Pareto Improving Coordination Policies in QueueingSystems: Application to Flow Control in Emergency

Medical Services

Hung Tuan DoSchool of Business Administration, The University of Vermont, Burlington, VT, hdo@uvm.edu

Masha ShunkoKrannert School of Management, Purdue University, West Lafayette, IN, mshunko@purdue.edu

One of the well known methods to improve performance in a queueing system is implementing coordination

policies that balance the load among servers. However, in decentralized queueing systems where each service

agent can decide whether to participate in the coordination mechanism or not, a sustainable policy has

to not only benefit the system, but also be beneficial to all agents. In particular, agents are willing to

participate in a coordination policy only if the performance of their individual queue is not hindered and if

their revenues are not decreased. As a motivating example, we use the emergency medicine setting, in which

emergency departments (EDs) act as independent agents and overcrowding in the EDs has direct impact on

the quality of service. In such setting, EDs are interested in seeing improvements in performance measures

that address the expected number of patients (or expected census, which is a widely studied and applied

metric in the emergency medicine literature) and the risk of having high census. We focus on reducing the

expected census, the variance of census, the probability of having high census, and the expected census in

the overcrowded state; and propose classes of coordination policies that provide improvement on all of these

measures for all agents. In addition, agents who receive revenue based on the processed load, are interested

in preserving the long-term average load. Hence, our proposed classes of coordination policies guarantee that

the expected arrival rate and hence, the expected revenue, is preserved for each agent in the system. We

include a discussion of the implementation issues and propose a policy that is easy to implement in practice.

Key words : Flow control, Queueing control, Ambulance coordination, Pareto improvement, Stochastic

orders

1. Introduction

Emergency medicine in the United States is in a critical condition: the number of emergency

departments (EDs) in the nation is decreasing while the number of ED visits is increasing, which

results in substantial patient crowding (Eitel and Samuelson 2011). One common approach to

reduce crowding in the EDs is to control the inflow of patients using coordination of ambulance

traffic, with an intent to balance the patient load between the EDs servicing a geographic area.

One of the well known coordination policies in the U.S. is ambulance diversion, which allows

individual EDs to signal to an emergency medical service (EMS) that the ED is crowded and

therefore, the ambulance traffic should be diverted to other EDs in the area. The EMS dispatcher,

in turn, passes these signals to the ambulance crews and advises the latter on better destination

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EDs. Since EMSs in the U.S. are largely governed by state regulations, there exists a large variety

in EMS structures and policies, including different variations of diversion policies implemented

in different states (Gundlach 2010). Not surprisingly, there has also been a lot of debate in the

media and in the emergency medical literature concerning the overall usefulness of ambulance

diversion and different diversion policies. In fact, the state of Massachusetts banned ambulance

diversion as of January 1st, 2009. Using data from Massachusetts before and after the ban, Burke

et al. (2013) show that ambulance diversion did not reduce the median length of stay in the ED

and conclude that there is thus no value in the ambulance diversion. Deo and Gurvich (2011)

provide one explanation for the non-effectiveness of the simple ambulance diversion mechanism: in

the presence of competition, equilibrium behavior of the EDs that attempt to minimize expected

wait time subject to a service level constraint is for all emergency departments to go on diversion

simultaneously, which is not helpful for load balancing.

We conducted interviews with emergency medicine professionals in the City of Pittsburgh and

found a similar aversion to diversion: The City of Pittsburgh has two hospital groups in the area

that operate several EDs each. Currently EDs do not actively use ambulance diversion and do

not coordinate traffic, even between the EDs that belong to the same hospital group; a decade

ago, however, diversion had been a common practice and the city still has an IT system in place

to implement signaling (Guyette 2010). We collected data on ED visits to three neighboring EDs

that belong to the same hospital group for a period of one year. We analyzed the data and found

the evidence of load imbalance between the EDs; in particular, we observe census in each ED at

the beginning of each hour during the year and compare it to the capacity of the ED (including

beds and treatment spaces), we find that in 22% of such observations, only one out of three EDs

was overcrowded; in 17 % - two EDs were overcrowded while the third ED had idle capacity. This

data illustrates that there exist multiple times when one ED is overcrowded while another ED has

idle capacity. During such times, load balancing should be beneficial to the system. Nevertheless,

discussion with the administrative personnel at the involved EDs revealed the following: One of

the reasons why the EDs in the area do not actively engage in diversion is independent decision

making by the involved EDs and EDs’ unwillingness to lose potential patients (Guyette 2010).

This observation motivates an interesting research question, which is novel in queueing control: in

a decentralized queueing system with independent agents that can decide whether to participate in

the coordination system or not, what coordination policy will be sustainable and beneficial to all

agents and the system as whole?

One of the challenges in identifying a good coordination policy for a queueing system is the fact

that there is no single commonly accepted performance measurement. We use the EMS context to

motivate different performance measurements that are based on the stochastic properties of the

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number of patients in the system. In EMS context, several studies have looked at ED coordination

using ambulance diversion from the perspective of the average wait time reduction (see e.g. Deo

and Gurvich (2011)), which is equivalent to the expected queue length reduction by Little’s Law.

In emergency medicine literature, crowding is commonly defined based on the total number of

patients in the ED (McCarthy 2011), commonly referred to as census, and we adopt expected

census as one of the performance metrics for our study. While this metric is undoubtedly important

from the patients’ and EDs’ perspectives, there are additional metrics that EDs need to consider:

• What is the probability that an ED is overcrowded? In the emergency medicine setting, many

metrics used to evaluate EDs’ performance are based on the ability of EDs to provide care within

a certain period of time, for example, the percentage of acute myocardial infarction patients with

door-to-balloon time less than 90 minutes as imposed by the American College of Cardiology,

and/or the percentage of time that the ED is on diversion (or overcrowded) (Hopp and Lovejoy

2012). We proxy such metrics using the probability of having high census in the ED.

• What is the variability of the number of patients in the system? Basic operations management

theory suggests that process performance may be improved if variability is reduced. Hence, one

goal of a coordination policy could be to reduce the variability of patient arrivals. Using a queueing

model of a supply chain, Do (2012) shows that variability in the order arrival process per se is

not the key, rather, performance of the system depends on correlation or synchronization of the

arrival process and the service process; and, control policies that reduce and smooth out the queue

improve the performance of the queueing system. Hence, we seek coordination policies that reduce

the variance of census in the system.

• What is the expected number of patients in the ED given that the ED is overcrowded? Lower

than usual census is not problematic for EDs, however, high census often leads to negative conse-

quences (Moskop et al. 2009, Sprivulis et al. 2006). This indicates that EDs are also interested in

improving the right-tail behavior of the number of patients in the system: reducing the frequency

and extent of extreme high observations of census. We capture this objective by measuring the

expected census in the ED conditional on being overcrowded.

All of the above mentioned objectives (expected census, probability of having high census, vari-

ance of census, and expected census conditional on being overcrowded) are important to queueing

system participants, and a single or composite measurement may not capture all of the objectives

in a fair way for all contexts. Hence, we consider multiple performance measures that address

the questions raised above and propose a class of coordination policies that guarantees Pareto

improvement on all objectives (or thus, on any polynomial combination of these objectives with

non-negative weights).

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Moreover, in a decentralized queueing system where the service agents receive revenue based on

the processed load and/or who compete based on market share, the agents’ are unlikely to accept a

coordination system that may decrease their long-term average load. Based on our interviews with

emergency medicine professionals in the City of Pittsburgh, this observation is consistent with their

view of the problem and the EDs are concerned about ensuring that the average arrival stream

of patients does not decrease as a result of a coordination mechanism (Guyette 2010). Hence, in

order to provide Pareto improvement, a good coordination policy should also preserve the average

arrival rate to each participating agent.

Our goal is to propose classes of Pareto improving coordination mechanisms that operate as

follows. Central controller (EMS) designs and announces the coordination policy. The agents (EDs)

have a choice whether to participate in the coordination policy or not; and we assume that if the

policy is Pareto-improving, the agents choose to participate. Participation in the system involves

the following: agents (EDs) reveal truthful information to the controller (EMS) and the con-

troller (EMS) uses this information to make coordinating decisions (in our case, decisions to divert

patients). Practically speaking, EDs may provide access to their census tracking information sys-

tem to the EMS or EMS may employ some monitoring mechanism to ensure truthful information

sharing (e.g. random inspections using independent observers similar to the ones used by the Joint

Commission to evaluate EDs and/or ambulance crews). Notice also that our goal is not to find

the optimal coordination policy (which will differ based on the particular measure combinations

that can be different for different EDs and EMS systems), but rather to propose a class of policies

that provides a structure that guarantees that all players are better off and, hence, the players

can find the optimal policy for their particular system and objectives within our defined class.

A pertinent issue is whether our proposed policy is sustainable; we will show in Section 2.2 that

neither individual EDs nor EMS have an incentive to withdraw from the coordination mechanism

and hence, all EDs participating in the coordination policy is an equilibrium.

Do and Iyer (2013) use flow control policy in a supply chain setting with two retailers facing

independent arrival streams and placing orders to a single manufacturer. In their setting, the

retailers can use price promotions to control their arrival streams and the authors show that given

information on the manufacturer’s queue size, under a simple promotion policy with threshold

structure, the queue at the manufacturer is stochastically smaller with smaller variance that leads

to cost savings for both retailers and the manufacturer. In our paper, we model and analyze a

queueing system (e.g., a network of EDs) with multiple single-server queues that face a shared total

arrival stream with exogenous rate, which means that a change in flow to one queue impacts the

flow to the other queues, and that is controlled by a central coordinator (EMS in our motivating

healthcare example). We propose a class of coordination policies with a threshold structure imposed

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on conditional expected arrival rates to each server and demonstrate that any policy in the class

reduces and smoothes each queue, and thus guarantees Pareto improvement on multi-dimensional

performance measures for each player in the queueing system. We demonstrate how the commonly

utilized ambulance diversion policy can be modified to satisfy our proposed threshold structure.

Then, we show how a policy in the proposed class can be enhanced to provide an even higher

improvement for each player based on the multi-dimensional performance criteria.

Deo and Gurvich (2011) analyzed a policy with full diversion with a focus on optimizing a single

objective function (expected wait time), they looked at a decentralized setting where diversion

threshold are set by the EDs and the centralized setting where EMS assigns diversion thresholds

to each ED in the network; this centralized full diversion policy is not Pareto improving as one of

the EDs may be worse off in terms of the expected arrival rate and/or in terms of the expected

wait time, and hence, is not sustainable. Moreover, as is clear from Deo and Gurvich analysis, the

decentralized full diversion policy analyzed in their paper has negative externalities: it can trigger

gaming behavior of the EDs, which leads to a poor equilibrium solution. We propose a policy class,

which is Pareto improving on multi-dimensional performance measure (and hence it works for any

individual objective function of EDs and EMS with general and reasonable properties presented in

Section 7.3) and does not incentivize gaming.

Gurvich and Perry (2012) consider a service network (e.g. a call center) operated under an

overflow mechanism where calls are routed to a dedicated service station with a finite buffer. When

the buffer is full, calls are forwarded to an overflow station. The authors provide an approximation

for the overflow processes via limit theorem and prove asymptotic independence between dedicated

stations and overflow station. Although, overflow is relevant to diversion flow, there are fundamental

differences between this model and ours: a) there is no dedicated overflow station that only processes

overflow demand in our model and b) diversions occur if the destination ED is crowded and there

is available capacity elsewhere in the network.

To summarize, we contribute to the queueing control and EMS coordination literature by finding

classes of coordination policies that preserve the long-term average arrival rate and guarantee that

the census in each ED is stochastically smaller, has smaller variance, and smaller expectation

conditional on being in overcrowding state. Using EMS setting as motivating example, we develop

a subclass of policies within our proposed class, which is based on the idea of an existing ambulance

diversion policy. As opposed to the existing ambulance diversion policy, which does not preserve

the average arrival rate and can hurt performance of one of the network participants (an ED), our

subclass guarantees that all participants are better off.

The rest of the paper is organized as following. In Section 2 we introduce our model of the ED

network and formulate the performance measurements. In Section 3 we formally introduce the

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class of coordination policies and provide an example of a practical policy within the proposed

class. In Section 4 we prove that the proposed class of policies is Pareto improving; we provide

numerical experiments to illustrate our findings and demonstrate the sensitivity of our results

in Section 5. In Section 6, we propose a policy that can be easily implemented in practice and

numerically evaluate the performance of this policy. In Section 7 we discuss different applications

and extensions for our policy class, also provide a technique for improving coordination policies,

and discuss implementation nuances. We conclude in Section 8.

2. Model

We consider a network of N non-symmetric EDs in a region. The region serviced by the EDs

faces a total arrival rate λ. Not all patients can be rerouted from their intended destinations:

e.g. most walk-in patients, patients insisting to go to a specific ED cannot be routed against

their will by the ambulance crews, patients whose condition does not favor rerouting, etc. Hence,

we split the arrival rate into two streams: controllable arrivals λc and uncontrollable arrivals λu,

such that λc + λu = λ. Similarly to Deo and Gurvich (2011), we assume that the arrivals follow

Poisson process and present the model and its analysis with this assumption for ease of exposition.

In emergency medical setting, however, arrivals may be more variable and/or more bursty than

Poisson, for example, due to accidents that result in multiple casualties. Our result is robust to any

memoryless arrival process and we demonstrate in Section 7.5 how our analysis can be extended

to accommodate Markov-Modulated Poisson Process (MMPP) for arrivals.

Without coordination, the arrival rate to each ED k is λok, which consists of two streams: uncon-

trollable stream with the rate λuk , and controllable stream with the rate λck such that λok = λuk +λck.

Let the total rate to the network be∑k

λok = λ which consists of the total uncontrollable stream rate

λu =∑k

λuk and controllable stream rate λc =∑k

λck. A controllable stream may include ambulance

patients, who are willing to change destination ED and/or walk-in patients who are willing to self

re-route upon getting (or observing) information about the ED congestion status. An uncontrol-

lable stream may include patients whose condition (or strong will) requires a certain destination

ED and/or walk-in patients who are not able or not willing to of self re-route. We assume that

λok is an equilibrium solution for the network - i.e. EDs are not able to manipulate λok anymore

without a policy change.

Let P be a Markov and stationary coordination policy that splits the controllable arrival stream

λc among different EDs based on the census at all EDs. Then, under any coordination policy P, the

arrival rate becomes a function of the census. Let ~LP = (LP1 ,LP2 , ...,L

PN) be the random vector of

the number of patients in each of the N EDs under the coordination policy P and ~l= (l1, l2, ..., lN)

be the realization of the number of patients at the N EDs. Then, Λk(~LP) represents the random

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variable for the arrival rate to the k’th ED under coordination policy P1. Service rate at ED k

equals µk and µ=∑k

µk. We assume that µk >λok ∀ k (which implies that µ> λ), this assumption

guarantees that the queueing system under no coordination is stable.

2.1. Comparing performance of coordination policies.

As mentioned in the introduction, a good coordination policy should perform well with respect to

multiple measures, and should provide incentives to the EDs to participate in the system. First,

we introduce a set of four performance measures on which we will compare coordination policies.

Then, we introduce a constraint, without which the EDs will not be willing to participate in the

coordination system.

1. One of the widely used metrics for measuring the EDs’ and hospitals’ performance is the

average length of stay in the ED (Hopp and Lovejoy 2012). This metric represents the expected

time in system, which is equivalent to the expected number of patients in the ED (using Little’s

Law). Hence, one of the performance metrics for a coordination policy is the expected number of

patients in the ED (or expected census), E[LPk ], and to say that policy P2 performs better than

P1, we require:

E[LP1k ]≥E[LP2k ]. (1)

2. As argued earlier, another important performance comparison measure is probability that the

ED is overcrowded or that the number of patients in ED k is larger than a critical threshold Sk

(e.g. Sk may represent ED’s capacity): P [LPk ≥ Sk]. To say that policy P2 performs better than

policy P1 with respect to the probability of exceeding the critical threshold, we require:

P [LP1k ≥ Sk]≥ P [LP2k ≥ Sk]. (2)

3. Next, we consider the variance of the number of patients in the ED (defining variance of

random variable X as V [X] = E[(X − E[X])2]). Smaller variability of the number of patients in

the ED is attractive for multiple purposes: for example, EDs can reduce their operational costs

because of smoother resource scheduling, and EMS can make better decisions since the crowding

information has less noise.In order to say that policy P2 is preferred to policy P1 with respect to

variance, we require:

V [LP1k ]≥ V [LP2k ] ∀ k. (3)

1 Notice that both the arrival rate Λ and the number of patients in the ED L depend on the policy P; for cleanerexposition, we suppress one superscript and use Λk(~LP) instead of ΛPk (~LP).

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4. Finally, we capture the patient’s delay risk by quantifying the expected census in the ED

given that the patient finds this ED overcrowded. Let α represent a service level, such that the

ED k is considered to be overcrowded if the number of patients in the ED exceeds Value-at-Risk

V aRα[LPk ], where V aRα[X] is the left-continuous inverse of FX (CDF of random variable X):

V aRα[X] = F−1X (α) =min{x : FX(x)≥ α}. Then, for a given value of α we use the following risk

measure that captures conditional expected census in overcrowded state:

CEoverα [LPk ] =E[LPk |LPk ≥ V aRα[LPk ]].

This measure is of particular importance from ambulances’ perspective because it addresses the

expected number of patients in the ED during the problematic (overcrowded) times - the times

when diversion decisions need to be made; and also from the quality of service perspective because

the risk of adverse care outcomes is higher during overcrowded times. Notice that the conditional

expectation in overcrowded state measure is equivalent to the definition of the Lower (or Tail)

Conditional-Value-at-Risk measure commonly used in financial risk analysis: CV aR−α [X] = [X|X ≥

V aRα[X]] (Rockafellar and Uryasev 2002). This measure has been used as a performance measure

of wait time in healthcare simulation studies, see e.g. (Dehlendorff et al. 2010). In order to say

that policy P2 is preferred to policy P1 with respect to conditional expected census in overcrowded

state, we require:

CEoverα [LP1k ]≥CEover

α [LP2k ] ∀ k. (4)

Inequality 4 is sufficient to provide weak improvement to the EDs and we will use this inequality

in the definition of Pareto improvement. Later, however, we will also show a stronger result that

under our proposed class of policies, the difference between the conditional expected census in

overcrowded state (CEoverα [LP1k ]−CEover

α [LP2k ]) is greater than the difference between the expected

census (E[LP1k ]−E[LP2k ]).

In addition to defining performance metrics, we also impose a condition that guarantees that the

players involved (the EDs) have an incentive to participate in the proposed coordination mecha-

nism. One key factor determining EDs’ market share and revenues is the number of patients served.

Hence, in order to provide an incentive for the EDs to participate in any coordination policy P, the

policy needs to guarantee that the expected arrival rate of patients E[Λk(~LP)] to ED k is at least as

large as the time-average arrival rate to ED k without coordination: E[Λk(~LP)]≥ λok. Notice that

we assume that all patients are similar and bring in the same amount of revenue, our work can be

easily extended to the case with multiple patient classes and we discuss this extension in Section

7.1. While some EDs may have preservation of the long-term arrival rate not as their primary

concern, this condition can also be viewed as preservation of long-term fairness in the system that

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is important from the system’s viewpoint to sustain participation in the policy in the long-term.

We refer to this constraint as our participation condition (in expected arrival rate). Notice that

since the total arrival rate to the network of all EDs is fixed, the only way to guarantee that the

participation constraint is satisfied for all EDs is to impose the following necessary condition:

Condition 1. Participation Condition. E[Λk(~LP)] = λok for all k.

Participation Condition 1 preserves the expected revenue as long as the revenue is an absolutely

bounded function of expected arrival rate (which is also bounded), which is the case in reality.

However, the difference between census at a specific ED under the proposed policy and census

under no coordination can go to infinity as time goes to infinity: Let NPk (t) be the number of

patients who have entered ED k by time t under policy P. The arrival rate to ED k is then limt→∞

NPk (t)

t

under policy P and limt→∞

Nok (t)

twithout coordination. Participation Condition 1 is then equivalent

to limt→∞

NPk (t)

t− lim

t→∞

Nok (t)

t= lim

t→∞

NPk (t)−Nok (t)

t= 0. It is then possible that NP

k (t) − N ok (t) = ±o(t),

where o(t) goes to infinity slower than t and hence, limt→∞

NPk (t) − NO

k (t) = limt→∞±o(t) = ±∞. If

this is a concern in practice, practitioner can impose the following restriction to avoid this issue:

|NPk (t)−N ok (t)| ≤M <∞ ∀ t, where M could be a large positive number.

All of the objectives introduced above and captured by inequalities 1, 2, 3, and 4 are important

along with the Participation Condition 1. Instead of creating a single composite performance

metric that encompasses these criteria, we focus on finding Pareto improving policies, where Pareto

improvement is defined as follows:

Definition 1. Pareto Dominance. A policy P2 is Pareto improving over policy P1, P2 ≥P P1, if

it satisfies Participation Condition 1 and inequalities 1, 2, 3, and 4, where at least one of these

inequalities is strict.

Later, in Section 7.3, we summarize the conditions under which our results extend to different

composite objective functions.

2.2. Decision making and participation

In our model, the diversion policy is announced by the central controller, EMS, who also makes

diversion decisions. EDs have a choice of whether to participate in the coordination system or not.

Participation in the system involves sharing truthful information with the EMS; namely, the EDs

report their diversion threshold to the EMS (diversion threshold is set once and cannot be changed

once announced) and continuously reveal census information. We further assume that the EDs

participate in the system if the coordination policy is Pareto improving: participation condition

is met and performance of the ED is at least as good on all performance measures and is strictly

better on one of the performance measures introduced above. It is important to note that it is

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incentive compatible for the EDs to participate in the coordination mechanism and that no ED

has an incentive to withdraw from the coordination system: Assume that out of N EDs, 1 ED

decides to withdraw from the system. Since the expected arrival rate stays the same within the

network of participating EDs (by Participation Condition 1), withdrawing ED will not gain any

advantage in terms of the expected arrival rate and moreover, will be strictly worse off in at least

one of the performance measures. Hence, as long as preservation of the expected arrival rates and

the defined performance measures are important for the EDs, there is no incentive to withdraw

from the system, which implies that our proposed policy is an equilibrium.

Participation in the system in our model also implies that the EDs reveal truthful information

about their congestion to the EMS; for example, through giving the EMS controller access to

their census tracking information system (this can be implemented in practice when, for example,

the central control is performed by the hospital group that owns the individual EDs in the area

and hence, can have access to the internal information system). If the EDs do not provide direct

information access, we assume that they truthfully reveal census status. It is theoretically possible

for the EDs to benefit from misrepresenting their census information in the short-term; however,

we assume that EMS can check and/or monitor the truthful behavior of the EDs and hence,

untruthful behavior will be punished and the ED will be removed from the coordination system,

which is not beneficial to the ED in the long-term. We leave this short-term issue outside of the

scope of our paper and assume truthful revelation of the census information, which is in the best

long-term interest of the ED. Moreover, from the social perspective, the ED that makes the census

information (or other information that can help improve the performance of the system and the

well-being of patients) public may be viewed as socially responsible, which creates an additional

incentive for the EDs to truthfully share information.

2.3. Benchmark policy

If the Participation Condition 1 is not satisfied or the policy does not provide a Pareto-improving

solution, we assume that the EDs are not willing to participate in the diversion mechanism and

share information (lk) and hence, coordination is not possible. This policy is representative of the

situation in the City of Pittsburgh in 2009 (Guyette 2010). We use such a case as a benchmark

and denote it as policy Po. Under policy Po, the arrival rates to each ED, λok are a result of

an equilibrium that ambulances and patients achieve over time without information lk. In the

numerical section, we also compare one of our proposed policies to the current policy of ambulance

diversion, denoted with PFD for Full Diversion, and demonstrate that Full Diversion does not

provide a Pareto improving solution to all players. Full Diversion policy PFD is identical to the

policy analyzed by Deo and Gurvich (2011).

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Now we define the class of all Pareto-improving coordination policies as:

P , {Pi :P i ≥P Po}. (5)

3. Coordination policies with threshold structure.

In this section we introduce a class of coordination policies for the queueing system controller.

Again, we use EMS setting as a motivating example. A commonly used ambulance diversion policy

has a threshold structure: when an ED reaches a certain state, typically expressed as a threshold

based on the number of patients in the ED, it announces diversion; and the EMS attempts to

route all traffic away from the ED on diversion. Such policy, however, does not satisfy Participation

Condition 1 because the resulting arrival rate to such ED will be altered and some of the EDs in

the network will see a decrease in revenue and in market share. Moreover, this policy may hinder

performance according to one or more performance measures for some of the EDs in the network.

Hence, such policy may be not sustainable in practice and we propose an enhanced coordination

mechanism, which still has threshold structure, but in addition, preserves the expected arrival rate

to each ED and guarantees improvement according to all performance measures.

Next, we define a condition on the expected arrival rate that has the following threshold structure:

For a given threshold mk (selected by the EMS), the condition ensures that the expected arrival

rate to ED k is greater than the time-average (equilibrium) arrival rate until this threshold and

lower than the time-average arrival rate above the threshold.

Condition 2. Threshold Condition. For all Pi ∈PT , there exists a finite threshold mk > 0 such

that: {E[Λk(~L

Pi)|Lk = l]≥ λok, ∀ l ≤ mk and ∃ l≤ mk s.t. E[Λk(~LPi)|Lk = l]>λok

E[Λk(~LPi)|Lk = l]≤ λok, otherwise.

Policy class PT represents coordinating mechanisms that satisfy Participation Condition 1 and

Threshold Condition 2. The class PT contains multiple policies; notice, however, that policy Po 6∈

PT because there is no l such that E[Λk(~LPo)|Lk = l]>λok.

Next, we describe a subclass of polices from class PT : policy subclass PCD, where diversion is

C ontrolled. The policies in this subclass are of special interest to us since they closely resemble

full diversion policy that is been used in practice and that has been analyzed in Deo and Gurvich

(2011) with a single objective to minimize expected waiting time; however, in contrast to the

full diversion policy, at least one of the diversion rates in subclass PCD is bound from above

such that the Participation Condition 1 is satisfied. We will later analyze policies within the

subclass using numerical experiments, which allow us to provide additional insights on setting

policy parameters and lets us quantify and assess potential performance improvements that can

result from implementing a policy in class PT .

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3.1. Policies with Controlled Diversion (Subclass PCD)

Consider a case with two EDs with finite diversion thresholds S1 and S2 that indicate some capacity

measurement. Assume that the diversion thresholds S1 and S2 are fixed by the EDs and are

truthfully revealed to the decision maker (EMS)2. The arrival rates to EDs 1 and 2 under the

current (equilibrium) scenario without coordination are λo1 and λo2. We propose a sample subclass

of control policies as follows. Let l1 and l2 be the censuses (observable by EMS) of ED 1 and ED

2, respectively. The diversion rates d12 ∈ [0, λc1] and d21 ∈ [0, λc2] are used as outlined in Table 1.

Since diversion rates can take multiple values, there are multiple policies in this class (we denote

a sample diversion policy from this class with P(d12,d21)D ), each policy is Markov and stationary,

P(0,0)D is equivalent to Po. We let PCD represent all policies that follow routing rules specified in

Table 1 and that have (d12, d21) controlled such that Participation Condition 1 is satisfied.

This class of policies resembles the ambulance diversion policy that is widely used in practice and

is analyzed in Deo and Gurvich (2011); in their paper EDs announce diversion when they reach the

threshold Sk and the EMS tries to divert all controllable traffic from these EDs, hence, we refer to

this policy as full diversion: PFD ≡P(λc1,λ

c2)

D . The difference in our policy is that the diversion rate

is controlled by the EMS: instead of diverting all traffic from ED 1 to ED 2, only d12 is diverted;

with this control, the policy ensures that each ED receives the same number of patients on average

as in the non-coordinated equilibrium state and hence, EDs maintain their revenue and market

share. We note that in some EMS systems, there are attempts to control the diversion rates: e.g. in

several counties in California, there are strict regulations on how long the EDs are allowed to stay

on diversion (California Healthcare Foundation 2009); such policy effectively adjusts the diversion

rate.

ED 2

ED 1

l2 <S2 l2 ≥ S2

l1 <S1Rate to ED1: λo1 Rate to ED1: λo1 + d21

Rate to ED2: λo2 Rate to ED2: λo2− d21

l1 ≥ S1Rate to ED1: λo1− d12 Rate to ED1: λo1Rate to ED2: λo2 + d12 Rate to ED2: λo2

Table 1 Sample diversion policy P(d12,d21)D .

The arrival rates to EDs depend on the diversion rates and the vector of the number of patients in

the ED: Λ1(d12, d21, ~LP(d12,d21)D ) and Λ2(d12, d21, ~L

P(d12,d21)D ). Since the rates are deterministic, we will

denote the arrival rate with λk(d12, d21, ~LP(d12,d21)D ) = Λk(d12, d21, ~L

P(d12,d21)D ). For a fixed diversion

2 Once EDs choose the diversion thresholds and announce them to the EMS, S1 and S2 are fixed and cannot bechanged. Our results hold for any pair of thresholds. Furthermore, our analysis can accommodate the case whereEMS also optimizes over S1 and S2: all results on Pareto improving policies remain the same.

13

rate d12, the rate d21 has to be chosen such that Participation Condition 1 holds for each ED.

Hence, we need to solve the following equation to find d21:

E[λk(d12, d21, ~LP(d12,d21)D )] = Pr(L1 <S1 And L2 <S2)λo1 +Pr(L1 ≥ S1 And L2 ≥ S2)λo1 +

(λo1 + d21)Pr(L1 <S1 And L2 ≥ S2) +

(λo1− d12)Pr(L1 ≥ S1 And L2 <S2) = λo1. (6)

It is clear that the system described by policy P(d12,d21)D is ergodic and hence, the stationary

distribution exists.

Lemma 1. 1. There exist d12 ∈ (0, λc1] and d21 ∈ (0, λc2], such that P(d12,d21)D ∈PCD;

2. PCD ⊂PT .

In Lemma 1, we show that there exists policy P(d12,d21)D that satisfies Participation Condition 1

and that all such policies satisfy Threshold Condition 2 (All proofs are provided in Section 9). We

will then use policies from class PCD to illustrate our results and to obtain additional insights

in Section 5. Notice that the parameters S1 and S2 in this sample policy are fixed by the EDs

because these thresholds depend on the internal processes and policies of the EDs, EMS knows the

thresholds (S1 and S2) and controls the diversion rates (d12 and d21). Next, we analyze stochastic

properties of the number of patients in the system under any policy in class PT (including subclass

of policies PCD) to show that the class PT is Pareto improving.

4. Analysis of policy class PT

In order to show that the proposed class of coordination policies PT is Pareto improving, we will

first derive stochastic properties of the number of patients in the system under any coordination

policy Pi in class PT . First, we obtain stochastic ordering of the censuses:

Theorem 1. Under any coordination policy from class PT , the number of patients in each ED is

stochastically smaller than the number of patients in the ED without coordination:

LPik ≤st LPok ∀ k and hence, ~LPi ≤st ~LPo ∀ Pi ∈PT . (7)

Stochastically smaller census has two desirable properties, which follow from the definition of

stochastic ordering of random variables (Shaked and Shanthikumar 2007).

Property 1. Under any coordination policy from class PT , the expected number of patients at an

ED is lower than under non-coordinated policy Po:

E[LPik ]<E[LPok ] ∀ k and ∀ Pi ∈PT .

14

This property implies that the average number of patients at an ED that participates in coordi-

nation is smaller, and consequently, by Little’s Law, the average length of stay is lower. Guttmann

et al. (2011) show in an empirical study of ED visits that each hour of average waiting time increases

the risk of patient death within seven days following the ED visit. Assuming that the service time

remains the same regardless of the coordination policy, Property 1 implies that participating in

the coordination policy can decrease this mortality risk via decreasing the expected wait time.

Property 2. Under any coordination policy from class PT , the probability that the number of

patients in ED k is larger than a critical threshold Sk is lower than under non-coordinated policy

Po:

P [LPik ≥ Sk]<P [LPok ≥ Sk] ∀ k and ∀ Pi ∈PT .

This property guarantees that the probability of being overcrowded for each ED is lower if the ED

participates in the coordination policy Pi ∈PT . Overcrowding has been linked to higher mortality

rates (Sprivulis et al. 2006) and other adverse patient outcomes (McCarthy 2011); hence, reducing

the probability of being in an overcrowded state may lead to improved health outcomes, which is

beneficial for all players in the healthcare system.

Before presenting the results on the risk measures of the number of patients in the system

(variance and expected census conditional on being overcrowded) under our coordination policy, we

present a technique that allows us to prove the stochastic properties of the number of patients in the

ED. The technique, which was introduced in Do (2012), transforms a discrete random variable into

a continuous random variable that has several desirable properties. The transformation operator

T is defined as follows:

Definition 2. (Do 2012) Given a discrete (non-negative integer) random variable Y with mass

function pY (z) where z ∈Z+ (set of non-negative integers), T (Y ) is a non-negative continuous ran-

dom variable with the right-continuous density function defined as: pT (Y )(t) =∑z∈Z+

pY (z)Iz≤t<z+1,

where Iz≤t<z+1 is an indicator function that equals to 1 when z ≤ t < z+ 1, and 0 otherwise.

To gain intuition about the transformation operator T (·), consider a simple example: Y is a

discrete random variable with the following probability mass function:

pY (z) =

12

for z = 2,12

for z = 4,

0 o/w.

Let FY (z) =∑k≤z

pY (k) and FT (Y )(t) =t∫

0

pT (Y )(v)dv be the cumulative distribution functions of

Y and T (Y ). As illustrated in Figure 1, this technique simply transforms a point function (Figure

1(a)) into a step function (Figure 1(b)), which, in turn, implies that the cumulative probability of

15

1" 2" 3" 4" 5"

½""

p Y(z)

1"

z"

(a) Probability mass function of Y .

1" 2" 3" 4" 5"

½""

p τ(Y)(t)

1"

t"(b) Probability density function of T (Y ).

Figure 1 Illustrating the transformation T (Y ) on the probability function.

½""

F Y(z)

1"

z"1" 2" 3" 4" 5"

(a) Cumulative distribution function of Y .

½""F τ

(Y)(t)

1"

t"1" 2" 3" 4" 5"

(b) Cumulative distribution function of T (Y ).

Figure 2 Illustrating the transformation operator T (Y ) on the distribution function.

the transformed random variable T (Y ) (Figure 2(a)) is equal to the cumulative probability of Y

(Figure 2(b)) at all integer points.

The transformed random variable T (Y ) has the following properties:

Lemma 2.

T (Y +C) = T (Y ) +C; (8)

E[T (Y )] = E[Y ] +1

2; (9)

V [T (Y )] = V [Y ] +1

12; (10)

CEoverα [T (Y )] = E[T (Y )|T (Y )≥ V aRα[Y ]] =E[Y |Y ≥ V aRα[Y ]] +

1

2=CEover

α [Y ] +1

2(11)

∀ α= FY [w] for w ∈Z+, where Z+ is the set of non-negative integers.

Lemma 2 demonstrates that the expectation, variance, and the conditional expected census in

overcrowded state of the transformed continuous random variable are equal to the corresponding

measures of the original discrete random variable shifted by a constant (equalities 8, 9, and 10 were

shown in Do (2012)). Denote the difference between the expected census at ED k under policy

16

Po and census under policy Pi ∈PT as ∆k,i = E[LPok ]− E[LPik ]. From Property 1, we know that

E[LPik ]≤E[LPok ] and hence, ∆k,i ≥ 0 ∀ k and Pi. Let ~∆i represent the vector of ∆k,i ∀ k. Next, we

show that the transformed random variable T (Q) is smaller in convex order under the coordination

policy Pi:

Theorem 2. Under any coordination policy from class PT , the transformed number of patients

in the system is smaller in convex order than under non-coordinated policy Po:

T (LPik ) + ∆k,i <cx T (LPok ) ∀ k and hence, T (~LPi) + ~∆i ≤cx T (~LPo) ∀ Pi ∈PT .

Using Lemmas 2 and 3 and Theorem 2, we can now show that the number of patients in the ED

under any coordination policy from class PT has the following desirable property:

Property 3. Under any coordination policy from class PT , the number of patients in the ED is

less variable than under the non-coordinated control policy Po:

V [LPik ]<V [LPok ] ∀ k and ∀ Pi ∈PT .

This property implies that the number of patients in the ED under the coordination policy is less

“risky” and hence, any risk-averse player has an interest in switching to this policy provided that

the expected number of patients in the ED is the same or smaller, which follows from Property

1. Moreover, lower variability has multiple benefits for the healthcare system: EDs are better off

because they can plan their internal processes better, improve scheduling in the ED and have

less need for expensive surge and flexible capacity; in addition, ambulance crews can make better

routing decisions because the information signal has less noise.

The queueing system participants, however, are also interested in the reduction of the right tail of

the number of patients in the ED, hence, we now analyze the CEoverα , which measures the expected

number of patients in the ED conditional on being overcrowded, or in the states in which re-routing

decision may need to be made. First, we show the result for the transformed random variable

T (LPik ), which allows us to prove the further result for LPik . Following from Theorem 2 and using

the definition of convex order (Shaked and Shanthikumar 2007), we can immediately conclude that

the CEoverα of the transformed random variable T (Q) is smaller under any coordination policy from

class PT :

Lemma 3. Under any coordination policy from class PT , the expected transformed number of

patients in the ED in an overcrowded state is smaller than under non-coordinated control policy Poby more than the difference between the expected censuses at ED k under policies Po and Pi (∆k,i):

CEoverα [T (LPik )]<CEover

α [T (LPik ) + ∆k,i]−∆k,i ≤CEoverα [T (LPok )]−∆k,i,

∀ k,∀ Pi ∈PT , and ∀ α∈ [0,1).

17

Now, using Lemma 2, Theorem 2, and Lemma 3, we show the following property of the ED

census:

Property 4. Under any coordination policy from class PT , the conditional expected census in the

ED in overcrowded state is smaller than under non-coordinated control policy Po by more than the

difference between the expected census at ED k under policy Po and census under policy Pi (∆k,i):

CEoverα [LPik ]<CEover

α [LPok ]−∆k,i ∀ k,∀ Pi ∈PT , and ∀ α= FLPik

[w] for w ∈Z+.

This property demonstrates that for a service level α, which corresponds to an integer census, the

conditional expected census in the ED in the overcrowded state is smaller under any coordination

policy from class PT . Notice that CEoverα [LPik ] ≤ CEover

α [LPok ] relationship follows directly from

Theorem 1: Theorem 1 implies that LPik ≤icx LPok (Muller and Stoyan 2002) and CEover

α is monotone

in increasing convex order (which follows from the results on CV aRα shown in (Pflug 2000)).

However, the result in Property 4 is stronger and shows that the improvement in the conditional

expected census is greater than the difference in the expected census at ED k, ∆k,i.

Finally, using Properties 1, 2, 3, and 4 we conclude that the coordination policy PT is Pareto-

improving.

Theorem 3. For all Pi ∈PT ,Pi ≥P Po, hence, PT ⊆P.

Being a Pareto-improving policy implies that all service agents in the queueing system have an

incentive to participate in the coordination policy as the policy performs better according to all

performance measures defined in Section 2 and preserves the long-term average arrival rate. In

practice, EMS controllers and EDs can use our result as a decision tool for identifying whether a

particular control policy will be sustainable and will provide benefits to the healthcare system.

5. Numerical analysis

In this section, we present numerical experiments using sample policies from subclass PCD that

achieves the following goals: a) We demonstrate numerically that the performance of policies within

subclass PCD is monotonically increasing in the diversion rate d12 (and consequently, in d21) and

hence, we can provide guidance on how to select diversion rates for policies within this subclass; b)

We calculate the percentage improvement in performance measures for sample cases to assess the

magnitude of potential improvement from using proposed policy with Controlled Diversion; and

c) We compare the improvement that the EDs can attain from adopting a policy with Controlled

Diversion as opposed to improvement that can be attained from Full Diversion.

To evaluate our performance measurements numerically, we first have to find the stationary

distribution for the described process, which involves solving the Markov chain that captures

18

transitions in policies from subclass PCD. For a network with two EDs, the Markov chain represents

a 2-dimensional birth-death process. Let state (i, j) represent i (j) patients in ED 1 (2). An example

in Figure 3 depicts the transition diagram for this policy, with diversion thresholds S1 and S2 equal

to 2 at each ED. To find the stationary distribution, we use the approach summarized in Deo and

Gurvich (2011): we truncate the state space at large numbers Mk, such that if we increase the

state space to Mk + 1, the stationary probabilities change by no more than 10−4. The stationary

distribution obtained by solving the set of balance equations for the truncated Markov chain using

Matlab is representative of the actual stationary distribution.

0,3$

0,2$

0,1$

λ 2

1,3$

1,2$

0,0$

λ 2

λ1

1,1$

2,3$

2,2$

3,3$

3,2$

λ1

2,1$

λ1

2,0$

3,1$

3,0$1,0$

μ1 μ1

μ 2

μ1 μ1

μ1 μ1

μ1 μ1

λ1-d12

λ1

λ1

μ1

μ1

μ1

μ1

λ 2

λ 2-

d 21

λ1-d12

λ 2+

d 12

λ 2+

d 12

λ 2

λ 2+

d 12

λ 2-

d 21

λ1+d21 λ1+d21

λ 2+

d 12

λ 2

λ1+d21 λ1+d21

λ1

μ 2

μ 2

μ 2

μ 2

μ 2

μ 2

μ 2

μ 2

μ 2

μ 2

μ 2

λ 2

μ 2

λ 2-

d 21

μ 2

…%

…%

…%

…%

λ 2

μ 2

μ 2

λ1-d12

λ1

λ1

μ1

μ1

μ1

μ1

λ1-d12

…%

…%

…%

…%

λ 2-

d 21

Figure 3 The underlying Markov chain for a sample policy from subclass PCD with S1 = S2 = 2.

We use two numerical settings to calculate diversion rates and percentage improvement in per-

formance measures in Sections 5.1 and 5.2 : Setting 1 is motivated by the utilization data from the

City of Pittsburgh where average utilization is 67% and we set µ1 = µ2 = 1, λo1 = λo2 = 0.67. One

of the limitations of our work is that each ED is modeled as a single-server system, which implies

that for utilization of 67% the average census in the ED is about 2 people. Hence, in this numerical

setting we set the diversion thresholds relatively low: S1 = 3 and S2 = 2 so that we achieve states

with census above the crowding thresholds with non-negligible probability. In order to explore a

scenario where the thresholds can be set high, we introduce Setting 2, in which utilization is 90%3

3 90% utilization was also the highest value used for the cases with symmetric utilization in the numerical study of(Deo and Gurvich 2011)

19

(µ1 = µ2 = 1, λo1 = λo2 = 0.9), which implies the average census of 9 and we then set high thresholds:

S1 = 20 and S2 = 15. In our dataset, 24% of patients arrive by ambulance and we assume that

the controllable portion of arriving patients is bound by λci = λoi ∗ 24%. This implies λci = 0.16 for

i∈ {1,2} in Setting 1 and λci = 0.216 for i∈ {1,2} in Setting 2. Note that the diversion rate cannot

exceed the controllable portion of arrivals: dij ≤ λci .

5.1. Diversion rates that satisfy Participation Condition 1

We first need to identify pairs of diversion rates (d12, d21) that guarantee that policy P(d12,d21)D

satisfies Participation Condition 1 and hence, belongs to class PCD. In Setting 1 (2), we let d12 take

values from {0.04,0.08,0.12,0.16} ({0.06,0.11,0.16,0.21}) and then we find d21 such that Equation

6 is satisfied within 10−3 tolerance. We plot the resulting diversion rate from ED 2 to ED 1 as a

function of the diversion rate d12 in Figure 4 for Settings 1 (left pane) and 2 (right pane). The

squares represent the case with S1 = S2 where the diversion rates at both EDs are equal, which is

intuitive because the EDs are symmetric and they reach their diversion thresholds with the same

average frequency. The circles represent the asymmetric cases: when diversion threshold at ED 2 is

lower than at ED 1, ED 2 reaches the overcrowded state more frequently, and hence, to maintain

fairness in the arrival rate, the diversion rate from ED 2 to ED 1 should be set lower than the

diversion rate from ED 1 to ED 2. Clearly, d21 is increasing in d12 because as ED 2 receives more

diverted arrivals from ED 1, they reach their diversion threshold fast and need to divert more

frequently as well.

0.04 0.08 0.12 0.16Diversion Rate from ED 1 to ED 2

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Diversion rate from ED 2 to ED 1

0.0178 0.0346

0.0513

0.0680

0.0400

0.0800

0.1200

0.1600

Setting 1: Utilization = 67%, S1=3, S2=2

0.06 0.11 0.16 0.21Diversion Rate from ED 1 to ED 2

0.03

0.06

0.09

0.12

0.15

0.18

0.21

Diversion rate from ED 2 to ED 1

0.0280 0.04900.0650

0.0950

0.0600

0.1100

0.1600

0.2100

Setting 2: Utilization = 90%, S1=20, S2=15

F1

Full Diversion No Diversion Our policy

0.65

0.70

Lambda:

2

5

E

0.4

0.6

Prob

10

30

Var

10

15

CVaR

Sheet 1F1 CVaR E Lambda: Prob Var

Full Diversion

No Diversion

Our policy 23.29

31.71

10.15

0.69

0.70

0.59

0.67

0.67

0.63

4.50

5.15

2.93

16.33

18.15

10.76

Sheet 5

Threshold settings:Asymmetric thresholds: S1 > S2 Symmetric thresholds: S1 = S2

Figure 4 Diversion rates that satisfy Participation Condition 1 under the policies in subclass PCD for Settings

1 and 2.

20

5.2. Comparison of policies within Controlled Diversion subclass PCD.

The numerical study in this subsection achieves two goals: a) it quantifies the magnitude of improve-

ment in performance measures that the system can achieve from adopting a policy with Controlled

Diversion, and b) it demonstrates that the improvement is monotonically increasing in diversion

rate, which provides an insight on how to set the diversion rates.

For each performance measure, we compute the percentage improvement relative to the No

Diversion policy (i.e. Po, which is equivalent to P(0,0)D ) as follows: for example, for Variance measure,

we calculateV [LPo

k]−V [L

P(d12,d21)D

k]

V [LPok

]for pairs of diversion rates (d12, d21) that were identified in Section

5.1. Similarly, for all other performance measures. We then show the sensitivity of the percentage

improvement relative to the diversion rate d12 in Figure 5 (top (bottom) pane illustrates Setting 1

(2) and left (right) pane illustrates ED 1 (2)). We observe that Policy P(d12,d21)D is improving over the

benchmark case Po for all tested pairs of (d12, d21) as is shown in our analysis in Section 4. Moreover,

the numerical results show that the improvement is monotone increasing in d12 and the highest is

achieved when d12 is at its highest feasible value: d12 = λc1. This observation obtained numerically

provides an important insight for policy designers: within the subclass PCD it is best to set one of

the diversion rates as high as possible (in this example, d12 = λc1) for any objective function that

satisfies the conditions outlined in Section 7.3. Notice that d21 will not be at its highest value of

d21 = λc2, but will be determined by Participation Condition 1. This is an important distinction of

our policy from the commonly used ambulance diversion policy (PFD =P(λc1,λc2)

D ) analyzed in Deo

and Gurvich (2011) with a single objective to minimize expected waiting time, which is not Pareto

improving and leads to the equilibrium solution of all-on-diversion or none-on-diversion and may

not be incentive compatible for the EDs.

5.3. Controlled Diversion versus Full Diversion

Next, we compare performance measures for each ED under our proposed Controlled Diversion

policy to No Diversion policy (Po) and to Full Diversion policy (PFD). For the numerical compar-

ison, we pick a case with asymmetric EDs from the numerical study of Deo and Gurvich (2011),

we match it on utilizations and on the controllable portion of arrivals and use it as our Setting 3.

Namely, the utilizations at EDs 1 and 2 are 0.846 and 0.954 correspondingly. To match utilization,

we set service rates at µ1 = µ2 = 1, and the arrival rates at λo1 = 0.846 and λ02 = 0.954 (the arrival

rates are different from the ones used in Deo and Gurvich (2011) because we model EDs a single-

server queues and hence, have to scale down their arrival rates used in the multi-server model by

the number of servers to match utilization). We match their assumption of 25% of patients arriving

by ambulance and hence, set λc1 = 25% ∗ 0.846 = 0.2115 and λc2 = 25% ∗ 0.954 = 0.2385. Finally, we

21

ED 1 ED 2

0.04 0.06 0.08 0.10 0.12 0.14 0.16Diversion Rate from ED 1 to ED 2

0.04 0.06 0.08 0.10 0.12 0.14 0.16Diversion Rate from ED 1 to ED 2

0%

10%

20%

30%

Per

cent

age

impr

ovem

ent i

npe

rform

ance

mea

sure

Setting 1

a.) Expected Censusb.) Probability of Overcrowding

c.) Variance of Censusd.) Expected Census in Overcrowded State

ED 1 ED 2

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20Diversion Rate from ED 1 to ED 2

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20Diversion Rate from ED 1 to ED 2

10%

20%

30%

40%

50%

Per

cent

age

impr

ovem

ent i

npe

rform

ance

mea

sure

Setting 2

Figure 5 Percentage improvement of performance measures for EDs 1 and 2 under Settings 1 and 2 implementing

policy from subclass PCD as a function of diversion rate d12.

match the diversion thresholds and set them at S1 = 14 and S2 = 11. For comparability, we keep

the diversion thresholds the same for both policies. Such comparison is practical when the EDs

truthfully share information with the EMS and set their thresholds, e.g. at capacity; the EMS then

compares the performance of Full Diversion and Controlled Diversion. Using our observation from

Section 5.2, we set the diversion rate d12 equal to its highest possible value: d12 = λc1 in Controlled

Diversion. First, we note that the expected arrival rate under Full Diversion is 0.87 to ED 1 and

0.92 to ED 2, which indicates that Full Diversion may have negative implications for the EDs: ED

2 may see reduced revenue and/or market share, ED 1 may see an undesired increase in patients

that exceeds planned capacity. Hence, Full Diversion Policy may be not incentive compatible for

the EDs to keep participating in the diversion system.

Next, we quantitatively assess performance of the EDs and show the result in Figure 6 where

we plot the performance measures for ED 1 (left pane) and ED 2 (right pane) under different

22

diversion policies: No Diversion policy Po in light-gray on the left, Full Diversion PFD in black

on the right, and Controlled Diversion P(0.2115,0.0135)D in dark-gray in the middle. We observe that

several performance measures, namely, expected census (pane a.) and probability of overcrowding

(pane b.) are higher for ED 1 than without diversion, which demonstrates that policy PFD is not

Pareto-improving and ED 1 may have an incentive to not participate in the diversion system. Notice

that Controlled Diversion mechanism, on the other hand, provides improvement on all performance

measures and preserves the expected arrival rate for both EDs. Moreover, for ED 1, Controlled

Diversion provides higher improvement in all performance measures.

6. Implementation

The sample policy presented in Section 3.1 may be hard to implement due to its probabilistic

nature: the policy prescribes to divert a rate of patients; however, it does not provide guidance

to the decision maker about what course of action to take with a particular patient. In practice,

there exist different diversion systems that try to control the rate: e.g. specifying a time interval,

during which to divert all patients and a time interval during which there can be no diversion,

implementing diversion for certain types of patients (e.g. divert all patients with neurological

problems) etc. Parameters of such policies are hard to calculate optimally and there is no empirical

evidence of success of such policies. Here, we propose an alternative sample policy, Policy IOU,

that is based on the idea of our threshold policy and is easy to implement in practice: this policy

deviates slightly from the definition of our proposed class PT , however, prescribes a specific course

of action in each state.

Consider a setting with 2 EDs with diversion thresholds S1 and S2. As opposed to the sample

policy described earlier in Section 3.1, we introduce an additional parameter: “IOU” counter K

that keeps track of the diverted patients in the following manner4. If a patient is diverted from ED

2 to ED 1, we increase the counter by one K =K + 1, which implies that ED 1 now “owes” one

more patient arrival to ED 2. Similarly, when a patient is diverted from ED 1 to ED 2, we decrease

the counter by one: K =K − 1, a negative K value implies that ED 2 “owes” one more patient to

ED 1. The ambulance crews can then use the census, diversion threshold, and counter information

as follows (the rules are summarized in Table 2): when only one of the EDs is overcrowded, all

controllable patients are diverted to the other ED and the counter is updated accordingly; when

both EDs are overcrowded, no diversion is made; when neither ED is overcrowded diversions are

made for all controllable patients according to the status of the counter: if K = 0, no diversion

is made, if K > 0, all controllable patients are diverted from ED 1 to ED 2, and if K < 0, all

4 In the case of three or more EDs, we will need to use multiple counters Kij .

23

ED 1 ED 2

NoDiversion

ControlledDiversion

FullDiversion

NoDiversion

ControlledDiversion

FullDiversion

a.) ExpectedCensus

b.) ProbabilityofOvercrowding

c.) Varianceof Census

d.) ExpectedCensus inOvercrowdedState

0.0

10.0

20.0

0.0

0.2

0.4

0.6

0.0

200.0

400.0

0.0

20.0

40.0

60.0

5.49 5.16 6.44

20.4518.66

7.56

0.080.130.10

0.580.59

0.26

28.19 34.3935.67

331.08

413.60

41.07

16.2818.49 17.76

66.9860.18

20.69

PolicyNo Diversion

Controlled Diversion

Full Diversion

Value as an attribute for each Policy broken down by ED vs. Measure. Color shows detailsabout Policy. The data is filtered on Utilization, which keeps Setting 3. The view is filtered onMeasure, which keeps d.) Expected Census in Overcrowded State, a.) Expected Census, b.)Probability of Overcrowding and c.) Variance of Census.

Figure 6 Performance measures for EDs 1 and 2 under Full Diversion policy (PFD), No Diversion policy (Po),

and Controlled Diversion policy with rates d12 = 0.16 and d21 = 0.03 (a policy from subclass PCD)

.

controllable patients are diverted from ED 2 to ED 1. Theoretically, as explained in Section 2.1,

number of “owed” patients can go to infinity: denoting number of “owed” patients at time t as

K(t), limt→∞|K(t)|=∞. If this is a concern in practice, practitioner can introduce an upper bound

on K(t): |K(t)| ≤M <∞, where M could be a large positive number. This policy can be refined

to accommodate multiple patient classes by introducing additional counters Kclass.

To confirm that this policy behaves similarly to the policy class analyzed above, we apply this

policy to numerical Setting 1 and plot the expected conditional arrival rate to EDs 1 (left pane)

24

ED 1 ED 2

-1 0 1 2 3 4 5 6 7 8 9 10Census

-1 0 1 2 3 4 5 6 7 8 9 10Census

0.55

0.60

0.65

0.70

0.75

Conditional Arrival rate

ED / Policy

ED 1 ED 2

Controlled diversion No Diversion Full Diversion Controlled diversion No Diversion Full Diversion

a.) Expectedarrival rate:

b.) ExpectedCensus:

c.) Probability ofOvercrowding:

d.) Variance ofCensus:

e.) ExpectedCensus inOvercrowdedState:

0.0

0.5

0

6

0.0

0.5

0

40

0

20

0.67 0.67 0.71 0.67 0.67 0.63

2.092.031.91

5.15

2.934.50

0.29 0.330.30

0.69 0.590.70

6.155.09 5.2710.15

23.2931.71

7.78 8.03 7.73

18.1516.3310.76

Comparison of three policies

Policy Controlled Diversion No Diversion Policy IOU

Figure 7 Expected arrival rate conditional on the ED census under Policy IOU (triangles) and Controlled Diver-

sion Policy P(d12=0.16,d21=0.068)D (squares) when λ1 = λ2 = 0.67, µ1 = µ2 = 1, S1 = 3, and S2 = 2.

ED 2Not crowded Overcrowded

(l2 <S2) (l2 ≥ S2)

ED 1Not crowded

(l1 <S1 )If K = 0, no diversion

If K > 0, divert from ED 1 to ED 2If K < 0, divert from ED 2 to ED 1

Divert from ED 2 to ED 1Update K=K+1

Rate to ED 1 is λ01 +λc2

Rate to ED 2 is λu2

Overcrowded(l1 ≥ S1 )

Divert from ED1 to ED2Update K =K − 1Rate to ED 1 is λu1

Rate to ED 2 is λ02 +λc1

Rate to ED1 is λo1Rate to ED2 is λo2

Table 2 Diversion decision rules under Policy IOU.

and 2 (right pane) in Figure 7 as a function of census in the ED and observe that it follows

a similar threshold pattern as the arrival rate in policy with Controlled Diversion5. Finally, we

compute all performance measures using Policy IOU and compare them to the benchmark policy

and the policy with Controlled Diversion for Setting 1 in Figure 8. Notice that performance under

Policy IOU deviates slightly from the performance under Controlled Diversion, but is comparable

in magnitude.

7. Policy applications and improvements

In this section, we discuss applications of our policy class PT to different realistic settings, discuss

improvements that can be made to an existing coordination policy to attain Pareto improvement,

and finally, introduce a subclass of policies PM within the class PT that Pareto-dominate all

policies in PT \PM .

5 Note that we do not have an analytical result that shows that the expected conditional arrival rate has thresholdstructure, rather we show it numerically in this section.

25

ED 1 ED 2

NoDiversion

ControlledDiversion

PolicyIOU

NoDiversion

ControlledDiversion

PolicyIOU

a.) ExpectedCensus

b.) Probability ofOvercrowding

c.) Variance ofCensus

d.) ExpectedCensus inOvercrowded State

0

1

2

0.0

0.2

0.4

0

2

4

6

0

2

4

6

8

1.8162.030

1.794 1.8362.030

1.664

0.277 0.2830.301

0.4250.449 0.438

4.269

6.152

4.331 4.6783.610

6.152

6.5177.030

5.5046.673

5.393

7.030

Figure 8 Performance measures under No Diversion, Controlled Diversion (P(d12=0.16,d21=0.068)D ), and Policy IOU

for Setting 1 (λ1 = λ2 = 0.67, µ1 = µ2 = 1, S1 = 3, and S2 = 2).

7.1. Multiple patient classes.

In the definition of our model, we assumed that there is one arrival stream of patients Λ. However,

this stream may consist of multiple classes of patients. For example, patients with different acuity

levels. To accommodate this practical setting, our definition of Pareto improvement can be extended

by creating duplicates of inequalities 1, 2, 3, and 4 for each patient class. For example, instead of

26

inequality 2, we will have P [LP1k,c ≥ Sk,c]≥ P [LP2k,c ≥ Sk,c] for each patient class c. Our result holds

after this modification and any policy in class PT will provide Pareto improvement according to

the modified definition.

7.2. Enhancing coordination policies.

Next, we show an auxiliary lemma that demonstrates how any Pareto improving coordination

policy implemented in practice can be enhanced to guarantee better performance.

Lemma 4. Let P1, P2 be two coordination policies satisfying Condition 1 and with the conditional

expected arrival rates sequence:{E[Λk( ~Q

P1)|Lk = l] = al

}∞l=0

and{E[Λk( ~Q

P2)|Lk = l] = bl

}∞l=0

,

where al = bl ∀ l ∈ {{0, .., n− 1}∪ {n+ 2, ..,∞}} and bn >an, then bn+1 <an+1 and P1 ≤P P2.

Lemma 4 shows that by re-allocating some conditional expected arrival rate towards a less

congested ED, a coordination policy P2 will Pareto-dominate policy P1. Using this re-allocation

principle from Lemma 4, next we define a class of policies PM that we will show to be a subset of

PT and to be Pareto-dominating over all policies in PT \PM .

Definition 3. Let PM be a set of coordination policies, where Participation Condition 1 holds

and the arrival rate to ED k satisfies the following condition:

Condition 3. Monotonicity Condition. Conditional expected arrival rate is monotone in the num-

ber of patients at ED k: For all k and P ∈PM , E[Λk(~LP)] = λok and

E[Λk(~LP)|Lk =m]=E[Λk(~L

P)|Lk =m+ 1] ∀ m= 0,1,2, ... (12)

and there exists m such that:

E[Λk(~LP)|Lk = m]>E[Λk(~L

P)|Lk = m+ 1] (13)

This condition implies that whenever the census at ED k increases, the conditional expected

arrival rate to ED k will decrease. This monotonicity condition is intuitive, however, such a policy

may be harder to implement in practice as compared to a threshold policy in class PT without

monotonicity. For example, in a case with symmetric EDs, join the shortest queue policy will satisfy

Monotonicity Condition 3.

Theorem 4. 1. PM ⊂PT ⊂P

2. For any policy P1 ∈PT \PM , there exists P2 ∈PM , such that P1 ≤P P2.

3. For any policy P1 ∈P \PT , there exists P2 ∈PT , such that P1 ≤P P2.

27

This theorem presents a strong nested classification of coordination policies: any Pareto improv-

ing policy with the monotone structure dominates policies that have threshold structure, but are

not monotone, which in turn, dominate all Pareto-improving policies that do not have threshold

structure. These policies along with the nested classification can be used as a decision tool by

EMS and EDs in constructing and improving coordination policies given their possible practical

constraints.

7.3. Composite performance measures.

We note that Pareto-improvement is a robust result that implies that any policy in class P is

beneficial for the EDs according to all measures defined in Section 2.1 treated equally and/or

according to any composite measure, which is a convex combination of the defined objectives

(expected census in the ED, probability of the census in the ED not exceeding a pre-specified

threshold, variance, and expected census conditional on being overcrowded) or even more generally,

any polynomial of the objectives with non-negative coefficients (weights). Furthermore, Pareto-

improvement also applies to any increasing function in each objective. As an example, consider

increasing cost functions that map our performance measures to operational cost of the ED k:

C1(E[LPk ]), C2(P [LPk ≥ Sk]), C3(V [LPk ]), and C4(CEoverα [LPk ]). We can create a composite cost mea-

sure by assigning different weights to the cost functions: C(LPk ) = γ1C1(E[LPk ]) + γ2C2(P [LPk ≥

Sk]) + γ3C3(V [LPk ]) + γ4C4(CEoverα [LPk ]). Consider two coordination policies P1 and P2, P1 ≤P P2

will imply C(LP1k ) ≤ C(LP2k ). Each ED can have a different set of weights and/or different cost

functions Ci(·). For a given composite measure, each ED can find a specific policy within class PM

that will be optimal for the particular ED.

7.4. What performance improvements do patients see?

Notice that we performed our analysis from the perspective of time-average distribution. If we

take the patient perspective, the distribution observed by patients differs from the time-average

because the arrival stream to each ED is an outcome of a coordination policy and hence, it is

no longer Poisson and PASTA (Poisson arrivals see time-average) does not apply. Consider the

flow of virtual patients who arrive to ED k according to the Poisson process with rate λok, but

only observe the system, never enter the system as opposed to the actual patients. Since virtual

patients arrive according to a Poisson process, they observe the time-average distribution πPik (n). In

healthcare practice, for example, The Joint Commission sends their inspectors to evaluate hospitals

and EDs based on the set of performance measures that includes observing congestion in the ED.

Assuming that the inspectors arrive according to the Poisson process, the performance observed

and reported by the inspectors, which is used to evaluate the ED performance, can be seen as

28

performance observed by the virtual customers. Hence, all measures calculated based on the time-

average distribution are representative of the ED perspective.

To evaluate the performance improvement observed by patients, first, we compare the time-

average distribution observed by the virtual patients, πPik (n), to the distribution observed by actual

patients, πPik (n), where n represents an aggregate state: probability that the number of patients

in ED k is n aggregating over the census possibilities at the other EDs. Then, we establish a

relationship between the two distributions. Let λPik (n) be the arrival rate while there are n patients

at ED k. First, we state the following relationship:

Lemma 5. πPik (n) =λPik

(n)

λokπPik (n) ∀ Pi, l, and k. 6

This intuitive relationship between the distributions allows us to show in the next Theorem

that the patients will see a distribution of the number of patients in the system that is better

than the stationary distribution in the analysis when a policy Pi ∈ PT is implemented because

the distribution observed by arriving patients is more balanced: patients are less likely to arrive

(because of lower arrival rates) during crowded times and more likely to arrive (because of larger

arrival rates) during non-crowded times. Before stating the next result, we define Pareto dominance

from patients’ perspective, where we use LPik to denote the number of patients in the system

observed by patients:

Definition 4. Pareto Dominance from patients’ perspective. A policy P2 is Pareto improving from

patients’ perspective over policy P1, P2 ≥P P1 if it satisfies inequalities 1, 2, 3, and 4, where all

the performance measures are defined using the distribution observed by patients (E[LPik ], V [LPik ],

P [LPik >Sk], and CEoverα [LPik ]) and at least one of these inequalities is strict.

Theorem 5. 1. The number of patients observed by patients is stochastically smaller than the

time-average number of patients in the ED: LPik ≤st LPik ∀ Pi ∈PT and ∀ k;

2. T (LPik ) + ∆Pik ≤cx T (LPik ) ∀ Pi ∈PT and ∀ k, where ∆Pik =E[LPik ]−E[LPik ];

3. For any policy P1 ∈PT \PM , there exists P2 ∈PM , such that P1 ≤P P2.

4. For any policy P1 ∈P \PT , there exists P2 ∈PT , such that P1 ≤P P2.

As a result, all the aforementioned measures as experienced by the actual patients will be lower

than the same measures calculated using the time-average distribution. Hence, the results in Theo-

rems 3 and 4 hold from the patients’ perspective as well as from the EDs’ perspective. Consequently,

the performance improvements perceived by patients are even higher than the improvements from

the ED’s perspective.

6 This relationship can be obtained using Palm Theory (see e.g. Shanthikumar and Zazanis (1999)), however, wepresent a simple intuitive proof using a counting argument in the Appendix, which is specific to our modeling setup.

29

7.5. Extension to Markov Modulated Poisson Processes

In this section, we demonstrate that our analytical model and approach can be extended to the

case where the total arrival process is modulated by some exogenous Markov process, i.e., Markov

Modulated Poisson Process (MMPP). For simplicity, we assume that there are two states of the

world: Let Z indicate the random variable that evolves between the two states zL and zH with γLH

(γHL) indicating transition probability from zL (zH) to zH (zL).

When there is no flow rate control between the EDs, we denote λk(Z|Z = zL) with λLk and

λk(Z|Z = zH) with λHk . The total arrival rates in each state are then λL =∑N

k=1 λLk and λH =∑N

k=1 λHk , where λLk (λHk ) consists of λLuk (λHuk ) and λLck (λHck ) for uncontrollable and controllable

streams of patients, similarly to our main model description. Let (lk,Z) be the aggregate state of

the system where the number of patients in the ED k is lk and the state of nature is Z taken across

all other patient numbers at the other EDs. As compared to the Markov diagram with Poisson

arrival rate analyzed in the main body of the paper (see Figure 9(a)), the diagram of the system

with MMPP will change as plotted in Figure 9(b).

l+1!l!

λk λkλk

µk µk µk

(a) Poisson arrival process.

l+1,"ZH"l,"ZH"

l+1,"ZL"l,"ZL"

λkH λk

HλkH

λkL λk

LλkL

µk

µk

µk

µk

µk

µk

γ LH γ LHγ HLγ HL

(b) MMPP arrival process.

Figure 9 Illustrating the change in Markov diagram for an ED with the MMPP arrival process.

Now, it is easy to see that if we adjust the participation condition to take expectation over all

states in Z: E[λk(~L(Pi),Z)] = λok, all the remaining analysis will remain the same. Hence, we can

define policy classes PT and PM with this adjusted participation condition and attain the same

results on Pareto improvement. In implementation, however, the central controller will now have

30

more control parameters and can decide the values of arrival rate to each ED given the state is zL

or zH . For example, in our sample policy PS, the controller will have to specify diversion rates for

each state of the world: dL12, dH12, dL21, and dH21.

As a special case of the MMPP described above, we can extend our result to a process where

the state of the world is represented by the time of day: the arrival rate changes throughout

the day, which is commonly observed in healthcare and ED applications. In such case, the state

transitions are controlled by the clock, rather than by transition probabilities γi and if we adjust

the participation condition accordingly, we will again be able to show that policy classes PT and

PM with the modified condition result in Pareto improvement for all participants.

8. Conclusion

ED overcrowding in the U.S. is a growing problem that requires robust solutions. Many EMS

across the U.S. are attempting to find a solution; for example, EMS policies in Kansas have a

sophisticated diversion system that allows EDs to announce diversion for specific types of patients

(for example, closed to trauma patients). In some counties in California, EMS imposes time limits

on diversion status; several EMS systems, including that of the City of Pittsburgh in 2004, use

multiple level diversion signals based on the extent of crowding in the EDs. However, there is no

consensus on what policy is best and no systematic way to evaluate and compare such policies.

We use this issue to motivate a general problem: how to coordinate flow in a queueing system with

independent agents that can decide whether to participate in the coordination system or not, that

have multiple performance measures, and that receive revenue based on the processed load.

We find classes of coordination policies that are Pareto improving for all agents in the system:

under the coordination policies within our defined classes, all agents receive the same expected

load, they see stochastically smaller number of patients with smaller variance and lower expected

number of patients in overcrowded situations. This analytical result proposes a convenient decision

tool that can help queueing system coordinators (such as EMS in healthcare practice) compare

and select coordination policies that guarantee performance improvement according to multiple

measures. We demonstrate that our solution is robust: our results hold under multiple patient

classes, under different composite measures, from the patients’ perspective, and under Markov

Modulated Poisson Process. Moreover, our result on policy improvements (Lemma 4) provides a

tool to refine any existing coordination policy, the performance of which will Pareto-dominate the

existing policy.

Numerous extensions of our work are possible by relaxing assumptions of the model, considering

multiple-server system, considering non-stationary arrival process, etc. Here, we also suggest a

direction that may lead to immediate practical implications for queueing system coordination. The

31

focus of our work has been on identifying a class of policies that provide Pareto improvement. Once

the participants select their performance measures, each participant (including service agents and

the system coordinator) can find an optimal policy within the defined classes. For example, in our

sample policy Ps, the service agents (EDs) select the diversion thresholds, while the coordinator

(EMS) selects the diversion rates. This problem may be formulated as a repeated game to find the

optimal policy for the EMS for the selected set of performance measures.

9. Mathematical Proofs of Results

Proof of Lemma 1 1. First, we show existence of d12 and d21:

The expected arrival rate to ED 1, E[λ1(d12, d21, ~LP(d12,d21)D )], is continuous in both d12 and d21.

Evaluating E[λ1(d12, d21, ~LP(d12,d21)D )] at three extreme points:

(a) When there is no coordination: E[λ1(0,0, ~LPo)] = λo1;

(b) ED 2 diverts all to ED 1 when it is crowded, but ED 1 never diverts to ED 2:

E[λ1(0, λc2, ~LP

(0,λc2)

D )]>λo1.

(c) ED 1 diverts all to ED 2 when it is crowded, but ED 2 never diverts to ED 1:

E[λ1(λc1,0, ~LP

(λc1,0)

D )]<λo1.

Note that E[λ1(λc1, λc2, ~L

P(λc1,λ

c2)

D )] could be larger, equal or smaller than λo1.

Since E[λ1(d12, d21, ~LP(d12,d21)D )] is continuous in both d12 and d21, E[λ1(d12, d21, ~L

P(d12,d21)D )] is a

surface w.r.t d12 ∈ [0, λc1] and d21 ∈ [0, λc2]. Consider a fragment of the plane: λ1(d12, d21, ~LP(d12,d21)D ) =

λo1 for d12 ∈ [0, λc1] and d21 ∈ [0, λc2], from the three points above, the surface has to intersect with

the part of the plane and the intersection must be a non-trivial continuous curve on the plane with

one point of (0,0).

Let this curve be C and define: d12 = sup{d12 : (d12, d21)∈C} and d21 = sup{d21 : (d12, d21)∈C}.

Clearly, d12 > 0 and d21 > 0 and there is no point on C other than (0,0) such that d12 = 0 or d21 = 0.

By the continuity of the curve, for each d12 ∈ [0, d12], there exists some d21 ∈ [0, d21] such that

E[λ1(d12, d21, ~LP(d12,d21)D )] = λo1.

Furthermore, if E[λ1(λc1, λc2, ~L

P(λc1,λ

c2)

D )] > λo1, since E[λ1(λc1,0, ~LP

(λc1,0)

D )] < λo1, there exists d21 < λc2

such that E[λ1(λc1, d21, ~LP

(λc1,d21)

D )] = λo1.

Similarly, if E[λ1(λc1, λc2, ~L

P(λc1,λ

c2)

D )] < λo1 then E[λ2(λc1, λc2, ~L

P(λc1,λ

c2)

D )] > λo2, there exists d12 < λc1

such that E[λ2(d12, λc2, ~L

P(d12,λ

c2)

D )] = λo2 (and hence, E[λ1(d12, λc2, ~L

P(d12,λ

c2)

D )] = λo1).

2. Let π(m, l) represent the stationary probability of being in state (m, l). By definition,

E[λk(~LP(d12,d21)D )] = λok ∀ P

(d12,d21)D ∈PCD. For ED 1, the conditional expected arrival rate when the

number of patients in the system is less than S1 is:

E[λ1(~LP(d12,d21)D )|L1 =m;m<S1] =

1

Pr{L1 =m}

∞∑l=0

π(m, l)λ1(m, l)

32

=1

Pr{L1 =m}

(S2−1∑l=0

π(m, l)λo1 +∞∑l=S2

π(m, l)(λo1 + d21)

)

≥ 1

Pr{L1 =m}

(S2−1∑l=0

π(m, l)λo1 +∞∑l=S2

π(m, l)(λo1)

)

≥ λo1Pr{L1 =m}

(S2−1∑l=0

π(m, l) +∞∑l=S2

π(m, l)

)= λo1

The equality occurs if d21 = 0, i.e., there is no diversion from ED 2 to ED 1. Similarly, we can

establish that E[λ1(~LP(d12,d21)D )|L1 =m;m≥ S1]≤ λo1. The proof for ED 2 is similar.

Proof of Theorem 1 We present the following proof for a pure policy, hence we use E[λk(~LPi)]

instead of E[Λk(~LPi)]. The proof can be easily modified for a non-pure policy.

Let ~l−k denote the vector of censuses with k-th component removed. Denote the arrival rate

to ED k with λk[~l|lk = m] = λk(m,~l−k), given the current census of this ED is m and the corre-

sponding stationary probability is π(m,~l−k). We use πk(m) =∑~l−k

π(m,~l−k) to denote the stationary

probability of having m patients in ED k aggregated across all censuses at other EDs.

Consider the N − 1 dimension hyperplane that separates the aggregate states (lk = m) and

(lk = m + 1)). The rate of up-crossing (transitioning from states with (lk 6 m) to states with

(lk >m+ 1)) is only from some state in the hyperplane (lk =m), which is:∑~l−k

λk(m,~l−k)π(m,~l−k)

The down-crossing (transitioning from states with (lk >m+ 1) to states with (lk 6m)) of this

hyperplane is only from the hyperplane (lk = m + 1) and the down-crossing rate is:∑~l−k

µk(m +

1,~l−k)π(m+ 1,~l−k) = µk∑~l−k

π(m+ 1,~l−k) = µkπk(m+ 1).

When the system is stationary, the up-crossing rate must be equal to the down-crossing rate, so

we have: ∑~l−k

λk(m,~l−k)π(m,~l−k) = µkπk(m+ 1)

Expanding πk(m), we have:∑~l−k

λk(m,~l−k)π(m,~l−k) = πk(m)1∑

~l−k

π(m,~l−k)

∑~l−k

λk(m,~l−k)π(m,~l−k) =

πk(m)∑~l−k

λk(m,~l−k)P (~L|Lk =m) = πk(m)E[Λk(~L)|lk =m]

Thus,

πk(m+ 1) = πk(m)E[λk(~L)|Lk =m]

µk

33

Now, consider the probability of having no patient in ED k, for an ergodic system under a policy

P, we have using Little’s Law:

πPk (0) = 1− E[λk(~LP)]

µk

Since we have: E[λk(~LPi)] =E[λk(~L

Po)] = λ0k, we have:

πPik (0) = πPok (0) = πk(0) = 1− λ0kµk

.

Let δPk (m) = πPk (m+ 1)− λokµkπPk (m) = πPk (m)

(E[λk(~LP )|Lk=m]−λok

µk

). Thus, πPk (m+ 1) =

λokµkπPk (m) +

δPk (m).

From Condition 2, for all Pi ∈PT :

E[λk(~LPi)|Lk = l]

{≥ λok, ∀ l ≤ m≤ λok, otherwise.

So, δPok (m) = 0 ∀ m∈Z+, and

δPik (m)

{≥ 0, ∀ m ≤ m< 0, otherwise.

We can show using contradiction that:

∃ l∗ s.t. πPik (m)

{≥ πPok (m), ∀ m ≤ l∗

≤ πPok (m), otherwise.

Because πPik (0) = πPok (0) = πk(0) and πk(m+ 1) = πk(m)E[Λk(~Q)|Lk=m]

µkfor any stationary control

policy, we have: πPik (1) =λokµkπPik (0) + δPik (0)>

λokµkπPik (0) =

λokµkπPok (0) + δPok (0) = πPok (1). Recursively,

∀ m ≤ m, πPik (m)>πPok (m).

Now we will prove that there must be a unique finite number l∗ such that πPik (l∗)> πPok (l∗) and

πPik (l∗+1)6 πPok (l∗+1) where one of the inequalities must be strict. First, there must exist a finite

l such that πPik (l + 1) 6 πPok (l + 1). Suppose not, then∑∞

m=0 πPik (m) >

∑∞m=0 π

Pok (m) = 1 which

is a contradiction. Next, we define: l∗ = min{l : πPik (l + 1) 6 πPok (l + 1) } and by this definition,

πPik (l∗)> πPok (l∗).

Apparently, we can see that l∗ > m, so δPik (m)< 0 ∀ m> l∗. Hence, πPik (m)< πPok (m)∀m>l∗ + 1

and that means the number l∗ is unique.

So, the interpolated line of the PDF of LPik crosses that of LPok from above at only one point. Now,

we prove that LPik ≤st LPok ∀ k using one of its definitions:

∑l

m=0 πPik (m)>

∑l

m=0 πPok (m) ∀ l . We

know from the above that∑l

m=0 πPik (m)>

∑l

m=0 πPok (m) ∀ l6 l∗. Suppose there exists a finite

m∗ ,min{l :∑l

m=0 πPik (m)<

∑l

m=0 πPok (m)}, then we must have m∗ > l∗+ 1 and hence, πPik (m)6

πPok (m)∀m>m∗. Consequently, the following must hold:

1 =∑∞

m=0 πPik (m) =

∑m∗

m=0 πPik (m) +

∑∞m=m∗+1 π

Pik (m) <

∑m∗

m=0 πPok (m) +

∑∞m=m∗+1 π

Pok (m) = 1,

which is a contradiction. �

Proof of Lemma 2 Show the shift in mean:

E[T (Y )] =

∞∫0

tpT (Y )(t)dt=

∞∫0

∞∑z=0

pY (z)Iz≤t<z+1tdt

34

Using the monotone convergence theorem, we can interchange the summation and integration:

E[T (Y )] =

∞∫0

∞∑z=0

pY (z)Iz≤t<z+1tdt=∞∑z=0

z+1∫z

tpY (z)dt=∞∑z=0

pY (z)

z+1∫z

tdt=∞∑z=0

pY (z)

(z+

1

2

)=

=∞∑z=0

pY (z)z+1

2=E[Y ] +

1

2

Similarly, we show the shift in the second moment:

E[(T (Y ))2] =∞∑z=0

z+1∫z

t2pY (z)dt=∞∑z=0

pY (z)

z+1∫z

t2dt=∞∑z=0

pY (z)

1∫0

(z+u)2du=

=∞∑z=0

pY (z)

(z2 + z+

1

3

)=E[Y 2] +E[Y ] +

1

3

Now, we can represent variance as:

V [T (Y )] =E[(T (Y ))2]−E2[T (Y )] =E[Y 2] +E[Y ] +1

3− (E[Y ] +

1

2)2 =

=E[Y 2] +E[Y ] +1

3−E2[Y ]−E[Y ]− 1

4=E[Y 2]−E2[Y ] +

1

12= V [Y ] +

1

12

Let α= FY [K] for some K ∈Z+ and note that the CDFs of T (Y ) and Y are equal at K. Then,

we can re-write CEoverα [T (Y )] as:

CEoverα [T (Y )] =E[T (Y )|T (Y )≥K] =

∞∫0

tpT (Y )(T (Y ) = t|T (Y )≥K)dt=

=

∞∫0

∞∑z=0

pY (z|Y ≥K)Iz≤t<z+1tdt=∞∑z=K

z+1∫z

tpY (z)

P (Y ≥ z)dt=

1

P (Y ≥ z)

∞∑z=K

pY (z)

z+1∫z

(t)dt=

=1

P (Y ≥ z)

∞∑z=K

pY (z)

(z+

1

2

)=

1

P (Y ≥ z)

∞∑z=K

(pY (z)z+

pY (z)

2

)=

=∞∑z=K

(pY (z)z

P (Y ≥ z)+

pY (z)

2P (Y ≥ z)

)=E[Y |Y ≥K] +

1

2=CEover

α [Y ] +1

2�

Proof of Theorem 2 Let FX(t) be the cumulative probability function of random variable X.

From Property 1, we have E[LPik ]≤ E[LPok ], combining it with property (2) from Lemma 2, we

get: ∆k,i =E[LPok ]−E[LPik ] =E[LPok ] + 1/2−E[LPik ]− 1/2 =E[T (LPok )]−E[T (LPik )].

Let X i = T (LPik ) + ∆k,i and Xo = T (LPok ). Thus, from the above: E[X i] = E[T (LPik )] + ∆k,i =

E[T (LPok )] =E[Xo]. This means that X i and Xo have the same mean.

Since FXi(t) = 0 < FXo(t) for 0 ≤ t ≤ ∆k,i and E[X i] = E[Xo], the curve of FXi(t) must cross

FXo(t) at some point t1 >∆k,i, otherwise FXi(t)≤ FXo(t) ∀ t≥ 0, i.e. X i >st Xo and this implies

E[X i]>E[Xo], which is a contradiction.

35

In the proof of Theorem 1, we have shown that there exists a finite integer l∗, such that πPik (m)>

πPok (m) ∀ m≤ l∗ and πPik (m)≤ πPok (m) ∀ m≥ l∗ + 1. This means there must also exist a point t

such that the slope of FXi(t) is larger than that of FXo(t) for t≤ t and is smaller otherwise. It can

be seen that t1 < t.

We argue that the curve of FXi(t) crosses that of FXo(t) at only one point, i.e., the point t1

is unique, and it is from below, by contradiction. Suppose FXi(t) crosses FXo(t) at a point t > t

from above then for t≥ t, the slope of FXi(t) must be smaller than the slope of FXo(t), i.e. t is the

last point FXi(t) crosses FXo(t). But when FXo(t)−FXi(t) is increasing in t for t≥ t there exists

t2 > t and ε > 0 such that FXo(t)−FXi(t)≥ ε ∀ t≥ t2. Therefore, 0< ε≤ limt→∞

(FXo(t)−FXi(t)) =

limt→∞

FXo(t)− limt→∞

FXi(t) = 1− 1 = 0, which is a contradiction.

According to Muller and Stoyan (2002), if ∃ t∗ s.t. FX(t)≤ FY (t) ∀ t≤ t∗ and FX(t)≥ FY (t) ∀ t≥t∗, then Y ≥icxX. Hence, we have established that X i is less than Xo in increasing convex order:

X i ≤icx Xo. Combining with the fact E[X i] = E[Xo], we have: X i ≤cx Xo, i.e., T (LPik ) + ∆k,i ≤cxT (LPok ). The last part of the theorem in vector forms follows immediately. �

Proof of Property 3 From Theorem 2, for all k and i s.t. Pi ∈PT , we have T (LPik ) + ∆k,i ≤cxT (LPok ). By definition of convex order, it implies: V [T (LPik ) + ∆k,i] ≤ V [T (LPok ))]. Hence, using

Lemma 2, we have:

V [LPik ] = V [T (LPik )]− 1

12= V [T (LPik ) + ∆k,i]−

1

12≤ V [T (LPok )]− 1

12= V [LPok ] �

Proof of Lemma 3 From Theorem 2, for all k and i s.t. Pi ∈PT , we have T (LPik ) + ∆k,i ≤cxT (LPok ). Since CV aR− is a convex measure (Uryasev 2000), the same applies to CEover

α , this

implies CEoverα [T [LPik ] + ∆k,i]≤CEover

α [T [LPok ]]. Also, CV aR− is translation-equivariant (Uryasev

2000), thus: CEoverα [T [LPik ] + ∆k,i] =CEover

α [T [LPok ]] + ∆k,i . �

Proof of Property 4 From Theorem 2, for all k and Pi ∈PT , we have T (LPik )+∆k,i ≤cx T (LPok ).

Using Lemmas 2 and 3, ∀ α= FLPik

[K],K ∈Z+, we have:

CEoverα [LPik ] =CEover

α [T [LPik ]]− 1/2 =

CEoverα [T [LPik ] + ∆k,i]−∆k,i− 1/2≤CEover

α [T [LPok ]]−∆k,i− 1/2 (14)

Let α+ , min{γ : γ ≥ α and γ = FLPok

[D],D ∈ Z+}. Since LPok is a discrete random variable,

CEoverα [Lok] =CEover

α+ [LPok ]. Let K1 = V aRα[T [LPok ]] and K2 = V aRα+ [T [LPok ]]. By definition of α+,

K2 ∈Z+ and K2− 1<K ≤K2.

CEoverα [T (LPok )] =E[T (LPok )|T (LPok )≥K1] =

1

P{T (LPok )≥K1}

∞∫K1

tdFT (LPok

)(t)

=1

P{T (LPok )≥K1}

∞∫K1

tpT (LPok

)(t)dt=

1

P{T (LPok )≥K1}

K2∫K1

tpT (LPok

)(t)dt+

∞∫K2

tpT (LPok

)(t)dt

36

Similarly,

CEoverα+ [T (LPok )] =

1

P{T (LPok )≥K2}

∞∫K2

tpT (LPok

)(t)dt

Note that pT (LPok

)(t) =

∞∑z=0

pLPok

(z)Iz≤t<z+1, so pT (LPok

)(t) = p

LPok

(K2 − 1) ∀ t ∈ [K2 − 1,K2]. Let

A, pLPok

(K2− 1). Then,K2∫K1

tpT (LPok

)(t)dt=A

K2∫K1

tdt=AK2

2−K21

2.

We introduce the following notation to simplify the exposition of the rest of the proof: Let

P{K1 ≤T (LPok )≤K2}=A1 and P{T (LPok )≥K2}=B and∞∫K2

tpT (LPok

)(t)dt=C. Now, P{T (LPok )≥

K1}= P{K1 ≤T (LPok )≤K2}+P{T (LPok )≥K2}=A1 +B.

We can rewrite: CEoverα [T (LPok )] = 1

A1+B

(AK2

2−K21

2+C

)and CEover

α+ [T (LPok )] = CB

. We will

show that CEoverα [T (LPok )] ≤ CEover

α+ [T (LPok )]. Multiplying both sides by (A1 + B)B, we get:(AK2

2−K21

2+C

)B ≤C(A1 +B)⇔

(AK2

2−K21

2

)B ≤CA1⇔ 1

2(K2

2 −K21 )B ≤ A1

AC.

It is obvious that A1A

= K2−K1K2−(K2−1)

=K2−K1, thus, the above inequality is equivalent to: 12(K2 +

K1)B ≤C.

Now, we have C =∞∫K2

tpT (LPok

)(t)dt >

∞∫K2

K2pT (LPok

)(t)dt = K2

∞∫K2

pT (LPok

)(t)dt = K2P{T (LPok ) ≥

K2}=K2B > 12(K2 +K1)B.

So, we have CEoverα [T (LPok )] ≤ CEover

α+ [T (LPok )] and note that the equality holds if and only if

K1 = V aRα[T (LPok )] =K2 ∈Z+.

From Equation 14, we obtain:

CEoverα [LPik ]≤CEover

α [T (LPok )]−∆k,i−1

2≤CEover

α+ [T (LPok )]−∆k,i−1

2=

CEoverα+ [LPok ]−∆k,i =CEover

α [LPok ]−∆k,i �

Proof of Lemma 4 Since both P1 and P2 satisfy Participation Condition 1, we have: πP1k (0) =

πP2k (0) =λokµk

. From the balance equations as shown in the proof of Theorem 1, we have:

πP1k (l) = πP1k (0) 1µk

Πlj=1aj−1 and πP2k (l) = πP2k (0) 1

µkΠlj=1bj−1

Thus, πP1k (l) = πP2k (l) ∀ l ∈ {0, .., n}. Now, consider the balance equations, we have: πP2k (n+ 1) =

bnµkπP2k (n)> an

µkπP2k (n) = an

µkπP1k (n) = πP1k (n+ 1).

We will show that bn+1 < an+1 by contradiction. Suppose bn+1 ≥ an+1 then πP2k (n + 2) =bn+1

µkπP2k (n+ 1)≥ an+1

µkπP2k (n+ 1)>

an+1

µkπP1k (n+ 1) = πP1k (n+ 2).

This implies πP2k (l)> πP1k (l) ∀ l≥ n+ 3 and hence,∞∑l=0

πP2k (l)>∞∑l=0

πP1k (l) = 1, which contradicts

the fact that∞∑l=0

πP2k (l) = 1. Thus, bn+1 < an+1 and πP2k (n + 2) < πP1k (n + 2). Since al = bl ∀ l ∈

0, .., n− 1∪n+ 2, ..,∞, using the balance equation as in the proof of Theorem 1, it implies that

πP2k (l)< πP1k (l) ∀ l ≥ n+ 2. This means that the cdf of LP2k crosses the cdf of LP1k from above at

37

only 1 point. Following the same method as applied in the proofs of Theorems 1 and 2, we conclude

that P1 ≤P P2. �

Proof of Theorem 4 (i) From the monotonicity of the conditional expected arrival rate of a

policy Pi in PM (equation 12) we have: E[Λk(~LPi)|Lk = 0] > E[Λk(~L

Pi)] = λok. Otherwise, if

E[Λk(LPi)|Lk = 0]≤ λok, then E[Λk(L

Pi)] = E(Lk)E[Λk(LPi)|Lk]<E[Λk(L

Pi)|Lk = 0]≤ λok, which is

a contradiction. Also, there must exist a finite m∗ such that:{E[Λk(~L

Pi)|Lk = l]>λok, ∀ l ≤ mE[Λk(~L

Pi)|Lk = l]≤ λok, otherwise.

Otherwise, if there is no such finite m∗, then E[Λk(LPi)|Lk = l]≥ λok ∀ l(with at least some strict

inequality), and hence, E[Λk(LPi)] =ELk [E[Λk(L

Pi)|Lk]]>λok, which is a contradiction.

Thus, Pi ∈ PT and hence, we have established that PM ⊂ PT . Combined with the result of

Theorem 3, we have PM ⊂PT ⊂P .

(ii) and (iii) follow directly from Lemma 4. �

Proof of Lemma 5 We take the perspective of one ED (hence, we suppress the subscript k in

the following proof), where l represents the aggregate state: number of patients in ED k taken

across all other censuses at other EDs. Let the number of virtual patients up to time t be NPiv (t)

and the number of actual patients be NPi(t). Observe, that both numbers have to satisfy the

following equality: limt→∞

NPiv (t)

t= lim

t→∞NPi (t)

t= λo. πPi(l) is then the percentage of time the system

spends in state l; during this time, the arrival rate is λPi(l). Let NPil (t) be the number of patients

arriving when the queue size is l counted up to time t. So, we have:∞∑l=1

NPil (t) =NPi(t).

Up to time t, the fraction of patients observing l number of patients in the system is:NPil

(t)

NPi (t)

and hence, πPi(l) representing the percentage of patients who observe l number of patients in the

system is: πPi(l) = limt→∞

NPil

(t)

NPi (t).

πPi(l) = limt→∞

NPil

(t)

NPi (t)= lim

t→∞

NPil

(t)

t

NPi (t)t

=limt→∞

NPil

(t)

t

limt→∞

NPi (t)t

. Note that limt→∞

NPi (t)t

= λo and hence, πPi(l) =

1λo

limt→∞

NPil

(t)

t.

Let tl be the total time the system (under control policy Pi) spends in aggregate state l. From

the time-average distribution πPi(l), we get: limt→∞

tlt

= πPi(l). So,

πPi(l) =1

λolimt→∞

NPil (t)

t=

1

λolimt→∞

NPil (t)

tl

tlt

=1

λolimt→∞

NPil (t)

tllimt→∞

tlt

Since limt→∞

NPil

(t)

tl= λPi(l) and lim

t→∞tlt

= πPi(l), πPi(l) = λPi (l)λo

πPi(l). �

38

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