This document describes a study conducted at Froedtert Hospital to develop a predictive model of emergency department operations and the effect of patient length of stay on ED diversion. The study analyzed patient length of stay data, developed an ED simulation model, and used the model to test scenarios with different upper limits on length of stay. The model predicted that ED diversion could be reduced to around 0.5% by limiting discharged patients' length of stay to 5 hours and admitted patients' length of stay to 6 hours.

1. PREDICTIVE MODELING OF EMERGENCY
DEPARTMENT OPERATIONS:
EFFECT OF PATIENTS’ LENGTH OF STAY ON ED
DIVERSION
ALEXANDER KOLKER, PhD
Froedtert Hospital
Milwaukee, Wisconsin

2. Froedtert Hospital, Milwaukee WI
• Primary teaching Hospital for MCW
• Tertiary Referral Center
• Level 1 Trauma Center for SE Wisconsin
• 433 staffed acute care beds
• 23,617 admissions/ 47,176 ED Visits
• 454,780 Outpatient Clinic Visits
• Surgeries - Inpt: 9,034/ Outpt: 5,711

3. PROBLEM STATEMENT:
• Froedtert Hospital ED ambulance diversion has become unacceptably high
due to no ED beds (average ~21% of time in 2007)
•There are two big groups of patients:
(i) admitted to the hospital, and
(ii) treated, stabilized and discharged home
• Among factors that affect ED diversion patients’ Length of Stay (LOS) in ED is
one of most significant one.
The Goal of this work was:
• develop a methodology that could quantitatively analyze and
predict an impact of patients’ LOS on ED diversion (both for
admitted and discharged home patients’ groups).
• identify the maximum LOS limit that will result in significant
reduction or elimination ED diversion.

4. STEPS THAT HAVE BEEN PERFORMED:
• Collected data on
- patients arrival in ED: day of week and time of day
- mode of transportation: walk-ins or ambulance
- discharge disposition: home / expired or admitted in the Hospital
• Analyzed LOS distribution and its functional approximation for
(i) discharged home, and
(ii) admitted patients
• Developed an ED Process Model aimed at modeling different scenarios
of LOS upper limits that will result in significant reduction (or elimination)
ED diversion
• Summarized the basics of modeling methodology: What did we learn ?

5. What is the LOS distribution for admitted and discharged home patients ?
181512963
Median
Mean
5.04.94.84.74.64.54.44.3
1st Quartile 3.3833
Median 4.4667
3rd Quartile 5.9333
Maximum 20.4333
4.7761 4.9548
4.3667 4.5667
A-Squared 32.74
P-Value < 0.005
Mean 4.8654
StDev 2.1305
Variance 4.5389
Skewness 1.22735
Kurtosis 3.14050
N 2185
Minimum 0.4667
Anderson-Darling Normality Test
95% Confidence Interval for Mean
95% Confidence Interval for Median
24.521.017.514.010.57.03.50.0
Median
Mean
3.23.13.02.92.82.72.6
1st Quartile 1.6167
Median 2.6667
3rd Quartile 4.2000
Maximum 25.8167
3.1039 3.2099
2.6000 2.7167
A-Squared 146.60
P-Value < 0.005
Mean 3.1569
StDev 2.1225
Variance 4.5051
Skewness 1.84310
Kurtosis 8.14424
N 6155
Minimum 0.0667
Anderson-Darling Normality Test
95% Confidence Interval for Mean
95% Confidence Interval for Median
95% Confidence Intervals
Summary for LOS admitted, hrs
95% Confidence Intervals
Summary for LOS home, hrs
Total number for Jan 07: 1133
Feb 07:1052
Total number for Jan 07: 3255
Feb 07: 2971

6. DO THE SAME DAYS OF WEEK FOR DIFFERENT WEEKS HAVE A SIMILAR
ARRIVAL PATTERN ?
Take away:
Same days for different weeks have very different patient arrival pattern.
Therefore arrivals for all Mondays, all Tuesdays, and so on should not be combined
A dm T ime_1 /1 /2 0 0 7
Frequency
2400230022002100200019001800170016001500140013001200110010009008007006005004003002001000
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A dm T ime_1 /8 /2 0 0 7
Frequency
2400230022002100200019001800170016001500140013001200110010009008007006005004003002001000
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101010
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Monday Adm Time_1/1/2007
Monday Adm Time_1/8/2007

7. ED structure and in-patient units
The high-level layout of
the entire hospital system:

8. Arrival pattern
wk, DOW, time
Mode of transp
Disposition
ED simulation model layout
Simulation
Digital clock

9. • ED diversion (closure) is declared when ED patients’
census reaches ED beds capacity.
• ED stays in diversion until some beds become available
when patients are moved out of ED (discharged home,
expired, or admitted as in-patients).
• % ED diversion = % time ED is at full capacity
• upper LOS limits (simulation parameters) are imposed on
the baseline original LOS distributions:
LOS higher than the limiting value is NOT allowed in the
simulation run.
Baseline LOS distributions should be recalculated as
functions of the upper LOS limits.
MODELING APPROACH
Take-away:

10. MODELING APPROACH (cont.)
Given original distribution density and the limiting value of the random variable T,
what is the conditional distribution of the restricted random variable T ?
121086420
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LOS, Hrs
Frequency
3-Parameter Gamma
Distributionof LOS_ home, Hrs
Imposed LOS limit 6 hrs
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LOS, Hrs
Frequency
Re-calculatedboundeddistributionof LOS_ home, Hrs
dTTf
Tf
LOSTf
LOS
original
original
new

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0
)(
)(
),(
LOSTif,0)( newTf
T, Hrs
LO
T, Hrs
Original unbounded distribution New re-calculated distribution
origTf )(
LOS limit

11. ED Simulation Run Example
Diversion time
& DOW
Diverted
Ambulance
Patients discharged Home
Patient admitted to the Hospital

12. • ED diversion could be negligible (~0.5%) if patients discharged home stay NOT more than 5
hrs and admitted patients stay NOT more than 6 hrs.
• Relaxing of these LOS limits results in low digits % diversion that still could be acceptable
SIMULATION SUMMARY & MODEL VALIDATION
Low single
digits
diversion
~4%24 hrs5 hrs3
Low single
digits
diversion
~ 2%6 hrs6 hrs2
Practically NO
diversion
~ 0.5 %6 hrs
Currently
24% with
LOS more
than 6 hrs;
5 hrs
Currently 17%
with LOS more
than 5 hrs;
1
Actual ED
diversion
was 21.5%
23.7%24 hrs24 hrsCurrent, 07
(Baseline)
NotePredicted ED
diversion, %
LOS for
admitted NOT
more than
LOS for discharged
home NOT more than
Scenario/option
Take-away:

13. Baseline ED census, January 2007
LOS <= 24 hrs. Capacity 30 beds.
Diversion 23.7 %
0
5
10
15
20
25
30
35
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744
time, hrs
EDcensus
ED census:
LOS_home<=6 hrs, LOS_adm<=6 hrs. Capacity 30 beds.
Diversion 1.8%
0
5
10
15
20
25
30
35
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744
time, hrs
census
Diversion is declared when ED census hits capacity limit.
The longer the census stays at capacity limit the higher is diversion %

14. What other combinations of upper limits LOS are possible to get low
single digits % ED diversion ?
Performed full factorial DOE with two factors (ULOS_home and ULOS_adm) at 6 levels each
using simulated % diversion as a response function.
Div %
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
2 45 6 8 1 81 0
1 2
1 21 8 1 08
6 ULOS_adm, hrs52 4ULOS_home, hrs
Surface Plot of Div % vs ULOS_adm, ULOS_home
We want to be here

15. ULOS_adm, hrs
MeanpredictedDiv%
241210865
24.0
22.5
21.0
19.5
18.0
16.5
15.0
13.5
12.0
10.5
9.0
7.5
6.0
4.5
3.0
1.5
0.0
5
6
8
10
12
24
ULOS_home, hrs
Simulated Div % as a function of upper LOS limits, hrs
What other combinations of upper limits LOS are possible to get low
single digits % ED diversion ?
Low single digits
% diversion
Performed full factorial DOE with two factors (ULOS_home and ULOS_adm) at 6 levels each
using simulated % diversion as a response function.

16. % of patients that stay longer than the upper LOS limit
242220181614121086420
USL = 6 hrs
LSL *
Target *
USL 6
Sample Mean 4.86544
Sample N 2185
Location 1.49891
Scale 0.246641
Process Data
% < LSL *
% > USL 24.03
% Total 24.03
Observed Performance
24.521.017.514.010.57.03.50.0
USL = 5 hrs
LSL *
Target *
USL 5
Sample Mean 3.1569
Sample N 6155
Location 0.955823
Scale 0.39521
Process Data
% < LSL *
% > USL 16.83
% Total 16.83
Observed Performance
LOS_adm Jan and Feb, hrs
C alculations Based on Loglogistic Distribution Model
LOS_home Jan and Feb, hrs
C alculations Based on Loglogistic Distribution Model
~17% exceed LOS limit 5 hrs
~24% exceed LOS limit 6 hrs

17. More than X patients
waiting for a bed ?
Wait time greater
than Y hours ?
YES
YES CONSIDER
CLOSING
NO
NO
Bed
Available?
NO
1. Maintain Safe Patient Care
2. Decrease ED Diversions
3. Decrease Length of Stay
4. Decrease Left Not Seens
Objectives
ALTERNATIVE CLOSURE CRITERIA
Stay open
Questions to be addressed using Process Model
simulation:
• What should X and Y be to get low percent diversion ?
• Are X and Y correlated ?
Evaluate options to create beds
expedite discharges
move dispo’d pts to off floor bed
move stable pts to non-monitored bed
Advanced Nurse initiatives
Investigate LOS greater than 5 hours
Consult delays?
Radiology delays?
Lab delays?

18. • The lower max number of patients in the waiting room and max waiting time the
lower is ED diversion
• Locations of peaks of the max number of patients in Waiting Room is strongly
correlated to locations of peaks of the max waiting time (see next slides)
SIMULATION OF ALTERNATIVE CLOSURE CRITERIA
Take-away:
1 hr12~4%24 hrs5 hrs3
43 min10~ 2%6 hrs6 hrs2
35 min7~ 0.5 %6 hrs5 hrs1
3 hr 30 min3123.7%24 hrs24 hrsCurrent state,
(Baseline)
Waiting time
for admitted
patients in
Waiting Room
Max number of
patients in
Waiting Room
Predicted ED
diversion, %
LOS for
admitted NOT
more than
LOS for
discharged
home NOT
more than
Scenario/option

19. Number of patients in Waiting Room
ULOS_home=24 hrs, ULOS_adm=24 hrs
0
2
4
6
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10
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16
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30
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0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744
time, hrs
number
Wait_time_adm in Waiting Room, hrs
ULOS_home=24 hrs, ULOS_adm=24 hrs
0
0.5
1
1.5
2
2.5
3
3.5
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744
time, hrs
waittime,hrs

20. Number of patients in Waiting Room
ULOS_home=5 hrs, ULOS_adm=6 hrs
0
1
2
3
4
5
6
7
8
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744
time,hrs
number
wait_time_adm in Waiting Room, hrs
ULOS_home=5 hrs, ULOS_adm=6 hrs
0
0.1
0.2
0.3
0.4
0.5
0.6
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744
time, hrs
waittime,hrs

21. N_WR
Div%
3231302928272625242322212019181716151413121110987654321
25.5
24.0
22.5
21.0
19.5
18.0
16.5
15.0
13.5
12.0
10.5
9.0
7.5
6.0
4.5
3.0
1.5
0.0
Scatterplot of Div % vs N_WR
Take-away:
The number of patients in waiting room 11 or less corresponds to single digits
diversion less than 3%

22. Time_adm_WR, hrs
Div%
3.53.02.52.01.51.00.50.0
25.5
24.0
22.5
21.0
19.5
18.0
16.5
15.0
13.5
12.0
10.5
9.0
7.5
6.0
4.5
3.0
1.5
0.0
Scatterplot of Div % vs Time_adm_WR, hrs
Take-away:
Waiting time for admitted patients in waiting room 1 hr or less corresponds to
single digits diversion less than 3%

23. Time_adm_WR, hrs
N_WR
3.53.02.52.01.51.00.50.0
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
Scatterplot of N_WR vs Time_adm_WR, hrs
Take-away:
• The number of waiting patients and the waiting time for admitted patients is strongly
correlated to each other. Pearson linear correlation coefficient is 0.996 ! !.
• Strong correlation indicates that either one or another criteria should be enough, not both.

24. Conclusions
• ED diversion could likely be negligible (less than 1 %) if patients discharged home stay
NOT more than 5 hrs and admitted patients stay NOT more than 6 hrs.
Currently:
-17% of patients discharged home stay above this limit up to 24 hrs;
-24 % of admitted patients stay above this limit up to 20 hrs.
This long LOS for large % of patients results in ED closure/diversion
• Some relaxing of these LOS limits will result in low single digits % ED diversion that still
could be acceptable
• Other combinations of LOS upper limits that result in low single digits % diversion
have been determined using full factorial DOE with two factors.
• An alternative diversion criteria could be used: the number of patients in waiting room.
The number of patients 11 or less corresponds to single digits diversion, less than ~3%

25. • All three components affect the flow of patients that the system can handle.
• A lack of the proper balance between these components results in the
system’s over-flow and closure/diversion
• Process Model Simulation methodology provides the only means of
analyzing and managing the proper balance
• Patient Throughput flow is an example of the general Dynamic Supply &
Demand problem .
Dynamic means that the system’s behavior depends on time (not a one-time snapshot)
• There are three basic components that should be accounted for in this type
of problems:
• The number of patients (or, generally, any items) entering the system at
any point of time
• The number of patients (any items) leaving the system at any point of
time
• Limited Capacity of the system which limits the flow of patients through
the system
What did we learn about simulation methodology?

26. APPENDIX

27. WHAT IS THE PROCESS MODEL ?
•It is a computer model that mimics the dynamic behavior of a real
process over the time in order to visualize and quantitatively analyze
its performance in terms of:
•Cycle times
•Throughput capacity
•Resources utilization
•Activities utilization
•It is a tool to perform ‘WHAT-IF’ analysis and play different
scenarios of the model behavior as conditions and process
parameters change.
This allows to make experiments on the computer model, and test
different solutions (changes) for their effectiveness before going to
the floor for the actual implementation.

28. WHAT ARE THE BASIC ELEMENTS OF THE PROCESS MODEL?
•Flow chart of the process: Diagram that depicts logical flow of a process
from its inception to its completion
•Entities: Items to be processed: patients, documents, customers, etc.
•Activities: Tasks performed on entities: medical procedures,
document approval, customer check out, etc
•Resources: Agents used to perform activities and move entities: service
personnel, operators, equipment, nurses, physicians.
•Connections:
•Entity arrivals: Define process entry points, time, and quantities of the
entities that enter the system to begin processing
•Entity routings: Define directions and logical conditions flow for entities

29. WHAT INFORMATION (DATA) IS REQUIRED TO FEED THE MODEL ?
•Entities quantities and arrival time: periodic, random, scheduled, daily
pattern, etc
•The time that the entities spend in the activities. This is usually not a
fixed time but a statistical distribution. The wider the time distribution the
higher the variability of the system behavior.
•The capacity of each activity, i.e. the max number of entities that can be
processed concurrently in the activity.
•The size of input and output queues for the activities
•The routing type or the logical conditions to take a specified routing.
•Resource Assignments: their number and availability, and/or resources
shift schedule