An Introduction to Classic DEVS

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by  Yentl Van Tendeloo

An Introduction to Classic DEVS
YentlVan Tendeloo∗ Hans Vangheluwe∗†‡
{Yentl.VanTendeloo,Hans.Vangheluwe}@uantwerpen.be
Abstract
DEVSisapopularformalismformodellingcomplexdynamicsystemsusingadiscrete-eventabstraction. Atthisabstraction
7 level,atimedsequenceofpertinent“events”inputtoasystem(orinternal,inthecaseoftimeouts)causeinstantaneouschanges
1
tothestateofthesystem. Betweenevents,thestatedoesnotchange, resultinginaapiecewiseconstantstatetrajectory. Main
0
advantagesofDEVSareitsrigorousformaldefinition,anditssupportformodularcomposition.
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ThischapterintroducestheClassicDEVSformalisminabottom-upfashion,usingasimpletrafficlightexample.Thesyntax

n andoperationalsemanticsofAtomic(i.e.,non-hierarchical)modelsareintrucedfirst. ThesemanticsofCoupled(hierarchical)
a modelsisthengivenbytranslationintoAtomicDEVSmodels. Asthisformal“flattening”isnotefficient,amodular abstract
J
simulator which operates directly on the coupled model is also presented. Thisis the common basis for subsequent efficient

6 implementations. Wecontinue toactual applicationsof DEVSmodelling andsimulation, asseen inperformance analysisfor
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queueingsystems. Finally,wepresentsomeoftheshortcomingsintheClassicDEVSformalism,andshowsolutionstothemin

theformofvariantsoftheoriginalformalism.
]
H
1 Introduction
O
.
s DEVS[17]isapopularformalismformodellingcomplexdynamicsystemsusingadiscrete-eventabstraction.Atthisabstraction
c
level, a timed sequence of pertinent“events” inputto a system cause instantaneouschanges to the state of the system. These
[
eventscanbegeneratedexternally(i.e.,byanothermodel)orinternally(i.e.,bythemodelitselfduetotimeouts).Thenextstateof

1 thesystemisdefinedbasedonthepreviousstateofthesystemandtheevent.Betweenevents,thestatedoesnotchange,resulting
v
in a piecewise constantstate trajectory. Simulation kernels must only consider states at which events occur, skipping over all
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intermediate points in time. This is in contrast with discrete time models, where time is incremented with a fixed increment,
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6 and onlyat these times is the state updated. Discrete eventmodelshave the advantagethat their time granularitycan become
7 (theoretically)unbounded,whereastimegranularityisfixedindiscretetimemodels. Nonetheless,theaddedcomplexitymakes
0 itunsuitedforsystemsthatnaturallyhaveafixedtimestep.
.
1 Main advantagesof DEVS comparedto other discrete eventformalismsare its rigoroursformaldefinition, and its supportfor
0
modularcomposition.ComparablediscreteeventformalismsareDESandStatecharts,thoughsignificantdifferencesexist.
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1 Compared to DES, DEVS offers modularity which makes it possible to nest models inside of components, thus generating
: a hierarchy of models. This hierarchy necessitates couplings and (optionally) ports, which are used for all communication
v
i betweentwocomponents.Incontrast,DESmodelscandirectlyaccessothermodelsandsendthemeventsintheformofmethod
X
invocation.Additionaly,DEVSoffersacleanersplitbetweenthesimulationmodelandthesimulationkernel.
r
a ComparedtoStatecharts,DEVSoffersamorerigorousformaldefinitionandadifferentkindofmodularity. Statechartsleaves
a lot of room for interpretation, resulting in a wide variety of interpretations[7, 6]. In contrast, DEVS is completely formally
defined,andthereisareferencealgorithm(i.e.,anabstractsimulator).WhilebothDEVSandStatechartsaremodularformalisms,
Statechartscreateshierarchiesthroughcompositestates,whereasDEVSusescompositemodelsforthispurpose.Bothhavetheir
distinctadvantages,makingbothvariantsusefulinpractice.
Thischapterprovidesan introductorytextto DEVS (oftenreferredto asClassicDEVS nowadays)throughtheuse of a simple
examplemodelinthedomainoftrafficlights. Westartfromasimpleautonomoustrafficlight,whichisincrementallyextended
uptoatrafficlightwithpolicemaninteraction.EachincrementservestointroduceanewcomponentoftheDEVSformalismand
thecorresponding(informal)semantics.Westartwithatomic(i.e.,non-hierarchical)modelsinSection2,andintroducecoupled
(i.e.,hierarchical)modelsinSection3. Anabstractsimulator,definingthereferencealgorithm,iscoveredinSection4. Section5
movesawayfromthetrafficlightexampleandpresentsamodelofasimplequeueingsystem. EventhoughDEVScertainlyhas
itsapplications,severalvariantshavespawnedtotacklesomeofitsshortcomings. Thesevariants,togetherwitharationaleand
thedifferences,arediscussedinSection6. Finally,Section7summarizesthechapter.
∗UniversityofAntwerp,Belgium
†FlandersMake,Belgium
‡McGillUniversity,Montréal,Canada
1
Red
60s
Yellow
3s
Green
57s e=0
Figure2:ModelgeneratingtraceinFigure1.
Figure1: Traceoftheautonomoustrafficlight.
2 Atomic DEVS models
We commence our explanation of DEVS with the atomic models. As their name suggests, atomic models are the indivisable
buildingblocksofamodel.
Throughoutthis section, we build up the complete formalspecification of an atomic model, introducingnew conceptsas they
becomerequired.Ineachintermediatestep,weshowandexplaintheconceptsweintroduce,howtheyarepresentintherunning
example,andhowthisinfluencesthesemantics.Thedomainwewilluseasarunningexamplethroughoutthischapterisasimple
trafficlight.
2.1 AutonomousModel
Thesimplestformofatrafficlightisanautonomoustrafficlight. Lookingatitfromtheoutside,weexpecttoseeatracesimilar
tothatofFigure1. Visually,Figure2presentsanintuitiverepresentationofamodelthatcouldgeneratethiskindoftrace.
TryingtoformallydescribeFigure2,wedistinguishtheseelements:
1. StateSet(S)
Themostobviousaspectofthetrafficlightisthestateitisin,whichisindicatedbythethreedifferentcoloursitcanhave.
Thesestates are sequential: the trafficlightcanonlybe inone ofthese states atthe sametime1. Theset ofstates is not
limitedtoenumerationstyleaspresentedhere,butcancontainanarbitrarynumberofattributes.
S:×n S
i=1 i
2. TimeAdvance(ta)
1Incontrastto,say,Statecharts.
2
Foreachofthestatesjustdefined,wenoticethetimeoutinthem. Clearly,somestatestakelongertoprocessthanothers.
Forexample,whereaswewillstayingreenandredarelativelylongtime,thetimeintheyellowstateisonlybrief. This
functionneedstobedefinedforeachandeveryelementofthestate set, andneedstodeterministicallyreturnaduration.
The durationcan be any positive real number, includingzero and infinity. A negativetime is disallowed, as this would
requiresimulationtogobackintime. DEVSallowsatimeadvanceofexactlyzero,eventhoughthisisimpossibleinreal
life. Twousecasesforthisexist:thedelaymightbeverysmallandirrelevanttotheproblemwearemodelling,orthestate
isanartificialstate,withoutanyreal-worldequivalent(e.g.,aspartofadesignpattern).NotethatDEVSdoesnotconsider
time bases, despite the use of secondsin our visualization. Simulation time is just a realnumber, and the interpretation
giventoitisuptotheuser. Whethertheseunitsindicateseconds,years,orevenp seconds,iscompletelyuptotheusers,
aslongasitisfixedthroughoutthesimulation.
ta:S→R+
0,+¥
3. InternalTransition(d )
int
Withthestatesandtimeoutsdefined,thefinalpartisthedefinitionofwhichisthenextstatefromagivenstate. Thisisthe
joboftheinternaltransitionfunction,whichgivesthenextstateforeachandeverystate. Asitisafunction,everystate
hasatmostonenextstate,preventinganypossibleambiguity.Notethatthefunctiondoesnotnecessarilyhavetobetotal,
norinjective:somestatesmightnothaveanextstate(i.e.,ifthetimeadvancewasspecifiedas+¥ ),andsomestateshave
thesamestateasnextstate. Uptonow,onlytheinternaltransitionfunctionisdescribedaschangingthestate. Therefore,
itisnotallowedforotherfunctions(e.g.,timeadvance)tomodifythestate: theirstateaccessisread-only.
d :S→S
int
4. InitialTotalState(q )
init
We also needto definethe initialstate ofthesystem. WhilethisisnotpresentintheoriginalspecificationoftheDEVS
formalism,weincludeithereasitisavitalelementofthemodel.Butnotethat,insteadofbeingan“initialstate(s )”,it
init
isatotalstate. Thismeansthatwenotonlyselecttheinitialstateofthesystem,butalsodefinehowlongwearealreadyin
thisstate. Elapsedtimeisthereforeaddedtothedefinitionoftheinitialtotalstate,toallowmoreflexibilitywhenmodelling
asystem. Tothesimulator,itwillseemasifthemodelhasalreadybeenintheinitialstateforsometime.
q :(s,e)|s∈S,0≤e≤ta(s)
init
WedescribethemodelinFigure2asa4-tupleofthesethreeelements.
hS,q ,d ,tai
init int
S={GREEN,YELLOW,RED}
qinit =(GREEN,0.0)
d int ={GREEN→YELLOW,
YELLOW→RED,
RED→GREEN}
ta={GREEN→delaygreen,
YELLOW→delayyellow,
RED→delayred}
Forthissimpleformalism,wedefinethesemanticsasinAlgorithm1. Themodelisinitializedwithsimulationtimesetto0,and
thestatesettotheinitialstate(e.g.,GREEN).Simulationupdatesthetimewiththereturnvalueofthetimeadvancefunction,and
executestheinternaltransitionfunctiononthecurrentstatetogetthenewstate.
2.2 AutonomousModelWithOutput
RecallthatDEVSwasamodularformalism,withonlytheatomicmodelhavingaccesstoitsinternalstate. Thisnaturallyraises
aproblemforourtrafficlight:othershavenowayofknowingitscurrentstate(i.e.,itscolour).
Wethereforewantthetrafficlighttooutputitscolour,inthiscaseintheformofastring(andnotastheelementofanenumera-
tion). Fornow,theoutputistightlylinkedtothesetofstate,butthisdoesnotneedtobethecase: thepossiblevaluestooutput
can be completely distinct from the set of states. Our desired trace is shown in Figure 3. We see that we now output events
3
Algorithm1DEVSsimulationpseudo-codeforautonomousmodels.
time←0
current_state←initial_state
last_time←−initial_elapsed
whilenottermination_condition()do
time←last_time+ta(current_state)
current_state←d (current_state)
int
last_time←time
endwhile
indicatingthestartofthespecifiedperiod.Recall,also,thatDEVSisadiscreteeventformalism:theoutputisonlyasingleevent
indicatingthetimeandisnotacontinuoussignal.Thereceiveroftheeventthuswouldhavetostoretheeventtoknowthecurrent
stateofthetrafficlightatanygivenpointintime. Visually,themodelisupdatedtoFigure4,usingtheexclamationmarkona
transitiontoindicateoutputgeneration.
Analysingtheupdatedmodel,weseethattwomoreconceptsarerequiredtoallowforoutput.
1. OutputSet(Y)
Similarly to defining the set of allowable states, we should also define the set of allowable outputs. This set serves as
aninterfacetoothercomponents,definingtheeventstheexpectstoreceive. Eventscanhavecomplexattributesaswell,
thoughweagainlimitourselftosimpleeventsfornow.Ifportsareused,eachporthasitsownoutputset.
Y :×l Y
i=1 i
2. OutputFunction(l )
With the set of allowable events defined, we still need a function to actually generate the events. Similar to the other
functions,theoutputfunctionisdefinedonthestate, anddeterministicallyreturnsanevent(ornoevent). Asseeninthe
Figureofthemodel,theeventisgeneratedbeforethenewstateisreached. Thismeansthatinsteadofthenewstate, the
output function still uses the old state (i.e., the one that is being left). For this reason, the output function needs to be
invokedrightbeforetheinternaltransitionfunction. Inthecaseofourtrafficlight,theoutputfunctionneedstoreturnthe
nameofthenextstate,insteadofthecurrentstate. Forexample,iftheoutputfunctionreceivestheGREEN stateasinput,
itneedstogenerateashow_yellowevent.
Similartothetimeadvancefunction,thisfunctiondoesnotoutputanewstate,andthereforestateaccessisread-only.This
mightrequiresomeworkarounds:outputtinganeventoftenhassomerepercussionsonthemodelstate,suchasremoving
theeventfromaqueueorincreasingacounter. Sincethestatecan’tbewrittento,thesechangesneedtoberemembered
andexecutedassoonastheinternaltransitionisexecuted.
Notethatitispossiblefortheoutputfunctionnottoreturnanyoutput,inwhichcaseitreturnsf .
l :S→Y∪{f }
Themodelcanbedescribedasa6-tuple.
hY,S,q ,d ,l ,tai
init int
Y ={show_green,show_yellow,show_red}
S={GREEN,YELLOW,RED}
qinit =(GREEN,0.0)
d int ={GREEN→YELLOW,
YELLOW→RED,
RED→GREEN}
l ={GREEN→show_yellow,
YELLOW→show_red,
RED→show_green}
ta={GREEN→delaygreen,
YELLOW→delayyellow,
RED→delayred}
4
Red
60s
!show_green
!show_red
Yellow
3s
!show_yellow
Green
57s e=0
Figure4:ModelgeneratingtraceinFigure3.
Figure3:Traceoftheautonomoustrafficlightwithoutput.
Thepseudo-codeisslightlyalteredtoincludeoutputgeneration,asshowninAlgorithm2. Recallthatoutputisgeneratedbefore
theinternaltransitionisexecuted,sothemethodinvocationhappensrightbeforethetransition.
Algorithm2DEVSsimulationpseudo-codeforautonomousmodelswithoutput.
time←0
current_state←initial_state
last_time←−initial_elapsed
whilenottermination_condition()do
time←last_time+ta(current_state)
output(l (current_state))
current_state←d (current_state)
int
last_time←time
endwhile
2.3 InterruptableModel
Ourcurrenttrafficlightspecificationisstillcompletelyautonomous.Whilethisisfineinmostcircumstances,policemightwant
to temporarily shut down the traffic lights, when they are managing traffic manually. To allow for this, our traffic light must
processexternalincomingevents: theeventfromthepolicemantoshutdownandtostartupagain. Figure5showsthetracewe
wishtoobtain.AmodelgeneratingthistraceisshowninFigure6,usingaquestionmarktoindicateeventreception.
WeoncemorerequiretwoadditionalelementsintheDEVSspecification.
1. InputSet(X)
Similartotheoutputset,weneedtodefinetheeventsweexpecttoreceive.Thisisagainadefinitionoftheinterface,such
5
?manual Red
60s !show_green
?automatic !show_red
Manual ?manual Yellow
s 3s
(cid:0)
!show_yellow
?manual Green
e=0
57s
Figure 6: Naive model that should generate
thetraceinFigure5(butdoesn’t).
Figure5: Traceoftheautonomoustrafficlight.
thatothersknowwhicheventsareunderstoodbythismodel.
X =×m X
i=1 i
2. ExternalTransition(d )
ext
Similartotheinternaltransitionfunction,theexternaltransitionfunctionisallowedtodefinethenewstateaswell. First
andforemost,theexternaltransitionfunctionisstilldependentonthecurrentstate,justliketheinternaltransitionfunction.
The externaltransitionfunctionhasaccess to two morevalues: the elapsed time, and the inputevent. The elapsed time
indicates how long it has been for this atomic modelsince the last transition (either internal or external). Whereas this
numberwasimplicitlyknownintheinternaltransitionfunction(i.e.,thevalueofthetimeadvancefunction),hereitneeds
tobepassedexplicitly. Elapsedtimeisanumberintherange[0,ta(s)],withsbeingthecurrentstateofthemodel. Note
thatitisinclusiveofboth0andta(s):itispossibletoreceiveaneventexactlyafteratransitionhappened,orexactlybefore
aninternaltransitionhappens. Thecombinationofthecurrentstateandtheelapsedtimeisoftencalledthetotalstate(Q)
ofthemodel.Wehavepreviouslyseenthetotalstate,inthecontextoftheinitialtotalstate. Thereceivedeventisthefinal
parameter to this function. A new state is deterministically defined through the combinationof these three parameters.
Sincetheexternaltransitionfunctiontakesmultipleparameters,multipleexternaltransitionsmightbedefinedforasingle
state.
d :Q×X→S
ext
Q={(s,e)|s∈S,0≤e≤ta(s)}
While we now have all elements of the DEVS specification for atomic models, we are not done yet. When we include the
additionalstate MANUAL, wealso needtosendoutanoutputmessageindicatingthatthetrafficlightisoff. Butrecallthatan
outputfunctionwas only invokedbeforean internaltransition function, so not beforean externaltransitionfunction. To have
an output nonetheless, we need to make sure that an internal transition happens before we actually reach the MANUAL state.
This can be done through the introduction of an artificial intermediate state, which times out immediately, and sends out the
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To !show_red Red
Automatic
0s 60s !show_green
?manual
?automatic !show_red
Manual Yellow
s 3s
(cid:0)
!show_black !show_yellow
?manual
To Green
Manual 57s e=0
0s ?manual
Figure8:ModelgeneratingtraceinFigure7.
Figure7:Traceoftheinterrupttrafficlightwithcorrectedartificialstates.
show_blackevent. Insteadofgoingto MANUAL uponreceptionofthemanualevent,wegototheartificialstate TOMANUAL.
Thetimeadvanceofthisstate issetto0, sinceitisonlyanartificialstate withoutanymeaninginthedomainunderstudy. Its
output function will be triggered immediately due to the time advance of zero, and the show_black outputis generated while
transferringtoMANUAL. Similarly,whenwereceivetheautomaticevent,weneedtogotoanartificialTOAUTOMATICstateto
generatetheshow_redevent. AvisualizationofthecorrectedtraceandcorrespondingmodelisshowninFigure7andFigure8
respectively.
Finally,wegivethefullspecificationofthetrafficlightasanatomicDEVSmodel,definedbya8-tuple.
hX,Y,S,q ,d ,d ,l ,tai
init int ext
7
X ={automatic,manual}
Y ={show_green,show_yellow,show_red,show_black}
S={GREEN,YELLOW,RED,TOMANUAL,TOAUTOMATIC,MANUAL}
qinit =(GREEN,0.0)
d int ={GREEN→YELLOW,
YELLOW→RED,
RED→GREEN,
TOMANUAL→MANUAL,
TOAUTOMATIC→RED}
d ext ={(GREEN,_,manual)→TOMANUAL
(YELLOW,_,manual)→TOMANUAL
(RED,_,manual)→TOMANUAL
(MANUAL,_,automatic)→TOAUTOMATIC}
l ={GREEN→show_yellow,
YELLOW→show_red,
RED→show_green,
TOMANUAL→show_black,
TOAUTOMATIC→show_red}
ta={GREEN→delaygreen,
YELLOW→delayyellow,
RED→delayred,
MANUAL→+¥ ,
TOMANUAL→0,
TOAUTOMATIC→0}
Algorithm 3 presents the complete semantics of an atomic model in pseudo-code. Similar to before, we still have the same
simulationloop,butnowwecanbeinterruptedexternally.Ateachtimestep,weneedtodeterminewhetheranexternalinterrupt
is scheduled before the internal interrupt. If that is not the case, we simply continue like before, by executing the internal
transition.Ifthereisanexternaleventthatmustgofirst,weexecutetheexternaltransition.
Algorithm3DEVSsimulationpseudo-codeforinterruptablemodels.
time←0
current_state←initial_state
last_time←−initial_elapsed
whilenottermination_condition()do
next_time←last_time+ta(current_state)
iftime_next_event≤next_timethen
elapsed←time_next_event−last_time
current_state←d ((current_state,elapsed),next_event)
ext
time←time_next_event
else
time←next_time
output(l (current_state))
current_state←d (current_state)
int
endif
last_time←time
endwhile
8
3 Coupled DEVS models
Whileourtrafficlightexampleisabletoreceiveandoutputevents,therearenootheratomicmodelstocommunicatewith. To
combinedifferentatomicmodelstogetherandhavethemcommunicate,wenowintroducecoupledmodels. Thiswillbedonein
thecontextofourprevioustrafficlight,whichisconnectedtoapoliceman.Thedetailsofthetrafficlightareexactlylikebefore;
thedetailsofthepolicemanareirrelevanthere,aslongasitoutputsautomaticandmanualevents.
3.1 BasicCoupling
The first problemwe encounterwith couplingthe trafficlightand policemantogetheris the structure: howdo we define a set
of modelsand their interrelations? Thisis the coredefinition of a coupledmodel: it is merelya structuralmodelthatcouples
modelstogether. Contrarytotheatomicmodels,thereisnobehaviourwhatsoeverassociatedtoacoupledmodel. Behaviouris
theresponsibilityofatomicmodels,andstructurethatofcoupledmodels.
Todefinethebasicstructure,weneedthreeelements.
1. Modelinstances(D)
Thesetofmodelinstancesdefineswhichmodelsareincludedwithinthiscoupledmodel.
2. Modelspecifications({M})
i
Apartfromdefiningthedifferentinstancesofsubmodels,wemustincludetheatomicmodelspecificationofthesemodels.
For each element defined in D, we include the 7-tuple specifying the atomic model. By definition, a submodel of the
coupled DEVS model always needs to be an atomic model. Later on, we will see how this can be extended to support
arbitrarilyhierarchies.
{hX,Y,S,q ,d ,d ,l ,tai|i∈D}
i i i init,i int,i ext,i i i
3. Modelinfluencees({I})
i
Apartfromdefiningthemodelinstancesandtheirspecifications,weneedtodefinetheconnectionsbetweenthem.Connec-
tionsaredefinedthroughtheuseofinfluenceesets:foreachatomicmodelinstance,wedefinethesetofmodelsinfluenced
by that model. There are some limitations on couplings, to make sure that inconsistent models cannot be created. The
followingtwoconstraintsareimposed:
• A modelshouldnot influenceitself. This constraintmakessense, as otherwise it would be possible for a modelto
influenceitselfdirectly. Whilethereisnosignificantproblemwiththisinitself,itwouldcausethemodeltotrigger
bothitsinternalandexternaltransitionsimultaneously. Asitisundefinedwhichoneshouldgofirst,thissituationis
notallowed. Inotherwords,amodelshouldnotbeanelementinitsownsetofinfluencees.
∀i∈D:i∈/I
i
• Only links within the coupled model are allowed. This is another way of saying that connections should respect
modularity. Modelsshould notdirectlyinfluencemodelsoutside of the currentcoupledmodel, nor modelsdeeper
insideofothersubmodelsatthislevel. Inotherwords,theinfluencedmodelshouldbeasubsetofthesetofmodels
inthiscoupledmodel.
∀i∈D:I ⊆D
i
Notethatthereisnoexplicitconstraintonalgebraicloops(i.e.,aloopofmodelsthathaveatimeadvanceequaltozero,
preventingthe progression of simulated time). If this situation is not resolved, it is possible for simulation to get stuck
atthatspecific pointintime. Thesituationisonlyproblematicifthe circulardependencynevergetsresolved,causinga
livelockofthesimulation.
Acoupledmodelcanthusbedefinedasa3-tuple.
hD,{M},{I}i
i i
3.2 InputandOutput
Ourcoupledmodelnowcouplestwoatomicmodelstogether. Andwhileitisnowpossibleforthepolicemantopasstheevent
tothetrafficlight,weagainlosttheabilitytosendoutthestateofthetrafficlight. Theeventscan’treachoutsideofthecurrent
coupledmodel. Therefore,weneedtoaugmentthecoupledmodelwithinputandoutputevents,whichserveastheinterfaceto
thecoupledmodel.ThisaddsthecomponentsX andY tothetuple,respectivelythesetofinputandoutputevents,resulting
self self
ina5-tuple.
hX ,Y ,D,{M},{I}i
self self i i
9
Theconstraintsonthecouplingsneedtoberelaxedtoaccomodateforthenewcapabilitiesofthecoupledmodel:amodelcanbe
influencedbytheinputeventsofthecoupledmodel,andlikewisethemodelscanalsoinfluencetheoutputeventsofthecoupled
model.Thepreviouslydefinedconstraintsarerelaxedtoallowforself,thecoupledmodelitself.
∀i∈D∪{self}:i∈/I
i
∀i∈D∪{self}:I ⊆D∪{self}
i
3.3 Tie-breaking
RecallthatDEVSisconsideredaformalandpreciseformalism. Butwhileallcomponentsarepreciselydefined,theirinterplay
is notcompletely definedyet: whathappenswhen the traffic lightchangesits state at exactly the same time as the policeman
performs its transition? Would the traffic light switch on to the next state first and then process the policeman’s interrupt, or
would it directly respond to the interrupt, ignoring the internal event? While it is a minimal difference in this case, the state
reachedafterthetimeoutmightrespondsignificantlydifferenttotheincomingevent.
DEVS solvesthisproblembydefininga tie-breakingfunction(select). Thisfunctiontakesallconflictingmodelsandreturns
theonethatgetspriorityovertheothers. Aftertheexecutionofthatinternaltransition,andpossiblytheexternaltransitionsthat
itcausedelsewhere,itmightbethatthesetofimminentmodelshaschanged. Ifmultiplemodelsarestillimminent,werepeat
theaboveprocedure(potentiallyinvokingtheselect functionagainwiththenewsetofimminentmodels).
select:2D→D
Thisnewadditionchangesthecoupledmodeltoa6-tuple.
hX ,Y ,D,{M},{I},selecti
self self i i
3.4 TranslationFunctions
Finally,inthiscasewehadfullcontroloverbothatomicmodelsthatarecombined. Wemightnotalwaysbethatlucky,asitis
possibletoreuseatomicmodelsdefinedelsewhere.Dependingontheapplicationdomainofthereusedmodels,theymightwork
withdifferentevents.Forexample,ifourpolicemanandtrafficlightwerebothpredefined,withthepolicemanusinggo_to_work
andtake_breakandthetrafficlightlisteningtoautomaticandmanual,itwouldbeimpossibletodirectlycouplethemtogether.
While it is possible to define wrapper blocks(i.e., artificial atomic models that take an eventas input and, with time advance
equaltozero,outputthetranslatedversion),DEVSprovidesamoreelegantsolutiontothisproblem.
Connectionsareaugmentedwithatranslationfunction(Z ),specifyinghowtheeventthatenterstheconnectionistranslated
i,j
before it is handed over to the endpoint of the connection. The function thus maps output events to input events, potentially
modifyingtheircontent.
Z :X →X ∀j∈D
self,j self j
Z :Y →Y ∀i∈D
i,self i self
Z :Z →X ∀i,j∈D
i,j i j
Thesetranslationfunctionsaredefinedforeachconnection,includingthosebetweenthecoupledmodel’sinputandoutputevents.
{Z |i∈D∪{self},j∈I}
i,j i
Thetranslationfunctionisimplicitlyassumedtobetheidentityfunctionif itisnotdefined. Incaseaneventneedsto traverse
multipleconnections,alltranslationfunctionsarechainedinorderoftraversal.
Withtheadditionofthisfinalelement,wedefineacoupledmodelasa7-tuple.
hX ,Y ,D,{M},{I},{Z },selecti
self self i i i,j
3.5 ClosureUnderCoupling
Similartoatomicmodels,weneedtoformallydefinethesemanticsofcoupledmodels. Butinsteadofexplainingthesemantics
from scratch, by defining some pseudo-code, we map coupled models to equivalent atomic models. Semantics of a coupled
modelisthusdefinedintermsofanatomicmodel.Inaddition,thisflatteningremovestheconstraintofcoupledmodelsthattheir
submodelsshouldbeatomicmodels:ifacoupledmodelisasubmodel,itcanbeflattenedtoanatomicmodel.
Inessence,foranycoupledmodelspecifiedas
<X ,Y ,D,{M},{I},{Z },select>
self self i i i,j
10