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Emergency Department (ED) Crowding�

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 morale
Negative influence on teaching mission in academic EDs
Reduced ability responding to mass casualty incidents

Negative Effects

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Queueing Theory (QT) �

Classical operations research methodology based upon mathematical models
Natural fit for modeling patient flow in a healthcare setting

Advantages
Closed-form analytical solution
Minimal data requirements
Easy implementation via spreadsheets

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Challenges In ED Queueing Modeling

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

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Descriptive Analysis By Year

Analytical QT articles focused exclusively on ED operations
39 articles published during 1970-2014
Limited publications before 2005
Increasing interest of researchers in this domain

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Descriptive Analysis By Publication Outlets

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

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Performance Measures�

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
Time	Expected Wait Time	9
	Average Length of Stay	3
	Length of Stay	4
	Expected Boarding Time	2
	Fraction of Time On Diversion	1
Queue	Average Queue Length	3
Queue	LWBS Rate	4
Probability	Wait Probability	6
	Area Overflow Probability	3
	Blocking Probability to Inpatient Unit	1
Resource	Resource Utilization	5
Resource	Marginal Resource (bed)	1

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Problem-Oriented Perspective

Recall ED Procedure:

Two Perspectives:

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Demand-Oriented Problems

Arrival Pattern
Problem 1: Time-varying arrival

Solution
Piecewise Stationary Approximation (PSA)
Stationary Independent Period by Period (SIPP)

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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 time

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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

�

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Ambulance Diversion

EDs requesting Emergency Medical Services (EMS) divert incoming ambulances to neighboring hospitals during periods of overcrowding.
Pros: Decrease the load on an ED
Cons: 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.

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Priority Queue: Triage-based

Pros: 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 rate
Split-flow saves resources by keeping the low-acuity patients vertical

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LWBS - reneging
Influenced by wait time, queue length and observed progress of other patients while waiting
In 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).

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

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Supply-Oriented Problems: Resource�

Studies 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

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Recall Kendall’s Notation

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
A	Probability distribution of the inter-arrival times
B	Probability distribution of the service times
m	Number of servers
K	Capacity of the system including patients in service (K ≥ m)
n	Size of the source population
D	Queueing discipline

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Modeling-Oriented Perspective

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 queue

Approximate general distribution by other distribution

Finite-capacity model

m/m/c/k model: used to make capacity decisions for the EDs
When c=k, Erlang loss formula can be used to calculate
overflow probability
Capacity requirement

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

Queueing Model		# of Articles
Infinite Capacity G(t)/G/c(t)	M/M/c	9
	M/M/c//n (infinite source)	1
	M/M/1	1
	M/G/1	1
	M/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
Markov Decision Process	2-Stage	3

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Modeling-Oriented Perspective

View the ED as an Independent Queueing System
View ED as a node in Queueing Networks
Hospital as the big network
ED + IU network

Article	QT	Assumption
[MM2012]	Inverted-V-shaped queueing system	A single centralized queue and k heterogeneous wards; each ward contains N_i 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/(N₁ − B) and M/M/N₂/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 ED

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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 purpose
Examine Papers that combine QT with simulation from the modeling perspective

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QT & Simulation Double Validation In the Same Paper

Achieved very close results from both models
Observed variations
Can help to compare different QT models

Papers	Simulation Purpose	QT/ Simulation Results Comparison
[YG2011]	Test reliability/assumption robustness of QT model; examine the impact of ALOS	QT 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 performance

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QT & Simulation Double Validation In the Same Paper

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 models

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QT in Combination with Simulation

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

Papers	Purpose	How QT is combined with simulation
[ZC2009]	Utilization Improvement	Square-root-staffing in conjunction with the M/M/s queue Set staffing level
[LW2012]	Congestion Alleviation	QT network with a heuristic iterative algorithm A specific delay probability simulation is used to estimate the percentage discharged during 4 hours
[HF2009]	AD routing Policy	First 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 network

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Conclusion

Queueing Models
Invaluable tools for ED design and management
The larger the system, the better performance
Cannot capture all of the characteristics of an actual ED, and may predict less variability than the real system experiments
Tend to simplify the system and underestimate delays and congestions, and thus obtain less accurate results than from simulation

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 settings

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Future Direction

Queueing models of the ED settings are still limited despite the abundance of established theoretical work

Future QT research in ED may consider incorporating state dependence into modeling

Combination 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

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