April 19, 20231

ORSIS 10/5/2009

Queues in Hospitals: Empirical Analysis

of Patients Flows through the Emergency

Department

Lecturer: Yariv Marmor, Industrial Engineering, Technion

• Joint work with Yulia Tseytlin, Galit Yom-Tov and Avishai Mandelbaum• This talk is based on a PhD thesis that started under the supervision of

David Sinreich (ז"ל) and is now guided by Avishai Mandelbaum. • Research has been partially conducted within the OCR research

project of Technion+IBM+Rambam, under the funding of IBM

April 19, 20232

Hospital Overview

Arrivals

Patients abandon

Patient discharge

IW1

IW5

MUn

Patient discharge

Services

Emergency Departmen

t

MU1

Joint project with Mandelbaum A., Yom-Tov G., and Tseytlin Y. :

Analyze ED, IW, and their interfaces, using simulation, empirical and theoretical models.

Yulia

Galit

April 19, 20233

Addressing the Following Questions:

• Given an ED environment, identify a suitable ED architecture (DEA / simulation / mathematics).

• Explain the difference in IW-LOS distributions, when plotted in resolutions of days vs. hours (+implication).

• Why is the distribution of ED lengths of stay (LOS) LogNormal? (no clear answer, yet)

• How to measure offered-load (congestion) in EDs? How to use it in support of staffing decisions (on-line, off-line)?

April 19, 20234

Given an ED Environment, Identify a Suitable ED Architecture: Data

Hospital Operating Model Average # Patients per month

1 Fast-Track 7,600

2 Internal / Surgical / Orthopedic 4,000

3 Fast-Track 10,300

4 Walking / Acute 10,000

5 Walking / Acute 8,000

6 Triage 4,400

7 Triage 4,000

April 19, 20235

Given an ED Environment, Identify a Suitable ED Architecture: Arrivals rate to the ED*

(from Non Homogeneous Poisson process)Patients Arrivals to Hospital ED

ILHospital, January 2000, Week days

0.00

2.50

5.00

7.50

10.00

12.50

15.00

17.50

20.00

22.50

25.00

27.50

0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00

Time (Resolution 60 min.)

Avera

ge n

um

ber

of

cases

Hospital1 Hospital2 Hospital3 Hospital4 Hospital5

*Via SEEStat

April 19, 20236

Given an ED Environment, Identify a Suitable ED Architecture: Parameters

• Outputs (Yjo):– Average length of stay.– Net patients throughput.– Number of patients waiting for a bed.

• Controllable inputs (Xio):– Manpower (70% of total costs).– Number of beds in the ED.– Number of hospitalizations.

• Uncontrollable inputs (Zko):– Number of children / elderly.– Number of patients with injury.– Number of patients arriving by ambulance.– Number of patients referred by physician

April 19, 20237

Given an ED Environment, Identify a Suitable ED Architecture: Mathematical Model (DEA)

1 1

1

1 1

1

max

.

1 ; 1,...,

0 , 1,..., ( )

0 , 1,..., ( )

0 , 1,..., (

s t

j jo k koj k

r

i ioi

s t

j jm k kmj k

r

i imi

j

i

k

w y u z

v x

s t

w y u zm n

v x

w j s weights for outputs

v i r weights for controlablle inputs

u k t weights for unconrtolablle in )puts

Efficiency

April 19, 20238

Given an ED Environment, Identify a Suitable ED Architecture: Comparison (DEA)

Operating Model Efficiency () LB [95%] UB [95%]

Fast Track 165.5 152.9 178.1

Internal/Surgical/Orthopedic 151.6 126.9 176.4

Walking/Acute 78.6 65.3 92.0

Triage 170.2 153.5 187.0

=> Best operating models: Fast Track (priority models)

Triage (routing control)

(Ongoing research)

April 19, 20239

Why is the Distribution of ED Lengths of Stay (LOS) LogNormal?

Patient length of stay in ED (hours)Rambam, January 2004, All days Total (Upper Quantile=99.5%)

0.00

2.50

5.00

7.50

10.00

12.50

15.00

17.50

20.00

22.50

25.00

27.50

0 3 5 8 10 13 15 18 20

Time(hours)( resolution 1)

Rela

tive f

req

uen

cie

s %

Empirical Lognormal (scale=1.03244 shape=0.7311677) Inverse gaussian (location=4.712321 scale=4.135055)

April 19, 202310

Why is the Distribution of ED Lengths of Stay (LOS) LogNormal: Phase Type Approximation

Each phase is exponential

Total LOS is LogNormal

Phase Type

April 19, 202311

Patient length of stay in Ward (days), Department of Internal MedicineHomeHospital, January 2004, All days

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

11.00

12.00

13.00

14.00

15.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Time(days)( resolution 1)

Rel

ativ

e fr

equ

enci

es

%

Explain the Difference in IW-LOS Distributions, When Plotted in Resolutions of Days vs. Hours

1 day resolution

April 19, 202312

Explain the Difference in IW-LOS Distributions, When Plotted in Resolutions of Days vs. Hours

2 hour resolution

April 19, 202313

Explain the Difference in IW-LOS Distributions, When Plotted in Resolutions of Days vs. Hours

(implication): Arrival / Departure Rate

April 19, 202314

How to Measure Offered-Load (Congestion) in

the ED: Offered-Load Model in Mt/G/∞

NurseArrivals

Dr

X-Ray

Lab

Exit

∞

L W

i – index of node.

=> Find the nominal resource level needed

duutWPutL i

t

ii )()()(

∞

April 19, 202315

How to Measure Offered-Load (Congestion) in the ED: Staffing Rule

duutWPutL i

t

ii )()()(

( , ) ( ) ( ). tn OF t L t L t

( ) ( )

1 { } { 0} { | 0}

( )

t i t i

q q q q

T L t L t

t

P W T P W P W T W

h e

h(t) is the Halfin-Whitt function (Halfin & Whitt 1981)

is the fraction of patients that start service within T time units (Quality of Service)

Staffing rule (Halfin & Whitt 1981; Borst, Mandelbaum, & Reiman 2004):

(We have used, and T=30 minutes for the first patient-physician encounter)

April 19, 202316

How to Measure Offered-Load (Congestion) in the ED: Results for On-Line Staffing Decisions

Hour Current RCCP OL 16-17 69% 78% 56% 17-18 52% 82% 36% 18-19 31% 86% 46% 19-20 40% 90% 38% 20-21 39% 91% 38% 21-22 23% 91% 42% 22-23 20% 91% 50%

• Current: The current staffing rule.• RCCP (Rough Cut Capacity Planning): Efficiency driven staffing rule.• OL: Offered-Load staffing rule.

Comparison of the predicted for each hour in the rest of the shift:

{ }qP W T

April 19, 202317

Question ?