Applying Queueing Theory and Simulation to the Modeling of - PowerPoint Presentation

Slide1

Applying Queueing Theory and Simulation to the Modeling of Emergency Departments

Summer (Xia) HuSean BarnesBruce GoldenUniversity of Maryland, College Park

1

Jul. 31 2015Slide2

Emergency Department (ED) Crowding2

Critical challenge

to operational

efficiency

Increase in ED visits, decrease in ED number

Patients

Providers

Higher

risks of morbidity and mortality

Prolonged

wait times

Higher

likelihood of leaving without being seen (LWBS)

Higher

rates of dissatisfaction

Higher

rates of medical errors

Miscommunication & stress Lower productivity and moraleNegative influence on teaching mission in academic EDsReduced ability responding to mass casualty incidents

Negative

EffectsSlide3

Queueing Theory (QT) 3

Classical

operations research methodology based upon mathematical models

N

atural

fit for modeling

patient

flow

in a healthcare setting

Advantages

Closed-form analytical solution

Minimal data requirementsEasy implementation via spreadsheetsSlide4

Challenges In ED Queueing Modeling 4

Time-varying demand

Various patient

flow

routes

ED

patients are prioritized and treated according to their assessed level of urgency, not according to their arrival time or a pre-determined schedule Slide5

Descriptive Analysis By Year5

Analytical QT articles focused exclusively

on ED operations

39

articles

published during 1970-2014

Limited publications before 2005

Increasing interest of researcher

s in this domainSlide6

Descriptive Analysis By Publication Outlets6

ED QT method are attracting increased attention from traditional healthcare areas, then from engineering, and finally from healthcare management and operations research

Operations

Research and Management

Science (ORMS) journals published the most QT-related ED articles, followed closely by the

Emergency, Health and Medicine Science

(EHM) journals

Slide7

Performance Measures7

Example:

Governmental policy evaluation

[

L. Mayhew & D.

Smith, 2008]

used a queueing model to evaluate the length of stay in

UK EDs in

light of the government-enforced target of completing and discharging 98% of patients within 4

hours

They demonstrated how the model could be used to assess the practicality of

ED targets in the future

ED Performance Measures

Number of Articles

TimeExpected Wait Time

9Average Length of Stay

3Length of Stay 4Expected Boarding Time

2Fraction of Time On Diversion 1Queue Average Queue Length

3

LWBS Rate

4

Probability

Wait Probability

6

Area Overflow Probability

3

Blocking Probability to Inpatient Unit

1

Resource

Resource Utilization

5

Marginal Resource (bed)

1Slide8

Problem-Oriented Perspective8

Recall ED Procedure:

Two Perspectives: Slide9

Demand-Oriented Problems9

Arrival Pattern

Problem 1:

Time

-varying

arrival

Solution

Piecewise

Stationary Approximation (PSA)

Stationary Independent Period by Period (SIPP) Slide10

10

Arrival Pattern

Problem 2:

Time

-lag between arrival and occupancy

(enduring effect)

Solution

Lag

-version of above models (i.e., Lagged-PSA, Lagged SIPP): shift the arrival rate to the right by the mean service timeSlide11

11

A Common Underlying Assumption

Arrival

and service rates

may depend on time, but do

not depend on the system state (e.g., occupancy

)

Future research should try incorporating this behavior Slide12

Ambulance Diversion EDs requesting Emergency Medical Services (EMS) divert incoming ambulances to neighboring hospitals during periods of overcrowding. Pros: Decrease the load on an EDCons: Put patients at risk of worse

outcomes; Lost revenue to the hospital Example:[G. Allon, S. Deo, et al.]: The capacity

of the inpatient unit is negatively correlated with the fraction of time that the ED diverts ambulances. Minimum number of

beds is positively

correlated with the fraction of time spent on active diversion.

12Slide13

Priority Queue: Triage-basedPros: Reduce the average wait time for all patients (e.g. lower-acuity patients in fast track)Cons: The wait time for higher priority patients is reduced while the lower-priority patients endured longer wait times on average.

Fast track decrease LWBS rateSplit-flow saves resources by keeping the low-acuity patients vertical13Slide14

LWBS - reneging Influenced by wait time, queue length and observed progress of other patients while waitingIn systems where demand exceeds server capacity, reneging is the only way that a system attains a state of dysfunctional equilibrium Example: [J.K

. Cochran, J. R. Broyles, 2010] explored the relationship between LWBS and ED utilization by approximating reneging using queueing with balking. They predicted future ED capacity based on patient safety (rather than congestion measures).

14

Such queue-based relationship is superior to the typical ad hoc regression relationships commonly foundSlide15

Supply-Oriented Problems: Resource

15Studies focused on the estimation of the necessary amount of ED resources can be classified into two types:

Steady state resource

requirements

- Use QT

to estimate the steady state resource requirement.

Then adjust

resource levels to meet the daily fluctuations in demand in specific ED Short-term resource adjustments - Use autoregressive integrated moving average models, Monte Carlo simulation and Markov Decision Processes to determine resource levels as a function of time Slide16

Recall Kendall’s Notation 16

Typical Values:A and

B: M - exponential,

D

- deterministic

,

PH - phase type, GI- general independent (i.i.d.), and G - general distribution When the final three parameters are not specified (e.g. M/M/1), it is assumed K = ∞, N = ∞ and D= first-in, first-out+G: abandonment is allowed with an arbitrary patience distribution

time

Queueing system

A/B/m/K/n/D

AProbability distribution of the inter-arrival times

BProbability distribution of the service timesmNumber of serversKCapacity of the system including patients in service (

K ≥ m)

nSize of the source populationDQueueing discipline Slide17

Modeling-Oriented Perspective 17

Infinite-capacity model

G/G/c queue

Popularity of

M/M/c

queue.

e.g., (

N. Yankovic, L.V. Green, 2011) modified M/M/c queueApproximate general distribution by other distribution Finite-capacity modelm/m/c/k model: used to make capacity decisions for the EDsWhen c=k, Erlang loss formula can be used to calculateoverflow probabilityCapacity requirement

e.g., (A.M. de Bruin et al., 2007)

Queueing Model

# of Articles

Infinite CapacityG(t)/G/c(t)

M/M/c

9

M/M/c//n (infinite source)

1M/M/11

M/G/11M/M/ ∞

2

M/G/c

3

D/G/1

1

M

t

/G/

c

t

1

G/GI/c/c

1

GI/G/

c

t

1

Finite Capacity

G/G/c/k

M/M/c/k

1

M/M/1/k

2

M/M/c/c

1

M/G/c/c

1

M/GI/c/c

1

Queue With Abandonment

M/M/n + G

1

M

t

/M/n + G

1

M/GI/r/s +GI

(Approximate to M/M/r/s + M(n))

1

Markov Decision Process

Bivariate

1

2-Stage

3Slide18

Modeling-Oriented Perspective View the ED as an Independent Queueing System View ED as a node in Queueing NetworksHospital as the big network

ED + IU network

18

Article

QT

Assumption

[MM2012]

Inverted-V-shaped queueing system

A single centralized queue and

k

heterogeneous

wards; each ward contains Ni

i.i.d servers (beds). Upon arrival, each patient is routed to one available pool if it has idle servers, or joins a centralized queue of infinite capacity if all the servers are busy

[LPL2014]

M/GI/ c1/∞ with priority, and G/GI/ c2/

c2 ED queue: five priority patients; high priority patients receive immediate service; Patient either discharge, or transfer to IUs after ED, depending on the availability of IU beds;

IU queue: no priorities or buffer.

Resource is bed capacity in the ED and IU. Each resource has a capacity of

one

[BC2011]

Two multi-server

M/M/c queue

in series

Service rate for ED, IU is unknown and estimated by statistical methods. Resource is bed for both

queues

[ADL]

M/M/(

N

1

− B)

and

M/M/

N

2

/K

queue (approximated)

Two priority patients as two queues; independent poison arrival rates to the ED and admission rates to the IU; each station has multiple servers (beds); hospital diverted patients if there were more than

K

boarded patients in the EDSlide19

ED QT & Simulation Simulation - Great flexibility in testing scenarios, hypotheses, policies, and re-engineering ideas Our Procedure

Examine ED QT papers that simultaneously implement simulation for double validation purposeExamine Papers that combine QT with simulation from the modeling perspective

19Slide20

QT & Simulation Double Validation In the Same PaperAchieved very close results from both models Observed variations

Can help to compare different QT models

20

Papers

Simulation Purpose

QT/ Simulation Results Comparison

[YG2011]

Test

reliability/assumption

robustness of

QT

model; examine the impact of ALOSQT

model’s staffing estimates are reliable under various

distribution

hypothesis, with occasional underestimation of delays when ALOS is short

[CR2009]Check performance measures

Performance measures are consistent[LPL2014]

Validate the impact of variables on necessary ED capacity Achieved very close results 

[SAG2013]

QT

model

to validate simulation model

The wait times predicted by the QT model are lower than the simulation model.

[YM2014]

Validate QT models in large and small system to pinpoint unfitness; Compare staffing

given by two QT models

In large system, the QT and simulation performance fit closely in the QED regime, but not necessarily for the efficiency driven system.

[XC2014]

Verify the insights generated by QT model on ED admission control

Simulation verified that the proactive policies based on QT are robust under the variation of

parameters

[AIM]

Simulation model as an acute measure of performanceSlide21

QT & Simulation Double Validation In the Same Paper21

Results: Compared with the

Mt/Mt/

∞

model which fits well only when ED patient number is small and

Mt/

Mi

/ ∞ model which is even less accurate, the state dependent model Mi/Mi/

∞ has an overall good fit with simulation and real system

performance

Example:[AIM]

Simulation Purpose:

Test how the number of patients in ED depends on time and state of the system for different QT modelsSlide22

QT in Combination with Simulation Combination of simulation with queueing techniques leads to theoretical insights and practical resultsQueueing models enables analytical formula derivation for the general casesSimulation models can validate, refine, or complement the results obtained by queueing

theory22

Papers

Purpose

How QT is combined with simulation

[ZC2009]

Utilization

Improvement

Square

-root-

staffing

in conjunction with the M/M/s

queueSet staffing level

[LW2012

]

Congestion Alleviation

QT network with a heuristic iterative algorithmA specific delay probability simulation is used to estimate the percentage discharged during 4

hours[HF2009]AD routing PolicyFirst derived a small scale QT model to generate the corresponding qualitative

solution

Then

applied DES and Agent-based simulation model to mimic the full-scale

networkSlide23

ConclusionQueueing ModelsInvaluable tools for ED design and managementThe larger the system, the better

performanceCannot capture all of the characteristics of an actual ED, and may predict less variability than the real system experimentsTend to simplify the system and underestimate delays and congestions, and thus obtain less accurate results than from simulation

23

Simulation

Models

Have the advantage of incorporating more detailed behavior

Can help to validate, refine or compare queueing models, and estimate missing parameters in queueing models, if necessary

Can be sensitive to specific ED settingsSlide24

Future DirectionQueueing models of the ED settings are still limited despite the abundance of established theoretical workFuture QT research in ED may consider incorporating state dependence into modelingCombination of

QT with simulation or statistics will help to solve many realistic problems (e.g., parallel/ sequential task modeling from care providers, time inhomogeneity of arrivals, ED re-visiting), while conserving QT’s advantage of generating analytical and generalizable results for tractable models

24Slide25

Thank you!Summer (Xia) HuUniversity of Maryland, College Parkxhu64@umd.edu

25