Jaén, Sebas ián
A icle
The dec ease o ED pa ien boa ding by implemen ing a
s ock managemen policy in hospi al admissions
Ope a ions Resea ch Pe spec i es
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Sugges ed Ci a ion: Jaén, Sebas ián (2024) : The dec ease o ED pa ien boa ding by implemen ing
a s ock managemen policy in hospi al admissions, Ope a ions Resea ch Pe spec i es, ISSN
2214-7160, Else ie , Ams e dam, Vol. 12, pp. 1-11,
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The dec ease o ED pa ien boa ding by implemen ing a s ock managemen
policy in hospi al admissions
Sebas ián Jaén
ALIADO Analy ics and Resea ch o Decision Making, Depa men o Indus ial Enginee ing, Uni e sidad de An ioquia, Calle 67 No.
53-108, Medellín 050010, Colombia
ARTICLE INFO
Keywo ds:
Inpa ien boa ding ime
Hospi al o e c owding
Bed capaci y planning
Inpa ien low
Sys em dynamics
Ope a ions esea ch
ABSTRACT
The p esence o conges ion is a common scena io in e ia y-le el hospi als wo ldwide. Cu en esea ch
sugges s ha an inc ease in hospi al bed capaci y is no a long- e m solu ion gi en ha pa ien demand adap s
o added capaci y. Recen li e a u e sugges s he need o he implemen a ion o a policy o in e -hospi al
ans e s o di e pa ien s o ou pa ien p io i y se ices o home ca e. This policy has p o en o be e ec i e
in educing ED boa ding wi hou comp omising pa ien sa e y. Howe e , de e mining he equi ed numbe o
pa ien s o be admi ed is key. The dynamic na u e o hospi al bed a ailabili y and discha ges equi es an
admission p ocess able o be in synch ony wi h hose a ia ions. A misma ch be ween pa ien demand and
hospi al admissions will esul in ei he ED boa ding o idle capaci y. The pu pose o his pape is o in oduce a
me hodology o suppo he p ocess o hospi al admissions by p o iding as an inpu a h eshold o he numbe
o pa ien s o be admi ed. The me hodology is es ed using a sys em dynamics model ha eplica es one yea
o ope a ions o a e ia y-le el hospi al. The simula ions e eal he po en ial o he me hodology o dec ease
he ED inpa ien boa ding a e as well as ED and hospi al leng h o s ay.
1. In oduc ion
The las wo decades ha e wi nessed an inc easing wo ldwide de-
mand o a en ion in eme gency depa men s (EDs) and hospi als [1].
As a consequence, o e c owding is a p oblem ha a ec s e e y ED
and hospi al ega dless o hei public o p i a e na u e [2]. The mos
p essing e ec o his p oblem is he inc ease in he numbe o boa ding
pa ien s in he ED. Tha is he numbe o pa ien s who canno be mo ed
o inpa ien uni s due o a lack o inpa ien bed a ailabili y in he
hospi al wa ds [3].
The inc easing numbe o boa ding pa ien s in he ED is conside ed
one o he mos impo an challenges ha hospi als ace [4]. In some
EDs in he U.S., 22% o he o al ED pa ien census a e boa ding
pa ien s, whe e 20% o hem boa ded o a leas eigh hou s o e en
days [5]. The pa ien boa ding ime is c ucial due o i s co ela ion
wi h inc eased ED leng h o s ay (LoS), mo ali y a es, nosocomial
in ec ions, and alls [4,6]. Addi ionally, i se es as he mos common
igge o ambulance di e sion [3]. No only does he ED boa ding
ime impac o e all hospi al pe o mance, bu i is also connec ed o
a ious socie al ac o s such as pa ien ca e sa e y, e iciency, esou ce
u iliza ion, heal hca e cos s, pa ien expe ience and sa is ac ion, public
heal h, communi y impac , and he quali y o ca e and ou comes [1,4,
7].
E-mail add ess: [email p o ec ed].
The ED boa ding p oblem has been associa ed wi h a ious ac o s
such as ‘‘unnecessa y ED isi s’’, delays in lab and adiology esul s,
inadequa e os e ing o en a ibu ed o unde s a ing, and he lack o
a ailabili y o des ina ion wa d beds [3,4,7]. Consequen ly, Boudi e al.
[4] sugges ha li e a u e on ED has expe ienced exponen ial g ow h
in ecen yea s, add essing his phenomenon om a ious disciplines,
including Ope a ional Resea ch/Managemen Science (OR/MS).
The pu pose o his esea ch is o add ess he ED boa ding p oblem
by conside ing a sys em dynamics-based app oach known as s ock
managemen policy. This app oach e e s o he s a egy used o con ol
and egula e he le els o s ocks o accumula ions wi hin a sys em [8],
in his case, ED boa de s. S ocks, in sys em dynamics SD, ep esen
accumula ions o en i ies ha change o e ime due o in lows and
ou lows. By elici ing he complexi y o he sys em unde s udy ( he
hospi al), his app oach has he p ima y goal o main aining op i-
mal le els o pa ien s o mee sys em equi emen s, a oid o e lows,
and manage he lows in o and ou o hese s ocks e ec i ely [8].
Ou esea ch p esen s indings om simula ing he applica ion o a
s ock managemen policy ha mi o s he ope a ional ai s o an
ac ual e ia y hospi al. The ou comes indica e p omising ad an ages
o his in e en ion, emphasizing he impo ance o c a ing p ac ical
in e en ions based on hese insigh s.
h ps://doi.o g/10.1016/j.o p.2024.100298
Recei ed 30 May 2023; Recei ed in e ised o m 6 Feb ua y 2024; Accep ed 7 Feb ua y 2024
Ope a ions Resea ch Pe spec i es 12 (2024) 100298
2
S. Jaén
This s udy p oposes a no el app oach o an exis ing body o li -
e a u e whose con ibu ions o his p oblem ange om ED-based
solu ions o hospi al-wide ini ia i es [2]. They consis in imp o emen s
in medical p ocedu es, ED layou s and mo e nu se aining [2,9]. O he
solu ions ha e as s a egy he inc ease o capaci y in ooms, beds, and
s a [10]. The inc ease in he low o inpa ien s a he downs eam
le el is ano he al e na i e ha is achie ed by pe o ming one o
he combina ion o he ollowing s a egies: imp o ing he classi i-
ca ion o inpa ien s [3,11], be e bed managemen [12], inc easing
he p edic ion o demand and he scheduling o discha ges [13], and
inally, he addi ion o special uni s such as acu e medical and su gical
uni s [10]. All he a o emen ioned app oaches ha e an immedia e im-
pac on educing he numbe o ED boa de s. Howe e , hese ini ia i es,
as well as hose p omo ing he inc ease o he low o inpa ien s a he
downs eam le el, p o ide sho - e m imp o emen s since he demand
o beds adap s o added capaci y [14,15]. Solu ions p e en ing o
di e ing he demand o ED o acu e hospi al se ices seem o ha e
long-las ing impac s [14,15].
Despi e he e is no a la ge body o OR/MS, di ec ly add essing he
ED boa ding p oblem, he e is a consis en body o esea ch modeling
he low o pa ien s in heal h ca e acili ies [3]. S udies sugges ha ED
boa ding is due in la ge pa because o he heal hca e p o ide s’ inca-
paci y o main aining pa ien low. High ED occupancy and p olonged
ED LoS a e linked o delays in discha ges and low se ice a es [3,16].
The e o e, OR/MS echniques ha e signi ican ly con ibu ed o imp o -
ing he ED pe o mance in se e al me ics such as ED LoS [3]. The mos
used me hodologies o modeling pa ien low a e Queuing Theo y,
Ma ko models, decision p ocesses, and simula ion [3,17].
The use o queuing heo y models is among he mos basic and
simplis ic app oaches o modeling he pa ien low [18]. Coch an
and Roche [19,20,21], p esen se e al examples in ended o inc ease
he capaci y o he ED o ea pa ien s. Rod íguez Jáu egui e al.
[22], ollow he Coch an and Roche app oach e alua ing capaci y and
designing s a ing policies in an ED. A di e en app oach is in oduced
by Zonde land e al. [23,24]. Thei wo k in oduces he use o queuing
heo y as a ool o e alua ing ED capaci y and in es iga ing he bes
possible s a egy o achie ing imp o emen s in e iciency. Ma ko
models a e a esou ce o ED and pa ien low modeling because heal h
ca e p ocesses can be ep esen ed by a numbe o in e dependen
wo k-s ages [3,17].
The g owing use o simula ion in heal h ca e e e s o he imple-
men a ion o ei he one, o he combina ion o se e al o he ollowing
echniques: Mon e Ca lo, disc e e e en simula ion (DES), SD and agen
based simula ion (ABS) [17,25–29]. Typically, he Mon e Ca lo simu-
la ion has been used as a me hod o p obabilis ic sensi i i y analysis
o analy ic o mula ions (e.g. a Ma ko Model). In he ew s udies
epo ed, i is mos ly used as a subsidia y echnique [28]. DES, on he
o he hand, is he mos commonly employed and widesp ead modeling
me hodology o suppo ing heal h ca e decisions o be aken a he
ope a ional le el [28,30]. This me hodology is ega ded as mo e lex-
ible and e sa ile abo e o he modeling me hods due o he ollowing
ea u es: DES models a e ee om bo h a i al assump ions (Poisson o
no ) and om ime se ice assump ions (exponen ial o no ) [3]. ABS
has been gaining popula i y among esea che s, su passing SD and e en
eplacing DES as a ool o ealis ic heal h ca e modeling [28,31]. ABS
sha es all o he ad an ages o using DES in e ms o elici ing sys ems
complexi y. I is claimed ha ABS models can also add ess ac ic and
s a egic le el p oblems ou pe o ming he DES each [31]. Howe e ,
he main indings o ABS models ocus on ope a i e changes o im-
p o ing pa ien h oughpu ime and he o he c i ical pe o mance
measu es [32].
The imp o emen o inpa ien low has been specially deal in
he sys em dynamics (SD) li e a u e [33]. The use o SD is especially
sui able o add essing he ED p oblem gi en i s e sa ili y ha allows
a mo e agg ega ed app oach o sys ems complexi y, and in many cases,
he modeling p ocess is mo e e icien han DES and ABS [34]. Also, SD
ope a es unde he pa adigm o elici ing eali y h ough he in e ac ion
o lows and s ock, which is pa icula ly sui ed o modeling pa ien
lows [35,36]. As a seminal esea ch, he wo k by Lane and Husemann
[37], ocuses on add essing he low o acu e pa ien s h ough he UK
heal h ca e sys em. Thei con ibu ion acknowledges he pe inence
o he SD me hodology o mapping he low o pa ien s in a way
ha he clinicians can bo h be e unde s and and become an ac i e
pa ne in he p ocess o policy making. Simila applica ions o sys em
mapping and policy design can be ound in wo ks by Wong e al.
[38], Vande by e al. [35], Pa ick e al. [39] and Home e al. [40].
Addi ionally, Esensoy and Ca e [41], use SD modeling o acili a e a
be e unde s anding o he sys em-wide e ec s o pa ien low ela ed
in e en ions.
Some SD li e a u e is ocused on s udying he low o pa ien s
in heal h ca e acili ies beyond a quali a i e app oach. Fo exam-
ple, B ails o d e al. [42], use SD o simula e pa ien lows and o
iden i y sys em bo le-necks in eme gency and on-demand heal h ca e
cen e s. Fu he mo e, Vande by and Ca e [36], alida e he use o
SD o modeling hospi al pa ien low. La e , Demi e al. [43], use
SD o modeling he LoS o a neona al uni , pa icula ly ocusing on
pa ien low o de e mine he majo d i e s o he sys em. Esensoy and
Ca e [44], implemen a whole-sys em, s a egic pe spec i e, designed
o e alua e he di ec ion and magni ude o pa ien low. A ecen
con ibu ion by G ida and Zeid [45] e alua ed he impac o hospi al
esou ces on inpa ien low and LoS using a SD app oach wi hin he
amewo k o he Theo y o Cons ain s.
The SD con ibu ions ha mo e closely add ess he inpa ien boa d-
ing p oblem a e hose by Wong e al. [38,46], Rashwan e al. [15],
and Mahmoudian-Dehko di and Sada [14]. The simula ions pe o med
by Wong e al. [46], showed ha by dis ibu ing discha ges o e
he cou se o a week, a dec ease in he numbe o ED boa de s is
achie ed. This wo k emphasizes he ole o he downs eam low on ED
boa ding. The second wo k by Wong e al. [38], sugges s he use o an
admission con ol policy as a use ul s a egy o alle ia e downs eam
occupancy a es and boa ding imes. Rashwan e al. [15], p opose a
SD me hodology o e alua e a compa ison be ween s ock (capaci y)
and low (demand) in e en ions o ease he bed blocking p oblem.
Thei conclusion sugges ha he impac o s ock in e en ions inde-
penden ly, such as inc easing pos -acu e capaci y, is ime-limi ed, while
he educ ion o in low has mo e las ing e ec s.
The main indings o SD pa ien - low li e a u e has e ol ed in wo
dis inc i e app oaches. The i s app oach ema ks he alue o he
down-s eam capaci y as a necessa y condi ion o imp o ing inpa ien
h oughpu [38,45–48]. The second app oach, a gues ha e en i he
sys em eaches he desi ed le el o capaci y, i gene a es a coun e -
in ui i e esul . Ini ially, he sys em shows a sho e m imp o emen
by educing p essu e in he se ices. Howe e , he consequence in he
long un is an inc ease in demand ha coun e ac s he impac o he
policy o e ime [15]. Thus, long las ing solu ions seem o be placed a
he up-s eam le el whe e pa ien demand and admissions a e he key
d i e s [14,15,38].
Se e al wo ks iden i y he pa ien admission p ocess as he main
d i e in con olling o e c owding [49–51]. Thus, some o he las
decade li e a u e has ocused speci ically on educing hospi al admis-
sions om he ED a guing h ee main easons: Fi s , mos o he a i als
o pa ien s occu because hey do no ha e access o hei p ima y ca e
doc o o specialis [52]. Second, some pa ien s a i e because hey,
and hei heal h ca e p o ide s, ha e he pe cep ion ha a comp e-
hensi e wo kup occu s mo e quickly when a pa ien is admi ed o
hospi al [52]. Thi d, he e is an inc easing e idence ha he numbe
o unnecessa y hospi aliza ions ange om 5%–82% wo ldwide [49,53,
54]. As example o he implemen a ion o such app oach is p o ided
by he Sunnyb ook Heal h Sciences Cen e hospi al in To on o [52].
By implemen ing a pa ien di e ing p o ocol called ‘‘ apid e e al
clinic’’ (RRC), which allows ED physicians o e e pa ien s o a gene al
Ope a ions Resea ch Pe spec i es 12 (2024) 100298
3
S. Jaén
in e nis o u gen ou pa ien assessmen , he hospi al educed admis-
sions, imp o ed ED boa ding imes, and inc eased pa ien low a he
downs eam le el. In consequence, wo ype o s a egies p edomina e
in he e o o educing admissions: p e en ion and in e -hospi al
ans e s o apid e e al p o ocols [52,55].
The main pu pose o he p esen wo k is o in oduce a me hodology
o suppo he in e -hospi al ans e p ocess s a egy. This is because
heal h ca e p o ide s need o conside when and how many pa ien s
need o be admi ed and ans e ed o his s a egy o s a o ake
place and be e ec i e. As an in-and-ou low policy, he a es o pa ien s
o be admi ed and ans e ed need o be calib a ed in acco dance
wi h a ailable capaci y and expec ed inpa ien discha ges, which a y
o e ime. Inadequa e a es o ans e s ( oo low o high) migh no
p oduce he desi ed e ec on he numbe o boa ding inpa ien s, o
hey will gene a e unnecessa y ans e s and idle capaci y. The p o-
posed me hodology does no in end o cons ain clinical c i e ia o
pa ien admission bu p o ides guidelines o manage inpa ien low and
o e c owding in acco dance wi h he hospi al’s pa ien - low dynamics.
Addi ionally, he me hodology inco po a es le els o policy adhe ence
ha p o ide lexibili y o he p oposed me hodology o be applied in
hose cases whe e medical c i e ia p io i ize pa ien admission o e ED
o e c owding.
Thus, his pape implemen s a SD model-me hodology which ad-
d esses se e al needs ound in he li e a u e: Sugges ac ic and s a e-
gic changes o suppo he admission p ocess [36]. Con ibu e o he
limi ed amoun o wo ks add essing he boa ding ime p oblem in he
ED. Finally, he wo k add esses he need o a combina ion o s ock
(capaci y) and low policies (admissions, in e -hospi al ans e s, and
discha ges) ha p e en coun e -in ui i e esponses o he sys em due
o adap a ions o new capaci y [15]. The modeling is es ed using he
inpu s o da a om a la ge hi d-le el. hospi al Thi d-le el hospi als,
also known as e ia y ca e hospi als, a e heal hca e acili ies ha o e
specialized and ad anced medical ca e beyond wha is p o ided in
p ima y and seconda y heal hca e se ings.
The selec ion o a e ia y hospi al o simula e he s ock manage-
men app oach conside s he unique cha ac e is ics o his ins i u ion
ype, gi en i s specialized ca e ha o en leads o inc eased demand,
ex ended leng hs o s ay, and ins ances o inpa ien boa ding. Addi-
ionally, i is wi hin hese ins i u ions ha he nega i e implica ions
o inpa ien boa ding a e mos p onounced [4,7].
2. Me hods and me hodology
2.1. Hospi al inpa ien low: p oblem desc ip ion
The pa ien boa ding p oblem a ises when ED doc o s decide ha a
gi en pa ien should be hospi alized and i s uni bed is no a ailable.
Then, he pa ien has o wai in he ED un il a bed is a ailable [3]. To
illus a e he p oblem, in Fig. 1, he e is an agg ega ed and simpli ied
ep esen a ion o he low o pa ien s in a hypo he ical hospi al. The
image depic s he mos common s eps aken upon a i al o almos all
pa ien s equi ing u gen ca e. Howe e , he e a e addi ional pa hs o
a i al o he hospi al, including e e als, ans e s om o he heal h-
ca e acili ies, and scheduled non-ambula o y su ge ies ha necessi a e
se e al days o supe ised eco e y. These pa hways a e no included in
Fig. 1 because pa ien s ollowing hese ou es ypically do no equi e
boa ding in he ED due o he scheduled na u e o hei a i al, which
accoun s o he a ailabili y o beds.
A e he a i al o he hospi al, he pa ien s wai in a wai ing a ea,
o p e- iage a ea which is a designa ed sec ion wi hin a hospi al’s ED
whe e pa ien s ini ially p esen hemsel es be o e o mal iage akes
place. This a ea se es as an ini ial poin o con ac o indi iduals
seeking u gen medical a en ion. The a ea is usually si ua ed nea he
en ance o wi hin he ED. In mos o he cases, pa ien s a e egis e ed
in his s age be o e being examined.
A e p e- iage and egis a ion, he pa ien aces an assessmen
( iage) pe o med by a physician which de e mines he pa hway o
a en ion. One pa h is o be discha ged a e ea men , he o he , is
o emain in he ED o ea men ha inco po a es he p esc ip ion
and a i al o diagnos ic aids. The nex s age de e mines i he pa ien
should be discha ged, admi ed o hospi aliza ion o ans e ed o
a di e en hospi al. The inpa ien s whose beds a e no a ailable a e
ep esen ed as queues o inpa ien s, and hey cons i u e he boa ding
inpa ien s. To be in a gi en queue depends on he wa d o des ina ion
in acco dance wi h he inpa ien pa h o ea men . Hospi aliza ion is
he nex s age. In Fig. 1, hospi aliza ion is ep esen ed as le els o
inpa ien s being ea ed in hei espec i e wa d. Wa ds in a hospi al
e e o speci ic uni s o sec ions wi hin he heal hca e acili y ha
a e dedica ed o p o iding ca e o pa icula ypes o pa ien s o
medical condi ions. The wa ds depic ed in Fig. 1, a e ep esen ing he
mos common wa ds ound in a hi d-le el hospi al: medical, su gical,
in ensi e ca e, pedia ic, ma e ni y, psychia ic, ge ia ic, and isola ed.
He e, inpa ien s ecei e he main ea men whose du a ion depends on
he wa d LoS. In acco dance wi h he ea men ou come, he inpa ien
can be ans e ed o a di e en wa d o discha ged.
2.2. Dynamic hypo hesis
The dynamic hypo hesis p esen ed he e sugges s ha he p ocess o
pa ien admissions does no conside a ailable hospi al bed capaci y,
o a leas , i is no one o he main d i e s o de e mine admissions.
Due o his lack o in o ma ion, a p oblema ic beha io a ises when
admissions do no ma ch a ailable bed capaci y and inpa ien s a e
equi ed o boa d in he ED (Fig. 2).
Unde his scena io, depic ed by Fig. 2, only when he a es o
Inpa ien s discha ge a e g ea e han he a es o Admi ed pa ien s, he
s ock o Boa ding inpa ien s is dec eased. When his si ua ion does
no occu , he inc ease in he s ock o Boa ding inpa ien s is expec ed
because he e exis s a di e ence be ween he lows o incoming and
ou going pa ien s.
2.3. Model equa ions
The p esen modeling cap u es he a o emen ioned dynamic hy-
po hesis in he con ex o a e ia y hospi al. The choice o such
ins i u ions impac s he modeling p ocess in de e mining he necessa y
le el o agg ega ion equi ed o elucida e he dynamic in e ac ions
among he wa ds and he ED. Consequen ly, SD eme ges as a sui able
and necessa y me hodology o modeling complex sys ems like e ia y
hospi als [35]. The modeling elici s he hospi al using he SD s ocks and
lows ep esen a ion (Fig. 3). Double lines indica e ha he a iable is
an a ay, whose dimension depends upon he numbe o 𝑈uni s.
The model elici s all o he s a es in which he inpa ien can be in he
hospi al: Eme gency depa men pa ien s (𝐸), Boa ding a ea pa ien s
(𝐵) and Hospi alized pa ien s (𝐻). The hospi aliza ion se ice 𝐻, is
di ided in o 𝑈uni s, and hus, he boa ding a ea has 𝑈s ocks o
inpa ien s wai ing o a bed. The boa ding a ea is no mally loca ed
inside he ED. Howe e , o simpli y he modeling, he la ge s ock o
ED is di ided in o wo s ocks: he s ock 𝐸con aining he numbe o
pa ien s admi ed, e ised, a ended and o be discha ged, and he
s ock 𝐵, which accoun s o all o he pa ien s wai ing o beds o
be assigned in he hospi aliza ion uni s. The le el 𝐸has one in low
ep esen ing pa ien a i als (𝐴), and h ee ou lows, which depic he
discha ge o pa ien s om he ED (𝑑), ans e s o di e en heal hca e
acili ies (𝑇), and admissions (𝑎), espec i ely. Admissions (𝑎), e e o
pa ien s necessi a ing hospi al s ays in he wa d due o hei medical
condi ions. The boa ding le el 𝐵, has an ou low (𝑡) o in a-hospi al
ans e s, indica ing he numbe o pa ien s mo ed om he ED o he
hospi al wa ds. A he hospi al 𝐻le el, he e a e wo se s o in lows
and ou lows o pa ien s. The i s in low co esponds o he in lux
o pa ien s ans e ed om he ED, p e iously deno ed as ou low 𝑡.
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Fig. 1. Pa ien low om he ED o hospi aliza ion.
Fig. 2. E ec o inpa ien discha ge on boa ding inpa ien .
Fig. 3. S ocks and lows diag am o he hospi al.
The second in low (𝑔) ep esen s he numbe o pa ien s a i ing om
di e en wa d uni s. Rega ding he ou lows, he i s one (𝑗) ep esen s
he numbe o pa ien s o be ans e ed o a di e en wa d uni , while
he second ou low (𝐷) co esponds o he numbe o pa ien s o be
discha ged om hei wa d. Flows 𝑎,𝑡,𝑔,𝑗and 𝐷, a e he espec i e
sum o 𝑎𝑖,𝑡𝑖,𝑔𝑖,𝑗𝑖and 𝐷𝑖 lows, whe e 𝑖= 1,…, 𝑈 uni s.
𝐸(𝑡)is a s ock o pa ien s in he ED, whe e a i als 𝐴(𝑡), discha ges
𝑑(𝑡), admissions (Admi ed pa ien s)𝑎(𝑡), and ans e s 𝑇(𝑡)a e i s lows
(Eq. (1)).
𝐸(𝑡) = ∫𝑡
𝑡0[𝐴(𝑡)−(𝑑(𝑡) + 𝑎(𝑡) + 𝑇(𝑡))]𝑑𝑡 +𝐸(0) 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (1)
The in low A i als 𝐴(𝑡)(Eq. (2)), is modeled as a GRAPH unc ion
e ie ing a gi en numbe o pa ien s pe hou . This unc ion p o-
ides ime- a ying exogenous inpu s e ie ing he numbe o pa ien s
a i ing o he ED pe hou .
𝐴(𝑡) = 𝐺𝑅𝐴𝑃 𝐻(𝑡, 𝑡𝑜, 𝛥𝑡, {𝑎𝑡0, 𝑎2,…, 𝑎𝑡})𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠
ℎ𝑜𝑢𝑟 (2)
The ou low discha ges 𝑑(𝑡), is modeled by Eq. (3), whe e 𝛽1is he
p opo ion o ED pa ien s o be discha ged, and 𝜏2is he a e age ED
LoS. Pa ien s discha ged in 𝑑(𝑡)a e hose no equi ing hospi aliza ion.
𝑑(𝑡) = 𝐸(𝑡)𝛽1
𝜏2
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (3)
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S. Jaén
The ou low o Admi ed pa ien s 𝑎(𝑡), (Eq. (4)), is modeled consid-
e ing 𝛽2as he p opo ion o ED pa ien s demanding beds in he 𝑈
uni s. The speci ic numbe o pa ien s demanding a bed in an uni 𝑖,
𝑎𝑖(𝑡), co esponds o Eq. (5), whe e he cons an 𝜌𝑖is he p opo ion o
pa ien s demanding a bed in a speci ic uni 𝑖.
𝑎(𝑡) = 𝐸(𝑡)𝛽2
𝜏2
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (4)
𝑎𝑖(𝑡) = 𝑎(𝑡)𝜌𝑖∨𝑖= 1...𝑈 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (5)
1 =
𝑈
∑
𝑖=1
𝜌𝑖(𝑡)𝐴𝑑𝑖𝑚𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 (6)
The hi d ou low 𝑇(𝑡)(Eq. (7)), co esponds o he numbe o pa-
ien s o be ans e ed o a di e en heal hca e acili y. These numbe
o ans e s a e no pa o a s a egy o educe boa ding pa ien s. They
ep esen he numbe o pa ien s o ha e o be ans e ed o a di e en
heal h ca e acili ies because he hospi al canno mee he pa ien needs
(i.e. echnology, le el o ca e, e c.).
𝑇(𝑡) = 𝐸(𝑡)𝛽3
𝜏2
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (7)
1 = 𝛽1+𝛽2+𝛽3𝐴𝑑𝑖𝑚𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 (8)
The s ock o boa ding inpa ien s 𝐵(𝑡)(Eq. (9)), co esponds o he
sum o he 𝑈s ocks (𝐵𝑖(𝑡)Eq. (10)), whose lows co espond o he
in low o Admi ed pa ien s 𝑎𝑖(𝑡), and he ou low o In a-hospi al ans e
o pa ien s 𝑡𝑖(𝑡)(Eq. (11)). 𝐻∗
𝑖is he capaci y in beds, and 𝐻𝑖(𝑡), he
numbe o occupied beds. The cons an 𝜏3co esponds o he a e age
In a-hospi al ans e ime om he boa ding a ea in he ED o he uni s.
𝐵(𝑡) =
𝑈
∑
𝑖=1
𝐵𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (9)
𝐵𝑖(𝑡) = ∫𝑡
𝑡𝑜
(𝑎𝑖(𝑡) − 𝑡𝑖(𝑡))𝑑𝑡 +𝐵𝑖(0) 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (10)
𝑡𝑖(𝑡) = 𝑀𝐼𝑁(𝐵𝑖(𝑡), 𝐻∗
𝑖(𝑡) − 𝐻𝑖(𝑡))
𝜏3
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (11)
The o al s ock o hospi alized pa ien s 𝐻(𝑡), is de ined by he sum o
he 𝑈s ocks 𝐻𝑖(𝑡)(Eq. (12)). 𝐻𝑖(𝑡)depends on he in lows 𝑡𝑖(𝑡)and 𝑔𝑖(𝑡),
modeling ans e s om he boa ding a ea and a hospi al uni 𝑗, o uni
𝑖. Also 𝐻𝑖(𝑡)depends on he ou lows 𝑗𝑖(𝑡)and 𝐷𝑖(𝑡), which de e mine
he numbe o ans e s o a di e en uni and discha ges om he uni
𝑖(Eq. (13)).
𝐻(𝑡) =
𝑈
∑
𝑖=1
𝐻𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (12)
𝐻𝑖(𝑡) = ∫𝑡
𝑡𝑜[𝑡𝑖(𝑡) + 𝑔𝑖(𝑡) − 𝐷𝑖(𝑡) − 𝑗𝑖(𝑡)]𝑑𝑡 +𝐻𝑖(0) 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡𝑠 (13)
The numbe o inpa ien s o be discha ged om he uni 𝑖,𝐷𝑖(𝑡)
is modeled by Eq. (14). Thus, 𝛽4𝑖co esponds o he p opo ion o
inpa ien s o be discha ged om he uni 𝑖, while 𝜏4𝑖, ep esen s he
uni 𝑖LoS. 𝑗𝑖(𝑡)is modeled by Eq. (15), whe e 𝛽5𝑖, is he p opo ion o
inpa ien s o be ans e ed om uni 𝑖 o an uni 𝑗. Finally, he low
o ans e ed inpa ien s be ween uni s 𝑔𝑖(𝑡), is de e mined by Eq. (16).
This is de e mined by he p oduc o he ou low 𝑗𝑖(𝑡)and he cons an
𝛬𝑖𝑗 . This cons an is a ma ix whe e each ow con ains he pe cen age
o inpa ien s o be ans e ed om he uni 𝑖 o he uni 𝑗. Thus,
∑𝑈
𝑗=1 𝛬𝑖𝑗 = 1, whe e 𝛬𝑖𝑗 = 0 when 𝑖=𝑗, meaning no inpa ien can
be ans e ed o he same uni o o igin.
𝐷𝑖(𝑡) = 𝐻𝑖(𝑡)𝛽4𝑖
𝜏4𝑖
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (14)
𝑗𝑖(𝑡) = 𝐻𝑖(𝑡)𝛽5𝑖
𝜏4𝑖
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (15)
𝑔𝑖(𝑡) =
𝑈
∑
𝑗=1
𝑗𝑖(𝑡)𝛬𝑖𝑗
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (16)
Table 1
Es ima ed pa ame e s and ini ial alues o le els.
Pa ame e Name Value
𝜌𝑖
A e age pe cen age o pa ien s in he
ED demanding a bed in he uni 𝑖0.04
A( ) A i als o pa ien s o ED pe hou (A e g.) 6.17
𝛽1
A e age % o pa ien s o be discha ged
om ED 36
𝛽2
A e age % o pa ien s equi ing
hospi aliza ion 63
𝛽3
A e age % o pa ien s o be ans e ed o a 1
di e en hospi al
𝜏2A e age ED LoS in hou 8.14
𝜏3A e age ans e ime om 𝐵(𝑡) o 𝐻(𝑡)in hou s 1
𝜏4𝑖A e age 𝐻𝑖(𝑡)LoS in hou s 118
𝛽4𝑖
A e age % o inpa ien s o be 91
discha ged om uni 𝑖
𝛽5𝑖
A e age o % inpa ien s o be 9
ans e ed om uni 𝑖
𝛬𝑖𝑗
A e age % o inpa ien s 3
in wa d 𝑖 o be ans e ed o wa d 𝑗
Le els Name Ini ial alue
𝐸Pa ien s no boa ding in he ED a ea 55
𝐵Pa ien s boa ding in he ED a ea 50
𝐻Hospi alized inpa ien s 592
2.4. Simula ion epo ing
So wa e: Powe sim S udio 10 Expe ®(10.14.5555.6) 64-bi e -
sion. Ha dwa e: OS Windows 10 home. P ocesso In el (R) Co e(TM)
i3-23-10M CPU @ 2.10 GHz 2.10 GHz. RAM 4.00 GB. Sys em ype
64-bi Ope a ing Sys em, x64-based p ocesso . Simula ion: Eule in-
eg a ion wi h s ep 1 h and ime ho izon 1 yea , using G ego ian
calenda .
3. Valida ion
3.1. Sou ce da a and analysis
The modeling is es ed using he inpu s o da a om a e ia y
le el hospi al. I s capaci y accoun s o 99 s e che s in he ED, 14
ope a ing ooms, 650 beds alloca ed in 24 wa ds o uni s. The hospi al
has an annual olume o hospi alized inpa ien s close o 32,000, wi h
an a e age LoS o 8.7 days, whe e 26% o hose pa ien s spend on
a e age 36 h boa ding in he ED.
The da a equi ed by he model o eplica e he his o ical beha -
io o he hospi al was ex ac ed om he hospi al’s da abase sys-
em. The so wa e e ie ed h ee main da abases con aining his o ical
in o ma ion om he 2019 yea o ope a ions:
•Admissions. Regis e s each admission o he hospi al.
•T ans e s. Regis e s he ans e o a pa ien be ween uni s and
p ocedu es.
•Discha ges. Regis e s he inpa ien s discha ged om wa ds.
A e p ocessing he h ee da abases i was possible o de e mine
he pa ame e s o he model p esen ed in Table 1, and he ime se ies
o ED and hospi al discha ges o compa e he simula ed hospi al e sus
he ac ual.
3.2. Tes o alida ion
The p ocess o alida ion was guided by he es s sugges ed by S e -
man [8,56]. The alida ion o he s uc u e was p esen ed in a wo k-
shop whe e he model was discussed among a g oup o hospi al s a
membe s. The alida ion o he beha io - ep oduc ion es was pe -
o med eplica ing he beha io o wo main g oups o ou pu a iables
du ing a ho izon ime o 12 mon hs (8640 h) o he hospi al ope a ions
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S. Jaén
Fig. 4. Simula ed and ac ual ED discha ge.
Table 2
Summa y s a is ics o assessing simula ed i o ac ual da a.
ED Hospi al
𝑅20.725 𝑅20.83
MAE 0.887 MAE 1.20
𝑋𝑆Mean 1.468 𝑋𝑆Mean 2.57
𝑋𝐴Mean 1.717 𝑋𝐴Mean 2.11
𝑆𝑆Mean 1.327 𝑆𝑆Mean 4.29
𝑆𝐴Mean 2.103 𝑆𝐴Mean 3.86
MSE 2.199 MSE 5.95
UM 0.028 UM 0.035
US 0.274 US 0.030
UC 0.698 UC 0.935
Sum 1Sum 1
in ime s eps o one hou . One g oup o main ou pu s co esponds o
he yea ly a e age occupancy a e o 24 wa ds and he ED. This es
e i ies i he s ocks o he model and he s ocks o he ac ual sys em
ha e simila le els o inpa ien s occupancy. Compa ing he simula ed
da a e sus he ac ual, he es pe o med he ollowing esul s: MAE:
0.0022821 and MSE: 3.82623E−05, which indica es ha he model is
able o ep oduce he yea ly a e age occupancy a e o 24 wa ds and
he ED.
The o he g oup co esponds o he discha ge a es o he uni s ( he
ED and he 24 hospi al wa ds). To simpli y he analysis, he alida ion
compa es he sum o he simula ed 24 wa d and ED discha ges, agains
he sum o he obse ed 24 wa d and ED discha ges. In Table 2, he
esul s compa e he abili y o he model o eplica e he ED and hospi al
discha ges. The 𝑅2sugges ha he modeling is be e a ep oducing
he beha io o he hospi aliza ion discha ges han ha o he ED, bu
hey a e s ill in an accep able ange. In addi ion, acco ding o he
Theil’s es [8,56], he la ge UC alues p esen ed in Table 2, indica e
ha despi e he poin -by-poin alues o he simula ed se ies do no
ma ch he ac ual da a, his e o is unsys ema ic and is no conside ed
a c i e ia o ejec ing he model S e man [56].
The Figs. 4 and 5illus a e he i o he simula ed s he ac ual
ou pu s o he hospi al wa ds and ED discha ges. The igu es only
compa e he i s 300 h o 8640, due o he lack o space, illus a ing
he beha io o simula ed and ac ual ou pu s. The cha s complemen
he analysis po ayed in Table 2.
Gi en he esul s o he analysis p esen ed in Table 2, hey suppo
he model as an app op ia e ep esen a ion o he hospi al beha io
unde s udy. The b oad alida ion o he model di ec ly s ems om
he ype o modeling (SD) and he complex na u e o he sys em i
ep esen s—a e ia y hospi al. Gi en he in ica e dynamics o he
e ia y hospi al en i onmen , i is impo an o no e ha he model
p o ides an agg ega ed ep esen a ion o he esul s. This alida ion
leads us o conclude ha despi e he complexi y and he need o
agg ega ion, he model adep ly desc ibes he agg ega ed dynamics o
bo h he ED and hospi al discha ges.
4. Policy analysis
4.1. Policy implemen a ion
In his s udy, o be e con ol he s ock o Boa ding inpa ien s,
we p opose a low-in al e na i e me hodology as opposed o a low-
ou me hodology which only ocuses on dec easing he wa d LoS o
inc easing he s ock o beds. The app oach p esen ed he e includes
conside a ion o he a e o Inpa ien s discha ge and he s ock o A ail-
able beds as addi ional inpu s needed by he physicians o de e mine
he h eshold o admissions. Fig. 6 illus a es how inco po a ing he
a iable A ailable beds o he a o emen ioned s uc u e (Fig. 2), as an
addi ional inpu o he calcula ion o he a iable Admi ed pa ien s,
c ea es wo addi ional pa ien admission eedback con ol loops (R1
and B3). These loops allow he hospi al o implemen a s ock manage-
men policy [8] by calcula ing an in low a e o Admi ed pa ien s, in
acco dance wi h bo h he ou low a e o Inpa ien s discha ge and he
s ock o A ailable beds.
4.2. Policy modeling
The modeling o a e ia y hospi al using an SD app oach de e mines
ha he policies o be conside ed ( he expe imen s) ope a e a he
ac ical-s a egic le el. This implies ha he esul s encompass a se
o guidelines implemen ed a he ope a ional le el wi h long- e m
impac s. The implemen a ion o he p oposed me hodology demands
he in oduc ion o a new a iable A ailable beds, able o de e mine he
numbe o beds a ailable in each uni 𝑖, o a oid ED o e c owding and
idle capaci y.
In Eq. (17),A ailable beds is ede ined by he addi ion o wo
componen s: expec ed ou low and adjus men o s ock. Expec ed ou -
pu is de e mined by he expec ed ou low o hospi al discha ges pe
hou (Inpa ien s discha ge), 𝐷𝑖(𝑡), (Eq. (14)). The adjus men o s ock
conside s he exp ession o ee capaci y 𝐻∗
𝑖(𝑡) − 𝐻𝑖(𝑡), di ided by a
e m 𝜏1𝑖. This exp ession, acco ding o he s ock managemen policy
de ined by S e man [8], ep esen s a i s o de ma e ial delay o he
ma e ial in ansi ( ee beds). The e m 𝜏1𝑖, es ablished in hou s o
his model, ep esen s he a e age delay ime o he ee beds in ansi
o be assigned. The alue o 𝜏1𝑖con ols he desi ed le el o hospi al
occupancy when bed demand is g ea e han hospi al capaci y. The
o al numbe o beds a ailable in he hospi al 𝑋(𝑡), is gi en by Eq. (18).
The new modeling is illus a ed in he s ocks and lows diag am in
Fig. 7.
𝑋𝑖(𝑡) = 𝐷𝑖(𝑡) + 𝐻∗
𝑖(𝑡) − 𝐻𝑖(𝑡)
𝜏1𝑖
𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (17)
𝑋(𝑡) =
𝑈
∑
𝑖=1
𝑋𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (18)
As 𝑋𝑖(𝑡)es ablishes he numbe o A ailable beds, he equa ions
de e mining he numbe o Admi ed pa ien s,𝑎(𝑡), (Eq. (4)), has o be
Ope a ions Resea ch Pe spec i es 12 (2024) 100298
7
S. Jaén
Fig. 5. Simula ed and ac ual hospi al discha ge.
Fig. 6. Discha ge-A ailable beds d i en admissions policy.
Fig. 7. Policy implemen a ion: s ocks and lows s uc u e.
ede ined o de e mine a limi in he numbe o admissions (Eq. (19)).
I 𝑋𝑖(𝑡)≥𝐸(𝑡)𝛽2𝜌𝑖∕𝜏2, all he pa ien s demanding beds in he uni
𝑖a e admi ed, o he wise, 𝑋𝑖(𝑡)pa ien s a e admi ed and 𝑙𝑖(𝑡)pa-
ien s (Eq. (20)), a e hose who need o be ans e ed o a di e en
heal hca e acili y o dec ease ED boa ding. Thus, he o al numbe
o in e -hospi al 𝑇(𝑡) ans e s also has o be ede ined acco ding wi h
Eq. (22).
𝑎𝑖(𝑡) = 𝑀𝐼𝑁(𝐸(𝑡)𝛽2𝜌𝑖
𝜏2
, 𝑋𝑖(𝑡)) ∨ 𝑖= 1...𝑈 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (19)
𝑙𝑖(𝑡) = 𝑀𝐴𝑋(𝐸(𝑡)𝛽2𝜌𝑖
𝜏2
−𝑎𝑖(𝑡),0) ∨ 𝑖= 1...𝑈 𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (20)
𝑙(𝑡) =
𝑈
∑
𝑖=1
𝑙𝑖(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (21)
Ope a ions Resea ch Pe spec i es 12 (2024) 100298
8
S. Jaén
Fig. 8. Boa ding inpa ien s be o e and a e policy.
Fig. 9. In e -hospi al ans e s.
Table 3
Base s Policy.
Indica o Be o e policy A e policy Change
To al ED discha ges 19,206 19,206 0%
To al in e -hospi al ans e s 533 1440 170%
To al hospi al discha ges 33,246 32,472 −2%
Boa ding inpa ien s a e 0.26 0.03 −88%
ED occupancy a e 1.99 1.32 −34%
Hospi al occupancy a e 0.93 0.91 −2%
𝑇(𝑡) = 𝐸(𝑡)𝛽3
𝜏2
+𝑙(𝑡)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (22)
The esul s o he simula ions a e p esen ed in Fig. 8 and Table 3.
The igu e compa es he wo le els o boa ding inpa ien s in he ED
be o e and a e he s ock managemen policy is implemen ed. The base
case shows how he le el o inpa ien boa ding is c i ical in he second
semes e o he yea eaching alues nea 60% o he o al ED pa ien s.
In con as , a e implemen ing he policy, he igu e illus a es how he
pe cen o boa ding inpa ien s dec eases om an a e age o 26% o an
a e age o 3% annually. Table 3 p esen s how he implemen a ion o
his policy impac s he ou pu and le el o hospi al indica o s.
The numbe o o al in e -hospi al ans e s is dis ibu ed du ing he
yea acco ding o Fig. 9.
4.3. Policy adhe ence
The success o he policy depends on he s ic assignmen o beds
o pa ien s in he ED gi en he numbe o beds de e mined by Eq. (17).
Howe e , i may be necessa y o allow le els o policy adhe ence. To
Table 4
Le els o policy adhe ence.
𝐿𝑖T ans e s Boa ding a e Boa ding ime (h)
1 906 3.2% 1.5
2 702 3.8% 1.51
3 541 4.5% 1.73
4 412 8% 3.8
5 288 14% 6.27
allow his, Eq. (23) can be modi ied o inco po a e hese le els o
adhe ence acco ding wi h he hospi al needs.
𝑎𝑖(𝑡) = 𝑀𝐼𝑁(𝐸(𝑡)𝛽2𝜌𝑖
𝜏2
, 𝑋𝑖(𝑡) ∗ 𝐿𝑖)𝑃 𝑎𝑡𝑖𝑒𝑛𝑡
ℎ𝑜𝑢𝑟 (23)
By using he 𝐿𝑖pa ame e , he equa ion can inc ease ‘‘a i icially’’
he numbe o a ailable beds allowing mo e inpa ien s o boa d, a oid-
ing hei ans e o ano he ins i u ion. In Table 4, he esul s o he
simula ions gi en di e en le els o policy adhe ence. This app oach
gi es he manage ial s a he possibili y o adjus he policy gi en he
needs o in e es o each hospi al.
4.4. S a is ical sc eening analysis o policy es ing
The s a is ical sc eening analysis is a echnique whose pu pose is
o iden i y he key inpu s o a model using a sensi i i y analysis and
he simple co ela ion coe icien CC [57]. The analysis es ablishes a
ange o unce ain y o each inpu . Then, a e pe o ming se e al
simula ions i calcula es he CC be ween he inpu s and he selec ed
ou pu . The esul ing alue o he CC anges om −1 o 1, which
indica es he s eng h o he linea ela ionship be ween he inpu and
