Jim Dai School of ORIE, Cornell University

(on leave from Georgia Institute of Technology)

Stochastic Network Models for Hospital Inpatient Flow Management

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Team members

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�  Georgia Tech � Pengyi Shi

�  University of International Business & Economics, Beijing � Ding Ding

�  National University of Singapore (NUS) �  Jame Ang, Mabel Chou

�  National University Hospital (NUH) �  Jin Xin, Joe Sim

Outline

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�  Part 1: Empirical observations �  Part 2: Stochastic network models �  Part 3: Two-time-scale framework �  Part 4: Managerial insights & future research

Empirical observation at NUH �  Average queue length curve over 547 days

� # of patients who are waiting for inpatient beds from the emergency department (ED)

� Can we build a model and find methods to predict the curve?

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Time

Avera

ge qu

eue l

ength

Waiting time statistics: Period 1 Average waiting time

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Bed request time

Aver

age

waitin

g (h

our)

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Bed request time

6−ho

ur s

ervic

e le

vel (

%)

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Fraction of patients who wait at least 6 hours

Can we flatten the curve?

Part 2: Stochastic network models

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�  Time-varying queues � Massey (1981), non-stationary queues � Whitt (1991) � Green-Kolesar (1991, 1997) � Massey, Mandelbaum and Reiman (1998) �  Feldman-Mandelbaum-Massey-Whitt (2008), “Staffing of Time-

Varying Queues to Achieve Time-Stable Performance.” �  Liu-Whitt (2011,2012) �  framework Mt/GI/N

A new stochastic network model �  Multi-server pools serving multi-class customers

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EL

ED-GW ICU-GW

Neuro Renal Gastro Surg Card Ortho

Onco Gen-Med

Respi

Gen-Med

Neuro

Surg

Ortho

Surg

Card

Respi

Surg

Overflow I Overflow II Overflow III

…

9 buīers …

9 buīers

…

9 buīers

SDA

…

9 buīers

New features �  Endogenous service times �  Allocation delays �  Overflow trigger times �  Missing any one of these features makes the model less relevant

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5%

10%

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20%

25%

LOS distributionlog−normal

Endogenous service times Service time = Discharge time – Admission time = LOS + Dis hour – Adm hour

Length-of-stay (LOS) = number of nights in hospital �  LOS distribution �  Average is ~ 5 days; admission source and medical specialty dependent

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0.1

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Discharge Time

Rel

ativ

e Fr

eque

ncy

Period 1Period 2

Checking the service time model

(a) Empirical (b) Simulation output

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.005

0.01

0.015

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0.025

Days

Rel

ativ

e fre

quen

cy

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0.005

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0.015

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0.025

Days

Rel

ativ

e fre

quen

cy

Allocation delays �  Getting a bed is a process

�  Pre-allocation delay �  Bed management unit searches/negotiates for beds

�  Post-allocation delay �  Delays in ED discharge �  Delays in transportation �  Delays in ward admission

�  In our model: each patient experiences a random delay T after a bed is allocated to her

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Overflow trigger times �  Wards usually accept patients from primary specialties

EL

ED-GW ICU-GW

Neuro Renal Gastro Surg Card Ortho

Onco Gen-Med

Respi

Gen-Med

Neuro

Surg

Ortho

Surg

Card

Respi

Surg

Overflow I Overflow II Overflow III

…

9 buīers …

9 buīers

…

9 buīers

SDA

…

9 buīers

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Entire hospital runs in the QED regime �  Quality- and Efficiency-Driven (QED) regime

� Waiting time is a small fraction of service time �  Average waiting time = 2.8 hours = 1/43 average LOS

�  Typical bed occupancy rate is 86% ~ 93%

�  Multi-server pools with certain flexibility

�  30 ~ 60 servers in each pool �  15 server pools (500-600 servers)

�  Trade-off between waiting time and overflow fraction

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Part 3: Two-time-scale framework

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�  Discrete-time queues � The LOS and daily arrival rate determine , the midnight

customer count, and thus determine the daily performance

�  Time-varying performance � The arrival rate pattern and discharge timing determine the time-

of-day behavior

{Xk}

A simplified single-pool model

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�  A single-pool model with N servers �  Arrival is periodic Poisson with rate function and period of 1 day �  LOS follow a geometric distribution with mean �  Discharge times follow a discrete distribution �  Allocation delay

�  Service times follow the non-iid model �  Performance measure: steady-state, mean queue length curve for

m�(t)

E[Q(t)] 0 t < 1

Step 1: daily customer count �  denotes the number of customers at midnight of day

�  Discrete time queue

�  Number of discharges only depends on and independent coin tosses since �  LOS is geometric �  LOS starts from 1 (no same-day discharge)

�  Number of arrivals is a Poisson random variable �  Independent of number of discharges

�  is a discrete time Markov chain (DTMC) �  Stationary distribution can be solved numerically

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Xk

Xk+1 = Xk �Dk +Ak

Ak

{Xk}⇡

Dk Xk

k

Step 2: hourly customer count � Conditioning on X(0), X(t) is a convolution between a

Poisson r.v. (arrival) and a Binomial r.v (discharge) �  The mean queue length

Mean customer count can be solved via fluid equation � 

� 

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X(t) = X(0)�D(0,t] +A(0,t]

E[Q(t)] = E[X(t)�N ]+

E[Q(t)]?= E[Q(0)] +

Z t

0�(s)ds� E[D(0,t]]

E[X(t)]=E[X(0)] +

Z t

0�(s)ds� E[D(0,t]]

Related work �  M. Ramakrishnan, D. Sier, P. Taylor (2005), “A two-time-scale model for

hospital patient flow”, IMA Journal of Management Mathematics. �  ED evolves in a much faster time scale than wards.

�  A. Mandelbaum, P. Momcilovic, Y. Tseytlin (2012), “On Fair Routing from Emergency Departments to Hospital Wards: QED Queues with Heterogeneous Servers”, Management Science. �  Two time scales: service times are in days; waiting times are in hours.

�  E. S. Powell et al. (2012), “The relationship between inpatient discharge timing and emergency department boarding”, The Journal of Emergency Medicine �  Affiliations: Department of Emergency Medicine, Northwestern University;

Harvard Affiliated Emergency Medicine Residency, Brigham and Women’s Hospital–Massachusetts General Hospital, …

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Numerical results �  Alloc delays follow a log-normal distribution

�  Mean alloc delay is 2.5 hours, CV=1

�  Discrete discharge distribution from NUH period 1 data �  N=525; m=5.3;

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⇤ = 90.95

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numerical" empirical"

Queue length curve from the FULL hospital model (Period 1)

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queues fail to capture �  Simulation results from an system

avg waiting time avg queue length

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Bed request time

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(hou

r)

SimulationEmpirical

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Time

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Mt/GI/N

Mperi/lognormal/N

Part 4: Insights & challenges

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Aggressive early discharge policy

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0%#

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1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14# 15# 16# 17# 18# 19# 20# 21# 22# 23# 24#

Discha

rge*distrib

u/on

*

NUH#per#1# NUH#per#2# aggressive#early#dis#

Insights from the simplified model

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�  Impact of discharge policy �  Steady-state, time-of-day mean waiting time

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Wai$n

g'$m

e'(hou

r)'

NUH"per"1" NUH"per"2" aggressive"early"dis"

Simulation results �  Simulation shows NUH early discharge policy has little improvement

(a) hourly avg. waiting time (b) 6-hour service level

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Aggressive early discharge + smooth allocation delay

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�  Waiting time performance can be stabilized (a) hourly avg. waiting time (b) 6-hour service level

Challenges

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�  For a multi-pool model with “state”-dependent overflow trigger time, develop an analytical theory for �  Performance analysis � Near optimal overflow policy (real time); impossible for

simulation � Optimal capacity allocation among different wards (once every 6

months?); time consuming for simulation �  Perry & Whitt (X-model); Pang & Yao (switch-over)

�  For a single-pool model, analyze the discrete time queue under � General LOS distribution � Day-of-week model � Matrix analytic method, diffusion approximations

Operational Challenges

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�  Push early discharge �  Reduce LOS

� AM- and PM-admissions � Using step-down care facilities

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Questions?