1. Emergency-Departments Simulation in Support of Service-Engineering: Staffing Over Varying Horizons
Yariv N. Marmor1, Avishai Mandelbaum1, Sergey Zeltyn2
1Technion – Israel Institute of Technology
2IBM Haifa Research Labs
Good morning.
My name is Yariv and I am a PhD student at the Technion.
I am going to talk about a joint project of IBM research lab in Haifa and the Technion, specifically the parts of my PhD from this project.
And the theme of the talk is “Toward Simulation-Based Real-Time Decision-Support System for Emergency Department”.
16th Industrial Engineering and Management Conference ( )

2. Motivation - ED overcrowding
Staff (re)scheduling (off-line) using simulation:
Sinreich and Jabali (2007) – maintaining steady utilization.
Badri and Hollingsworth (1993), Beaulieu et al. (2000) – reducing Average Length of Stay (ALOS).
Raising also the patients' view: Quality of care
Green (2008) – reducing waiting times (also the time to first encounter with a physician).
Introduction
As a background, it is a well known fact that Emergency Departments are over-crowded and struggling to provide reasonable care in a chaotic environment.
There exists plenty of research dealing with ED overcrowding, for example:
Off-line Staff (re)scheduling by using simulation to maintain steady utilization, or by aiming at reducing Average Length of Stay.
Another possibility would be to look on Alternative operational ED designs, again, aiming mostly at reducing Average Length of Stay.
A more recent approach, arising from the Service Engineering field, raises also the patients‘ point of view or Quality of care, which is manifested through reducing waiting times (or alternatively, the time to first encounter with a physician). 2

3. Simulation support: short- to long-term
Part 1 (short-term): Decision-Support system
(ED Staffing) in real-time (hours, shift).
Part 2 (medium) (Staffing) Over mid-term (weeks).
Introduction
To put my talk in an appropriate perspective, here I am listing the main topics of my PhD research, which is supervised by Prof. Avi Mandelbaum from the Technion:
Part 1, introduces a decision-support system for supporting staffing of the ED in Real time (hour resolution, say up to a single shift).
Part 2 extends the staffing problem to the medium horizon, namely weeks, to account, as an example, for flu epidemics.
And Part 3,deals with long-time horizon – , namely the strategic problem of choosing the most efficient operating model, given an ED environment.
My lecture will focus on Part 1. If time permits, or if questions will be asked, I shall touch also on the other two parts. 3

4. Part 1: Decision-Support system for Intraday staffing in real-time
The staffing problem seeks to assign medical staff, doctors and nurses, at the appropriate times in appropriate numbers.
Staffing schedules are determined per day, which is referred to as intraday staffing.
We wish to do that in real-time, which means that we can update our schedules in response to short-term changes in the state of our ED.
For example, if our forecast predicts a surge in arrivals in the evening, we wish to prepare for that in the morning. 4

5. Objectives [Gather real data in real-time regarding current state]
Complete the data when necessary via simulation.
Predict short-term evolution (workload) via simulation.
Corrective staffing, if needed, via simulation and mathematical models.
All the above in real-time or close to real-time
Part 1: Intraday staffing in real-time
For real-time staffing we need the following:
First – accurate data in real time regarding the current state of our ED. Such data is crucial for any near-future staffing decisions.
However, rarely if at all, full accurate data is available, and we are likely to have only partial ED data. As will be explained, we complete the missing data with the help of simulation, .
Then, one must predict the short-term evolution of the ED, most importantly the future workload.
If we don’t like what we see, we let the this workload determine corrective staffing levels, using mathematical models and simulation again.
And, as stated, we wish to achieve all that in real-time, or close to real-time. 5

6. Estimation of current ED state
Goal – Estimate current ED state (using simulation):
For each patient: type (e.g. internal, ….) and status in the ED process (e.g. X-ray, Lab,…)
[status un-extractable from most currently installed ED IT systems]
Data description:
Accurate data - arrival and home-discharge processes.
Inaccurate (censored) data - departure times for delayed ED-to-Ward transfers (recorded as departures but are still in an ED bed)
Unavailable data – all the rest (e.g. patients status).
Method to estimate present state:
Run ED simulation from “t=-∞”; keep replications that are consistent with the observed data (# of discharged)
Part 1: Intraday staffing in real-time
As I already said, the first objective is to estimate the current ED state, using the limited data available.
For each patient, we need the type (if it is internal, walking or acute, etc.), and his or her status in the ED process (for example, is he waiting for X-ray, are the results of her lab tests due to arrive).
Such detailed data is unavailable in most currently installed ED systems, hence we infer it via simulation. (Let me comment that we are hoping to install, at least as a pilot, an RFID system that, when implemented, could provide such information in the future.)
To be more specific, for each patient we know the type and arrival time;
Then the system looses track of the patient during the ED process.
And for those patients who are supposed to be hospitalized, there are many who are first delayed in the ED due to unavailable beds, or some other reason. Such patients are recorded as already hospitalized in their receiving ward, (there are economic and other reasons for that), however they still occupy a bed in the ED.
To infer the ED present state, we run the simulation from a “remote past”, using actual observed data;
then we keep only the replications that are consistent with the observed data, that is the number of discharges from the ED.
These replications give us a distribution on the present ED state, which then serves us in simulating the future ED evolution. 6

7. Required staffing level – short-term prediction
Staffing models:
RCCP (Rough Cut Capacity Planning) – Heuristic model aiming at operational-efficiency (resource utilization level).
Arrival Time 15 15
15 minutes t
OL (Offered Load) - Heuristic model aiming at balancing high levels of service-quality (time till first encounter with a physician) and operational-efficiency (resource utilization). 15 t
Part 1: Intraday staffing in real-time
For assisting the staffing problem in the ED we consider two heuristics.
The first on is borrowed from manufacturing staffing models. It is called Rough Cut Capacity Planning model (abbreviated to RCCP). This heuristic aims at maintaining reasonable levels of resource utilization.
RCCP loads the entire workload from a patients at the moment of arrival. For instance, suppose that, on average, patients need 3 meetings with a physician, each taking 15 minutes. Then, all 45 minutes of work are loaded at the time of arrival.
In contrast, the Offered Load heuristic aims at balancing the operational with service-quality points of view. For example, these 3 meetings will be loaded at their expected times. And to do this properly, one needs simulation.
In short, RCCP tracks the arrivals to the system, while OL tracks the workload within the system, which we shall show is a better way of doing things. 7

8. OL: Offered-Load (theory)
In the simplest time-homogeneous steady-state case:
R - the offered load is:
l – arrival rate,
E(S) – mean service time,
“Square-Root Safety Staffing" rule: (Erlang 1914, Halfin & Whitt ,1981):
b > 0 is a “tuning” parameter.
Part 1: Intraday staffing in real-time
In its basic form, the Offered Load theory is known to most engineers as Little’s law: in steady-state, the Load R (or the average number of patients in the system) would be the arrivals rate lambda, multiplied by the average length of stay E(S) (E of S).
The “Square-Root Safety Staffing" rule, already known to Erlang 100 years ago, recommends a staffing level n, which adds to the load a tuning parameter beta, multiply by the square-root of the workload. It turns out that this carefully balances service quality with resource utilization, both achieving high levels.
The resulting operational regime is now called the Quality- and Efficiency-Driven Regime, or QED for short.
[QED in Mathematics = Quad Erat Demonstrandum = mashal
QED in Physics = Quantum Electro Dynamics, which is what Richard Feynman got the Nobel prize for.]
Gives rise to Quality and Efficiency-Driven (QED) operational performance: carefully balances high service-quality (time to first-encounter) with high resource-efficiency (utilization levels). 8

9. Offered-Load (theory), time-inhomogeneous
Arrivals can be modeled by a time-inhomogeneous Poisson process, with arrival rate l(t); t ≥ 0:
OL is calculated as the number of busy-servers (or served-customers), in a corresponding system with an infinite number of servers (Feldman et al. ,2008). For example,
S - (generic) service time.
Part 1: Intraday staffing in real-time
However, in the ED, arrivals are better modeled by an inhomogeneous Poisson process, with arrival rate that depends on the time of day. For this scenario, the offered load formula must be re-interpreted.
One way is to calculate it as the number of busy-servers (which is equals to the number of served-customers), in a system with an infinite number of servers, namely no capacity constraints. A little thought will convince you that this is a reasonable definition for the workload that arrives to the system.
For simple models, we have mathematical expressions for the offered load – here are two of them. (These can be interpreted as time-varying versions of Little’s Law). However, in the complex ED world, one can only calculate it by/through/via simulation with infinite number of servers. 9

10. Offered Load (theory): time-inhomogeneous
QED-staffing approximation, achieving service goal a:
nr(t) - recommended number of resource r at time t, using OL.
a - fraction of patients that start service within T time units,
Wq – patients waiting-time for service by resource r,
h(bt) – the Halfin-Whitt function (Halfin and Whitt ,1981),
Part 1: Intraday staffing in real-time
Now we can link the service goad alhpa, manifested by the fraction of patients that are first seen by a physician within T time units (or that their waiting time Wq is less than T), with the recommended number of units of resource r, n_r, at time t.
Here we are using asymptotic approximations for 1 minus alpha, coming out of the work by Halfin-Whitt.
This provides us with tuning parameter beta, which finally determines staffing leves. 10 10

11. Offered Load methodology for ED staffing
∞ servers: simulation run with “infinitely-many” resources (e.g. physicians, or nurses, or both).
Offered-Load: for each resource r, and each hour t, calculate the number of busy resources (= total work).
Use this value as an estimate for the offered load R(t) of resource r at time t (averaging over simulation runs).
Staffing: for each hour t we deduce a recommended staffing level nr(t) via the formula:
Part 1: Intraday staffing in real-time
How do we apply the OL methodolody?
First, we use “infinitely-many” resources in our simulation model.
Resources are physicians or nurses, in our case.
We run the simulation, calculating for each hour t the number of busy resources (which equals the total workload that requires processing).
We use this value as our estimation for the offered load R(t) for resource r at time t. (The final value of R(t) is a function that iss calculated by averaging over simulation runs).
Then, given the offered load, staffing levels are determined, for each hour t, by the formula we saw in the previous slide. 11

12. Methodology for short-term forecasting and staffing
Our simulation-based methodology for short-term staffing levels, over 8 future hours (shift):
Initialize the simulation with the current ED state.
Use the average arrival rate, to generate stochastic arrivals in the simulation.
Simulate and collect data every hour, over 8 future hours, using infinite resources (nurses, physicians).
From Step 3, calculate staffing recommendations, both nr(RCCP,t) and nr(OL,t).
Run the simulation from the current ED state with the recommended staffing (and existing staffing).
Calculate performance measures.
Part 1: Intraday staffing in real-time
Our simulation-based methodology for short-term staffing levels, over 8 future hours is as follows:
We first Initializing the simulation with the current ED state, that we estimate as explained before.
Then we use the average arrival rate, to generate time-varying stochastic arrivals in the simulation.
We simulate and collect data every hour, over 8 future hours, using infinite resources (nurses and physicians).
Then, from Step 3, we calculate staffing recommendations, both for RCCP and for the Offered Load method.
We run the simulation from the current ED state with the recommended staffing (and existing staffing).
Finally, we calculate performance measurement. 12

13. Simulation experiment – current state (# patients)
n=100 replications, Avg-simulation average, SD-simulation standard deviation, UB=Avg+1.96*SD, LB=Avg-1.96*SD, WIP-number of patients from the database
Comparing the Database with the simulated ED current-state (Weekdays and Weekends)
Part 1: Intraday staffing in real-time
Let me now describe examples of our simulation experiments.
Here we run simulations that estimate the current state – we validated it against real data, and found out it worked fine:
The red line represents the simulation average number of patients, as it evolves in time.
The blue is the number of patients in the database.
The other lines are confidence intervals.
And you can see that, most of the time, the simulation captures the ED data very well. 13

14. Experiment – performance of future shift
Utilization:
Ip - Internal physician
Sp - Surgical physician
Op - Orthopedic physician
Nu - Nurses.
Used Resources (avg.):
#Beds – Patient’s beds,
#Chairs – Patient’s chairs.
Service Quality:
%W - % of patients getting physician service within 0.5 hour from arrival (effective of a).
Part 1: Intraday staffing in real-time
The simulation is used, of course, to estimates more than just the number of patients, for example utilization levels of physicians, the number of beds being used, and so on.
Recall that service quality is measured by the fraction of patients that see a physician within 30 minutes of arrivals: we denote it by %W (percent W).
Here we summarize simulations with real data until 16:00, and then we continued to simulate for 1 shift, which is 8 hours more.
Staffing levels are those used at the ED.
We see that the internal physician-utilization is the highest.
here it is varying from 7% in the morning to 77% at noon, then it declines back towards midnight. 14

15. Simulation experiment – staffing recommendations
Staffing levels (current and recommended)
Part 1: Intraday staffing in real-time
In this table, we present the summary of the current ED staffing levels, that we obtained from our data.
The next block is the Offered Load, coming out of simulating infinitely many resources.
This block is the recommended number of resources, physicians and nurses, using the Offered load heuristic, and here is the RCCP heuristics. 15

16. Simulation experiments – comparison
Part 1: Intraday staffing in real-time
RCCP is a very simple methodology. It is therefore interesting to investigate if OL has any advantage over it.
Here we compare these two methods.
First, we note that both methods achieved their goals.
RCCP reached utilization levels of near 90%, but only for multiple resources – here Internal Physicians and Nurses.
For the Offered Load method, we see that the quality of service is stable over time, at the level of around 40 percent. This is very significant, since the arrival rates do vary significantly over time.
[Recall Boaz+Edward: they also measured the average waiting time beyond 30 minutes, for those experiencing such long delays.]
OL method achieved good service quality: %W is stable over time.
RCCP method yields good performance of resource utilization - near 90%. 16

17. Comparing RCCP and OL given the same average number of resources
Simulation experiments – comparisons
Comparing RCCP and OL given the same average number of resources
Part 1: Intraday staffing in real-time
But the OL uniformly dominates RCCP.
More precisely, at the same average staffing levels, OL conclusively achieves a significantly higher level of service, as you can clearly see on this graph:
[The way we have done it was to change the alpha in the Offered Load model gradually and by that we got a different amount of resources per hour.
We then forced the utility target in the RCCP model to match the Offered Load average resources per hour.]
The simulation results are conclusive – OL is superior, implying higher quality of service, with the same number of resources, for all values of a. 17

18. Part 2: Intraday staffing over the mid-term
In my PhD, I wanted to make sure that the simulation results applicable in general, so that they apply to shifts other than the specific one that I just showed.
For that, I ran (NOT RUN) simulated a period of 100 weeks, and here is the comparison over an average week: 18

19. Mid-term staffing: Results
%W (and #Arrivals) per Hour by Method in an Average Week (a = 0.3)
Part 2: Intraday staffing in mid-term
The X-axis is the hour in a full week.
Each such part corresponds a single day of the week.
The red line is the arrival rate per hour.
The black line is the service quality under RCCP.
And the blue line is service quality under the OL.
As you can see, both methods suffer from the worst service quality during evenings.
RCCP is matched against the arrival rate, not the workload in the system. Hence, it is over overstaffing when arrival rates increase, and it is understaffing when they decrease. One can then expect very low service quality when the arrival rates are descending.
And indeed, this occurs in the evening, as you can see here.
Note that the OL also suffers from a relatively poor service level in the evening, still it is much better than RCCP and, in fact, it is relatively stable over the day. 19

20. Conclusions and future research
Developed a staffing methodology for achieving both high utilization and high service levels, over both short- and mid-term horizons, in a highly complex environment (e.g. ED)
More work needed:
Refining the analytical methodology (now the a is close to target around a = 50%).
Accommodate constrains (e.g. rigid shifts).
Incorporate more refined data (e.g. from RFID).
Parts 1+2: Intraday staffing
Let me summarize the article.
We developed a staffing methodology for achieving both high utilization and high service level, over both short- and mid-term horizons, in a highly complex environment. We used it to support the staffing procedures in the Emergency Department.
More work is needed:
First, we must refining the analytical methodology (now the alpha is close to target just around alpha that is 50%. For other alphas, we stabilize service levels over time, but we are off target.
We would also like to introduce constrains into our staffing methodology, because we know that ED staffing is not flexible in the real world, as we assume in our simulations.
Finally, it will be useful to incorporate more detailed data into the current ED state (for example from an RFID system) and check if the models achieve better results then. 20

21. Questions? 21