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European Journal of Operational Research 0 0 0 (2017) 1–15

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Innovative Applications of O.R.

Fluid approximations and control of queues in emergency

departments

Jerome Niyirora

a , ∗, Jun Zhuang

b

a SUNY Polytechnic Institute, Utica, NY, USA b University at Buffalo, SUNY, Buffalo, NY, USA

a r t i c l e i n f o

Article history:

Received 10 September 2016

Accepted 2 March 2017

Available online xxx

Keywords:

OR in health services

Emergency department

Queues

Square root staffing

Optimal control

a b s t r a c t

Long queues in emergency departments (EDs) lead to overcrowding, a phenomenon that can potentially

compromise patient care when medical interventions are delayed. There are several causes of this prob-

lem, one of which is inadequate resource allocation. In this paper, we propose using a modified version

of the square root staffing (SRS) rule to satisfy the probability of delay target. We use the concepts of

kinetics and biological modeling to approximate the fluid behavior of the queueing process. We are then

able to estimate the offered load and the appropriate service grade necessary to construct a staffing pol-

icy that meets the target. Additionally, we show how to utilize Pontryagin’s maximum principle to find

the optimal number of providers that minimizes delay and staffing costs. Finally, we demonstrate the

implementation of our model using data from a hospital in upstate New York.

© 2017 Elsevier B.V. All rights reserved.

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. Introduction

Long queues in emergency departments (EDs) lead to over-

rowding, a phenomenon that can potentially lead to adverse pa-

ient care when medical interventions are delayed ( Warden et al.,

006 ). Additionally, increased mortality has been linked to over-

rowding ( Hoyle, 2013; Pines et al., 2011 ). Recent surveys in the

nited States (US) have indicated that, due to overcrowded EDs,

bout 50 0,0 0 0 ambulances are diverted each year ( McCaig, Xu, &

iska, 2009; Pitts, Niska, Xu, & Burt, 2008 ). Moreover, an estimated

million patients leave EDs, annually, without receiving medical

are ( Warden et al., 2006 ). Some of the proposed solutions to this

roblem include: alleviating delays in processing lab and radiology

ests (American College of Emergency Physicians – ACEP, 2004 ), re-

ucing delays in patients transportation ( Au et al., 2009; Green &

all, 2006 ), and minimizing delays in the schedules of operating

ooms for ED patients ( Litvak, Long, Cooper, & McManus, 2001 ). It

s also believed that solving the problem of boarding admitted pa-

ients in the ED ( Shi, Chou, Dai, Ding, & Sim, 2015 ) and transferring

ental health patients in a timely manner could lessen long delays

Government Accountability Office – GAO, 2009 ).

In this paper, we focus on the proposed solution to overcrowd-

ng that relates to efficient allocation of resources ( Green & Hall,

006; Hall, 2012 ). To this end, we introduce a variation of the

quare root staffing (SRS) rule for cost-effective allocation of re-

∗ Corresponding author.

E-mail address: jerome.niyirora@sunyit.edu (J. Niyirora).

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ttp://dx.doi.org/10.1016/j.ejor.2017.03.013

377-2217/© 2017 Elsevier B.V. All rights reserved.

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

ources in the ED. Additionally, we establish optimal control of

ueues to minimize delay and staffing costs.

. Literature review

For many service systems, including EDs, resources can effi-

iently be allocated using the SRS rule. The general form of this

ule is c = R + β√

R , where c represents the number of servers, R

s the offered load (or resource demand), and β is the service grade

Garnett, Mandelbaum, & Reiman, 2002; Halfin & Whitt, 1981; Jen-

ings, Mandelbaum, Massey, & Whitt, 1996; Mandelbaum & Zel-

yn, 2004; Whitt, 2007 ). In stationary systems, where the cus-

omer arrival rate is constant, the offered load may be obtained

sing Little’s law ( Little, 1961 ). For non-stationary systems, esti-

ations of the offered load include the Pointwise Stationary Ap-

roximation (PSA), for slowly changing arrival rate ( Green & Kole-

ar, 1991; Whitt, 2007 ), and the infinite server (IS) approximation

see analysis in Eick, Massey, & Whitt, 1993a; 1993b ). Additionally,

he Modified Offered Load (MOL), an extension of the IS first an-

lyzed in Jagerman (1975) , has been proposed Massey and Whitt

1994) , Massey and Whitt (1996) , Massey and Whitt (1997) , Grier,

assey, McKoy, and Whitt (1997) . Applications of the IS method

o stabilize performance can be found in Jennings et al. (1996) ,

eldman, Mandelbaum, Massey, and Whitt (2008) , Yom-Tov and

andelbaum (2014) . For other proposed methods to stabilize per-

ormance see Liu and Whitt (2012) , Massey and Pender (2013a) ,

assey and Pender (2013b) , Liu and Whitt (2014) , Pender (2014) ,

ender (2015) , Liu and Whitt (2017) .

d control of queues in emergency departments, European Journal

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2 J. Niyirora, J. Zhuang / European Journal of Operational Research 0 0 0 (2017) 1–15

ARTICLE IN PRESS

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Traditionally, the SRS rule has been analyzed in the context of

call centers ( Gans, Koole, & Mandelbaum, 2003; Hampshire, Jen-

nings, & Massey, 2009; Mandelbaum & Zeltyn, 2005; Whitt, 1999;

20 05; 20 07 ). However, this rule is increasingly being studied for

staffing decisions in healthcare ( Defraeye & Van Nieuwenhuyse,

2016 ). For example, SRS is used in Izady and Worthington (2012) to

find staffing levels to meet the quality of service target that limits

the number of patients in the ED. In Yom-Tov and Mandelbaum

(2014) , the queueing process is modeled using Erlang-R whereas

SRS is used to allocate resources in various medical units, includ-

ing the ED. Zaied (2012) also proposes SRS for resource allocation

in the ED where the queueing process is modeled as a fork-join-

network. Zeltyn et al. (2011) use simulation to evaluate resource

needs in EDs and apply SRS for staffing decisions. In Sinreich and

Jabali (2007) , SRS is also used to allocate resources in the ED.

The literature that relates to optimal control of queues includes

Seidl, Kaplan, Caulkins, Wrzaczek, and Feichtinger (2016) where

optimal control theory is applied to terror queues. In their model,

Pontryagin’s maximum principle is utilized to evaluate the govern-

ment’s counter-terrorism staffing policies. In Matveev, Feoktistova,

Bolshakova, and Ishchenko (2016) , a periodic control of a time-

invariant polling system is established. In Bhandari, Scheller-Wolf,

and Harchol-Balter (2008) , a constrained dynamic optimization

problem is considered to find the optimal number of permanent

versus temporary servers in call centers. In Fu, Marcus, and Wang

(20 0 0) , dynamic programing over a finite horizon is applied

to study optimal staffing in call centers. An admission control

problem is considered in Koça ̆ga and Ward (2010) for stationary

Erlang-A ( M/M/c + M) queues. The authors use Markov Decision

Process (MDP) and Diffusion Control Problem (DCP) methods for

staffing decisions. In Weerasinghe and Mandelbaum (2013) , DCP is

also considered but for a finite horizon where the queueing type

is G/M/n/B + GI and the objective is to minimize costs by trading

off blocking versus abandonment. In Bassamboo, Harrison, and

Zeevi (2005) , an admission control problem is also considered but

for queues with multiclass customers and different server skills.

In Borst, Mandelbaum, and Reiman (2004) , optimal staffing in call

centers is pursued where the manager’s goal is to minimize delay

and staffing costs in Erlang-C ( M / M / c ) queueing systems. In Baron

and Milner (2009) , SRS, based on optimal control, is devised to

maximize profit in outsourced call centers. In Nobel and Tijms

(1999) , optimal control is applied to a two nodes M

X / G /1 queue

with the objective of minimizing the long-run mean number of

customers in the system. In Shioyama (1991) , optimal control is

applied to a network of queues and the problem is formulated

as an undiscounted semi-MDP. The objective in the model is to

minimize the hourly costs. In both Hampshire et al. (2009) and

Rudolph (2011) , variational calculus is employed to find the opti-

mal number of servers via Lagrangian mechanics. Additionally, in

Hampshire et al. (2009) a fluid version of the MOL is proposed

for staffing. In Stolletz (2008) , a stationary backlog-carryover

approach is proposed for approximating the M ( t )/ M ( t )/ c ( t ) queue.

In Koeleman, Bhulai, and van Meersbergen (2012) , MDP is used

to establish optimal policy for personnel scheduling at a care-at-

home service facility. In Ramirez-Nafarrate, Hafizoglu, Gel, and

Fowler (2014) , ambulance diversion in EDs is studied and MDP is

used to find control policies that minimize the average time that

patients wait beyond their recommended safety time threshold. In

Tirdad, Grassmann, and Tavakoli (2016) , MDP is also used, but this

time the aim is for optimal control of an M ( t )/ M / c / c queue. Xiang

and Zhuang (2016) consider a queueing network and introduce

optimization methods for allocating medical resources to emer-

gency victims with deteriorating health conditions. Other relevant

optimization methods for queues can be found in Li and Stanford

(2016) , Wang, Zhang, and Huang (2017) , Dimitrakopoulos and Bur-

netas (2016) , Yarmand and Down (2013) , Izady and Worthington

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

2012) , Niyirora and Pender (2016) . For applications of optimal

ontrol to biological models see Lenhart and Workman (2007) .

Besides SRS and optimal control, other methods suggested for

llocating resources in service systems such as EDs, include ma-

hine learning ( Lee et al., 2015 ), simulation ( Abo-Hamad & Ar-

sha, 2013; Ahmed & Alkhamis, 2009; Brenner et al., 2010; Draeger,

992; Kumar & Kapur, 1989; Mielczarek, 2014; Rossetti, Trzcinski,

Syverud, 1999 ), optimization/scheduling methods ( Beaulieu, Fer-

and, Gendron, & Michelon, 20 0 0; Brunner, Bard, & Kolisch, 20 09;

arter & Lapierre, 2001; Ferrand, Magazine, Rao, & Glass, 2011;

uscombe & Kozan, 2016; Rousseau, Pesant, & Gendreau, 2002; Sin-

eich & Jabali, 2007 ), dynamic programing ( Vassilacopoulos, 1985 ),

terative staffing algorithms ( Defraeye & Van Nieuwenhuyse, 2013 ),

nd social network analysis ( Niyirora & Klimek-Yingling, 2016 ). In

reen, Soares, Giglio, and Green (2006) the lag stationary indepen-

ent period by period (SIPP) model is suggested.

. Contributions

Our work introduces new fluid approximations of queues. Ad-

itionally, new methods are presented to approximate the offered

oad and to find efficient resource allocation policies for EDs.

To the best of our knowledge, the main contributions of this

aper are as follows:

1. We use the notions of kinetics and biological modeling to con-

ceptualize a model of the change in the patient’s welfare state,

from sick to well, via the provision of medical care. With this

model, we are able to obtain fluid limits that approximate the

mean number of patients in the queue and the mean number

of idle resources.

2. We introduce new approximations of the offered load, the ser-

vice grade, and the probability of abandonment. Subsequently,

we provide an algorithm to derive staffing policies that meet

the probability of delay target. Numerical examples confirm

that our proposed method provides stable performance, espe-

cially when the probability target is small.

3. When no delay target is specified, we show how to apply Pon-

tryagin’s maximum principle to find resource allocation policies

that minimize delay and staffing costs.

4. We demonstrate the potential implementation of our model in

an actual ED setting. We use data from a hospital in upstate

New York and provide insights into staffing options under dif-

ferent scenarios.

The organization of the rest of the paper follows. In Section 4 ,

e derive our model. In Section 5 we discuss the model perfor-

ance. In Section 6 we introduce resource allocation methods. In

ection 7 we show how our model can be implemented. Lastly, we

rovide concluding remarks in Section 8 .

. Model

.1. A motivating problem

A providers group has a contract with a hospital to provide

mergency care. In this contract, a congestion clause specifies the

robability of delay target. The goal for the manager of this group

s to find an efficient resource allocation policy to satisfy the terms

f this contract.

.2. Model derivation and assumptions

We use the notions of kinetics ( Holmes, 2009 ) and biological

odeling ( Lenhart and Workman, 2007 ; Smith, 2008 ) to conceptu-

lize a model of the change in the patient’s welfare state, from sick

d control of queues in emergency departments, European Journal

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J. Niyirora, J. Zhuang / European Journal of Operational Research 0 0 0 (2017) 1–15 3

ARTICLE IN PRESS

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Fig. 1. A kinetic model depicting the change in the state of the patient’s welfare

from sick ( S ) to well ( W ) via the provision of medical evaluation ( SP ) by the idle re-

source ( P ). The arrival and the abandonment rates are respectively λ and θ , whereas

the evaluation and the service rates are respectively γ and μ.

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Q

S ) to well ( W ). This change is facilitated by the availability of re-

ource P (e.g. a medical provider, ED bed, etc.) and the provision of

ome type of medical service (e.g. evaluation), symbolized by the

unction f ( S , P ). It is assumed that:

f (S, 0) = f (0 , P ) = 0 , (1)

eaning that without resource P , a patient cannot move from state

into state W . Likewise, having resource P without a patient in

tate S is functionally zero. Kinetically, this relationship can be

haracterized as:

+ P → W (2)

e now use the second-order Maclaurin’s series expansion to ap-

roximate f ( S , P ), as follows:

f (S, P ) = f (0 , 0) + f S (0 , 0) S + f P (0 , 0) P

+

1

2! [ f SS (0 , 0) S 2 + 2 f SP (0 , 0) SP + f PP (0 , 0) P 2 ] + · · · (3)

iven Eq. (1) and the kinetic relationship in Eq. (2) , the following

roduct approximation results:

f (S, P ) ≈ SP, (4)

ince the only non-zero coefficient of the series is likely f SP (0, 0).

ee Holmes (2009) for similar approximations in biological mod-

ls. For the purposes of modeling EDs, it should be remarked that

ther approximations are possible, as dictated by the kinetic rela-

ionship in the medical evaluation process. For example, if, in ad-

ition to sick patients, there is also a constant resource demand,

hen f (S, P ) ≈ (S + 1) P would be a candidate approximation since

he f P (0, 0) coefficient in Eq. (3) would be non-zero.

For the arrival process, we assume that patients arrive in the

D, independently, at a rate λ ≡ { λ( t )| t ≥ 0}, where t is for

ime. Additionally, we assume no retrials. Upon arrival, patients

re triaged into various levels of severity ( Gilboy, Tanabe, Travers,

Rosenau, 2012 ). For mathematical simplification, we assume in-

tantaneous triage, one level of severity, and a queueing discipline

f first-come, first served (FCFS). Likewise, for simplification, we

ssume that only one type of resource P is needed to move the

atient from state S into state W .

For the abandonment (impatience) process, we assume that sick

atients who become impatient abandon the ED at rate θ or oth-

rwise move into the well state ( W ) at rate γ . The implication of

his simplified assumption is that a patient (more generally a cus-

omer) can only abandon while in the queue, which is a common

ssumption in Erlang-A queueing systems (e.g. see Garnett et al.,

0 02; Hampshire et al., 20 09; Rudolph, 2011 ). Although, in reality,

atients can abandon the ED at any stage of the medical care pro-

ess, including before triage ( Johnson, Myers, Wineholt, Pollack, &

usmiesz, 2009 ).

The final modeling assumption relates to the medical care ser-

ice process. After a medical evaluation, at rate γ , the patient is

ared for and is discharged at rate μ. Naturally, as patients are

ischarged (after service), resources become available to serve pa-

ients who are still in state S . The implication of this assumption

s that a patient who is waiting to be discharged, in state W , takes

riority over a patient who is still in state S .

A graphical depiction of our kinetic model is portrayed in Fig. 1 .

.3. Kinetic fluid approximations

We refer to Holmes (2009) , to generate the following fluid ap-

roximations of our kinetic model in Fig. 1 :

d

dt

(

S P

W

)

=

(

1 −1 −1 0

0 −1 0 1

0 1 0 −1

)

⎛

⎜ ⎝

λγ SP θS μW

⎞

⎟ ⎠

(5)

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

careful inspection of the matrix in Eq. (5) reveals that dP dt

+

dW

dt

s a conserved quantity since d dt

(P + W ) = 0 . This means that if we

now P , we can easily find W or vice-versa. That is, given the initial

onditions P (0) = P 0 and W (0) = W 0 , where P 0 + W 0 equals some

onstant c , we obtain the following conservation law:

+ W = c (6)

rom Eq. (6) , c can be interpreted as the amount of resources ini-

ially assigned. In essence, P represents the fluid level of idle re-

ources. Then W , the fluid level of patients in the well state, equals

he amount of busy resources, c − P . It is implied that one resource

an only serve one patient. In Section 7 , we show how this sugges-

ion can be relaxed when productivity is accounted for.

In the rest of this paper, the discussion of c and P is exclusive

o medical providers, as opposed to other resources such as beds.

Next, we explore the queueing analysis of our kinetic model.

.4. Queueing analysis

From a queueing perspective, our kinetic model resembles a

andem queue with two nodes ( S and W ), where λ( t ) is a rate

f a non-homogeneous Poisson arrival process, and θ and μ are,

espectively, abandonment and service rates. Both the time-to-

bandon and the service time are exponentially distributed. It fol-

ows that node S , in the tandem queue, behaves like an Erlang-

queue while node W behaves like an infinite server queue (see

ig. 2 ). The total number of providers initially assigned, c , remains

onstant.

Now, we let f ( Q 1 ( t ), Q 2 ( t )) represent the total number of pa-

ients in the system, as the sum of Q 1 ( t ) and Q 2 ( t ), where Q 1 ( t )

s the number of patients at node S and Q 2 ( t ) is the number of

atients at node W . Then, the conservation law in Eq. (6) can be

ranslated as:

(t) + Q 2 (t) = c (7)

We now refer to Grier et al. (1997) to derive the following func-

ional Kolmogorov forward equations to characterize the jumps in

he queueing process:

d

dt E[ f (Q 1 (t) , Q 2 (t))]

= λ(t) · E[ ( f (Q 1 (t) + 1 , Q 2 (t)) − f (Q 1 (t) , Q 2 (t)) ) ]

+ γ · E[ ( f (Q 1 (t) − 1 , Q 2 (t)) − f (Q 1 (t) , Q 2 (t)) ) ]

+ θ · E[ ( f (Q 1 (t) − 1 , Q 2 (t)) − f (Q 1 (t) , Q 2 (t)) ) ]

+ γ · E[ ( f (Q 1 (t) , Q 2 (t) + 1) − f (Q 1 (t) , Q 2 (t)) ) ]

+ μ · E[ ( f (Q 1 (t) , Q 2 (t) − 1) − f (Q 1 (t) , Q 2 (t)) ) ] , (8)

here E [ ·] is used to symbolize expectation. The sample paths of

1 ( t ) and Q 2 ( t ) follow:

1 (t) = Q 1 (0) + �1

(∫ t

0

λ(s ) ds

)− �2

(∫ t

0

γ · Q 1 (s ) P (s ) ds

)

−�3

(∫ t

0

θ · Q 1 (s ) ds

)

d control of queues in emergency departments, European Journal

13

4 J. Niyirora, J. Zhuang / European Journal of Operational Research 0 0 0 (2017) 1–15

ARTICLE IN PRESS

JID: EOR [m5G; March 25, 2017;13:45 ]

Fig. 2. A queueing view of our kinetic model where Q 1 ( t ) is the number of patients in the queue, at node S , and Q 2 ( t ) is the number of patients in service, at node W .

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θ

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ρ

T

o

(

n

E

Q 2 (t) = Q 2 (0) + �2

(∫ t

0

γ · Q 1 (s ) P (s ) ds

)(9)

−�4

(∫ t

0

μ · Q 2 (s ) ds

)Here �i ≡ { �( t ): t ≥ 0} for i = 1 , . . . , 4 are mutually independent

time-homogeneous Poisson processes with rate 1.

Following Mandelbaum, Massey, and Reiman (1998) , we use the

factor η > 0 to scale both the arrival rate, η · λ( t ), and the number

of providers, η · c , such that:

Q

η1 (t) = Q

η1 (0) + �1

(∫ t

0

λ(s ) ds

)− �2

(∫ t

0

γ Q

η1 (s ) P η(s ) ds

)

−�3

(∫ t

0

θQ

η1 (s ) ds

)

Q

η2 (t) = Q

η2 (0) + �2

(∫ t

0

γ Q

η1 (s ) P η(s ) ds

)

−�4

(∫ t

0

μQ

η2 (s ) ds

)(10)

Given, lim

η→∞

Q η

η = Q(0) ≡ { q (t) : t ≥ t} , the following limit holds:

lim

η→∞

sup

0 ≤t≤T

∣∣∣Q

η

η− q (t)

∣∣∣ = 0 a.s (11)

Then, (E[ Q 1 (t)] , E[ Q 2 (t)]) = (q 1 (t) , q 2 (t)) and:

d

dt q 1 (t) = λ(t) − γ q 1 (t ) P (t ) − θq 1 (t) (12)

d

dt q 2 (t) = γ q 1 (t) P (t) − μq 2 (t) (13)

By Eq. (7) , we can rewrite Eq. (13) as:

d

dt (c − P (t)) = γ q 1 (t) P (t) − μ(c − P (t)) (14)

After rearrangement, we obtain:

d

dt P (t) = μ(c − P (t)) − γ q 1 (t ) P (t ) (15)

Additionally, since by the conservation law, in Eq. (7) , we can

solved for Q 2( t ) if we know P ( t ), it follows that the next coupled

dynamical system is sufficient to characterize the fluid behavior of

the queue in our model:

•q = λ − γ qP − θq (16)

•P = μ(c − P ) − γ qP, (17)

where q 1 ( t ) ≡ q and P ( t ) ≡ P . Eq. (16) models the rate of change

in the number of patients in the queue whereas Eq. (17) models

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

he rate of change in the number of idle providers. The time de-

endence t was suppressed to simplify notation and the • symbol

s used to signify the time derivative.

In the next section, we discuss the performance of the dynam-

cal system in Eqs. (16) and ( 17 ).

. Model performance

.1. Stationary solutions

To achieve a steady state (when λ is constant), the dynamical

ystem in Eqs. (16) and ( 17 ) must equal zero. When this happens,

he nullclines of q (when

•q = 0 ) and P (when

•P = 0 ) are respec-

ively given by:

=

αλ

μP + αθ(18)

=

α(c − P )

P , (19)

here α ≡ μγ , for the purposes of this model, is the probability of

elay target.

As a precursor to our next theorems, we define the offered load

s:

λ

μ≡ ρ, (20)

here ρ < c ( Little, 1961 ).

heorem 5.1. When the abandonment rate θ = 0 , the limiting mean

umber of patients in the queue, q , is given by:

= αρ

c − ρ(21)

nd the limiting number of idle providers is given by:

= c − ρ (22)

The proof of Theorem 5.1 follows from solving for the equilib-

ium points of the dynamical system in Eqs. (16) and ( 17 ) when

= 0 .

orollary 5.2. The offered load in Eq. (20) is equivalent to:

= c − P (23)

The proof of Corollary 5.2 follows from rearranging Eq. (22) in

heorem 5.1 . The implication of this corollary is that the number

f patients at node W , in our model, represents the offered load

see Fig. 2 ).

As a result of both Theorem 5.1 and Corollary 5.2 , the expected

umber of patients in the system, E [ N ], can be expressed as:

[ N] = q + c − P (24)

d control of queues in emergency departments, European Journal

13

J. Niyirora, J. Zhuang / European Journal of Operational Research 0 0 0 (2017) 1–15 5

ARTICLE IN PRESS

JID: EOR [m5G; March 25, 2017;13:45 ]

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q

e

t

r

T

n

q

a

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l

t

C

E

ρ

T

μ

b

P

5

5

f

λ

A

a

δ

W

t

r

l

m

n

5

l

E

6

c

dditionally, by Little’s law ( Little, 1961 ) the average time in the

ueue can be obtained by q / λ and the time spent in service can be

xpressed as 1/ μ. Accordingly, the expected time spent in the sys-

em is q/λ + 1 /μ, which is equivalent to the traditional queueing

esults of an Erlang-C model (e.g. see Harchol-Balter, 2013 ).

heorem 5.3. When the abandonment rate θ > 0, the limiting mean

umber of patients in the queue, q , is given by:

=

ρ − c − α θμ +

√ (ρ − c − α θ

μ

)2 + αρ 4 θμ

2 θ/μ(25)

nd the limiting number of idle providers is given by:

=

c − ρ − α θμ +

√ (c − ρ − α θ

μ

)2 + αc 4 θμ

2

(26)

The proof of Theorem 5.3 also follows from solving for the equi-

ibrium points of the dynamical system in Eqs. (16) and ( 17 ), but

his time when θ > 0 (see Appendix A ).

orollary 5.4. As a result of Theorem 5.3 , the offered load in

q. (20) can be expressed as:

= c − P + q θ

μ(27)

The proof of Corollary 5.4 follows from rearranging Eq. (26) in

heorem 5.3 and solving for ρ (see Appendix D ).

As a consequence of Corollary 5.4 , it is evident that when θ =, the offered load in an Erlang-A model is equivalent to the num-

er of patients in the system in an Erlang-C model (see Eq. (24) ).

roposition 5.5. The limiting behaviors in both Theorems 5.1 and

.3 are asymptotically stable.

See proof in Appendix B .

.1.1. Insights into the probability of abandonment

We let δ be the probability of abandoning the system. The ef-

ective arrival rate can then be expressed as:

(1 − δ) ≡ λ − θq (28)

Fig. 3. Model dynamics when the probability of delay target α = 0 . 1 , λ = 10 + 5 s

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dditionally, using the nullcline in Eq. (18) , δ can be approximated

s:

≈ θq

λ=

θ

γ P + θ(29)

e notice that ∂ δ/ ∂ γ > 0 and ∂ δ/ ∂ θ > 0, which indicates that

he abandonment probability δ increases in both the abandonment

ate and in the probability of delay target α ≡ μ/ γ . It also fol-

ows that ∂ δ/ ∂ μ < 0 and

∂δ∂P

∂P ∂c

< 0 , which, operationally speaking,

eans that to decrease δ, either the service rate μ or the initial

umber of providers c must be increased.

.2. Non-stationary solutions

When the arrival process is time varying, we observe the fol-

owing:

1. The limiting behavior converges to limit cycles. See

Appendix C for the graphical sensitivity of these cycles, as

parameters are varied. Following Corollary 5.4 , we speculate

that the stability of these cycles occurs when:

1

T

∫ T

0

λ − θq dt < μc (30)

2. Per Corollary 5.4 , the time-varying offered load can be approx-

imated by:

c − P +

θ

μq, (31)

where both P and q are time dependent but c is still constant.

3. The patient evaluation term, γ qP , peaks at about the same time

as the arrival rate λ. However, there is a lag until the peak in

the time-varying offered load (see Fig. 3 ).

Next, we explore efficient resource allocation methods for the

D based on our model.

. Resource allocation

We propose using a variation of the SRS rule for efficient allo-

ation of resources in the ED. For our purposes, we are interested

in (t) , and θ/μ = 0 . 5 . The initial conditions are P(0) = S(0) = 0 and c = 16 .

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Table 1

A numerical example of finding the dynamic policy c(t) .

Target θ/μ = 0 . 1 θ/μ = 1 θ/μ = 10

α ˜ β Garnet t ( ̃ β) ˜ δ c ˜ β Garnet t ( ̃ β) ˜ δ c ˜ β Garnet t ( ̃ β) ˜ δ c

0.1 1.383863 0.10329 0.002228 15 1.264238 0.103072 0.022233 14.6 0.835603 0.102745 0.173903 13.2

0.2 1.001909 0.208694 0.005994 13.7 0.81828 0.206599 0.054826 13.1 0.195395 0.203749 0.322742 11.1

0.3 0.77094 0.30293 0.011172 12.9 0.519607 0.301669 0.091953 12.1 −0.32676 0.304845 0.446268 9.4

0.4 0.573416 0.403326 0.018543 12.2 0.249791 0.401375 0.134209 11.2 −0.82058 0.405879 0.555689 7.8

0.5 0.382008 0.516215 0.029852 11.5 −0.02098 0.508369 0.183177 10.3 −1.31662 0.504979 0.65773 6.2

0.6 0.224978 0.616472 0.045345 10.9 −0.26248 0.603523 0.234 4 49 9.5 −1.84626 0.602412 0.759212 4.5

0.7 0.074976 0.713206 0.067339 10.3 −0.53511 0.703711 0.295884 8.6 −2.47323 0.702202 0.872539 2.5

0.8 −0.0913 0.812913 0.10013 9.6 −0.86973 0.807776 0.376233 7.5 – – – –

0.9 −0.27373 0.90112 0.144685 8.8 −1.298 0.902856 0.48844 6.1 – – – –

6

d

P

c

A

a

λ

p

t

α

A

p

a

θ

i

c

6

n

s

e

d

m

p

s

u

a

n

a

i

l

1

f

in finding the appropriate amount of c , the initial total number

of providers. Subsequently, we can prescribe static and dynamic

staffing policies to meet the contractual probability of delay target.

When no target is stipulated, we introduce optimal control meth-

ods to find c that minimizes the overall delay and staffing costs.

6.1. Formulating the SRS rule

The general formulation of the SRS rule is as follows:

c = ρ + β√

ρ, −∞ < β < ∞ , (32)

where ρ is the offered load and β is the service grade. When

θ = 0 , the relationship between β and the probability of delay

target α follows the Halfin–Whitt (HW) delay function such that

HW (β) = α ( Halfin & Whitt, 1981 ). When θ > 0, this relation-

ship follows the Garnett delay function such that Garnet t (β) = α( Garnett et al., 2002 ). The following proposition introduces novelty

in the estimation of β based on our model.

Proposition 6.1. The service grade β for the stationary model is

given by:

β =

P − qθ/μ√

c − P + qθ/μ, −∞ < β < ∞ (33)

For the non-stationary model, β ≈ ˜ β, where

˜ β =

1

T

∫ T

0

P − qθ/μ√

c − P + qθ/μdt, −∞ <

˜ β < ∞ (34)

See Appendix D for proof.

Using Proposition 6.1 , we are able to approximate the probabil-

ity of abandonment δ as follows:

δ ≈ θ

γ(β√

c − P + qθ/μ + qθ/μ)

+ θ(35)

The proof of Eq. (35) follows from solving for P in the

numerator of Eq. (33) and then substituting the result ( P =β√

c − P + qθ/μ + qθ/μ) into Eq. (29) .

Fig. 4. Simulated probability versus

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.2. Finding c to meet the probability of delay target

The static staffing policy that meets the probability of

elay target α is obtained by searching for optimal c in

roposition 6.1 whereas the dynamic policy c ( t ) is obtained by:

(t) = c − P + qθ/μ +

˜ β√

c − P + qθ/μ (36)

lgorithm Appendix E.1 presents the steps to follow to find both c

nd c ( t ).

In Table 1 , we present results of a numerical example when

(t) = 10 + 5 sin (t) , P (0) = S(0) = 0 , θ/μ = { 0 . 1 , 1 , 10 } , and c ( t )

olicy is constructed per Algorithm Appendix E.1 . Figs. 4 –6 por-

ray the stability of the proposed dynamic policy when the target

= 0 . 1 . Additional simulation results are portrayed in Appendix F .

general observation from our graphical results is that the pro-

osed dynamic policy meets the probability of delay target α in

ll cases, but the best stability occurs when α is small or when

/μ = 1 . It should also be noted that when the abandonment rate

s already high, setting the probability target too high (e.g. α = 0 . 9 )

ould lead to infeasible solutions (see Fig. F.13 in Appendix F ).

.3. Finding optimal c to minimize staffing and delay costs.

In the absence of any contractual delay target α, there is a busi-

ess case to be made that the manager of the providers group

hould seek to minimize staffing costs in order to increase the op-

rating income. Additionally, the manager should seek to minimize

elay costs so customers (in this case patients) are satisfied. To

otivate the corresponding staffing problem, let us assume that

atients arrive in the system when providers have not been as-

igned yet (meaning that the probability of delay α = 1 ). Also, let

s assume that no abandonment is allowed. To find the appropri-

te staffing policy, the manager decides to steadily increase the

umber of providers c (from zero) until both the costs of delay

nd that of staffing are minimized. Clearly, the manager’s approach

s tedious, especially when the system is non-stationary. As a so-

ution, we recommend using optimal control theory ( Pontryagin,

987 ) to find c . The objective function, �( c ), is formulated as

ollows:

the target: α = 0 . 1 , θ/μ = 0 . 1 .

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Fig. 5. Simulated probability versus the target: α = 0 . 1 , θ/μ = 1 .

Fig. 6. Simulated probability versus the target: α = 0 . 1 , θ/μ = 10 .

�

H

l

a

t

&

p

t

s

T

m

b

c

C

s

L

β

β

P

s

g

t

l

n

m

0

1

c

s

d

7

7

w

Y

e

a

t

o

m

T

b

2

μ

a

p

a

b

t

#

c

F

t

t

f

#

T

l

t

(c) = max c

−∫ T

0

(1

2

wc 2 + dq

)dt (37)

ere, the minus sign indicates that we are minimizing both the de-

ay costs w and the staffing costs d . We use the quadratic c term to

void boundary solutions since these types of solutions are opera-

ionally unreasonable for service systems such as the ED ( Niyirora

Pender, 2016 ). Besides, a lower boundary of zero would lead to

atient safety concerns since the implication is that no provider is

o be allocated.

To set the stage for the next theorem, we let p 1 and p 2 be the

hadow prices of q and P , respectively.

heorem 6.2. Given the initial conditions q (0) and P (0), and the ter-

inal conditions p 1 (T ) = 0 and p 2 (T ) = 0 , the optimal mean num-

er of providers, c ∗, that maximizes �( c ) is given by:

∗ =

μ

wT

∫ T

0

p 2 dt (38)

For proof see Appendix G.1 .

orollary 6.3. For the stationary system, Theorem 6.2 implies that:

d

w

q = c ∗P (39)

The proof of Corollary 6.3 results from the zeros of the neces-

ary conditions in Appendix G .

emma 6.4. Following Corollary 6.3 , when α ≈ 1, the service grade

is approximated by:

≈√

d

wc ∗(40)

The proof of Lemma 6.4 follows from rewriting

roposition 6.1 using the results of Corollary 6.3 and the steady

tate results in Theorem 5.1 .

As a general remark, we have observed that the numerical inte-

ration of our optimal control problem, using the objective func-

ion in Eq. (37) , can be unstable. Fortunately, Corollary 6.3 al-

ows us to approximate c ∗, in Theorem 6.2 , by solving a more

umerically stable optimal control problem given by �(c) =ax c −

∫ T 0

1 2 c

2 dt and an isoperimetric constraint of ∫ T

0 d w

q − cP dt = (see Appendices G.2 and G.3 ).

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Table 2 shows results of a numerical example when λ(t) =0 + 5 sin (t) , μ = 1 , and P (0) = S(0) = 0 . The service grade β is

omputed using Lemma 6.4 . We use c ∗ to construct a dynamic

taffing policy following Eq. (36) .

We next explore implementation strategies of our model using

ata obtained from a hospital in upstate New York.

. Model implementation

.1. Case 1: meeting the probability of delay target

To show how our model can be implemented in an actual ED,

e use the December 2012 data from a hospital in upstate New

ork. We use the idea in Hall (1991) and Green and Hall (2006) to

stimate the non-homogeneous Poisson rate by fitting the average

rrival data on a function. Fig. 7 a shows the average of arrival pat-

erns by the day and by the hour. Fig. 7 b shows the overall average

f arrivals fitted to a sinusoidal function. To estimate the abandon-

ent rate θ , we use the formula in Eq. (29) and obtain θ = 0 . 026 .

he overall abandonment probability δ of this hospital happens to

e 3.1%, which is about the same as the national average ( GAO,

009 ). We did not have enough data to estimate the service rate

. As a result, we conveniently set μ = 1 . Additionally, the prob-

bility of delay target is conveniently set to α = 0 . 15 . The staffing

olicy c ( t ) is then determined following Algorithm Appendix E.1 . To

chieve the exact integer values, which would represent the num-

er of providers to be allocated, we use the ceiling function such

hat

prov iders (t) ≈ c(t) � (41)

Fig. 8 a portrays both the static and the dynamic staffing poli-

ies. The corresponding probabilities of delay are portrayed in

ig. 8 b. Clearly, the static (constant) staffing policy fails to stabilize

he probability of delay performance. To account for productivity,

he manager could refine the dynamic staffing policy in Eq. (41) as

ollows:

prov iders (t) ≈⌈

c(t)

productivity rate

⌉(42)

he caution is that using Eq. (42) leads to over-staffing. Nonethe-

ess, this estimation is simple and can easily be implemented in

he ED (see Fig. 9 ).

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Table 2

A numerical example of c ∗ .

d/w 0.5 1 2 4 8 16 32 64 128 256 512

c ∗ 9.985638 10.49743 11.053 11.75027 12.64001 13.75319 15.07902 16.72294 18.83566 21.50356 24.99898

β 0.223768 0.308644 0.425378 0.583453 0.795557 1.078595 1.456762 1.956292 2.606841 3.450363 4.525576

HW ( β) 0.747118 0.664874 0.563101 0.4 4 4693 0.317135 0.193877 0.09272 0.029945 0.005116 0.0 0 0301 3.15E −06

a b

Fig. 7. Averages of arrivals in the ED at a hospital in upstate New York, December 2012. The fitted sinusoidal function λ( t ), in Fig. 7 b, follows 4 . 20961 − 0 . 351744 sin (9 . 5278 −1 . 04835 t) − 0 . 77649 sin (5 . 16532 − 0 . 637895 t) − 2 . 77031 sin (2 . 00299 − 0 . 24639 t) .

a b

Fig. 8. Staffing policies to meet the probability of delay target α.

Fig. 9. Staffing policies using Eq. (42) .

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a b

Fig. 10. Simulated delay probability versus HW (β) = 2 .

Fig. 11. A graphical proof that c is optimal. �( c ) is solved following Appendices G.2 and G.3 .

7

t

d

e

c

I

c

H

r

i

f

8

t

O

a

v

s

d

m

t

m

u

o

t

i

s

t

p

t

d

s

c

s

.2. Case 2: optimal policy to minimize staffing and delay costs

In this section, we show how to apply Theorem 6.2 to find c ∗

hat minimizes delay and staffing costs. It is difficult to estimate

elay costs d . But, given the data, d may be estimated using the

xpected financial liability of delayed care. In turn, staffing costs w

an easily be estimated using the average hourly wage of providers.

n this example, we arbitrarily choose the ratio d/w to be 8, which

orresponds to the service grade of β ≈ 1.047 and consequently,

W ( β) ≈ 0.2 (see Eq. (6.4) for the estimation of β). For the service

ate, we again use μ = 1 . See Fig. 10 for graphical results.

The dynamic policy is constructed following Eq. (36) . In Fig. 11 ,

t is verified that the chosen c does indeed maximize the objective

unction �( c ).

. Concluding remarks

Long queues in the ED lead to overcrowding, a phenomenon

hat potentially compromises the care and the safety of patients.

ne of the ways to alleviate this problem is through the efficient

llocation of resources to reduce delay.

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

Using new fluid approximations of queues, we introduced the

ariation of the SRS rule and provided an algorithm to construct

tatic and dynamic staffing policies to meet the probability of

elay target. Numerical examples confirmed that our proposed

ethod stabilizes performance. We also showed how optimal con-

rol methods could be utilized to find the number of providers that

inimize both the costs of delay and staffing.

Our model has the potential to be applied in an actual ED. We

sed data of a hospital in upstate New York and estimated vari-

us parameters in our model. We then constructed staffing policies

hat seemed reasonable for this particular ED.

Given new fluid approximations and control methods that we

ntroduced, we have made several mathematically simplifying as-

umptions. Our goal was to first get insights into the feasibility of

he simple model. For example, we have assumed homogeneous

atients and a queueing discipline of first-come, first-served. In ac-

uality, ED patients have differing severity and thus the queueing

iscipline is likely to be priority based. Additionally, we have as-

umed instantaneous triage, one type of resource, and did not ac-

ount for the possibility of retrials. We intend to relax these as-

umptions in the future research.

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B

ρ

G

f

c

P

β

F

t

β

A

A

A

A

G

(

1

H

W

−

−G

t

m

Acknowledgment

We are deeply grateful to the referees of this paper for their

constructive comments and suggestions.

Appendix A. Proof of Theorems 5.1 and 5.3

Using the nullclines in Eqs. (18) and ( 19 ), we derive:

λ

γ P + θ=

α(c − P )

P (A.1)

and:

λ − θq

(μ/α) q =

μc

μ + (μ/α) q (A.2)

By solving for P , in Eq. (A.1) , and then dividing the result by μγ ,

we obtain:

P 2 −(

c − ρ − αθ

μ

)P − αc

θ

μ= 0 (A.3)

P =

{

c − ρ θ = 0

c−ρ−α θμ +

√

( c−ρ−α θμ )

2 + αc 4 θμ

2 θ > 0

(A.4)

By solving for q in Eq. (A.2) and then dividing the result by μγ ,

we obtain:

θ

μq 2 −

(ρ − α

θ

μ− c

)q − ρα = 0 (A.5)

q =

⎧ ⎪ ⎨

⎪ ⎩

αρc−ρ θ = 0

ρ−c−α θμ +

√

( ρ−c−α θμ )

2 + αρ 4 θμ

2 θ/μθ > 0

(A.6)

Appendix B. Proof of stability of the equilibrium points in

Theorems 5.1 and 5.3

To verify that the steady state solutions in Theorems 5.1 and

5.3 are stable, we first derive the Jacobian matrix corresponding to

the dynamical system in Eqs. (16) and ( 17 ).

J =

[−γ · P − θ −γ · q

−γ · P −μ − γ · q

]Since we are interested in the solutions where { q ≥ 0, P ≥ 0}, we

notice that the trace Tr of this Jacobian matrix, −γ · P − μ − γ · q,

is negative. Furthermore, the determinant D is positive since:

D = (−γ · P ) · (−μ − γ · q ) − (−γ · P ) · (−γ · q ) (B.1)

= γ · μ · P + γ · θ · q + θ · μ (B.2)

Then, by Blanchard, Devaney, and Hall (2007 , p. 344), the eigenval-

ues corresponding to this Jacobian matrix are real and both nega-

tive, an indication of a sink and asymptotically stable equilibrium

point (see also Holmes, 2009 , chapter 3).

Appendix C. Limit cycles of the non-stationary fluid model

Fig. C.12

Appendix D. Proof of Corollary 5.4 and β approximation

From Eq. (A.3) , we solve for ρ and obtain:

ρ = c − P +

θ

μα(

c − P

P

)(D.1)

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y substituting α(

c−P P

)by the nullcline in Eq. (19) , we obtain:

= c − P +

θ

μq (D.2)

iven c = ρ + β√

ρ, we use the results from Eq. (D.2) to derive βor the stationary system, as follows:

= c − P +

θ

μq + β

√

c − P +

θ

μq (D.3)

− θ

μq = β

√

c − P +

θ

μq (D.4)

=

P − θμ q √

c − P +

θμ q

(D.5)

or the non-stationary system, β ≈ ˜ β, where ˜ β is the average of

he right-hand side in Eq. (D.5) , obtained by:

˜ =

1

T

∫ T

0

P − θμ q √

c − P +

θμ q

dt (D.6)

ppendix E. Algorithm for determining c ( t )

lgorithm E.1.

1. Initialize all variables and set parameters according to the

known system performance. The probability of delay target

must be set to 0 < α < 1. Initial staffing levels c can be set

to zero.

2. Numerically integrate the dynamical system in Eqs. (16) and

( 17 ) (see algorithm in Appendix G.1 ).

3. Compute β according to Proposition 6.1 .

4. Increment c until Garnett ( β) ≈ α for θ > 0, or HW ( β) ≈ α for

θ = 0 .

5. Construct c ( t ) following Eq. (36) .

ppendix F. Meeting the probability of delay target α

ppendix G. Proof of Theorem 6.2

1. Optimal control problem 1

With the objective function in Eq. (37) coupled with state Eqs.

16) and ( 17 ), we use Pontryagin maximum principle ( Pontryagin,

987 ) to obtain the following Hamiltonian function: H

= −1

2

wc 2 − dq + p 1 ( λ − γ qP − θq ) + p 2 ( μ(c − P ) − γ qP )

(G.1)

e then obtain the following necessary conditions:

∂H

∂ p 1 ≡ •

q = λ − γ qP − θq

∂H

∂ p 2 ≡ •

P = μ(c − P ) − γ qP

∂H

∂q ≡ •

p 1 = (p 1 + p 2 ) γ P + d

∂H

∂P ≡ •

p 2 = (p 1 + p 2 ) γ q + μp 2

iven the initial conditions, q (0) and P (0), and the terminal condi-

ions p 1 (T ) = 0 and p 2 (T ) = 0 , the optimal value of c that maxi-

izes H and �( c ) is obtained by:

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Fig. C.12. The dynamics of the limit cycles when λ(t) = 10 + 5 · sin (t) , μ = 1 , γ ≡ μ/ α. When parameters are not varied, they are set to θ = 1 , α = 0 . 9 , c = 13 .

T

∫ W

i

d

∂H

∂c = −c + μ · p 2 = 0

hen, the optimal mean value c ∗ is obtained by:

c ∗ =

μ

w

p 2

T

c ∗ dt =

μ

w

∫ T

p 2 dt

0 0

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c ∗T =

μ

w

∫ T

0

p 2 dt

c ∗ =

μ

wT

∫ T

0

p 2 dt

e prove the sufficient conditions of our optimal results by show-

ng that the Hamiltonian function in Eq. (G.1) is negative semi-

efinite. We start by deriving the Hessian matrix A and the

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Fig. F.13. Simulated probability versus the target when λ(t) = 10 + 5 sin (t) , θ / μ varied, and P(0) = S(0) = 0 .

|

|

G

s

b

T

corresponding determinant | A | as follows:

| A | =

∣∣∣∣∣H cc H cq H cP

H qc H qq H qP

H Pc H Pq H PP

∣∣∣∣∣=

∣∣∣∣∣−1 0 0 P 0 0 −γ (p 1 + p 2 ) P 0 −γ (p 1 + p 2 ) 0 P

∣∣∣∣∣ = 0

The leading minors are given by:

| A 1 | = −1

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

A 2 | =

∣∣∣∣−1 0

0 0

∣∣∣∣ = 0

A 3 | =

∣∣∣∣∣−1 0 0

0 0 −γ (p 1 + p 2 ) 0 −γ (p 1 + p 2 ) 0

∣∣∣∣∣ = 0

iven | A 1 | ≤ 0, | A 2 | ≥ 0, | A 3 | ≤ 0, we conclude that H is negative

emi-definite, thus concave in ( c ∗, S ∗, P ∗). But, the solution may not

e unique since H is not strictly concave in these variables (see

heorem 3.1 in Caputo, 2005 ).

d control of queues in emergency departments, European Journal

13

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ARTICLE IN PRESS

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m

0

b

H

w

t

�

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a

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s

x

A

S

1

−

−

F

c

G

s

a

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o

A

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t

a

c

m

a

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A

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2. Optimal control problem 2

If the control problem is formulated such that: �(c) =ax c −

∫ T 0

1 2 c

2 dt and an isoperimetric constraint of ∫ T

0 d w

q − cP dt = is added, the corresponding Hamiltonian function is obtained

y:

= −1

2

c 2 + p 1 ( λ − γ qP − θq )

+ p 2 ( μ(c − P ) − γ qP ) − x

(d

w

q − cP

)(G.2)

here x is a penalty multiplier of some auxiliary variable � such

hat:

= −∫ T

0

d

w

q − cP dt = 0 (G.3)

• = − d

w

q − cP (G.4)

nd �(T ) = 0 . Since � does not appear in the Hamiltonian

q. (G.1) , then

•x = −∂ H/∂ � = 0 , meaning that x is a constant that

atisfies the following complementary of slackness equation:

·[

0 −∫ T

0

d

w

q − cP d t d t

]= 0 (G.5)

ccordingly, x = 0 when

∫ T 0

d w

q − cP dt � = 0 , else x > 0. As in

ection G.1 , we use Pontryagin’s maximum principle ( Pontryagin,

987 ) to obtain the following necessary conditions:

∂H

∂ p 1 ≡ •

q = λ − γ qP − θq

∂H

∂ p 2 ≡ •

P = μ(c − P ) − γ qP

∂H

∂q ≡ •

p 1 = (p 1 + p 2 ) γ P +

d

w

x

∂H

∂P ≡ •

p 2 = (p 1 + p 2 ) γ q + μp 2 − xc

inally, the mean value c ∗ is approximated by:

∗ ≈ 1

T

∫ T

0

μp 2 + xP dt (G.6)

3. Numerical integration algorithm

We use the 4th order Runge Kutta method to derive numerical

olutions of c ∗. Our algorithm is related to the Forward–Backward

lgorithm in Lenhart and Workman (2007) except for Step 3, which

elates to the computation of the complementary of slackness in

ur model.

lgorithm G.1 (Modified Forward–Backward algorithm) .

Step 0: Set initial conditions such that q (0) = q 0 and P (0) ={ P 0 | 0 ≤ P 0 ≤ c} . Also, set terminal conditions such that

p 1 (T ) = p 2 (T ) = 0 , for all t , 0 ≤ t ≤ T . Initialize the multi-

plier penalty x = 0 , and the constant control policy c = 1 .

The initial number of iterations n is set to 1.

Step 1: Given { q n −1 (t) , P n −1 (t) | 0 ≤ t ≤ T } , solve backward in

time the dynamical system { •p 1 (t) = − ∂H

∂q ( p 1 n , p 2 n , q n −1 ,

P n −1 )(t) , •p 2 (t) = − ∂H

∂P ( p 1 n , p 2 n , q n −1 , P n −1 )(t) } , starting

with the terminal conditions { p 1 n (T ) = p 2 n (T ) = 0 } , for

all 0 ≤ t ≤ T .

Step 2: Using values of { p 1 n ( t ), p 2 n ( t )|0 ≤ t ≤ T }, solve for-

ward in time the dynamical system { •q (t) =

∂H

∂ p ( p 1 n (t) ,

1

Please cite this article as: J. Niyirora, J. Zhuang, Fluid approximations an

of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.03.0

p 2 n (t) , P n −1 (t)) , •P (t) =

∂H

∂ p 2 ( p 1 n (t) , p 2 n (t) , q n −1 (t)) } , start-

ing with the initial conditions { q n (0) = q 0 , P n (0) = { P 0 | 0 ≤P 0 ≤ c} , for all 0 ≤ t ≤ T .

Step 3: Determine the control policy c n by Eq. (G.6) such that:

c n =

1

T

∫ T

0

μ · p 2 n + x n P n dt

If ∫ T

0 d w

q n − c n P n dt � = 0 :

(a) n + = 1

(b) x n + = εHere ε is a very small number that increments the penalty

x .

Step 4: Repeat Steps 1–3 until the following convergence condi-

tions are met: ∫ T

0

d

w

q n − c n P n dt ≈ 0 and

‖

−→

c ‖ n − ‖

−→

c ‖ n −1

‖

−→

c ‖ n

≤ τ,

where τ is the accepted convergence tolerance.

Step 5: After determining c ∗, construct c ( t ) according to Eq. (36) .

emark G.2. The Forward–Backward algorithm works well for op-

imal control problems where the state is fixed at the initial time

nd free at the terminal time ( Lenhart & Workman, 2007 ). The

ontrol problem we consider in this paper fits this class of opti-

al control problems. For further discussion of Forward–Backward

lgorithms see Lv, Tao, and Wu (2016) , McAsey, Mou, and Han

2012) , Tseng (20 0 0) .

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