Queueing Models of Patient Flow in Hospitals:What Does the Data Tell Us?
Mor Armony
Joint work with: Avi Mandelbaum, Yariv Marmor,Yulia Tseytlin and Galit Yom-Tov
NYU, Technion, Mayo, IBM, Columbia
November 2010
Mor Armony INFORMS 2010
Patient Flow in Hospitals as a Queueing Network
Questions:How to model arrivals, departures and transitions?Who are the servers?What are the service and arrival rates?What are the relevant performance measures?
Our data:Anonymous Israeli hospital with 1000 beds and 45 medicalunits75,000 patients are admitted annuallyYears data collected: 2004 - 2008Individual patient level data, time stamps (admission,transfers and discharge)Our focus: ED, IW and transfers ED –> IW
Mor Armony INFORMS 2010
Arrival rates into ED
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Hour
Ave
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f pat
ient
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Average number of patients per hour (Jan 2005, weekdays)
Mor Armony INFORMS 2010
ED: Distribution of the number of occupied beds
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
0 10 20 30 40 50 60 70 80 90 100L (Number of Occupied Beds)
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abili
ty
Mor Armony INFORMS 2010
ED: Hourly distribution of the number of occupied beds
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Am
ount
of h
ours
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ach
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H(0) H(1) H(2)H(3) H(4) H(5)H(6) H(7) H(8)H(9) H(10) H(11)H(12) H(13) H(14)H(15) H(16) H(17)H(18) H(19) H(20)H(21) H(22) H(23)
Mor Armony INFORMS 2010
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23Hour of the Day
avgL
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Average number of beds per hour of the day
Mor Armony INFORMS 2010
ED: Number of occupied beds as an Mt/M/∞queueing model
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0 10 20 30 40 50 60 70 80 90 100 110L (Number of Occupied Beds)
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EmpiricTheoretic
Mor Armony INFORMS 2010
ED: Number of occupied beds as Birth & Death model
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0 10 20 30 40 50 60 70 80 90 100 110L (Number of Occupied Beds)
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Mor Armony INFORMS 2010
ED: Arrival and departure rates
Observation: Both arrival and departure rates are statedependent.Consistent with Diwas and Terwiesch (2008)
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[pa
tie
nts
pe
r h
ou
r]
λ µ
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LMor Armony INFORMS 2010
Internal Wards: Capacities and ALOS
Ward A Ward B Ward C Ward D Ward EStandard capacity(# beds) 45 30 44 42 24Maximal capacity(# beds) 52 35 46 44 27ALOS(days) 6.368 4.474 5.358 5.562 4.11
Mor Armony INFORMS 2010
Routing: ED to IW
Single line system is more efficientReality requires multiple linesPatients require care even when in queuePush versus PullJustice table is meant to ensure fairness
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Fairness and Dis-economies of Scale
Ward A Ward B Ward C Ward DALOS (days) 6.368 4.474 5.358 5.562Mean Occupancy Rate 97.8% 94.4% 86.8% 91.1%Mean # Patients per Month 205.5 187.6 210.0 209.6Standard capacity 45 30 44 42Mean # Patients per Bedper Month 4.57 6.25 4.77 4.77Return Rate (within 3 months) 16.4% 17.4% 19.2% 17.6%
* Data refer to period 1/05/06 - 30/10/08 (excluding the months1-3/07)
Mor Armony INFORMS 2010
Fork-Join networks revisited
ED physician
IW nurse, Help force
Stretcher Bearer
IW nurse in charge
General NurseReceptionist
ED nurse in charge
IW physician
Hospitaliza-tion
decision
Patient allocation request
Transferal time
decision
Patient’s status
updating
Coordination with the IW
Running the Justice Table
Request skipping?
Approve skipping?
Initial measurements
collection
Patient’s transferal
Availability check
Bed preparation
Initial medical check
Yes
No
YesNo
Resource Queue - Synchronization Queue -
Availability check
Ventilated patient
- Ending point of simultaneous processes
Transferal time
decision
“Walking patient” Ward E
Ward E
Ward E
Zviran (2008): Diffusion limits and controlZaied (2010): Offered load
Mor Armony INFORMS 2010
Operational Regime of IW
Beds: QED regime.Prediction: Erlang-B in QED:N ' R + β
√R ⇒ P(block) ' 1√
Nφ(β)Φ(β) and
ρ ' 1− β+φ(β)/Φ(β)√N
.For our data, the QED regime predicts: P(block) ' 2.9%and ρ ' 91.7%.Actual numbers: P(block) = 3.54% and ρ = 93.1%.
Doctors: ED regime.Average handling time for patient admission: 30 minutes.Average wait for admission (once a bed is ready): 2.5hours.
Mor Armony INFORMS 2010
Operational performance measures
Focus: Quality of CarePatient surveysRate of returnMortality rates
Mor Armony INFORMS 2010
Conclusions
Hospital as a queueing networkArrival and departure rates are state dependentPush versus Pull in routingFairness: Occupancy + FluxEconomies and dis-economies of scaleFork-Join networksQED and ED regime co-exist in a single system.Operational measures should be in line with quality of caremeasures
Mor Armony INFORMS 2010