The potential of patient-based nurse staffing – a queuing theory application in the neonatal intensive care setting

Sülz, Sand a; Fügene , And eas; Becke -Pe h, Michael; Ro h, Be nha d
A icle — Published Ve sion
The po en ial o pa ien -based nu se s a ing – a queuing
heo y applica ion in he neona al in ensi e ca e se ing
Heal h Ca e Managemen Science
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Sülz, Sand a; Fügene , And eas; Becke -Pe h, Michael; Ro h, Be nha d (2024) :
The po en ial o pa ien -based nu se s a ing – a queuing heo y applica ion in he neona al
in ensi e ca e se ing, Heal h Ca e Managemen Science, ISSN 1572-9389, Sp inge US, New Yo k,
NY, Vol. 27, Iss. 2, pp. 239-253,
h ps://doi.o g/10.1007/s10729-024-09665-8
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315268
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Vol.:(0123456789)
Heal h Ca e Managemen Science (2024) 27:239–253
h ps://doi.o g/10.1007/s10729-024-09665-8
The po en ial o pa ien ‑based nu se s a ing – aqueuing heo y
applica ion in heneona al in ensi e ca e se ing
Sand aSülz1 · And easFügene 2 · MichaelBecke ‑Pe h3 · Be nha dRo h4,5
Recei ed: 4 Ma ch 2021 / Accep ed: 11 Janua y 2024 / Published online: 30 Janua y 2024
© The Au ho (s) 2024
Abs ac
Faced by a se e e sho age o nu ses and inc easing demand o ca e, hospi als need o op imally de e mine hei s a ing
le els. Ideally, nu ses should be s a ed o hose shi s whe e hey gene a e he highes posi i e alue o he quali y o
heal hca e. This pape de elops an app oach ha iden i ies he inc emen al bene i o s a ing an addi ional nu se depending
on he pa ien mix. Based on he easoning ha imely ul illmen o ca e demand is essen ial o he heal hca e p ocess and
i s quali y in he c i ical ca e se ing, we p opose o measu e he inc emen al bene i o s a ing an addi ional nu se h ough
educ ions in ime un il ca e a i es (TUCA). We de e mine TUCA by elying on queuing heo y and pa ame ize he model
wi h eal da a collec ed h ough an obse a ional s udy. The s udy indica es ha using he TUCA concep and applying
queuing heo y a he ca e e en le el has he po en ial o imp o e quali y o ca e o a gi en nu se capaci y by e icien ly
ading si ua ions o high e sus low wo kload.
Keywo ds Nu sing demand· Flexibili y· Quali y· Queuing· S a ing· Ope a ions esea ch· Ope a ions managemen
Highligh s
• The pape ocuses on sho - e m s a ing decisions in lu-
enced by sho - e m a ia ions in pa ien mix and illus-
a es and app oach when and whe e s a ing an addi-
ional nu se gene a es he highes alue.
• We apply queuing heo y in a neona al in ensi e ca e uni
and collec da a on ca e e en le el o pa ame ize he
queuing model.
• The case s udy discusses he implica ions o di e en
s a ing le els and e lec s on challenges and implica ions
o p ac ice.
1 In oduc ion
De eloped coun ies a e acing a se e e sho age o nu ses.
S a ing acancies emain un illed despi e a g owing end
o “shop” o he coun ies o lu e nu ses om ab oad [1, 2].
Meanwhile, heal h policy egula ions ha e imposed u he
manda o y s a ing le els and a e gaining dominance [3–7].
While hese heal h policy egula ions a e implemen ed wi h
he bes in en ions, heal hca e p o ide s a e s uggling o
comply wi h hese equi emen s amid a nu se sho age.
Heal hca e p o ide s hus need o s a hese sca ce pe son-
nel esou ces op imally, and he speci ied numbe o nu ses
is c ucial he e. I p o ide s had an unlimi ed supply o
* Sand a Sülz
[email p o ec ed].nl
And eas Fügene
and eas. uegene @uni-koeln.de
Michael Becke -Pe h
m.becke [email p o ec ed]
Be nha d Ro h
Be nd.Ro [email protected]
1 E asmus School o Heal h Policy & Managemen , Bu g.
Oudlaan 50, 3062PARo e dam, TheNe he lands
2 Depa men o Supply Chain Managemen & Managemen
Science, Uni e si y o Cologne, Albe us-Magnus Pla z,
50923Cologne, Ge many
3 Ro e dam School o Managemen , Bu g. Oudlaan 50,
3062PARo e dam, TheNe he lands
4 Depa men o Neona ology andPaedia ic In ensi e Ca e,
Child en’s Hospi al, Uni e si y Hospi al Cologne, Ke pene
S . 62, 50937Cologne, Ge many
5 Depa men o Business Adminis a ion andHeal h Ca e
Managemen , Uni e si y o Cologne, Albe us-Magnus Pla z,
50923Cologne, Ge many
240 S.Sülz e al.
nu ses, hey could ma ch nu se supply wi h nu sing demand
o ul ill mos pa ien ca e.
Ye , supply–demand imbalance p e ails. E en when
heal hca e p o ide s enlis a lex pool o nu ses o bu e
sho - e m demand su ges o nu se supply sho ages, his
lex-pool me hod s ill has i s limi s. S a ing nu ses om
a lex pool in ol es a ade-o : when nu ses a e s a ed in
one clinical a ea, hese nu ses canno se e al e na i e clin-
ical a eas. Simila conside a ions also complica e s a ing
decisions wi hin a clinical depa men : wi h ixed os e o
depa men nu ses, mo e nu ses s a ed o wo k he ea ly
shi means ewe assigned he la e shi . S a ing an indi-
idual nu se can hus no only be exp essed as di ec pe son-
nel cos s, bu also as he oppo uni y cos o no being able
o s a he nu se elsewhe e.
F om a planning and s a ing pe spec i e, nu ses should
be s a ed o shi s whe e hey gene a e he highes posi i e
alue o he quali y o heal hca e. This equi es iden i y-
ing he inc emen al bene i o s a ing an addi ional nu se.
Relying on an inno a i e applica ion o queuing heo y, his
pape shows how his quan i ica ion can be done. We con-
duc a case s udy in a neona al in ensi e ca e uni (NICU),
show how o u ilize queuing heo y o imp o e s a ing deci-
sions, discuss he po en ial o such an app oach, and e lec
on p ac ical implica ions and challenges he eo .
Queuing heo y has enjoyed no able applica ion in he
ope a ions communi y whe e a i als and depa u es o
pa ien s a e ypically modeled as s ochas ic p ocesses (see,
o ins ance, he e iews by Bai e al., 2018 [8] o Lak-
shmi and Si akuma , 2013 [9]). A mony e al. (2017), o
ins ance, analyze condi ions whe e s ep-down uni s as in e -
media es add alue i se up be ween c i ical ca e uni s and
gene al wa ds. These au ho s elied on queuing heo y o
model a i al and discha ge a he indi idual pa ien le el
o in o m s a egic capaci y decisions [10]. In he con ex
o nu se s a ing, ea ly applica ions o queuing heo y a e
p o ided by he wo k o De Vé icou and Jennings (2011)
who model excessi e delays expe ienced by pa ien s [11]
and Yanko ic and G een (2011) who model he in e ela-
ions be ween he demand o inpa ien beds and nu ses [12].
The case s udy p esen ed in his pape complemen s his
wo k by ocusing on sho - e m s a ing decisions in luenced
by sho - e m a ia ions in pa ien mix. We analyze a mo e
ope a ional scheme and apply queuing heo y a he le el o
indi idual-ca e e en s. This modelling choice is impo an
in he con ex o c i ical ca e whe e a delayed ul ilmen o
an indi idual-ca e e en may p o e a al.
Real-wo ld se ings usually in ol e s a ing le els ixed
o e ime based on he a e age wo kload. Ye , such models
usually do no inco po a e dynamic aspec s, e.g., pa ien -
based di e ences o e ime. This is p oblema ic since a mis-
ma ch be ween nu se capaci y and pa ien needs is ine i a-
ble, ei he leading o (oppo uni y) cos s om idle nu se
capaci y in cases o o e s a ing o o high wo kload le els in
cases o unde s a ing. Excessi e wo kload and unde s a ing
no only nega i ely a ec pa ien ou comes [13–19]. They
also co ela e wi h o e loaded nu ses, which incu s wo k
dissa is ac ion, bu n-ou [14] and nu se u no e [20, 21].
The e o e, p ope alignmen o nu se capaci y and pa ien
demand po en ially can also boos he quali y o ca e deli -
e y indi ec ly h ough be e wo king condi ions.
Fo he c i ical ca e se ing in gene al and he NICU spe-
ci ically, imely ul illmen o ca e demand is essen ial o
he heal hca e p ocess and i s quali y. We he e o e p opose
o measu e he inc emen al bene i o s a ing an addi ional
nu se h ough educ ions in ime un il ca e a i es (TUCA).
We assume ha pa ien -gene a ed ca e e en s can be mod-
eled as a s ochas ic p ocess, se up a queuing model, and
pa ame ize his wi h eal da a. Ou case s udy speci ically
in ends o cap u e in o ma ion abou po en ial sys em-
a ic empo al a ia ion in nu sing equi emen s o he wise
obscu ed in a e age wo kload s a is ics equen ly used in
pa ien classi ica ion sys ems [20, 22]. As such, da a needs
o e lec he inhe en demand a ia ion bo h among pa ien s
and du ing pa ien ajec o y [23] bu his in o ma ion is
no di ec ly a ailable om heal h in o ma ion sys ems. To
pa ame ize he queuing model wi h eal da a, we he e o e
collec ed his da a manually h ough an obse a ional s udy.
Based on his we quan i y he po en ial o lexibly espond-
ing o a ia ions in he expec ed ca e demand and analyze
coun e ac ual scena ios o es ablish an e iciency on ie
ha can in o m s a ing decisions when nu ses a e sca ce.
We close he pape wi h e lec ing on challenges and impli-
ca ions o p ac ice.
2 Analyzing heNICU asaqueuing sys em
Assuming ha pa ien -gene a ed ca e e en s can be modeled
as a s ochas ic p ocess, we analyze he NICU as a queuing
sys em. In his sec ion we discuss he main model assump-
ions and b ie ly lis model inpu and ou pu pa ame e s. We
subsequen ly pa ame ize he queuing model wi h eal da a
in Sec ion3.
2.1 Model inpu : a i al andse ice a es
The model elies on basic queuing heo y and makes he ol-
lowing assump ions. Each pa ien issues ca e-demand e en s
h ough a andom p ocess. The expec ed numbe o ca e
e en s issued du ing a de ined ime pe iod pe pa ien
p
is
deno ed as
λp
. In queuing heo y, his comp ises he a i al
a e. The a e age du a ion o a ca e e en issued by pa ien
p
is deno ed as
1∕μp
. Thus,
μp
desc ibes he expec ed numbe
o ca e e en s one nu se could handle du ing one ime pe iod
– deno ed as se ice a e in queuing heo y.
241
The po en ial o pa ien ‑based nu se s a ing – aqueuing heo y applica ion in heneona a…
In his line o easoning, he NICU consis ing o a se
o pa ien s
p∈P
will issue ca e e en s a a e
λ
wi h an
expec ed du a ion o
1
/𝜇
using:
While he a i al a e o he uni simply equals he sum
o he a i al a es o all pa ien s, he expec ed du a ion o a
ca e e en is he expec ed du a ion o ca e e en s pe pa ien
weigh ed by he espec i e a i al a e [24].
We de elop he model based on he assump ion ha bo h
ca e-e en a i als and he du a ion o p o iding ca e can be
modelled as Ma ko p ocesses: he numbe o ca e e en s
pe ime pe iod ollows a Poisson dis ibu ion wi h s a ion-
a y a i al a es, and he du a ion o ca e e en s ollows an
exponen ial dis ibu ion. We u he p opose an app oxima-
ion o accoun o de ia ions om hose assump ions.
While he i s assump ion (Poisson dis ibu ion o occu -
ences) has been widely a o ed in many eal-wo ld s ochas-
ic se ings, he second (exponen ial dis ibu ion o du a-
ions) has o en aced challenge. Exponen ially dis ibu ed
du a ions a e cha ac e ized by high a iance (coe icien o
a ia ion = 1), a hea y p opo ion o b ie du a ions, and
ew e y long du a ions – a ibu es ha may no ma ch
ypical se ice p ocesses ha ins ead exe (log) no mal
dis ibu ions.
C i ical ca e as p o ided in he NICU, howe e , migh
be one example whe e exponen ial du a ions i easonably
well. Mos ca e p ocesses a e e y b ie in na u e (such
as quick checks o ou inely moni o ed ca dio- espi a o y
pa ame e s o espi a o se ings), while some ake a medium
le el o ime (such as blood sampling, adminis a ion o
d ugs, and o al o endo acheal suc ioning). Ra ely do ca e
e en s demand ex ended imes (such as assis ing endo a-
cheal in uba ion, ches d ainage inse ion, o es ablishing
a cen al enous access). Thus, exponen ial du a ions seem
ap opos o he c i ical ca e se ing (and we alida e his
o ou empi ical se ing). Assumed Poisson occu ences
and exponen ial du a ions ea u e qui e commonly in he
esea ch o c i ical ca e se ings (see, o example, [25–27]
o a summa y in [8]). Please no e, howe e , ha hose s ud-
ies analyzed da a a he pa ien le el spo ligh ing he a i al
o pa ien s and hei leng hs o s ay in a c i ical ca e uni ,
whe eas ou s udy models occu ence and du a ion o ca e
e en s.
As NICUs se e many pa ien s wi h a a ie y o ca e
demand, hese (a i al) a es and ca e du a ions o each
pa ien need o be es ima ed and agg ega ed o de i e o e -
all sys em a es (Eqs.1 and 2). To ensu e easibili y, i may
(1)
𝜆
=
∑p∈P
𝜆
p
(2)
1
�
𝜇=
∑
p∈P
�
𝜆p
�
𝜇p
�
𝜆
.
be use ul o ca ego ize pa ien s in o ypes whe e a es can
be es ima ed mo e easily. An easy ca ego iza ion o pa ien
ypes needs c a ing bu esul s in he ollowing ade-o .
The g ea e he pa ien - ype di e en ia ion, he mo e unce -
ain y may be educed wi hin a i als and se ice imes, bu
his complica es he se -up and ope a ion o he esul ing
policy. Also, pa ien ypes mus be easily obse able be o e
making he s a ing decision. In Sec ion3, we will use es-
pi a o y suppo o g oup pa ien s when es ima ing ca e-
a i als and ca e-du a ions in a eal NICU se ing.
Ano he common assump ion is s a iona i y o he a i al
p ocess. In he NICU ca e p ocess, we see wo main d i e s
o po en ial iola ions o his assump ion: Fi s , he clinical
si ua ion o newbo n in an s is a ec ed by ex e nal ac o s
such as gene al NICU ca e p ocesses and acous ic le els,
wi h he consequences o di e en le els o need o ca e a
day and nigh [28]. Second, wa d policies may change wi hin
he shi s, o example, ce ain ca e e en s (such as cleaning
ac i i ies) a e pe o med mo e o en du ing day shi s han
du ing nigh shi s. In his pape , we app oach his issue by
di e en ia ing be ween shi s, and only assume s a iona i y
o a i als wi hin each shi .
2.2 Model ou pu : ime un ilca e a i es
Assuming Poisson a i als and exponen ially dis ibu ed
du a ions, he sys em is equi alen o an M/M/c sys em,
whe e
c
is he numbe o nu ses wo king in he uni [24].
Based on he a i al a e o ca e e en s
λ
, he a e age e en
du a ion
1
/𝜇
, and he numbe o s a ed nu ses
c
, we calcula e
he p obabili y ha a ca e e en canno be ea ed imme-
dia ely (whe e all nu ses a e busy) by applying E lang’s
C-Fo mula,
F om his, we de i e he expec ed wai ing ime o a ca e
e en , i.e., he expec ed ime un il ca e a i es (TUCA) as:
Using his equa ion, we calcula e he TUCA o each
a i al a e and expec ed du a ion o ca e e en s, and he
ally o nu ses s a ed (i.e., o any po en ial si ua ion). As in
o he queuing models, wai ing ime con exly dec eases in
c
(i.e., numbe o nu ses – see Dye and P oll 1977 o o mal
p oo [29]) while con exly inc easing in bo h
λ
(i.e., a i al
a e o ca e e en s) and
1
/𝜇
(i.e., du a ion o ca e e en s).
Taking
λ
and
1
/𝜇
as gi en, we de e mine he TUCA o each
(3)
C
�c, λ
μ�=⎡⎢
⎢⎢⎣
1+�1−λ
cμ�⎛⎜
⎜⎜⎝
c!
�
𝜆
𝜇
�
c⎞⎟
⎟⎟⎠
�c−1
k=0
�
𝜆
𝜇
�
k
k!⎤⎥
⎥⎥⎦
−1
.
(4)
TUCA
(c,𝜆,𝜇)=
C
(
c, λ
μ
)
c𝜇−𝜆.
242 S.Sülz e al.
s a ing si ua ion
c
, hus o ming he basis o he app oach
o e alua e po en ial s a ing policies.
One o he bene i s o he Ma ko assump ion is ha i
allows o an exac calcula ion o ou pe o mance measu e
(TUCA). Howe e , in some si ua ions, a i al and se ice
p ocesses migh no exe hese p ope ies. Fo hese si ua-
ions, we p opose a simple adap ion o he model based on
he Kingman App oxima ion Equa ion [30] (an ex ension o
he o iginal Kingman’s Fo mula s a ed in [31]) ha allows
o gene al dis ibu ions o a i al and se ice p ocesses:
Thus, highe (lowe ) coe icien s o a ia ion o he
a i al and se ice p ocess lead o highe (lowe ) expec ed
alues o TUCA. Acco ding o Webe (1980), ma ginal anal-
ysis as desc ibed in he ollowing sec ion is e icien o G/
GI/c sys ems as well [32]. The e o e, he app oach o e alu-
a e po en ially s a ing policies may be applied equally well
based on he app oxima ed alues o TUCA, e en hough
his gene alized model p o ides only an app oxima ion.
We acknowledge ha TUCA is an agg ega ed pe o -
mance indica o ha does no di e en ia e be ween c i i-
cal and non-c i ical ca e e en s. In he NICU se ing, some
ca e needs will induce p io i iza ion – an u gen need o
espi a o y suppo will be handled immedia ely, whe eas
a less u gen e en will be handled a e wa ds. A queuing
model ha di e en ia ed be ween c i ical and non-c i ical
ca e needs would show ha dec easing s a ing le els would
ba ely a ec he wai ing imes o c i ical ca e e en s bu
d ama ically a ec he wai ing imes o non-c i ical ca e
e en s. As an agg ega ed pe o mance indica o , highe
le els o TUCA indica e s ongly inc easing wai ing imes
o non-c i ical ca e e en s wi h nega i e e ec s on quali y
o ca e, and o a lesse ex end an inc easing isk o ha ing
o wai o c i ical ca e e en s. Thus, we ea TUCA as a
single-dimensional, easy o unde s and measu e o quali y
o ca e wi hin he NICU se ing.
2.3 E alua ing s a ing policies h oughTUCA
educ ion
As desc ibed abo e, nu ses a e a sca ce esou ce, and we
mus s a ca e ully o achie e he bes le el o ca e o
he gi en capaci y. This helps balancing needs be ween
depa men s o wi hin depa men s o e ime. We p o-
pose an e icien p ocedu e o de e mine s a ing le els o
gi en pa ien mix scena ios, whe e e icien means ha
he e is no al e na i e s a ing policy wi h he same num-
be o equi ed nu ses and a lowe a e age alue o TUCA
o e all possible pa ien mix scena ios. Such an e alua ion
(5)
E
(TUCA)=
c
2
𝜆+c
2
𝜇
2
⋅
𝜌
√
2(c+1)−1
c(1−𝜌)
⋅
1
𝜇
assumes ha he a i al a es a e known in ad ance and
ha he numbe o nu ses can be adap ed acco dingly.
Ob iously, hese a e s ong equi emen s. In p ac ice, we
will no ace ull in o ma ion and we would ha e o deal
wi h impe ec o ecas s since he c i ical ca e se ing is
cha ac e ized by demand unce ain y. Howe e , he a i al
o some pa ien s is known some ime in ad ance, e.g. i
complex p ocedu es a e planned ha equi e obse a ion
a he c i ical ca e uni a e wa ds. And in he neona al
ca e se ing i holds ha he mos ulne able pa ien s ha e
a easonably high leng h o s ay a he uni – ex emely
low bi h weigh and e y-low bi h weigh in an s ha e an
a e age leng h o s ay o mo e han 5weeks (SD: 29days)
[33]. The pa ien s and hei heal h ajec o ies a e hus (a
leas pa ially) known o he uni . The second assump ion
ela es o lexible adap a ions o s a ing le els and while
his is challenging, i is no in easible. Floa pool o nu ses
and on-call schemes a e means o add ess his and we will
come back o his while discussing p ac ical implica ions.
To a ain e icien alloca ion o nu ses, hospi al manage-
men should de ine he alue TUCA educ ion pe addi-
ional nu se s a ed. Since
TUCA(c,𝜆,𝜇)
is non-inc easing
and con ex in
c
, alloca ion o esou ces based on ma ginal
alues [34] adds he nex esou ce o he sys em whe e i
p o ides he mos addi ional alue o achie e he high-
es o al educ ion in TUCA pe gi en esou ce consump-
ion. Gi en ha he numbe o nu ses
c
is an in ege , we
can de ine he educ ion o TUCA pe addi ional nu se
di ec ly:
This measu e can de e mine he ma ginal bene i o
an added nu se o a gi en se ing. In o he wo ds, i he
educ ion in TUCA, i.e.
ΔTUCA(c,𝜆,𝜇)
, is highe han a
p e-de ined h eshold
ΔTUCAmin
, adding a nu se is wo h-
while. The app oach we p opose alloca es esou ces based
on ma ginal alues, i.e. i adds he nex esou ce o he sys-
em whe e i p o ides he mos addi ional alue o achie e
he highes o al educ ion in TUCA pe gi en esou ce
consump ion. This is no he same as minimizing he maxi-
mum wai ime. Fo ins ance: I adding one nu se could
educe TUCA om 1.0 o 0.9 insi ua ion 1 and could
educe TUCA om 0.9 o 0.4 insi ua ion 2, he app oach
p oposed in he pape alloca es he addi ional nu se o
si ua ion 2 (lea ing he maximum wai ime a 1.0). Bu
since
TUCA(c,𝜆,𝜇)
is non-inc easing and con ex in c,
bo h app oaches a e likely o esul in compa able s a -
ing solu ions o si ua ions ha do no di e subs an ially
in demand. No e ha his app oach equi es a
ΔTUCAmin
h eshold ha s a es un il which educ ion in TUCA an
addi ional nu se should be s a ed. Ano he p ac ical
app oach (gi en sca ce esou ces) would be o alloca e
(6)
ΔTUCA(c,𝜆,𝜇)=TUCA(c,𝜆,𝜇)−TUCA(c+1, 𝜆,𝜇)

243
The po en ial o pa ien ‑based nu se s a ing – aqueuing heo y applica ion in heneona a…
nu ses o si ua ions un il he o al numbe o nu ses s a ed
caps ou . Such an app oach adds a nu se o he si ua ion
wi h he highes
ΔTUCAs
un il he expec ed numbe o
nu ses (weigh ed o si ua ion likelihoods) eaches he
a ge capaci y.
3 Pa ame izing hemodel wi h eal da a—
he empi ical case
3.1 S udy se ing
The s udy se ing is a pu e NICU wi hin a e ia y pe ina-
al cen e in Ge many ha p o ides ca e o p e e m and
sick newbo n in an s. This NICU is an open-bay s a ion
using oom di ide s spaced app oxima ely h ee me e s
apa whe e all in an beds a e cen ally moni o ed. Regu-
la capaci y o he NICU is 11 beds, and up o wo addi-
ional beds may be p o ided when needed. The NICU uses
ixed s a ing policies wi h he numbe o nu ses s a ed
depending on he shi ime. Ea ly shi s un om 6:30 am
o 2:30pm wi h NICU s a ing o i e nu ses; la e shi s
un om 1:30pm o 9:30pm wi h i e nu ses s a ed, and
ou nu ses a end he nigh shi om 9:00pm o 7:00
am. The o e all nu sing s a o ou s udy uni comp ised
52 nu ses (equi alen o 31.5 FTE) – all quali ied in line
wi h Neona al Nu se P ac i ione s (NNPs) and subjec o
join planning.
3.2 Da a collec ion
As da a on nu sing ca e demand is no a ailable wi hin in o -
ma ion sys ems, i had o be collec ed p ospec i ely wi hin
an obse a ional s udy ha had been app o ed by he e hics
e iew commi ee and he s a council o he pa icipa -
ing ins i u ion. Following Milligan e al. (2008) and Pillay
e al. (2012) [35, 36], we collec ed nu sing ca e da a h ough
passi e obse a ions. These obse a ions we e conduc ed
by s udy nu ses wi h su icien medical expe ise and back-
g ound knowledge. Using a able PC and a sel -modi ied
so wa e applica ion, he s udy nu ses assigned ime s amps
o nu se ac i i ies using he ollowing ou ca ego ies: (1)
di ec ca e ac i i ies, de ined as ac i i ies pe o med while
ca ing o he in an (e.g., eeding, adminis a ion o d ugs,
and documen ing he pa ien ’s da a); (2) indi ec ca e
ac i i ies (e.g., cleaning equipmen ), which we e de ined
as ac i i ies ha we e no di ec ed o an indi idual in an ;
(3) adminis a i e ac i i ies such as scheduling and cen al
documen a ion; and (4) o he ac i i ies, such as pe sonal
needs and s a b eaks. The axonomy is based on he o igi-
nal classi ica ion used in [35, 36] and only sligh ly adap ed
whe e equi ed in he Ge man con ex . The comple e lis is
p o ided in Appendix A1.
Ou analysis ocuses on di ec ca e because hese a e
ime-c i ical ac i i ies ha equi e nu ses o espond in
a imely manne and because di ec ca e ac i i ies a e
s ochas ic and can be nei he scheduled no an icipa ed
en i ely.
To limi dis u bances in he ca e p ocess and o con-
o m wi h he da a collec ion equi emen s posed by he
pa icipa ing ins i u ion, we e ained om con inuous
obse a ion and ins ead elied on andom sampling ech-
niques and andomized obse a ion in e als [35, 36]. We
designed obse a ion blocks o h ee hou s, each block
con aining 12 obse a ion in e als o 10min each, wi h
5-min b eaks in be ween ( o allow he s udy nu se o es ).
Fo each wo kday, we andomly selec ed one o he h ee
shi s (ea ly, la e, nigh ) as well as a andom s a ing poin
wi hin he shi ensu ing ha each obse a ion block alls
comple ely in o he shi o exclude hando e ac i i ies.
This ga e us a andom sample o each ype o shi as well
as andom imes wi hin each shi .
Each obse a ion block con ained mul iple obse a ion
in e als. Fo each obse a ion in e al, one o he nu ses
on du y (and who ga e consen o pa icipa e in ou s udy)
was andomly selec ed o being obse ed. This design
allowed us o obse e a ep esen a i e andom sample o
nu sing wo kload a he NICU in ou pa ne ing hospi-
al. This esul ed in obse ing one nu se a a ime, and
he eco ded asks ep esen ed he asks pe o med by an
“a e age” nu se. Acco ding o ou app o ed s udy p o-
ocol, we aimed o 80–90 obse a ion days. Since he
obse a ions had o be done by s udy nu ses who also had
egula nu sing du ies, we had o gi e he s udy nu ses he
possibili y o combine hei esea ch and egula nu sing
ac i i ies. The e o e, we gene a ed a sample wi h po en-
ial obse a ion days and andomized s a ing imes (as
desc ibed abo e) o which s udy nu ses could egis e on
a i s -come- i s -se ed basis. Due o sho - e m una ail-
abili y o s udy nu ses, we ended up ha ing 84 obse a ion
days be ween 06/2015 and 11/2015. Obse a ions ook
place du ing each day o he week and s a ed 33.33% o
he ime in ea ly shi s, 45.24% in la e shi s, and 21.43%
in nigh shi s yielding an unbiased sample. On some
obse a ion days, s udy nu ses s opped he obse a ions
be o e he h ee-hou block was eached due o sickness,
being called o om hei esea ch ac i i ies, o echnical
ailu e o he equipmen . In some ins ances, he obse -
a ion o indi idual nu ses was no pe mi ed because
he nu se did no p o ide consen , o he nu se p o ided
pallia i e ca e and he obse a ion was no pe mi ed o
e hical easons. Fo hese easons, he o al obse a ion
ime amoun ed o 155h. Du ing he s udy pe iod, he a e -
age mon hly demand was almos iden ical o he a e age
mon hly demand o he yea be o e. The e o e, we a e
con iden ha o he s udy NICU, he choice o he s udy
244 S.Sülz e al.
pe iod did no unde mine i s gene alizabili y beyond he
s udy pe iod.
3.3 Pa ien ‑based ca e demand anddu a ions
To pa ame e ize he model, we need he NICU’s a i al
a e and he expec ed du a ion o ca e e en s (Eqs.1 and
2). Ca e e en s a y be ween and wi hin pa ien s. Ideally,
we would enlis pa ien -speci ic, ime- a ying pa ame e s
we e his no p ac ically in easible. To sides ep ime- and
pa ien -speci ic a es, we equi e a pa ien indica o easily
obse able ex an e ha well disc imina es a ying le els o
ca e in ensi y. F om a medical pe spec i e, pa ien isk and
se e i y indica o s migh se e bes since pa ien s su e ing
mo e se e e condi ions demand g ea e nu se a en ion and
may igge mo e and/o longe ca e e en s ha co ela e
wi h mo e ime p o iding di ec ca e.
Wi hin he NICU, se e al pa ien isk and se e i y indi-
ca o s exis , such as bi h weigh , ges a ional age [37–39],
CRIB sco e [40], and Apga sco e [41]. These ac o s a e
ou inely collec ed a e bi h and o e easy access. Ye , in
his case, hei s a ic-pe -pa ien na u e limi s p edic ion o
ca e in ensi y a ia ion o e ime. We hus decided o use
espi a o y-suppo ype as an indica o o a pa ien ’s le el o
ca e in ensi y, which is dynamic and easily obse able [42].
Rega ding espi a o y suppo , he highes medical se e i y
is eco ded when in an s equi e in uba ion and mechanical
en ila ion.
While mechanical en ila ion has been applied mo e o en
in he pas , nonin asi e o ms a e cu en ly eplacing hei
in asi e coun e pa s in medical ou comes [43, 44]. Non-
in asi e o ms, such as nasal con inuous posi i e ai way
p essu e (nCPAP) and high- low cannulas, suppo in an
espi a ion wi h he p e equisi e an in an can spon ane-
ously b ea he on he own. The e o e, nonin asi e o ms o
espi a o y suppo , in gene al, e lec a be e ca dio espi-
a o y condi ion and lowe se e i y le els han mechanical
en ila ion.
In he NICU, a nonin asi e espi a o y-suppo ype is
he i s -line he apy. Whe e in uba ion is ine i able, he
p ima y goal is expedi ed ex uba ion and ans e o nonin-
asi e espi a o y suppo . Consequen ly, we may assume
in an espi a o y-suppo ype o indica e ca dio espi a o y
condi ion ha e lec s se e i y di e ences o bo h be ween
and wi hin in an s along hei heal h ajec o ies. This s udy
dis inguishes ou o ms o espi a o y suppo : (1) mechani-
cal en ila ion (i.e. in asi e wi h endo acheal in uba ion o
non-in asi e), (2) nCPAP (nasal o pha yngeal CPAP), (3)
high- low cannula, and (4) a miscellaneous ca ego y desc ib-
ing all in an s wi hou espi a o y suppo .
Beyond pa ien - ela ed ac o s, s uc u al ea u es inhe -
en in he NICU’s cu en os e , such as shi ypes (ea ly,
la e, and nigh ), likely a ec ca e demand when shi s di e
ega ding hei ca e p ocedu es. Planned p ocedu es, such as
exchange o espi a o y suppo ma e ial, ou ine ul asound,
blood sampling, and scheduled C-sec ions, occu mo e o en
in ea ly shi s. Also, dayshi noise om many heal h ca e
p ac i ione s and isi o s on he wa d migh i i a e in an s o
he poin o igge ing ca e e en s, i.e. apnea o b adyca dia
[28]. An empi ical analysis demons a ing ha a pa ien ’s
o al ca e demand di e s be ween shi s and espi a o y ype
can be ound in he Appendix A2.1 and A2.2. The case s udy
hus inco po a es wo aspec s ele an o ca e demand, i.e.
shi s (ea ly, la e, nigh ) and pa ien ypes ( ou ypes o
espi a o y suppo ) and he e o e ea u es 3 × 4 di e en
pa ame e s each o a i al a e and expec ed du a ion o
ca e e en s se ing as inpu s o he model.
To de i e he pa ame e s o he model we need o look
a he obse a ion p ocedu e in de ail. The da a collec ion
p ocess and he esul ing da a pa e n a e illus a ed in Fig.1.
Nu ses a e obse ed o e p ede ined in e als. Fo exam-
ple, he obse a ion o Nu se 1 s a s a s1 and ends a e1. A
s1, he obse e eco ds ha Nu se 1 is cu en ly p o iding
di ec ca e o pa ien D1. A 1, Nu se 1 swi ches o pa ien
D2 and p o ides ca e un il 2, when she p oceeds wi h indi-
ec ca e ac i i ies. A 3, Nu se 1 swi ches ac i i ies again
and p o ides ca e o pa ien D1. This obse a ion pa e n
has he consequence ha no e e y ca e e en is obse ed
in i s en i e y and ha some a e subjec o censo ing. Fo
ins ance, a s1, we obse e Nu se 1 p o iding di ec ca e o
Fig. 1 Obse a ion pa e n
245
The po en ial o pa ien ‑based nu se s a ing – aqueuing heo y applica ion in heneona a…
pa ien D1; howe e , he e en s a ed be o e he obse a ion
in e al, i.e., he e en is le -censo ed. As is he case a
e1, he obse a ion in e al s ops, bu he ca e con inues, in
which case he e en is igh -censo ed. Finally, we may ha e
cases whe e nu ses con inuously p o ide ca e o one pa ien
(D5) h oughou he en i e obse a ion in e al, as illus a ed
in he case o Nu se 3. He e, we obse e nei he he exac
s a no he exac ending o he ca e e en . Al hough i is
challenging o iden i y he exac du a ion o ca e e en s, his
obse a ion pa e n allows us o de e mine he p opo ion o
ime ha each nu se spends on di ec ca e p o ision.
To de i e he a i al a es o ca e e en s, we mus calcu-
la e he numbe o ca e e en s pe minu e and pa ien . To do
so, we s a by es ima ing he numbe o ca e e en s wi hin
he obse ed imespan. We equi e one ixed poin o iden i y
indi idual ca e e en s. As a s aigh o wa d ixed poin , we
may conside ei he he s a o he ca e e en o he end
o he ca e e en . As desc ibed abo e, we do no obse e
nu ses and he ca e ha hey p o ide in a con inuous ashion;
hence, we do no obse e e e y ca e e en comple ely bu ,
o some o hem, only he s a o he end. The e o e, we
conside he a i hme ic mean o he obse ed s a s and ends
as an unbiased es ima e o he numbe o ca e e en s ea ed
by one nu se. To ob ain he a e age numbe o ca e e en s
pe minu e and pa ien , i.e., he a i al a e, we di ide his
igu e by he o al numbe o obse ed minu es (leading o
he a e age numbe o ca e e en s o one nu se pe minu e),
mul iply by he a e age numbe o nu ses (leading o he
numbe o ca e e en s o he comple e NICU), and di ide
by he a e age numbe o pa ien s (leading o he a e age
numbe o ca e e en s pe minu e and pa ien ).
We illus a e his app oach wi h an example: Fo in an s
wi h nCPAP espi a o y suppo in he ea ly shi s, we
obse ed 131 s a s and 117 ends du ing he obse ed ime
in e al o 2,847min ( he o al obse a ion ime in ea ly
shi s). To de e mine he expec ed numbe o ca e e en s
pe nu se and minu e, we di ide 124 (i.e., he a i hme ic
mean o 131 and 117) by 2,847min and ob ain an a e age
o 0.04 ca e e en s om nCPAP in an s pe minu e ea ed
by one nu se. On a e age, 4.78 nu ses we e p esen du ing
ou obse a ion pe iods in ea ly shi s; hus, he comple e
NICU expe ienced on a e age 0.21 ca e e en s om nCPAP
in an s pe minu e. Gi en ha 7.04 nCPAP in an s we e p e-
sen on a e age, one nCPAP in an leads o 0.03 ca e e en s
pe minu e. The le pa o Table1 summa izes he occu -
ence a e o ca e e en s o each pa ien i di e en ia ed by
shi and ype o espi a o y suppo . We assume ha he
NICU’s a i al o ca e e en s ollows a Poisson p ocess wi h
an a i al a e o λ equal o he cumula ed a i al a e ac oss
all in an s (
𝜆
=
∑p∈P
λ
p)
.
Analogous o he a i al a es, we de e mine he a e -
age du a ion o one ca e e en o each combina ion o
espi a o y suppo and shi . Fo his pu pose, we di ide
he cumula ed obse ed du a ion o ca e e en s in min-
u es by he es ima ed numbe o ca e e en s wi hin he
obse ed imespan. Fo ins ance, as s a ed abo e, o
in an s wi h nCPAP espi a o y suppo in ea ly shi s, we
iden i ied 124 ca e e en s, encompassing a o al imespan
o 1,051min. Thus, he a e age du a ion o one ca e e en
equals o 8.48min. We p oceed simila ly o he o he
shi - espi a o y suppo ype combina ions and es ima e
he a e age se ice imes o ca e e en s and summa ize
hose in he igh pa o Table1. We assume ha he du a-
ions o NICU’s ca e e en s ollow an exponen ial dis i-
bu ion wi h an expec ed du a ion o 1/ µ ac oss all in an s
(
1
�
𝜇=
∑
p∈P
�
𝜆p
�
𝜇p
�
𝜆)
. As s a ed in Sec ion2, NICU migh
be one o a e se ings whe e he assump ion o exponen-
ially dis ibu ed se ice p ocesses is jus i ied due o a
high numbe o e y b ie p ocesses. Following he logic
o Li ak e al. [25], we assume he numbe o NICU’s ca e
e en s pe ime uni o ollow a Poisson dis ibu ion, while
he du a ion o NICU’s ca e e en s ollows an exponen ial
dis ibu ion o each pa ien mix pe shi .
As indica ed in Table1, depending on shi ype and
espi a o y suppo , pa ien ca e demand a ied 0.8 o 2.3
imes pe hou (a i al a es o 0.014 o 0.038 ca e e en s
pe minu e), and expec ed ca e-e en du a ion an 6.47 o
30.11min. These pa ame e s se e as basic inpu pa ame e s
o ou queuing model ha agg ega es a pa ien ype’s ca e-
e en a i al a es (
λp
) and du a ions (
1∕μp
) on he uni le el
applying Eqs.1 and 2. In he ollowing sec ion, we de e -
mine he pa ien mix, i.e., he numbe o pa ien s pe ype.
Table 1 Inpu pa ame e s o queuing model
A e age numbe o ca e e en s (a i al a e) pe in an pe minu e A e age du a ion o ca e e en s in minu es
Respi a o y
suppo /
Shi
Mechanical nCPAP High- low
cannula
Miscellane-
ous
Mechanical nCPAP High- low
cannula
Miscellaneous
Ea ly 0.034 0.030 0.022 0.017 10.76 8.48 7.81 30.11
La e 0.037 0.025 0.017 0.020 8.93 8.54 9.66 15.28
Nigh 0.038 0.022 0.014 0.029 6.87 6.47 12.25 7.30
246 S.Sülz e al.
3.4 Pa ien mix – agg ega ion ac osspa ien s
andpa ien dis ibu ion
Pa ame e s in Table1 ep esen he a i al a e and du a ion
o ca e e en s ini ia ed by an indi idual pa ien o a speci ic
espi a o y ype du ing a speci ic shi . To analyze he NICU
as a queuing sys em, we need o conside he cumula i e
a i al (and se ice) a e ac oss all pa ien s pe gi en pa ien
mix.
The da a shows wide a ie y in u iliza ion and pa ien
mix. Gene ally, we see po en ial s a es o he NICU ang-
ing om 0/0/0/0/13 (0 mechanical, 0 nCPAP, 0 high- low, 0
miscellaneous, 13 emp y beds) o 13/0/0/0/0 (13 beds occu-
pied by mechanically en ila ed pa ien s). Fo each pa ien
mix, we can calcula e he NICU’s a i al a e and expec ed
du a ion ia he speci ied pa ien - ype a es in Table1. When
alida ing he Ma ko assump ions o occu ences and
du a ions o ca e e en s (see Appendix A3), we conclude
ha he Ma ko ian ea u es seem o apply o ea ly and la e
shi s. Fo nigh shi s, we ely on he gene al app oxima ion
s a ed in Sec ion2.2.
To op imally alloca e nu ses ac oss mul iple shi s and
days, we need o es ima e he likelihoods ha a gi en pa ien
mix occu s. Using he 84 obse a ion days, we es ima e he
dis ibu ion o numbe o occupied beds, and he p obabili y
o each possible espi a o y ype. Assuming a mul inomial
dis ibu ion o hose inpu pa ame e s, we can es ima e he
p obabili y o each possible pa ien mix. The mos equen
si ua ions u ilize 11 beds o se en o nine pa ien s unde
non-in asi e espi a o y suppo , and he emaining wo o
ou pa ien s unde in asi e espi a o y suppo . This dis i-
bu ion allows de e mina ion o he o e all pe o mance o
a ce ain s a ing policy by calcula ing he a e age TUCA
and numbe o s a ed nu ses.
3.5 Analysis o TUCA imp o emen s
Based on he model discussed in Sec ion2 and he inpu
pa ame e s om Sec ion3.3 and 3.4, we now assess he
impac o di e en s a ing scena ios o he NICU. We
engage he ollowing ques ions o he case s udy: How
much gain in quali y o ca e is achie able using a mo e lex-
ible s a ing policy? O , al e na i ely, can he NICU use a
mo e lexible s a ing policy ha main ains he cu en qual-
i y o ca e le el wi h ewe nu sing hou s? Analyzing hese
wo scena ios maps ou an e iciency on ie .
The i s e e ence poin is he s a us quo a NICU whe e a
ixed numbe o nu ses a ends ea ly shi s (c = 5), la e shi s
(c = 5), and nigh shi s (c = 4). Fo his s a us quo scena io
(5–5-4) and he es ima ed pa ien -mix dis ibu ion (see p io
Sec ion3.3), he expec ed ime un il ca e a i es (TUCA)
ac oss all si ua ions is 1.15min. We scaled all subsequen
ou comes acco ding o his s a us quo.
We nex p oceed wi h al e na i e s a ing le els o
ixed s a ing scena ios. He e, we use a cons an numbe
o nu ses s a ed pe gi en shi ega dless o he uni ’s
pa ien mix (shaded g ey in Fig.1). Unsu p isingly, adding
esou ces, i.e., inc easing he o al nu sing hou s, imp o es
expec ed quali y o ca e, while wi hd awing esou ces
impai s quali y. A 6–5-4 policy, o ins ance, cu s a e age
TUCA by 41% compa ed o he s a us quo, while a 5–4-4
policy inc eases a e age TUCA by 61%. No e ha we dis-
play only he lowes a e age TUCA o he same numbe
o s a ed nu ses ac oss shi s (e.g., we do no epo plans
like 2–5-6 yielding highe a e age TUCA s a ing he same
o al nu ses (13) as 5–4-4).
In he lexible s a ing app oach, he numbe o s a ed
nu ses depends on he pa ien mix du ing ha speci ic shi .
We assume ha he in o ma ion on he numbe o occupied
beds and in an espi a o y ype is known be o e he shi
onse (no e ha his assump ion is elaxed in Sec ion4) and
ha he numbe o nu ses can be adap ed acco dingly. We
ollow he h eshold app oach o Sec ion2.3 whe e an indi-
idual nu se is added o a speci ic shi as long as he new
nu se sa es mo e TUCA han a h eshold alue. To ep e-
sen a la ge ange o po en ial s a ing le els, we a ied he
h eshold alue om 0.65min o 3.60min, and d a an e i-
ciency on ie . Due o he con exi y o TUCA in he numbe
o s a ed nu ses, he e is no alloca ion o nu ses ha leads
o a lowe TUCA wi hou inc easing he numbe o s a ed
nu sing hou s. The esul o he lexible nu se alloca ion sce-
na ios is depic ed in ligh g ey in Fig.2.
As a esul , he lexible s a ing alloca ion can achie e
he a e age s a us-quo TUCA using 97% o i s nu sing
hou s (Poin B). In o he wo ds, by lexibly adjus ing he
numbe o nu ses o he pa ien mix, he uni could ha e
main ained he same quali y o ca e le el incu ing 97%
o he di ec nu sing cos s. Fewe nu sing hou s ha main-
ain quali y o ca e is in e es ing no only om a cos -con-
ainmen iew, bu also because he cu en skill sho age
impai s hospi al s a ing. Al e na i ely, we also see ha
he uni could ha e cu bed a e age TUCA by 18% while
main aining i s cu en nu se-hou s (see Poin A). No e ha
e e y poin be ween A and B ep esen s a lexible s a ing
policy ha o e s be e ca e a less di ec nu sing cos ,
ha is, a lowe a e age TUCA plus lowe expec ed nu sing
hou s. The eason o he lexible s a ing ou pe o mance
lies in a e ing si ua ions o ex emely high TUCAs. O
cou se, a oiding high-TUCA si ua ions by s a ing nu ses
who would ha e o he wise been squande ed in shi s wi h
ela i ely low TUCAs means ha lowe TUCAs mus ise
in esponse o hese less busy si ua ions. No e ha he cos
sa ings desc ibed abo e only pe ain o he di ec nu sing
cos s. Decision-make s wonde ing whe he o implemen
such a policy migh a ibu e alue o a ull cos –bene i
analysis. In ou case, he bene i o dec ease o TUCA is
253
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Publishe 's No e Sp inge Na u e emains neu al wi h ega d o
ju isdic ional claims in published maps and ins i u ional a ilia ions.