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Policies for Physician Allocation to Triage and Treatment in
Emergency Departments
immediate
January 9, 2017
Abstract
In the emergency department (ED), low-acuity patients divert resources from more critical patients.
To facilitate flow, EDs are experimenting with new care models, such as the Triage-Treat-and-Release
program at the Lutheran Medical Center (LMC) ED in New York, where physicians handle both phases
of service for low-acuity patients. Our goal is to determine how physicians in such settings should pri-
oritize triage versus treatment, to balance initial delays with timely discharges. Triage and treatment are
modeled as a two-phase stochastic service system, for which an essential feature is patients may leave
without receiving treatment. Because patients may choose to leave without receiving treatment, this in-
creases the importance of the second phase. We introduce K-level threshold policies which prioritize
treatment unless there are K or more patients in triage. The effect is a class of policies that are flexible
enough to capture a decision-maker’s valuation of the importance of each activity; lower K values sig-
nifies triage priority. Sufficient conditions are provided to ensure these policies yield a stable system in
the sense that the average queue lengths are finite. A heuristic is presented for choosing K in systems
with abandonments. Using LMC data, K-level threshold policies, compared to other practical policies,
consistently perform well with respect to average rewards and waiting times over a range of parameters.
These policies promise physicians an effective and simple way to allocate their time between triage and
treatment, without the need for complex model formulation and calibration.
Keywords: ED triage and treatment process, tandem queue, abandonments, simulation.
1 Introduction
Emergency departments (EDs) in the United States received around 129.8 million visits in 2010, a number
which has since increased annually at about a 3% rate [1]. Meanwhile, the number of ED beds has decreased,
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leading to overcrowded departments, long waiting times, overworked staff, and patient dissatisfaction. Many
patients in EDs, however, present with low-acuity conditions that do not require hospitalization. These
patients still have to be treated, thereby diverting resources from more critically ill patients.
To handle lower-acuity patients, EDs are experimenting with new care models. One such model is
the Triage-Treat-and-Release (TTR) program developed in 2010 by the Lutheran Medical Center (LMC)
in Brooklyn, New York. At the LMC, patients arrive to the ED, register, and then proceed to triage on a
first-come-first-served basis. After triage, high acuity patients who are likely to be admitted to the hospital
are assigned to another part of the ED for testing and/or treatment, while low acuity patients are treated in a
treatment room adjoining the triage area. Patients rarely leave before being seen while awaiting triage, but
may abandon the system before receiving final treatment. After treatment, patients are discharged.
An important aspect of the TTR program is that physicians and other highly trained providers, i.e. physi-
cian assistants (PAs) or nurse practitioners (NPs), are used for both phases of serving low-acuity patients.
This contrasts the traditional approach where nurses are responsible for “triaging” patients while highly
trained providers are responsible for treating patients. Including highly trained providers in triage was mo-
tivated by empirical studies that show physicians, PAs, and NPs are more reliable than nurses in assessing
patients during triage (c.f. Subash et al. [25], Medeiros et al. [17], Han et al. [9], Soremkun et al. [24],
Burstr om et al. [4], and Burstr om et al. [3]). For example, in a retrospective study of three Sweedish EDs,
the authors in Burstr om et al. [4] analyze the effect of three different triage models: senior physician-lead
team; nurse first, emergency physician second; and nurse first, junior physician second. They show that the
first model (i.e. physician triage) outperforms the others leading to reductions in the time to first doctor en-
counter, length of stay, and rate at which patients leave without being seen. Additionally, the TTR program
is also related to other care models of the ED such as the “fast-track system”, which differs from the TTR
mainly in that the person performing triage is not doing the treatment. However, if a highly trained provider
is to treat patients, then it may be important for them to also triage patients.
Assuming a TTR system is in place, our goal is to determine how to allocate physicians and other highly
trained providers between triage and treatment when they are responsible for both phases of service. The
central dilemma is that, on the one hand, prioritizing triage leads to patients leaving without treatment and
on the other, prioritizing treatment leads to patients waiting a long time for triage, which should be quick to
ensure truly emergent patients receive timely care.
We model the triage and treatment process as a two-phase stochastic service system with multiple
servers. We assume that rewards are received for patients who complete a phase of service and not for pa-
tients who abandon the system. The reward structure reflects not simply the accrual of actual cash rewards
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by the hospital—hospitals rarely charge for triage but do charge for treatment—but rather the trade-off be-
tween prioritizing one phase of service over the other. As noted, prioritizing triage can lead to the loss of
rewards from patients who abandon before receiving treatment at the second phase of service. We compare
service disciplines with respect to long-run average rewards accrued.
Policies for this model have been analyzed using a continuous-time Markov decision process (CTMDP)
in Zayas-Caban et al. [28] (c.f. Theorem 3.3 in [28]). It is shown there that in a two phase stochastic system
with abandonments from the second stage only, prioritizing treatment is average reward optimal no matter
what the rewards for triaging patients are, so long as this policy yields a stable system in the sense that the
average queue lengths are finite. This policy is the current practice in the LMC ED and benefits both the
patient and provider since it means that once a patient receives triage, (s)he immediately receives treatment.
By contrast, physicians want more emphasis on triage so to safeguard against delayed care to truly emergent
patients. Moreover, if the system is unstable or significantly deviates from reality, then optimal policies are
difficult to obtain and compute because of abandonments and require precise model calibration, which is
usually difficult to do.
In this paper, we would like to determine if these competing objectives can be addressed with a sim-
ple policy where the number of patients in triage determine when to switch from prioritizing treatment to
prioritizing triage. To the best of our knowledge, we are the first to model the TTR system as a two-stage,
tandem queueing system with abandonments and to provide insights for when simple policies may be used.
Our work includes the following contributions in this area:
• We propose a class of easy-to-use policies that we call K-level threshold policies, which are a com-
promise between the two priority rules.
• We show that these policies yield stable Markov chains under the weakest conditions possible, i.e.
when prioritizing phase one service is stable; that is to say, when the long-run average number of
patients in the system is finite.
• We provide a simple rule-of-thumb for choosing a suitable value for K.
• Lastly, and most importantly, we compare all the aforementioned service disciplines and other practi-
cal service disciplines in a simulation study with respect to different performance metrics and under
general assumptions using data from the LMC as inputs. Results from the study are used to provide
insights for when to use each.
The rest of this paper is organized as follows. Section 2 contains a summary of the literature relevant
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to our work. Section 3 describes the queueing dynamics in detail. In Section 4, we introduce the K-
level threshold policies and provide stability conditions assuming exponential inter-arrival, service, and
abandonment times. We then relax the exponential assumptions and compare different threshold policies
and other practical policies in a simulation study in Section 5. We discuss possible extensions to our work
and conclude in Section 6.
2 Literature Review
We review the most recent relevant literature from two perspectives: the analysis of new care models for
EDs using control and simulation and the analysis of service policies for flexible servers in multi-phase
stochastic service systems. The latter provide useful insights into the types of service policies one might
implement in practice. In what follows, we will refer to triage (treatment) as station 1 (2).
In many hospitals, admitted patients typically consist of severe cases whose timely care is of primary im-
portance, and thus require short times to first treatments (TTFT). Patients that are discharged typically have
short lengths of stay (LOS). A new care model for EDs, called patient streaming, separates, or “streams”,
patients into two groups based on predictions on whether they will be discharged or admitted. Streaming
aims to provide resources that better match patient need. Recently, Saghafian et al. [20] use a combination
of analytic (i.e. scheduling) and simulation techniques to compare patient streaming with complete resource
pooling. Their benchmarks are TTFT for admitted patients and LOS for discharged patients. They propose a
modified version of streaming which they call virtual streaming, where patients are streamed without physi-
cally separating patients. This allows physicians to serve both groups of patients. The study in Saghafian et
al. [20] differs from the present study in its focus on ED design after triage, where we are interested in both
triage and treatment.
The authors of Huang et al. [10], Dobson et al. [6], and Saghafian et al. [21] use queueing models,
control techniques, or simulation for allocating physicians to patients at different phases of care. In Huang
et al. [10], the allocation dilemma is deciding between prioritizing patients that need to be treated for
the first time versus already treated patients who require additional care. The authors consider a queueing
model in the heavy traffic regime and seek allocation policies that minimize a combination of delay and
waiting costs. Dobson et al. [6] also consider when to prioritize new patients versus existing patients in
the presence of service interruptions. Their objective is to maximize throughput. Similar to Huang et al.
[10], the authors in Saghafian et al. [21] focus on when to prioritize patients immediately after triage versus
patients further along in their treatment who require additional care. In contrast to Huang et al. [10], they
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study allocation policies for prioritizing different patient classes when a complexity-based triage system is
used. They use a Markov decision process formulation that minimizes the risk for adverse events for patients
waiting immediately after triage and LOS for patients waiting downstream.
The present work differs from any of the previous work in four significant ways. First, we are con-
cerned with the care of patients before triage, in addition to low-acuity and low-complexity patients after
triage. Low-acuity patients make up the majority of patients in EDs. Second, we address an important
clinical consideration: patients who leave, or abandon, without receiving medical care. Third, our model
does not assume heavy-traffic, and thus, applies to low-traffic and high-traffic settings. Fourth, we provide
implementable, yet systematic guidelines for allocating providers without resorting to complex mathemati-
cal formulations or difficult model calibration. We believe that such an approach makes this approach more
likely to be accepted by the medical community.
Motivated by this triage and treatment system, Zayas-Caban et al. [28] analyzed optimal service disci-
plines in a two-phase service system with abandonments using a continuous-time Markov decision proces
(MDP) formulation. The authors show that the optimal policy ensures that all providers stay busy. That is
to say, when there are not enough patients to serve at one station, they work at the other. They also show
that when there is enough work to do, providers need not be split between stations. We make both of these
assumptions in our simulation model. Importantly, the authors provide sufficient conditions for when to
prioritize treatment. In the discounted model, the condition is closely related to the classic result of the cµ
rule Buyukkoc et al. [5]. In the average case, this inequality is not needed. Intuitively, it does not matter
when patients arrive in the average case, the reward can be collected upon arrival (instead of after triage).
This is not true for patients moved to treatment, because they might abandon. The way to maximize reward
is by prioritizing station 2.
For a triage and treatment system, there are two drawbacks of the study in Zayas-Caban et al. [28]
that motivate considering a policy other than prioritizing treatment. The first has already been alluded to:
clinicians may want to limit delays for triage rather than simply treating patients whenever there are patients
to treat. The second is that prioritizing treatment compared to any other policy has the most restrictive
condition to ensure the system is stable in the sense that the average queue lengths are finite. When the ED
is busy, prioritizing treatment may lead to an unstable system. Thus, this policy may no longer be optimal,
but more importantly, should not be implemented.
One way to overcome the drawbacks of a policy that prioritizes treatment is to consider simple rules for
switching between triage and treatment. Nair [19] considers one with Poisson arrivals and general service
times under a non-zero switching rule for phase-one and a zero switching rule for phase two. A non-zero
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switching rule is one where the provider continues to serve in a phase until some specified number of
consecutive services have been completed and then switches to the other phase, while a zero switching rule
continues to serve until the phase is empty before switching to the other phase. Taube-Netto [26] considers
the same model but with a zero switching rule at each phase. Katayama [11] analyzes the system under a
zero switching rule at each phase but with non-zero switchover times. A K-limited policy has the provider
visit a phase and continue serving that phase until either it is empty or K patients are served, whichever
occurs first. A gated service policy is one in which, once the provider switches phases, (s)he serves only
patients who are in that phase at the time immediately following the switch. In [12] and [13], Katayama
extends his previous work to consider gated service and K-limited service disciplines, respectively. In both
papers, intermediate finite waiting room is allowed. Katayama and Kobashi [14] analyze the sojourn time
under general K-decrementing service policies; policies in which once the server visits phase one, (s)he
continues serving that phase until either this phase becomes empty or K patients, are served, whichever
occurs first, and then serves at phase-two until it is empty.
3 Model Description
We approximate the triage and treatment system with the following model. Patients arrive to the system
according to a Poisson process of rate λ and immediately join the triage queue. After a patient is triaged, a
reward of R1 is accrued. They then join the treatment queue with probability p, independent of the service
time and arrival process . Otherwise, they leave the system. If the patient joins the treatment queue, their
(random and hidden) abandonment time is generated. Service times at triage are exponential with rate
µ1 > 0, service times at treatment follow a phase-type distribution with rate µ2 > 0 and abandonment times
are exponential with rate β > 0. If the patient is not treated before the abandonment time ends, the patient
leaves the system without being treated. Otherwise, the patient accrues a reward ofR2 after treatment. There
areN ≥ 1 highly trained providers, each of which can be assigned to either triage and treatment. As alluded
to in Section 2, we assume that all providers stay busy. That is to say, when there are not enough patients to
serve at one phase, they work at the other. We also assume that when there is enough work to do, providers
need not be split between stations.
3.1 Model Justification
Justification for the arrival process, service times, and abandonments are based on empirical data and previ-
ous work.
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Arrival process — For many health care systems including the ED, the arrival process can be accurately
described by a non-homogeneous Poisson process with a piece-wise constant arrival rate function (c.f. Shi
et al. [23], Xie et al. [27], and Kim and Whitt [15]. For each interval where the arrival rate function
is constant, the system is akin to a homogeneous Poisson process. Thus, our model can provide insights
specifically tailored to each interval. This methodology has been shown very useful in supporting ED
staffing (c.f. Green et al. [8]).
Service times — For our work at Lutheran Medical Center, we collected data on service times for triage
and treatment. We found that for the triage phase, the mean time was 7.25 minutes and the standard deviation
was 6.46 minutes, so that the coefficient of variation was close to 1, making the exponential assumption
reasonable. For the treatment phase, the mean was 13.2 and the standard deviation was about 7, raising
some concerns about an exponential assumption. In our simulation, we use a more general phase-type
distribution for treatment times (c.f. Fackrell [7]).
Abandonments — With regard to abandonments, our understanding based on work with several hospi-
tals, including Lutheran, is that there are virtually no abandonments before triage. This is because waiting
times for triage are generally far shorter than waiting times to be seen by a provider. For example, at LMC,
the staffing goal for triage at Lutheran is that no more than 10% of patients wait more than 10 minutes.
Their automated registration system, which records patient arrivals as soon as they walk through the door,
substantiates that there are no abandonments between arrival and triage. On the other hand, it is well-known
that low-acuity patients abandon after triage while waiting for treatment (c.f. Sharieff et al. [22]).
4 K-Level Threshold Policies
We define a K-level threshold policy as a non-idling policy in which all of the providers prioritize treatment
except when there are K or more patients in triage. Specifically, the threshold policy has providers work at
the treatment queue until either it is empty or the number of patients at triage reaches K. In the first case,
they continue to prioritize treatment. In the second case, the providers work at triage until it is empty and
then return to treating patients. The threshold value ranges from K = 1 to K = ∞. Policies at the two
extremes are the simple priority policies, i.e. the policy that prioritizes triage (K = 1) and the policy that
prioritizes treatment (K =∞) except to avoid unforced idling.
The providers spend increasingly less time (on average) at the triage queue as the threshold value in-
creases. For example, the policy that prioritizes triage (K = 1) has physicians triage patients whenever
possible. Thus, it spends the most amount of time working at triage out of any non-idling policy. At the
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other extreme, the policy that prioritizes treatment (K =∞) begins treating a patient immediately after they
are triaged. Thus, it spends the least amount of time working at triage (on average) out of any non-idling
policy. Since triage should take place quickly so truly emergent patients receive timely care, it is desirable
to minimize the number of patients awaiting triage, while at the same time ensuring that patients awaiting
treatment do not abandon.
4.1 Stability condition
Whenever possible, policies should be used that yield stable systems to avoid unbounded queue lengths and
unbounded waiting times. For example, the policy that prioritizes treatment (K = ∞) leads to an unstable
system when λ(
1Nµ1
+ pN(µ2+β)
)≥ 1. On the other hand, λ
(1
Nµ1+ p
N(µ2+β)
)< 1 turns out to be more
restrictive than is needed to ensure stability under any policy. By simply prioritizing triage (K = 1), the
system is stable when λNµ1
< 1. This latter condition is, in fact, the best we can do. That is, the system is
unstable for any policy when λNµ1≥ 1.
Prioritizing triage is stable over a larger range of parameter values than prioritizing treatment because
patients abandon while waiting for treatment. Their abandonments ensure that the average length of the
treatment queue and the average waiting time for treatment are finite, regardless of whether the provider
spends time treating patients. Thus, the system is stable provided the triage queue is bounded. When triage
is prioritized, this requires that patients are triaged faster than they arrive (on average), i.e. that λNµ1
< 1.
The natural question then is under what conditions do the other K-level threshold policies yield a stable
system. The next result shows that a K-level threshold policy for finite K is stable if and only if the policy
that prioritizes triage is stable, provided that service times at treatment are exponential. This is detailed in
Proposition 1 below whose proof is shown in Appendix A.1. For the sake of completeness, we have also
included a proof of the stability condition for prioritizing treatment in Appendix A.1 (c.f. Proposition 2).
Proposition 1 Suppose that treatment times are exponential with rate µ2. In this case, theK-level threshold
policy (with 1 ≤ K <∞) is stable if and only if λ < Nµ1.
Proof. The proof of this result is given in Appendix A.1.
4.2 Choosing the Threshold Level K
To choose K, a decision-maker must consider the trade-off between delay times for patients awaiting triage
and abandonments at the treatment queue. For example, one target is to have patients wait an average of
10 minutes for triage. The threshold K could be chosen to correspond to this objective. Suppose that a
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decision-maker does not want patients to wait more than an average of t minutes for triage. If a threshold
level of K is used, then all of the medical service providers switch to the triage queue at the end of the
current treatment service upon the arrival of theK-th patient to the triage queue (i.e. there is an arrival to the
system withK−1 patients waiting for triage and at the end of the current treatment service when said arrival
occurs). Assume that K is strictly greater than the number of medical providers N . Since by assumption,
we never split the servers between the two phases of service, N patients will be triaged immediately once
medical providers move to triage. So there are at least K − 1 −N patients waiting before the Kth patient
arrives. In this case, the expected service rate is Nµ1, and the expected waiting time of the Kth patient is
roughly (K − N)/Nµ1. If the decision makers decide to set this time equal to t, then we may choose the
value of K using the following formula:
K = btNµ1 +Nc , (4.1)
where b·c is the floor function. For the LMC, there are 1 to 5 physician assistants depending on the time of
day, and the average time for triage is approximately 7 minutes. If N = 3 and t = 10, say, then the above
formula yields a value K around 7. If N = 5 and t = 10, then K ≈ 12.
5 Simulation
We use a discrete-event simulation of the triage and treatment system at the Lutheran Medical Center (LMC).
Broadly, we want to predict how K-level threshold policies would perform in practice when compared
against other practical policies. These predictions can serve to guide how the LMC and other hospitals allo-
cate highly trained medical providers between triage and treatment. The central question that is addressed is
whether K-level policies can achieve relatively high rewards from service, while safeguarding against long
waits in triage for truly emergent patients. We also determine how often providers switch between triage and
treatment for each policy, as providers may want to avoid too much switching between triage and treatment.
Lastly, we want to justify our procedure for choosing a particular K by showing agreement between the
target waiting time in triage and the simulated waiting time in triage.
There are seven policies that are compared. Two policies are the policies that either prioritize triage or
prioritize treatment, which as noted are equivalent to K-level threshold policies for K = 1 and K = ∞,
respectively. Four policies are K-level threshold policies having threshold values of 5, 10, 15 and 20. The
final policy is an exhaustive policy (E). Note that prioritizing treatment is the current policy at the LMC. For
each set of parameters that we consider, the simulation length is one year (24 hours per day) per replication
with 30 replications.
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Parameter Value(s)
µ1 8.57
µ2 4.62
N 1,3,5
p 0.2, 0.4, 0.6, 1
β 0.15, 0.3, 0.5, 0.8
R2 20
R1 10, 15
λ 4.2–23.4a
Table 1: Parameters used in discrete-event simulation. Parameters are chosen to model the triage and treat-
ment system at the Lutheran Medical Center.
a Range of λ across all simulation runs. Specific values of λ varied depending on N , p, and β; see main text for
details.
While maximizing the average reward is a primary concern, some providers may prefer to forgo some
average reward in favor of having the provider prioritize triage more often. One reason for this is that
minimizing the number of patients awaiting triage may be desirable given that triage is supposed to take
place rather quickly to safeguard getting timely care to those who have truly emergent problems. Moreover,
it seems that the model may have higher average rewards, but more patients in the system on average; and
thus, longer average wait times. This trade-off is also examined. The basic insights obtained from the
numerical study follow with details in the following sections.
5.1 Model Parameters
The Lutheran Medical Center (LMC) located in Brooklyn is a 468-bed academic teaching hospital. Model
parameters are chosen to reflect the triage and treatment system at LMC’s emergency department (Table 1):
Service times—The average service time for triage is approximately 7 minutes and the average time for
treatment is approximately 13 minutes. We fix service rates µ1 = 607 ≈ 8.57 per hour and µ2 = 60
13 ≈
4.62 per hour. We also assume that the triage time is exponential while the treatment time has an Erlang
distribution with parameters (3, 3µ2). This last assumption keeps the average service time equal to 1µ2
.
Number of providers—The number of providers ranges from 1 to 5 at LMC depending on the time and
day. We thus vary the number of providers over a range of values between 1 and 5.
Probability of joining treatment queue—Consistent with national data, the fraction of patients at the
LMC that are not admitted to the hospital is approximately 0.7. However, not all patients that are not
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admitted to the hospital receive treatment via the TTR program. During the time of the study, it was observed
that only 40% of the patients triaged were treated in the TTR program. As a result, we vary p over a range
of values between 0.2 to 1.
Rewards—Rewards are used to capture a decision-maker’s relative preference for timely service at triage
versus timely service at treatment. We fix a reward of 20 after a patient is treated (i.e. R2 = 20), and consider
two scenarios for R1. In the first scenario, R1 = R22 = 10 so that µ1R1 < µ2R2. In the second scenario,
R1 = 15 so that µ1R1 > µ2R2.
Abandonments—Hospitals do not generally track the time that a patient spends waiting for treatment
before leaving the system. As a result, we do not have direct estimates for β. We propose the following
for estimating a range for β and then parameterize β over that range. Assuming that abandonment time
correlates with service time, let θ = µ
β, where β is our estimate of the abandonment rate. This is the average
patience measured in units of average service time. We use θ to estimate the abandonment rate. That is to
say, vary θ over a sensible range of values, and for each value of θ in this range, determine the corresponding
value of β. For this study, we assume that θ ∈ (5, 30). This implies that the mean abandonment (actually
patience) time is from 5 to 30 times that of the mean service time (i.e. about one to 6.5 hours). This yields
that the range for the abandonment rate β is approximately (0.15, 0.92).
To justify the range we have proposed for θ, assume there is a linear relationship between the fraction
of abandonments, P(Ab), and average waiting time, E[W ], (see for example Mandelbaum and Zeltyn [16]
where it is shown that this relationship holds for the Erlang A model):
P(Ab) = β E[W ]. (5.1)
The relation (5.1) between the average wait in queue, E[W ], and the fraction of patients who left without
being seen, P(LWBS), provides a method of estimating β as follows
β =P(LWBS)E[W ]
=%LWBS
Average wait. (5.2)
Using the data in Table 1 of Batt and Terwiesch [2], we use (5.2) to obtain estimates for the abandonment
rate β (see Table 2). The Emergency Severity Index (ESI), which is used in triage, has 5 levels of severity
with levels 1 and 2 signifying patients requiring prompt attention and in need of hospitalization. The TTR
is designed for patients who will not need to be hospitalized and generally only includes those in levels 3-5.
We have extended this to include level 2 to expand the range of consideration for our study. If we compute
the corresponding θ estimates we get a range for θ that is approximately (5,20). We extend this to (5,30)
(which leads to (.29,.92)) which fall in the range of (0.15, 0.92); a proper subset of the range for β we have
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proposed. In other words, we have proposed a range of values for θ that includes the range of values for
β we have estimated using the data from Batt and Terwiesch [2] and an estimation procedure based on the
Erlang-A model from Mandelbaum and Zeltyn [16]. We consider a slightly wider interval to account for the
possibility of a smaller hospital than that considered in Batt and Terwiesch [2].
ESI 2 ESI 3 ESI 4 ESI 5
Population 27,538 65,773 39,878 10,509
%LWBS 1.70% 9.50% 4.70% 7.40%
Average wait (hr.) 1.0 1.9 1.3 1.3
Service time (hr.) 3.7 4.0 1.8 1.2
β 0.02 0.05 0.04 0.06
Table 2: Estimation on abandonment rate
Arrival process—Let S(µ1, µ2, β, p) := 1µ1
+ pµ2+β
and we recall a result from Zayas-Caban et al. [28],
which states that under exponential service times and single provider system assumptions, λ < 1S(µ1,µ2,β,p)
implies that the provider should prioritize station 2. The intuition (as has already been alluded to) is that this
policy minimizes the number of abandonments. However, this policy is unstable when λ ≥ 1S(µ1,µ2,β,p)
. The
analogous condition when there are N servers is λN < (≥) 1
S(µ1,µ2,β,p). The policy that prioritizes station
1 leads to a stable system whenever λN < µ1. Thus we would also like to study the region, 1
S(µ1,µ2,β,p)≤
λN < µ1.
We consider four cases for our choice of λ. In the main case, which we focus on in the subsequent
section, we test all values of β and R1, fix p = 1and N = 3, and test λ = 6, 7, 8, 9, 10, 11, 12, so the system
under the policy that prioritizes station 2 is unstable for the given values of λ ≥ 10 and stable otherwise.
The remaining three cases consider other values of p and N . These cases yield similar qualitative trends
regarding differences in rewards and waiting times across threshold policies and with changes in λ. Thus,
we report these cases in the Appendix to simplify subsequent exposition (Figures 7–9). In the second case,
we consider β = 0.15, 0.3, 0.5 with p = 0.4 and N = 3 and let λ = 10.3, 11.8, 13.3, 14.8, 16.2, 17.7, so
the system under the policy that prioritizes station 2 is unstable for the given values of λ ≥ 16.2 and stable
otherwise. In the third and fourth case, we focus on when the system under the policy that prioritizes station
2 is stable ( λN S(µ1, µ2, β, p) is about 0.8–0.95). The third case considers N = 1, 3, 5 with p = 0.4 and
β = 0.3 and lets λ = 4.2N ; 4.7N . The fourth case considers p = 0.2, 0.4, 0.6 with N = 3 and β = 0.3 and
lets λ = 15.9, 17.8 when p = 0.2, λ = 12.5, 14.0 when p = 0.4, and λ = 10.3; 11.6 when p = 0.6.
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Figure 1: Comparison of reward for a policy over the policy that prioritizes triage (K = 1). Here, R1 = 10,
N = 3, and p = 1, and all policies yield a stable system.
5.2 Comparison of Rewards when Prioritizing Treatment is Stable
As long as it yields a stable system, prioritizing treatment was proven to be optimal in Zayas-Caban et
al. [28] for exponential service times. The numerical study agrees with this finding for treatment times
that are Erlang distributed (Figure 1–2). This stands to reason as this policy minimizes the number of
abandonments from the system. The policy that prioritizes triage (K = 1) yields the lowest rewards when
compared to the other policies. This also is expected since this is the policy that spends the least amount of
time treating patients, yielding the most abandonments from the system. Threshold policies have rewards
that lie in between those two priority policies.
Threshold policies with “high” threshold values perform well. The threshold policy with highest thresh-
old value K = 20 yields comparable rewards to the “optimal” policy that prioritizes treatment (K = ∞),
independently of the abandonment rate β. Indeed, it is near identical to prioritizing treatment when λ ≤ 6.
When λ = 9, it has rewards that are within 4% of the rewards from prioritizing treatment (Tables 3–4 in
Appendix). In short, the higher threshold appears to alleviate most of the loss of average reward compared
to lower thresholds.
Higher arrival and abandonment rates lead to less average rewards. This is not surprising since a higher
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Figure 2: Comparison of reward for a policy over the policy that prioritizes triage (K = 1). Here, R1 = 15,
N = 3, and p = 1 and all policies yield a stable system.
abandonment rate β results in more abandonments at the treatment queue. A higher arrival rate λ also
leads to more abandonments at the treatment queue, because the provider spends more time at triage. In
particular, note that the threshold policy with K = 5 is within 7% of the rewards from prioritizing treatment
when λ = 6 — in fact, for each β, lightly loaded systems (with λ ∈ {6, 7}) are comparable since all of the
policies are within 2% of the policy that prioritizes treatment. However, this gap significantly increases to
5.19% when λ = 9. Likewise, when λ and β are highest, i.e. λ = 9; β = 0.15, the gaps for the policy that
prioritizes triage and the exhaustive policy increase to 6.23% and 4.76%, respectively.
Lastly, the exhaustive policy sits between the policy that prioritizes triage and the threshold policy with
K = 5.
5.3 Comparison of Rewards when Prioritizing Treatment is Unstable
When prioritizing treatment does not yield a stable Markov chain, then this policy does not always yield the
highest rewards when compared to other policies (Figure 3–Figure 4). In fact, prioritizing triage in certain
cases yields the highest rewards (e.g, when R1 = 15 and λ = 12).
In the unstable cases, the threshold policies with K = 5, 10, 15, 20 performed reasonably well. Of the
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Figure 3: Comparison of reward for a policy over the policy that prioritizes triage (K = 1). Here, R1 = 10
and prioritizing treatment (K =∞) is unstable.
policies we consider, for instance, prioritizing treatment yields the highest rewards when R1 = 10, but the
threshold policies with K = 5, 10, 15, 20 yielded average rewards that were, at worst, 5% less than the
average reward from prioritizing treatment (see Table 5). In this same case, the threshold policy with K
= 20 yield average rewards that were only, at worst, 4.4% less than the average reward from prioritizing
treatment. When R1 = 15, it is no longer the case that the average rewards for prioritizing treatment are
the highest. In this case, the threshold policies K = 5, 10, 15, 20 have average rewards that are comparable
to one another for all instances and no more than 3.5% away from the highest average rewards of all the
policies (see Table 6).
Lastly, the average reward of the system under threshold policy generally improves as K increases. For
example, the threshold policy with K = 20 performs the best out of all K ∈ {1, 5, 10, 15, 20}.
We also note that, as expected, higher abandonment rates β yields less rewards. For example, Tables 5
and 6 show that when λ = 10, the average reward rates tended to decrease as β increases. As alluded to,
when λ > 10, the average rewards for all of the policies except for when triage or treatment are prioritized
are approximately the same and are very close to the policy that prioritizes triage, especially as the load (λ)
increases. Since the average queue length in each case except when treatment is prioritized is finite, this
suggests that enough time is spent at triage to stabilize the system, but then those patients are eventually
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Figure 4: Comparison of reward for a policy over the policy that prioritizes triage (K = 1). Prioritizing
treatment (K =∞) is unstable and R1 = 15.
allowed to abandon the system adding nothing to the reward.
Lastly, the exhaustive and threshold policy with K = 5 are comparable, albeit a slight improvement
when the threshold policy is used.
5.4 The case for K-Level Threshold policies
The previous two sections compare rewards among the different policies and confirm that the threshold
policies are “good” policies with respect to maximizing rewards. Next, we study how well the threshold
policies perform with respect to important considerations — average waiting times in triage and treatment.
5.4.1 Average Waiting Time in Triage
As illustrated in Figure 5, the challenge with prioritizing treatment (K = ∞) is that it may lead to long
waiting times in triage.
Decreasing the threshold from K = ∞ alleviates this issue by decreasing the average wait at triage. In
fact, using some of the finite K-level threshold policies yield very reasonable average waits at triage. For
example, when the system load is low (i.e. λ ∈ {6, 7, 8}), the average wait for the threshold policies with
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AverageWait,hours
Waiting Time at Triage● K=1
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● E
Figure 5: Comparison of average waits at triage. Here, R1 = 10, N = 3, p = 1, and β = 0.15.
K = 5, 10, 15, 20 ranges from being as quick as 4.5 minutes (K = 5, λ = 6, and β = 0.15) to 26 minutes
(K = 20, λ = 8, and β = 0.15). The latter average wait of 26 minutes may be too high, but this is primarily
due to our conservative assumption that p = 1. In particular, when λ = 10.3, β = 0.15, and p = 0.4, the
same policy K = 20 yields an average wait of approximately 6 minutes (see Figure 8 in the Appendix). In
comparison when the system load is low, the exhaustive policy yields the lowest average waits at triage: 3.4
minutes when λ = 6 and 11.5 minutes when λ = 8.
When the load increases to λ = 9, the policy that prioritizes treatment still yields a stable system and
is still optimal, but leads to an extremely long wait at triage, averaging 20 hours. In contrast, the threshold
policies with K = 5 and K = 20, respectively, lead to average waits of 13 and 51 minutes. The exhaustive
policies has average waits between the threshold policies withK = 7 andK = 10. Thus, even though stable
and optimal, prioritizing treatment would not adequately safeguard against long waits for truly emergent
patients that arrive to triage and should not be recommended. The threshold for the threshold policies can
be chosen to achieve reasonable target wait times at triage.
The problem with prioritizing treatment is further exacerbated when the load increases to λ > 9, and
prioritizing treatment is no longer stable. The waiting times at triage become even more extreme when
treatment is prioritized. Yet, the finite threshold policies remain reasonable. For example, when λ = 12,
the threshold policies with K = 5 and K = 20, respectively, yield average waits for triage of 14 minutes
and 52 minutes. We remark that when p is decreased from 1 to 0.4, the latter two average waits decrease to
approximately 7.8 minutes and 10.8 minutes, respectively (see Figure 7 in the Appendix). Next to the policy
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1
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AverageWait,hours
Waiting Time at Treatment● K=1
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● E
Figure 6: Comparison of reward for a policy over the policy that prioritizes triage (K = 1). Here, R1 = 10
and prioritizing treatment (K =∞) is unstable.
that prioritizes treatment, the exhaustive policy yields the highest average waits. The threshold policies
are able to maintain low waiting times, because the physicians spend increasingly more at triage as more
customers arrive. Thus, it adapts to the increase in demand.
It is also important to note that for all policies, higher arrival rates lead to higher average waits for triage.
In addition, higher abandonment rates lead to lower waits for triage.
5.4.2 Average Waiting Time in Treatment
If using a finite threshold policy over prioritizing treatment can dramatically reduce average waits at triage,
then it is natural to question whether average waits at treatment become unreasonable. Figure 6 shows this
does not happen. That is, decreasing threshold fromK = 1 toK =∞ decreases average waits at treatment,
but in all instances, all policies yield very reasonable average waits at triage. As noted before, the reason
for the asymmetry between triage and treatment is that patients abandon at treatment, thereby leading to low
average waits even when treatment is not prioritized.
Specifically, when the arrival rate is low (i.e. λ ∈ {6, 7, 8}), the average wait at treatment for the
threshold policies with K = 5, 10, 15, 20 range from a minimum of 0 minutes (K = 20, λ = 20, and
β = 0.15) to a maximum of 14 minutes (K = 5, λ = 8, β = 0.15). The policy that prioritizes triage has
an average wait of 20 minutes when λ = 8. The exhaustive policy has average waits at treatment that sit
between those for prioritizing triage and the K = 5 threshold policy.
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The waiting times at treatment increase with higher arrival rates and lower abandonment rates. Yet, the
threshold policies have comparable average waits to the policy that prioritizes treatment. When the arrival
rates increases to λ = 9, the policy that prioritizes triage yields an average wait for treatment of 40 minutes.
The threshold policies with K = 5 and K = 20, respectively, yields average waits of 34 and 18 minutes.
Again, the exhaustive policy sits between the threshold policies.
When λ > 9, the exhaustive policy and threshold policies have high waiting times at treatment, but
they are only marginally higher than the average waiting time for the policy that prioritizes treatment, as
illustrated in Figure 6. For instance, when λ = 12, all policies yield an average wait at treatment of
approximately 2.6 hours.
6 Conclusion
In this paper, we consider how to facilitate delivery of care by highly trained providers such as physicians,
physician assistants, or nurse practitioners, in a two-phase stochastic service system. This is motivated by
patient care in health care service systems like the triage and treatment process in EDs, and in particular,
by the TTR Program at the LMC in Brooklyn, NY. We model the triage and treatment operation as a multi-
server, two-stage tandem queueing system. In contrast to the existing literature on tandem systems, our
model considers an important, yet less studied phenomenon: that patients can leave, or abandon, while
waiting for treatment. For this system, the policy that prioritizes treatment has the benefit that once the
patient is diagnosed at triage, she/he proceeds to treatment. This makes the policy that prioritizes treatment
the most likely to be implemented in the TTR program so long as it yields a stable system. On the other
hand, it is desirable to minimize the time that patients wait for triage, but this policy neglects patients who
may leave without receiving treatment.
The main contribution of this paper is that we propose and analyze a new class of service policies, which
we call K-level threshold policies. In particular, a 1 (∞)-level threshold policy is the same as prioritizing
triage (treatment) with more time being spent at treatment with increasing values of K. We contend that a
threshold policy provides a reasonable compromise between the two priority service disciplines. Moreover,
prioritizing treatment leads to an unstable system when λ ≥ NS(µ1,µ2,β,p)
. In this case, if waiting times are of
concern, alternative service disciplines must be used. Finally, these threshold policies have the added benefit
that they can be implemented with the service provider only having a sense of the magnitude of the number
of patients for triage. In particular, whether or not they switch from treatment to triage when the number
of patients at triage is 15 or 16 does not significantly affect any of the performance measures we evaluated
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(average rewards, waits at triage, waits at treatment, and switching rate between triage and treatment).
We show that these policies yield stable Markov chains under the weakest conditions possible, i.e.
when prioritizing triage is stable. We also provide a simple rule-of-thumb for choosing a suitable value for
K that depends the decision-maker’s valuation of waits for phase-one service. Using data reported from
the TTR program at the LMC, we compare these and other practical service disciplines in a simulation
study. Our results support the use of K-level threshold policies in practice. They show that the average
rewards generated between one that maximizes long-run average rewards and all of the threshold polices
are comparable. Moreover, the threshold policies yield very reasonable, and at time outperform priority
rules and other practical policies, with respect to important practical considerations such as average waits
for triage and treatment, and average number of switches between the two phases of service. In short, these
policies promise physicians an effective and simple way to allocate their time between triage and treatment.
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A Appendix
A.1 Stability conditions for K-Level Threshold Policies
Our first result gives a sufficient condition for the stability of prioritizing treatment (i.e. threshold policy
with K =∞).
Proposition 2 Under each fixed stationary policy, the Markov process generated by said policy has the
following properties:
1. There is at most one recurrent class.
2. Under the assumption that λN
(1µ1
+ 1µ2+β
)< 1 all states that communicate with (0, 0) are positive
recurrent. Moreover, transient states reach (0, 0) in finite expected time.
3. For a fixed a stationary policy f ∈ F , define (η(i,j),(k,`)(f)) := limt→∞ P (t, f). If λN
(1µ1
+ 1µ2+β
)<
1, then for any initial state (i, j) ∈ X we have∑
(k,`)∈X
η(i,j),(k,`)(f) = 1.
Proof.Observe that under each fixed stationary policy, the Markov process generated by said policy has at
most one recurrent class. To see this, let XN = {(i, j)|i+ j ≤ N} and note that under any stationary (non-
idling) policy XN is reachable from all states and that all states within XN communicate. This gives the first
statement. In fact, if π is a stationary policy and G(π) is the set of states that communicate with (0, 0) for
the Markov process generated by π, then XN ⊆ G(π). With s(i, j) = iµ1
+ i+jµ2+β
+ 1 and (i, j) ∈ X \ XNwe have for any non-idling policy∑
(k,`)∈X
q((k, `)|(i, j), a)s(k, `) = λ[ 1
µ1+
1
µ2 + β
]− jβ
[ 1
µ2 + β
]−min{i, n1}µ1
[(1− p)
( 1
µ1+
1
µ2 + β
)+ p( 1
µ1
)]−min{j, n2}µ2
[ 1
µ2 + β
]= λ
[ 1
µ1+
1
µ2 + β
]− 1
µ2 + β
[min{j, n2}µ2 + jβ
]−[( (1− p)n1µ1
µ2 + β
)]−min{i, n1}
23
Page 24
Now since (i, j) ∈ X \ XN and the action is chosen so as not to idle, i ≥ n1, j ≥ n2∑(k,`)∈X
q((k, `)|(i, j), a)s(k, `) = λ[ 1
µ1+
1
µ2 + β
]− 1
µ2 + β
[n2µ2 + jβ
]−[( (1− p)n1µ1
µ2 + β
)]− n1
≤ λ[ 1
µ1+
1
µ2 + β
]− 1
µ2 + β
[n2µ2 + jβ
]− n1
≤ λ[ 1
µ1+
1
µ2 + β
]− 1
µ2 + β
[n2µ2 + n2β
]− n1
≤ λ[ 1
µ1+
1
µ2 + β
]− n2 − n1.
Since X \ XN we have n1 + n2 = N . The upper bound on the right hand side is λ[
1µ1
+ 1µ2+β
]−N < 0,
where the inequality holds by assumption. Applying an analogue to Foster’s criterion for continuous-time
processes (see Meyn and Tweedie [18]; Theorem 4.2 with f = 1) yields that the Markov process associated
with any stationary policy has XN and all states that communicate with say (0, 0) as positive recurrent.
Theorem 4.3(i) of Meyn and Tweedie [18] (again with f = 1) implies that under any stationary policy,
the Markov process generated starting in any initial transient state reaches (0, 0) in finite expected time;
Statement 2 of the proposition holds. Statement 3 is a direct consequence of the first two.
Next we prove Proposition 1. That is to say, we derive conditions under which the other threshold policies
(i.e. those with 1 ≤ K < ∞) are stable. For a threshold policy with 1 ≤ K < ∞, since all states
communicate, they are reachable from one another. As a result, said policies have at most one recurrent
class. We will show that λ/µ1 < N implies that the threshold policy is stable. To this end, let XK,N =
{(i, j)|i ≤ max{K,N}, j ≤ λ/β}} and let
s(i, j) =
iµ1
if i > max{K,N};
i+jµ2
if i ≤ max{K,N} and j > λ/β;
1 otherwise,
For (i, j) ∈ X \ XK,N we have that i > max{K,N} or j > λ/β . In the former case, min{i, n1} = n1 = N and
min{j, n2} = 0 , so that by assumption, we have that∑(k,`)∈X
q((k, `)|(i, j), a)s(k, `) = λ[ i+ 1
µ1
]+ jβ
[ iµ1
]+min{i, n1}µ1
[ i− 1
µ1
]+min{j, n2}µ2
[ iµ1
]− (λ+ jβ +min{i, n1}µ1 +min{j, n2}µ2)
[ iµ1
]= λ
[ 1
µ1
]−min{i, n1}µ1
[ 1
µ1
]=
λ
µ1−N
< 0,
where the last inequality follows by assumption. On the other hand, suppose j > λ/β , then by assumption,
24
Page 25
we have that∑(k,`)∈X
q((k, `)|(i, j), a)s(k, `) = λ[ i+ j + 1
µ2
]+ jβ
[ i+ j − 1
µ2
]+min{i, n1}µ1
[ i+ j
µ2− 1− p
µ2
]+min{j, n2}µ2
[ i+ j − 1
µ2
]− (λ+ jβ +min{i, n1}µ1 +min{j, n2}µ2)
[ i+ j
µ2
]<
1
µ2
[λ− jβ
]−min{j, n2} ≤
1
µ2
[λ− jβ
]< 0.
In all cases,∑
(k,`)∈X q((k, `)|(i, j), a)s(k, `) < 0 . This yields that the Markov process associated with a thresh-
old policy with 1 ≤ K <∞ has all of the states as positive recurrent and the result holds.
25
Page 26
A.2 Tables and Figures
β λ P1 P2 E T (K=5) T (K=10) T (K=15) T (K=20)
0.15
6 178.4517 179.7341 178.9335 179.6013 180.0949 180.3528 179.9564
7 206.2711 209.9587 207.9912 208.1551 209.4038 210.0716 209.921
8 231.8722 240.0443 234.8398 234.8684 237.0368 238.1768 239.4585
9 252.1312 268.9099 256.1025 254.9438 257.6872 260.2266 261.4919
0.30
6 177.2805 179.7409 178.152 179.2352 180.0179 180.3527 179.955
7 203.943 209.9456 206.2745 207.0621 209.1499 210.0245 209.8278
8 227.8969 240.0348 231.7491 232.4374 235.9233 237.7251 239.23
9 246.8668 268.9219 251.6235 251.2036 255.3407 258.6933 260.4984
0.50
6 176.04 180.0699 177.4857 178.599 179.6891 179.8839 179.7881
7 201.7158 209.6013 204.341 206.4371 208.8696 209.7769 209.9979
8 224.1751 240.1627 228.9238 230.3749 234.8455 237.2153 238.8935
9 242.0059 269.0646 248.1879 248.0326 253.9984 257.6847 259.7667
0.80
6 174.6479 179.9755 176.2873 178.2194 179.9079 179.9403 179.7538
7 199.3424 210.1486 202.5909 204.8829 208.5326 209.5644 209.9739
8 220.6935 240.0538 225.4578 228.1392 233.9821 237.6244 238.8104
9 237.9069 269.2373 244.0736 245.773 252.4881 256.5224 259.0877
Table 3: R1 = 10, λN < 1
S(µ1,µ2,β)
26
Page 27
β λ P1 P2 E T (K=5) T (K=10) T (K=15) T (K=20)
0.15
6 208.4457 209.6898 208.9062 209.6195 210.1218 210.4124 209.9493
7 241.2414 244.9521 243.017 243.1372 244.3826 245.1061 244.9119
8 271.8592 280.052 274.8605 274.9229 277.0319 278.1137 279.4955
9 297.1322 313.7285 301.0157 299.9508 302.6946 305.2701 306.4613
0.30
6 207.2728 210.0856 208.3064 208.9182 209.7535 209.7975 209.6906
7 238.9219 244.5557 240.9672 242.5763 244.0378 244.6942 244.9824
8 267.6386 280.1866 271.8697 272.5009 275.8073 277.7983 279.139
9 291.4367 313.9329 297.1669 295.8413 300.5698 303.7909 305.3512
0.50
6 206.1213 209.8769 207.387 208.609 209.784 209.8587 209.6538
7 236.7737 245.1857 239.6681 241.0241 243.8179 244.5824 244.945
8 264.2993 280.048 268.6471 270.2846 274.8295 278.2274 279.0665
9 287.4199 314.1159 292.9373 293.7982 298.8993 302.5556 304.7765
0.80
6 204.4198 209.5937 206.445 208.385 209.7627 209.9944 209.8859
7 234.4955 244.973 237.8259 240.1903 243.8089 244.4071 244.5889
8 260.7965 280.6859 265.6655 268.1664 274.4627 277.3925 278.5732
9 282.6156 314.4225 288.8644 290.3595 297.1911 301.3531 303.8358
Table 4: R1 = 15, λN < 1
S(µ1,µ2,β)
27
Page 28
β λ P1 P2 E T (K=5) T (K=10) T (K=15) T (K=20)
0.15
10 264.1832 270.1749 266.9549 265.8106 266.7882 267.9229 268.3895
11 267.7788 269.756 267.9964 267.7298 268.2715 268.6664 268.5616
12 267.7385 269.924 267.7906 267.4819 267.4418 267.8994 267.6528
0.30
10 259.5048 270.1641 263.6992 262.6019 265.0161 266.9223 267.7915
11 265.3646 269.7563 267.3369 266.3042 267.6979 268.41 268.4586
12 266.9945 269.9207 267.7441 267.1266 267.3465 267.8509 267.6708
0.50
10 254.9315 270.0799 260.4891 259.4356 263.8898 265.7719 267.2392
11 262.075 269.9198 266.058 264.7655 267.0858 267.7064 268.3378
12 265.2852 270.1384 267.1985 266.7269 267.2808 267.5242 267.4176
0.80
10 250.3276 269.9386 256.6657 257.0211 262.0346 265.2454 266.3109
11 258.2901 270.4351 263.6008 263.0475 266.259 267.4597 267.6129
12 263.0253 269.8848 266.0939 265.4593 266.7312 267.2667 267.6124
Table 5: R1 = 10, λN ≥
1S(µ1,µ2,β)
β λ P1 P2 E T (K=5) T (K=10) T (K=15) T (K=20)
0.15
10 314.2633 315.2044 316.85 315.8805 316.7435 317.9649 318.3715
11 322.7397 314.7157 323.0906 322.7443 323.2504 323.7314 323.5028
12 327.7061 314.9118 327.7572 327.4048 327.4944 327.8507 327.6203
0.30
10 309.3195 315.1148 313.8774 312.2225 315.2938 316.5383 317.8128
11 320.158 314.9142 322.7129 321.4492 322.9374 323.008 323.4487
12 326.8466 315.151 327.6407 327.5436 327.5874 327.5391 327.424
0.50
10 304.7074 314.8364 310.4797 309.7309 313.4142 316.2141 316.8164
11 317.02 315.4523 321.0363 320.0101 322.0006 322.9374 322.8326
12 325.4446 314.9515 327.09 326.6732 327.1008 327.3568 327.6705
0.80
10 300.4782 315.2616 306.8317 306.7718 312.5421 314.9058 316.6695
11 313.5819 314.8859 318.6392 318.3116 321.2595 322.1448 323.0151
12 322.8836 315.0143 326.0246 325.2271 326.6779 327.6544 327.3765
Table 6: R1 = 15, λN ≥
1S(µ1,µ2,β)
28
Page 29
β λ P1 P2 E T(K=5) T(K=10) T(K=15) T(K=20)
0.15 6 0.029 0.096 0.057 0.075 0.094 0.097 0.097
7 0.041 0.198 0.102 0.117 0.170 0.189 0.195
8 0.055 0.511 0.191 0.167 0.291 0.377 0.434
9 0.069 19.748 0.394 0.217 0.419 0.632 0.852
0.3 6 0.028 0.097 0.054 0.074 0.094 0.096 0.097
7 0.040 0.196 0.094 0.113 0.168 0.189 0.194
8 0.053 0.499 0.161 0.158 0.278 0.368 0.420
9 0.066 20.653 0.281 0.201 0.393 0.597 0.805
0.5 6 0.028 0.097 0.052 0.072 0.094 0.097 0.097
7 0.039 0.199 0.086 0.110 0.167 0.187 0.195
8 0.051 0.513 0.139 0.150 0.269 0.357 0.418
9 0.063 18.044 0.221 0.189 0.373 0.570 0.766
0.8 6 0.027 0.098 0.049 0.072 0.093 0.097 0.097
7 0.038 0.195 0.077 0.106 0.164 0.188 0.192
8 0.049 0.504 0.119 0.143 0.261 0.350 0.412
9 0.061 18.693 0.177 0.177 0.356 0.544 0.734
Table 7: Average waits for triage, λN < 1
S(µ1,µ2,β)
29
Page 30
β λ P1 P2 E T(K=5) T(K=10) T(K=15) T(K=20)
0.15 10 0.08 438.48 0.80 0.25 0.48 0.74 1.03
11 0.09 797.07 1.33 0.25 0.47 0.71 0.95
12 0.09 1094.83 1.80 0.24 0.45 0.66 0.87
0.3 10 0.08 433.96 0.47 0.23 0.46 0.72 1.01
11 0.09 797.42 0.71 0.24 0.46 0.71 0.98
12 0.09 1093.30 0.95 0.24 0.44 0.66 0.89
0.5 10 0.07 431.66 0.33 0.22 0.44 0.69 0.97
11 0.08 796.08 0.47 0.23 0.45 0.70 0.98
12 0.09 1093.79 0.60 0.23 0.44 0.67 0.91
0.8 10 0.07 441.51 0.25 0.20 0.41 0.66 0.92
11 0.08 798.48 0.33 0.22 0.44 0.69 0.95
12 0.09 1098.75 0.42 0.23 0.43 0.66 0.90
Table 8: Average waits for triage, λN ≥
1S(µ1,µ2,β)
30
Page 31
β λ P1 P2 E T(K=5) T(K=10) T(K=15) T(K=20)
0.15 6 0.086 0.000 0.051 0.027 0.003 0.000 0.000
7 0.172 0.000 0.102 0.082 0.023 0.006 0.001
8 0.338 0.000 0.217 0.227 0.125 0.062 0.032
9 0.657 0.000 0.501 0.563 0.449 0.371 0.305
0.3 6 0.075 0.000 0.046 0.023 0.003 0.000 0.000
7 0.141 0.000 0.091 0.067 0.019 0.004 0.001
8 0.254 0.000 0.173 0.164 0.085 0.042 0.021
9 0.426 0.000 0.332 0.347 0.266 0.209 0.174
0.5 6 0.066 0.000 0.043 0.020 0.002 0.000 0.000
7 0.117 0.000 0.078 0.055 0.015 0.003 0.001
8 0.195 0.000 0.141 0.122 0.063 0.030 0.016
9 0.306 0.000 0.244 0.243 0.179 0.138 0.113
0.8 6 0.056 0.000 0.038 0.017 0.002 0.000 0.000
7 0.096 0.000 0.067 0.044 0.012 0.003 0.001
8 0.151 0.000 0.112 0.093 0.046 0.023 0.011
9 0.222 0.000 0.181 0.170 0.123 0.093 0.076
Table 9: Average waits for treatment, λN < 1
S(µ1,µ2,β)
31
Page 32
β λ P1 P2 E T(K=5) T(K=10) T(K=15) T(K=20)
0.15 10 1.195 0.000 1.096 1.149 1.099 1.068 1.045
11 1.891 0.000 1.877 1.875 1.864 1.870 1.862
12 2.556 0.000 2.550 2.559 2.559 2.556 2.563
0.3 10 0.683 0.000 0.599 0.627 0.583 0.550 0.534
11 0.980 0.000 0.936 0.962 0.947 0.934 0.930
12 1.290 0.000 1.285 1.287 1.282 1.281 1.279
0.5 10 0.451 0.000 0.393 0.403 0.362 0.336 0.329
11 0.618 0.000 0.586 0.589 0.574 0.567 0.564
12 0.789 0.000 0.772 0.777 0.769 0.771 0.768
0.8 10 0.309 0.000 0.270 0.270 0.235 0.218 0.210
11 0.407 0.000 0.377 0.382 0.365 0.356 0.353
12 0.507 0.000 0.491 0.495 0.484 0.483 0.482
Table 10: Average waits for treatment, λN ≥
1S(µ1,µ2,β)
32
Page 33
● ●●● ●●
● ●●● ●●●
●●
● ●●
4.2 4.3 4.4 4.5 4.60
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
N=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ●●● ●●● ●●● ●●●
●●
● ●●
12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.00
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
N=3
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ●●● ●●● ●●● ●●●
●●
● ●●
21.0 21.5 22.0 22.5 23.00
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
N=5
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
● ●●● ●●● ●●● ●●●
●●
● ●●
16.0 16.5 17.0 17.50
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
p=0.2
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ●●● ●●● ●●● ●●●
●●
● ●●
12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.00
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
p=0.4
● K=5
● K=10
● K=15
● K=20
● K=∞
● E ● ●●● ●●● ●●●
●●●
●●
● ●●
10.4 10.6 10.8 11.0 11.2 11.4 11.60
2
4
6
8
10
12
14
Arrival Rate λ, hours-1ChangeinReward,%
p=0.6
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●●
● ● ●●
●●●
●
●
●●
● ● ● ● ●●
11 12 13 14 15 16 170
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
β=0.15
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ● ● ● ● ●● ● ● ● ● ●●
● ● ●●
●●
●● ●
●●●
●
●
●
●
●
● ● ● ● ●●
11 12 13 14 15 16 170
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
β=0.3
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ● ● ● ● ●● ● ● ● ●
●●
● ● ●●
●●
●● ●
●●●
●
●
●
●
●
● ● ● ● ●●
11 12 13 14 15 16 170
2
4
6
8
10
12
14
Arrival Rate λ, hours-1
ChangeinReward,%
β=0.5
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
Figure 7: Comparison of reward for a policy over the policy that prioritizes triage (K = 1) when varying
the number of providers N , the probability of requiring treatment after triage p, and the abandonment rate
β. Here, R1 = 10 and unless otherwise specified p = 0.4, β = 0.3, and N = 3.
33
Page 34
●●●
●●●●●●●●●●●●●
●●
●●●
4.2 4.3 4.4 4.5 4.60.2
0.5
1
2
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, N=1
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E ●●●
●●●●●●●●●●●●●
●●
●
●●
12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0
0.1
0.5
1
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, N=3
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E ●●●
●●●●●●●●●●●●
●
●●
●
●●
21.0 21.5 22.0 22.5 23.0
0.05
0.10
0.50
1
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, N=5
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●●●●●
●●●
●●●●●●●
●●
●
●●
16.0 16.5 17.0 17.50.1
0.2
0.5
1
2
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, p=0.2
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E ●●●
●●●●●●●●●●●●●
●●
●
●●
12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0
0.1
0.5
1
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, p=0.4
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E ●●●
●●●●●●●●●●●●
●
●●
●●●
10.4 10.6 10.8 11.0 11.2 11.4 11.6
0.1
0.5
1
5
Arrival Rate λ, hours-1AverageWait,hours
Waiting Time at Triage, p=0.6
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●
●● ● ● ● ●
●●
●
●
● ●
● ●●
●● ●
11 12 13 14 15 16 17
0.1
1
10
100
1000
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, β=0.15
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●
● ● ● ● ●
●●
●
●
● ●
● ● ● ● ● ●
11 12 13 14 15 16 17
0.1
1
10
100
1000
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, β=0.3
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●
● ● ● ● ●
●●
●
●
● ●
● ● ● ● ● ●
11 12 13 14 15 16 17
0.1
1
10
100
1000
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Triage, β=0.5
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
Figure 8: Average waiting time for triage when varying the number of providers N , the probability of
requiring treatment after triage p, and the abandonment rate β. Here,R1 = 10 and unless otherwise specified
p = 0.4, β = 0.3, and N = 3.
34
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●●
●●
●
●
●
●
●
●●
●
4.2 4.3 4.4 4.5 4.60.05
0.10
0.50
1
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, N=1
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●
●
●
●
●
●
●
●●●
12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0
0.050.10
0.501
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, N=3
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●
●
●
●
●
●
●
●
●
●●
●
21.0 21.5 22.0 22.5 23.0
0.050.10
0.501
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, N=5
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●
●
●
●
●
●
●
●●
●
16.0 16.5 17.0 17.5
0.050.10
0.501
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, p=0.2
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●
●
●
●
●
●
●
●●●
12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0
0.050.10
0.501
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, p=0.4
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●
●
●
●
●
●
●
●●●
10.4 10.6 10.8 11.0 11.2 11.4 11.6
0.050.10
0.501
5
Arrival Rate λ, hours-1AverageWait,hours
Waiting Time at Treatment, p=0.6
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●●
●●
●●
●●
●●
●
●
●●
●●
●
●
●
●●
●
●
●
●
●●
●
●●
●●
●●
11 12 13 14 15 16 17
0.0050.010
0.0500.100
0.5001
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, β=0.15
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●●
● ●
●●
●●
●●
●
●●
●●
●
●
●
●●
●●
●
●
●
●●
●
●●
●●
●●
11 12 13 14 15 16 17
0.0050.010
0.0500.100
0.5001
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, β=0.3
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
●●
●● ● ●
●●
●●
● ●
●●
●●
● ●
●
●
●●
●●
●
●
●
●●
●
●●
●●
● ●
11 12 13 14 15 16 17
0.0050.010
0.0500.100
0.5001
5
Arrival Rate λ, hours-1
AverageWait,hours
Waiting Time at Treatment, β=0.5
● K=1
● K=5
● K=10
● K=15
● K=20
● K=∞
● E
Figure 9: Average waiting time for treatment when varying the number of providers N , the probability
of requiring treatment after triage p, and the abandonment rate β. Here, R1 = 10 and unless otherwise
specified p = 0.4, β = 0.3, and N = 3.
35
