Modelling Patient Flow in an
Emergency Department
Modelling Patient Flow in an
Emergency Department
Mark Fackrell
Mark Fackrell
Department of Mathematics and Statistics The University of Melbourne
Motivation
Motivation
•
Eastern Health closes more beds•
Bed closures to cause ‘ambulance delays’•
Hospital targets cost lives•
Girl ‘waited for hours for emergency care’Motivation
Motivation
Public hospitals will be required by 2015 to ensure that 90% of all patients spend no more than 4 hours in the emergency department.
Currently (2012)
•
54% – major metropolitan hospitals•
63% – major regional hospitals•
67% – large metropolitan hospitalsThe Australian Triage Scale
The Australian Triage Scale
ATS Category Description Treatment Acuity % Adherence
1 Resuscitate Immediate 100%
2 Emergency 10 minutes 80%
3 Urgent 30 minutes 75%
4 Semi-urgent 60 minutes 70%
The Emergency Department
The Emergency Department
Desk Triage Treatment Queue Assessment and Treatment Queue Bed Ward 3 Ward 2 Ward 4 Ward 1 x x Do Not Wait Discharge
Modelling the Bed Queue
Modelling the Bed Queue
For each six hour block (00-06, 06-12, 12-18, 18-24), and each day of the week, we assumed that the
•
arrival rate to the bed queue is constant, ie.λ
, and•
departure rate from the bed queue depends on the number of patientsn
waiting in the bed queue, ie.µ
(
n
)
.Data available from 1 January 2001 to 18 April 2005 (200,000 entries).
Modelling the Bed Queue
Modelling the Bed Queue
We model the bed queue (service centre) with a continuous-time Markov chain.
•
C
beds (servers)•
State spaceS
=
{0
,
1
,
2
, . . . , C
}
•
λ
– Arrival rate•
n
– current number of beds occupied (number of customers being served)•
µ
(
n
)
– Departure rate•
No queueing (ie. if the bed queue becomes full the emergency department goes on bypass.)Transition rates
Transition rates
The transition rates for the Markov chain are
α
nn+1=
λ,
0
≤
n
≤
C
−
1
0
,
n
=
C
α
nn−1=
0
,
n
= 0
µ
(
n
)
,
1
≤
n
≤
C
Transition Probabilities
Transition Probabilities
The Markov chain stays in state
n
for a random timeT
n and then moves to either staten
+ 1
orn
−
1
with probabilitiesP
(
n
→
n
+ 1) =
λ
λ
+
µ
(
n
)
,
0
≤
n
≤
C
−
1
P
(
n
→
n
−
1) =
µ
(
n
)
Time Until an Arrival or Departure
Time Until an Arrival or Departure
In state
n
let•
T
n+ be the time until the next arrival, and•
T
−n be the time until the next departure.
Note that
T
n+∼
exp(
λ
)
andT
n−∼
exp(
µ
(
n
))
. We have thatT
n=
T
0+,
n
= 0
min(
T
n+, T
− n)
,
1
≤
n
≤
C
−
1
T
− C,
n
=
C
Density Function
Density Function
The density function of
T
n isf
n(
t
) =
λe
−λt,
n
= 0
(
λ
+
µ
(
n
))
e
−(λ+µ(n))t,
1
≤
n
≤
C
−
1
µ
(
C
)
e
−µ(C)t,
n
=
C
Probability of Reaching Capacity
Probability of Reaching Capacity
•
p
n(
t
)
– probability of moving fromn
toC
patients in the time interval[0
, t
]
•
p
n(
t
|
x
)
– probability of moving fromn
toC
in[0
, t
]
given that the first transition fromn
occurs at timex
p
n(
t
|
x
) =
0
,
n < C, x > t
λ
λ
+
µ
(
n
)
p
n+1(
t
−
x
)
+
µ
(
n
)
λ
+
µ
(
n
)
p
n−1(
t
−
x
)
,
n < C, x
≤
t
1
,
n
=
C
Probability of Reaching Capacity
Probability of Reaching Capacity
For
0
≤
n
≤
C
p
n(
t
) =
Z
∞ 0p
n(
t
|
x
)
f
n(
x
)
dx
.
Thus,p
0(
t
) =
Z
t 0p
1(
t
−
x
)
λe
−λxdx
p
n(
t
) =
Z
t 0[
λp
n+1(
t
−
x
) +
µ
(
n
)
p
n−1(
t
−
x
)]
e
(λ+µ(n))tdx
p
C(
t
) = 1
Laplace Transforms
Laplace Transforms
Taking Laplace transforms gives
b
P
0(
s
) =
λ
s
+
λ
P
b
1(
s
)
b
P
n(
s
) =
λ
s
+
λ
+
µ
(
n
)
P
b
n+1(
s
) +
µ
(
n
)
s
+
λ
+
µ
(
n
)
P
b
n−1(
s
)
b
P
C(
s
) =
1
s
To find
p
n(
t
)
invertP
b
n(
s
)
numerically – Euler method, Abate and Whitt, 1995.Probability of Ambulance Bypass
Probability of Ambulance Bypass
Probability of Bypass on Monday 12-18 (capacity of 20 cubicles) Hours P ro b a b ili ty 0 1 2 3 4 5 6 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 20 19 18 17 16 15
The ED Capacity Prediction Tool
The ED Capacity Prediction Tool
Model Validation
Model Validation
•
To validate the model we used data observed from 19 April 2005 to 26 September 2006 (48,000 entries).•
Choose a thresholdC
.•
At the beginning of each 6 hour block calculate the probability of reaching capacityC
using the model.•
Group the blocks into 10 bins of equal sizeN
i according to the probabilities.•
For each bini
, calculate the mean probabilityp
¯
i, and the variance of the probabilitiesV
i.Model Validation
Model Validation
•
Fori
= 1
,
2
, . . . ,
10
, record the number of times capacityC
is reached,O
i.•
Calculate the expected number of times capacityC
is reached,E
i=
N
ip
¯
i.•
The goodness-of-fit statistic isG
2=
10X
i=1(
O
i−
E
i)
2 var(
O
i)
where
