic engineers. We simply measure what is easiest to measure.
By increasing the sophistication of our methodology for understanding flow, and by subsequently educating administrators about these new paradigms, we can instill a new way of thinking about how we manage ED flow. Paradoxes notwithstanding, the effort will be worth it–for our hospitals, for ourselves, and for our patients.
Phillip Stephens, DHSc, PA-C, is the associate practitioner site director for Emergency Medical Associates at Southeastern Regional Medical Center, Lumberton, N.C. He is adjunct faculty at A.T. Still University in Mesa, Ariz., and has practiced as an emergency medicine physician assistant for more than 20 years.