Generally Distributed Transition Times
Andrea Bobbioand MiklosTelek
Dipartimentodi Elettroniaperl'Automazione
Universita di Bresia, 25123Bresia, Italy
Department of Teleommuniations
Tehnial University of Budapest, Budapest, Hungary
February5, 2001
Abstrat
Theanalysisofstohastisystemswithnon-exponentialtimingrequiresthedevelopmentofsuit-
ablemodelingtools. Reently,someeorthasbeendevotedtogeneralizetheoneptofStohasti
Petrinets,byallowingtheringtimestobegenerallydistributed. TheevolutionofthePNintime
beomesastohasti proess,for whih ingeneral,noanalytialsolutionis available. Thepaper
desribessuitablerestritionsofthePNmodelwithgenerallydistributedtransitiontimes,thathave
appearedintheliterature,andomparesthesemodelsfromthepointofviewofthemodelingpower
andthenumerialomplexity.
1 Introdution
Thedesigner andtheanalystofasystemareinrstinstaneinterestedinthesolutionofthemodeling
problem, notin howthis solutionisatuallyderived. Theyshouldbeableto desribetheirsystemin
suhawaythatitiseasyandnaturaltouse. Themodeler'srepresentationshouldinludeenoughinfor-
mationtobuildupananalytialrepresentationsuitablefornumerialsolution,andshouldalsoontain
thespeiationofthe measuresofinterests. Themodeler'srepresentationshould thenautomatially
be transformed into the analytial representation. Finally the numerial results should be again au-
tomatiallymapped bak into themodeler's representation, so that the userof thetoolan interpret
themin that ontext. ForMarkoviansystemsseveraltoolshavebeendevelopedinreentyears,based
onvariousspeiationparadigms,assurveyedin[26℄.
Thereare,however,situationsthatarenotoveredbythesetools. Onetypialsituationourswhen
therandomtimesharateristiofthesystemarenotexponential. Aseondsituationourswhenthe
analyst requires the omputation of stohasti measures (like the distribution funtion of umulative
measures[34, 7℄)whosenumerialevaluationannotbeperformedbysolvingaset oflinearrstorder
equationstypialofMarkoviansystems.
InreentyearsseverallassesofStohastiPetriNet(SPN)modelshavebeenelaboratedwhihin-
orporatesomenon-exponentialharateristisintheirdenition. ThesemantisofSPN'swithgenerally
distributed transition timeshasbeendisussedin [1℄. Wereferto thismodel asGenerally Distributed
Transition SPN(GDTSPN).Ingeneral,thestohasti proessunderlying aGDT SPNdoesnothave
anumeriallytratableanalytialformulation,whileasimulativesolutionhasbeeninvestigatedin[24℄.
With theaim of providinga modeler's representation ableto automatially generate an analytial
representation,variousrestritionsofthegeneralGDT SPNmodelhavebeendisussedintheliterature.
Dugan et al. have studied the onditions under whih the stohasti realization of the GDT SPNis
a semi-Markovproess [21℄. Cumani [20℄ has realized a pakage in whih eah PN-transition anbe
assignedaPHdistributedringtime. Werefertothismodelin thefollowingasPHSPN.
Apartiularaseofnon-MarkovianSPN,isthelassofDeterministiandSPN(DSPN).ADSPNis
denedin[3℄asaMarkovianSPNwhere,ineahmarking,asingletransitionisallowedtohaveassoiated
state algorithm waspresentedin [29℄, and some strutural extensionswere investigatedin [15℄. Choi
et al. [13℄havedevelopedatehniqueforthetransientanalysisof thestateprobabilitiesof theDSPN
model. Reently,Choietal. [12℄andGermanandLindemann[23℄haveextendedthepotentialityofthe
modelbyallowingthepresenein eahmarkingofatransition withagenerallydistributedringtime.
In[12℄, theauthors haveshown that theunderlying stohastiproess isaSemi Markov Regenerative
proess, forwhih atransientaswellasasteady statesolutionanbegiven. Forthisreason,Choiet.
al. refertothismodelasMarkov RegenerativeSPN(MRSPN)[12℄. A lassiationofGDT SPNsand
oftherelatedunderlyingstohastiproessesisinCiardoet al. [14℄.
TheaimofthispaperistoomparetheavailableGDT SPNmodelsreentlyappearedinthelitera-
turefromtwodistintandonitingpointsofview: themodellingpowerandtheanalytialtratability.
Tothisend,themainfeaturesofthevariousrestritionsonsideredintheliteraturearebrieydesribed
withtheintentofstressingthebasimodelingassumptionsandtheomplexityoftherelatedanalytial
solution.
Analexample,basedonthetransientanalysisofalosedqueuingsystemwithdeterministiservie
timeandvariouskindsofpreemptiveserviepoliies,isdevelopedinlengthinordertoputinevidene
thelimitsandthepotentialitiesofthedierentapproahes.
TheGDT SPNmodelisformallydenedinSetion2. InSetion3abriefsurveyofthemostreent
restritionsappearedintheliteraureisreported. Tworestritionsaredesribedinmoredetails,namely
thePHSPN modelimplemented byCumani in [20℄ and theDSPN model desribed by Choiet al. in
[13℄. A omparativedisussion ofthe modeling powerofthe onsidered models isreportedin Setion
4. InSetion 5,startingfrom asimplequeuingsystem,inreasingmodelling omplexitiesareaddedin
ordertoshowhowtheonsideredmodelsreattotheseaddedstrutures. Thealgorithmialomplexity
ofthenumerialsolutionsisdisussedinSetion6.
2 Generally Distributed Transition SPN
AmarkedPetriNet(PN)isatuplePN =(P;T;I;O;H;M);where:
P =fp
1
;p
2
;:::;p
np
gisthesetofplaes(drawnasirles);
T =ft
1
;t
2
;:::;t
nt
gisthesetoftransitions(drawnasbars);
I,OandH aretheinput,theoutputandtheinhibitorfuntions,respetively. Theinputfuntion
I provides the multipliities of theinput ars from plaes to transitions; theoutput funtion O
providesthemultipliitiesoftheoutput arsfromtransitions toplaes;theinhibitorfuntion H
providesthemultipliityoftheinhibitorarsfrom plaestotransitions.
M =fm
1
;m
2
;:::;m
np
gisthemarking. Thegenerientrym
i
isthenumberoftokens(drawnas
blakdots) inplaep
i
,inmarkingM.
Inputandoutputarshaveanarrowheadontheirdestination,inhibitorarshaveasmallirle. A
transitionisenabledinamarkingifeahofitsordinaryinputplaesontainsatleastasmanytokensas
themultipliityoftheinputfuntionI andeahofitsinhibitorinputplaesontainsfewertokensthan
themultipliityoftheinhibitorfuntionH. An enabledtransitionresbyremovingasmanytokensas
themultipliityof theinputfuntion I fromeahordinaryinputplae,andadding asmanytokensas
themultipliity of theoutput funtion O to eah output plae. Thenumberoftokens in aninhibitor
inputplaeisnotaeted.
A marking M 0
is said to be immediately reahable from M, when is generated from M by ring
a singleenabled transition t
k
. The reahability set R(M
0
) is theset of allthe markingsthat anbe
generatedfromaninitialmarkingM
0
byrepeatedappliationoftheaboverules. IfthesetT omprises
bothtimedandimmediatetransitions,R(M
0
)ispartitionedintotangible(noimmediatetransitionsare
enabled)andvanishingmarkings,aordingto[2℄.
AtimedexeutionsequeneT
E
isaonnetedpathinthereahabilitygraphR(M
0
)augmentedbya
non-dereasingsequeneofrealnon-negativevaluesrepresentingtheepohsofringofeahtransition,
suhthatonseutivetransitionringsorrespondto orderedepohs
i
i+1 in T
E .
E 0
(0) 1
(1)
i
(i)
The time interval
i+1
i
betweenonseutive epohs representsthe period of time that thePN
sojourns inmarkingM
(i) .
A variety oftiming mehanismshavebeenproposed in the literature. The distinguishingfeatures
ofthetimingmehanismsarewhetherthedurationoftheeventsismodeledbydeterministivariables
or random variables, and whether the time is assoiated to the PN plaes, transitions or tokens. If
a probability measure is assigned to the duration of the events represented by a transition, a timed
exeutionsequeneT
E
is mappedintoastohastiproessX
T
(t);(t0),alled theMarking Proess.
PN'sin whihthetiming mehanismisstohastiare referredtoasStohastiPN(SPN).
ASPNwithstohastitimingassoiatedtothePNtransitionsandwithgenerallydistributedring
times was dened in [1℄, with partiular emphasis to thesemantial interpretation of the model. We
refertothismodelasGenerally DistributedTransitionSPN(GDTSPN).
Denition3 -A stohasti GDT SPNisamarkedSPNinwhih:
To any transition t
k
2 T is assoiated a random variable
k
modeling the time needed by the
ativityrepresentedby t
k
toomplete, whenonsideredinisolation.
Eah random variable
k
isharaterizedby the (possibly marking dependent) Cumulativedistri-
butionfuntion G
k (xjM).
A set of speiations are given for univoally dening the sstohasti proess assoiated to the
ensembleof allthetimedexeutionsequenesT
E
. Thissetofspeiationsisalledthe exeution
poliy.
A initial probability isgiven onthe reahability set.
An exeutionpoliy is a set of speiationsfor univoally dening the stohasti proess underlying
theGDT SPN, given thePN topologystruture andthe set ofCdf's G
k
(xjM). Indeed, theinlusion
of non-exponentialtimings destroysthe memorylessproperty and fores to speifyhowthe systemis
onditioneduponthepasthistory. Thesemantis of dierentexeutionpoliies hasbeen disussedin
[1℄. Theexeutionpoliyomprisestwospeiations: ariteriontohoosethenexttimedtransitionto
re(the ringpoliy), andariterionto keepmemoryofthepasthistoryofthe proess(the memory
poliy). Anaturalhoietoseletthenexttimedtransitiontoreisaordingtoaraepoliy: ifmore
thanonetransitionisenabledinagivenmarking,thetransitionreswhoseassoiatedrandomdelayis
statistially theminimum. TheMemory Poliyis thepartofthe setof speiationsof theexeution
poliythat deneshowtheproessisonditioneduponthepast. Weassoiatetoeahtransitiont
k an
agevariablea
k
. Thewayinwhiha
k
isrelatedtothepasthistoryZ
(j)
determinesthedierentmemory
poliies. Weonsider threealternatives:
Agememory -Theagevariablea
k
aountsfortheworkperformedbytheativityorresponding
to t
k
fromitslast ringuptotheurrentepoh. Theringdistribution dependsontheresidual
timeneededforthisativitytoompletegivena
k .
Enabling memory - Theage variable a
k
aountsfor the work performed by the ativity orre-
spondingtot
k
fromthelastepoh inwhih t
k
hasbeenenabled. Theringdistribution depends
ontheresidualtimeneededforthisativitytoompletegivena
k
. Whentransitiont
k
isdisabled
(evenwithoutring)theorrespondingenablingagevariableisreset.
Resampling-Theagevariablea
k
isresettozeroatanyhangeofmarking. Theringdistribution
dependsonlyonthetimeelapsedinthepresentmarking.
At the entrane in anew tangible marking, the residual ring time is omputed for eah enabled
timedtransition givenitsagevariable. Thenextmarkingisdeterminedbytheminimalresidualring
time among the enabled timed transitions (rae poliy). Under anenabling memory poliy thering
timeofatransitionisresampledfromtheoriginaldistributioneahtimethetransitionbeomesenabled
underlyingstoastiproessannotetendbeyondasingleyleofenable/disableofthetransition with
enabling memory poliy. On the ontrary, if a transition is assigned an age memory poliy, the age
variableaountsforalltheperiodsoftimein whihthetransitionhasbeenenabled,independentlyof
thenumberofenable/disableyles. Thememoryoftheproessextendsuptotherstepohinwhih
thetransitionhasbeenenabledforthersttimeafteraring.
3 Computational Restritions
The markingproess X
T
() does not have, in general, an analytially tratable formulation, while a
simulativeapproahhasbeendesribedin[24,25℄. Variousrestritionsofthegeneralmodelhavebeen
disussed in theliteraturesuhthat theunderlying markingproessX
T
()is onnedto belongto a
knownlassofanalytiallytratableproblems.
3.1 Exponentially Distributed SPN
Whentherandomvariables
k
assoiatedto thePN transitionsareexponentiallydistributed,thedy-
namibehaviourofthenetanbemappedintoaontinuoustimehomogeneousMarkovhain(CTMC),
withstatespaeisomorphitothereahabilitygraphofthenet. Thisrestritionisthemostpopularin
theliterature[31,22,2℄,andanumberofpakagesarebuiltonthismodel[11,16,30,28℄.
3.2 Semi-Markov SPN
When all the PN transitions are assigned a resampling poliy the marking proess beomes a semi-
Markovproess. Thisrestritionhasbeenstudiedin[32,5℄butisoflittleinterestinappliationswhere
itisdiÆulttoimagineasituationwheretheringofeahtransitionofthePNhastheeetofforing
aresamplingresettingtoalltheothertransitions. Onlytheaseinwhiheahtransitionisompeting
withalltheother onesseemsto beappropriateforthismodel.
A more interesting semi-Markov SPN model has been disussed in [21℄. In this denition, the
transitionsarepartitionedinto threelasses: exlusive,ompetitiveand onurrent. Providedthat the
ringtimeofallonurrenttransitionsisexponentiallydistributedandthatompetitivetransitionsare
resampledatthetimethetransitionisenabled,theassoiatedmarkingproessbeomesasemi-Markov
proess.
3.3 Phase Type SPN (PHSPN)
Anumeriallytratablerealizationof theGDT SPN,isobtainedbyrestritingtheringtime random
variables
k
tobePHdistributed[33℄, aordingtothefollowing:
Denition1 A PHSPNisaGDT SPNinwhih:
Toanytransitiont
k
2T isassoiatedaPHrandomvariable
k
withCdfG
k
(xjM). ThePHmodel
assigned to transition t
k has
k
stages with a single initial stage numbered stage 1 and a single
nal stage numberedstage
k .
Toany transition t
k
2T is assigned amemory poliy amongthe threedenedalternatives: age,
enabling orresamplingmemory.
Thedistinguishingfeatureofthismodel,isthatitispossibletodesignaompletelyautomatedtool
that responds to therequirementsstated in [26℄, and, at thesame time, inludes all the issues listed
in Denition 4. The non-markovian proess generated by the GDT SPN is onverted into a CTMC
denedoveranexpandedstatespae. Themeasurespertinenttotheoriginalproessanbeevaluated
bysolvingtheexpandedCTMC.
TheprogrampakageESP[20℄ realizesthePHSPNmodelaordingto Denition 4. Theprogram
allowstheusertoassignaspeimemory poliytoeahPNtransitionso thatthedierentexeution
poliies an be put to work. In the ESP tool, the expanded CTMC is generated from the model
algorithmisdrivenbythedierentexeutionpoliiesthattheuserassignesto eahtransition.
The expandedCTMC isrepresentedbyan orientedgraphH = (N
H
;A
H
) whereN
H
is thesetof
nodes(states oftheexpanded CTMC)andA
H
is thesetof orientedars(transitionsof theexpanded
CTMC). Thenodes in N
H
are pairs (M;W), where M 2 R (M
0
) is amarking and W is an integer
n
t
-dimensionalvetor,whosekthentryw
k
(1 w
k
k
)representsthestageofringoft
k
in itsPH
distribution.
Arsin A
H
are representedby5-tuples(N; N 0
; k;i;j), whereN is thesourenode,N 0
thedesti-
nationnode, and (i;j)is an ar in the PH model of transition t
k
. Therefore, (N;N 0
;k;i;j) 2 A
H
meansthatintheexpandedgraphtheproessgoesfromnodeN tonodeN 0
whenthestageofringof
t
k
goesfromstageito stagej.
The expanded graphH is generated by an iterative algorithm illlustrated in details in [20℄. The
markingM (`)
oftheoriginalreahabilityset,ismappedintoamarostateM (`)
formedbytheunionof
allthenodesN
H
(M;W)oftheexpandedgraphsuhthatM=M (`)
. Thismappingallowstheprogram
toredenethemeasuresalulatedassolutionofthemarkovequationovertheexpandedgraphinterms
ofthemarkingsoftheoriginalPN.
Theardinalityn
H
oftheexpandedstatespaeisoftheorderofmagnitudeoftherossprodutof
theardinality ofthereahabilityset of thebasiPN timestheardinalityofthe PH distributionsof
then
t
randomvariables
k .
An alternativeapproah for theimplementationof aPHSPN model ouldonsist in inluding the
PHmodelsforeahtransitionatthePNlevel,thusexpandingthePN.Thisapproahhasbeenstrongly
disouragedin[1℄onthebasisofthefollowingmotivations:
Theinlusionof asubnetforeahtransitionmakestheexpandedPNveryintriguedanddiÆult
tounderstandjust beausesomeprimitiveelements(plaes,transitionsandars)areadded,that
onlyrefertothestohastibehaviourofasingletransitionandhidenthegeneralstrutureofthe
model. The fasinating simpliity of thePN languageto representomplex logialinterations
betweenobjetsisdestroyed.
It seemshardly possibleto automatize aproedure forgenerating thePHSPN model exapnding
the basiPN andtaking into aountallthe possibleinteration amongtheintrodued memory
poliies.
3.4 Deterministi SPN
The Deterministi and Stohasti PN model has been introdued in [1℄, with the aim of providing a
tehniquefor onsideringstohastisystemsin whih sometimevariables assumeaonstantvalue. In
[1℄onlythesteady statesolutionhasbeenaddressed. An improvedalgorithmfortheevaluationofthe
steadystateprobabilitieshasbeensuessivelypresentedin [29℄. Reently,the DSPNmodelhasbeen
revisitedin [14℄and[13℄wherethetransientsolutionisprovided.
Denition5 -A DSPNisaGDT SPNin whih:
Toany transition t
k
2T isassoiatedanexponentiallydistributedrandom variable
k .
Atmost,asingledeterministitransition (DET)isallowedtobeenabledineahmarkingandthe
ringtimeof the deterministitransitionismarkingindependent.
The timeelapsed inaDET annot berememberedwhenthe transitionbeomes disabled; the only
allowedexeutionpoliyisthe raepoliywith enablingmemory.
Inorder toprovethatthe markingproess assoiatedto aDSPNis aMarkovregenerativeproess
(MRP),Choietal. [13℄haveintroduedthefollowingmodiedexeutionsequene:
T
E
= f(
0
;M
(0) );(
1
;M
(1)
);::: ;(
i
;M
(i)
); :::g (2)
Epoh
i+1
isderivedfrom
i
asfollows:
1. If noDET transition isenabled in markingM
(i)
,dene
i+1
to betherst timeafter
i
that a
statehangeours.
2. IfaDETtransitionisenabledinmarkingM
(i)
,dene
i+1
tobethetimewhentheDETtransition
resorisdisabled asaonsequeneoftheringof aompetitiveexponentialtransition.
Aordingto ase2) ofthe abovedenition,during [
i
;
i+1
),thePN anevolvein thesubsetof
R(M
0
)reahablefromM
(i)
,throughexponentialtransitionsonurrentwiththegivenDETtransition.
The markingproess during this time intervalis a CTMC alled the subordinated CTMC of marking
M
(i)
. Therefore, if a DET transition is enabled in M
(i)
, the sojourn time is given by the minimum
betweenthe rstpassagetime outof thesubordinated CTMC andthe onstantring timeassoiated
totheDETtransition.
Choi et al. show that the sequene [
i
forms a sequene of regenerativetime points, so that the
marking proess X
T
() is a Markovregenerative proess MRP. Aording to [12, 17℄, we dene the
followingmatrixvaluedfuntions:
V (t) = [V
ij
(t)℄ suhthat V
ij
(t) = PrfX
T
(t)=jjX
T
(0)=ig
K(t) = [K
ij
(t)℄ " K
ij
(t) = PrfM
(1)
=j;
1 tjX
T
(0)=ig
E(t) = [E
ij
(t)℄ " E
ij
(t) = PrfX
M
(t)=j;
1
>tjX
M
(0)=ig
(3)
Matrix V (t) is the transition probability matrix and provides the probability that the marking
proessX
T
(t)isinmarkingj attimetgivenitwasiniatt=0. ThematrixK(t)istheglobalkernelof
theMRPandprovidesthedf oftheregenerationintervalgiventhatthenextregenerationmarkingis
j,startinginmarkingi att=0. Finally,thematrixE(t)istheloal kernelanddesribesthebehavior
ofthe markingproessinside twoonseutiveregenerationtimepoints. The transientbehaviorof the
DSPNanbeevaluatedbysolvingthefollowinggeneralizedMarkovrenewalequation(inmatrixform)
[17,12℄:
V (t) = E(t) +K V (t) (4)
Equation(4) anbesolvednumeriallyin thetimedomain. An alternativeapproahsuggestedby
theauthorsonsists in transformingthetransientsolutionin theLaplaetransformdomain, andthen
derivingthetimesolutionbyanumerialinversiontehnique. ThepaperproposestousetheJagerman's
method[27℄, asadaptedbyChimento andTrivedi[10℄.
3.5 Markov Regenerative SPN (MRSPN)
Afurtherextension,alledMarkov RegenerativeSPN,hasbeendevelopedin [12℄andalassiationof
thestohastiproessunderlyingaGDT SPNhasbeendisussedin [15℄.
Denition2 A MRSPNisaGDT SPNinwhih:
Toany transition t
k
2T isassoiatedanexponentiallydistributedrandom variable
k .
At most, asingletransitionwith generally distributedringtimeisallowedtobe enabled ineah
marking.
The only allowed exeutionpoliyis the rae poliy with enablingmemory. This meansthat the
ring timeof the generally distributedtransition is sampled atthe timethe transition isenabled
andannothange untilthetransition eitherres orisdisabled.
The ringtimedistributionmaydependuponthe markingatthe timethe transitionisenabled.
The onvolution equation (4) still holds; however, the analyti kernel expressions depend on the
spei Cdf's assumed in the model. In [12℄, losed form expressions are derived when the Cdf of
thegenerally distributed transitions is theuniform distribution. A further approah, resortingto the
method of supplementary variables proposed by Cox [18℄, is disussed in [23, 15℄, where the use of
polyexponomialdistributions isinvestigated.
Theonsidered models dierbeauseof thedierent lassesofdistribution funtions they areableto
support,andbythewayinwhihthehistoryoftheproessistakenintoaounttoonditionthefuture
evolutionofthenet.
Under theenabling memorypoliythetimeaumulatedbyaPNtransitionisresetassoonasthe
transitionisdisabled,whileundertheagememorypoliythetimeaumulateswheneverthetransition
is enabled before ring. The enabling memory poliy is suited to realize the interation mehanism
among tasks in servie that in queueing theory orfault-tolerantsystems isalled apreemptive repeat
dierent (prd) poliy. Whenever the task in servie is preempted a orresponding PN transition is
disabledresettingtheaumulatedtime. Hene,whenthepreemptedtaskrestartsitsworkrequirement
should be resampled from the same distribution [6℄. On the other hand, the age memory poliy is
suitedtorepresentaninterationmehanismusuallyreferredtoaspreemptiveresumepoliy: theserver
doesnotloosememoryoftheworkalreadydoneevenifthetaskispreempted(andtheorresponding
PN transition disabled). When thetask isenabled againthe exeutionrestarts from thepointit was
interrupted.
TheDSPN model, ombining onstanttimes with exponentialrandom times, oersan innovative
approahinmanypratialappliations. ThemainlimitationoftheDSPNmodelinthepresentstateof
theart,isthatonlyenablingmemorypoliyissupported. Heneonlysystemswithaserviedisipline
ofpreemptivedierenttypeanberepresentedwiththisapproah. Moreover,there notoolsavailable
forthe automatigeneration ofthe matriesV (t), K(t) andE(t), and thesolutionoftheonvolution
equationisperformedbymeansofstandardpakagesforsymbolimanipulation.
Ontheontrary,thePHSPNmodelfullysupportsallthedenedmemorypoliies,and,inpartiular,
theagememorypoliy. Themodelerisallowedtorepresentinanaturalwayprsinterationmehanisms.
Moreover,iftherandomvariablesofthesystemtobemodeledarereallyofPHtype,thePHSPNprovides
exat results. Otherwise, a preliminary step is needed in whih the random times of the systemare
approximated by PH random variables resorting to asuitable estimation tehnique [8, 9℄. A tool is
oneivable[20℄forsupportingthegenerationandanalysis ofthemodelaordingto therequirements
speiedin[26℄. Theexpansionofthestatespaeis,ofourse,aauseofnonnegligiblediÆulties,sine
it worsens theproblem of the exponential growth ofthe statespae bothwith themodel omplexity,
andwiththeorderofthePHdistributionassigned toeahtransition.
5 Example - Finite Queue with Preemption
We arry ona omparison between themodeling power and the numerialresults obtainedfrom the
DSPNand thePHSPNmodelsthroughtheanalysis ofasimplenite queueingsystemswithdierent
kindsofpreemption. Weonsider,asabase example,theM/G/1/2/2(alosed queueingsystemwith
twobuerpositionsandtwoustomers)introduedin [3℄. Thenon-preemptiveserviemehanismhas
beenalreadyanalyzedin[3℄forwhatonernsthesteadystatemeasuresandrevisited in[13℄forwhat
onernsthetransientbehavior. Weinitially omparetheresultsobtainedbyapproximatingaDSPN
bymeansof aPHSPNandthenweintroduevariouskindsofpreemptivemehanisms.
5.1 Non Preemptive Queue
ThePNfortheM/G/1/2/2system,proposedin[3℄,isreportedinFigure1. Plaep
1
ontains"thinking"
ustomers (i.e. awaiting to submit a job) and transition t
a
represents the submission of jobs. Jobs
queueingforserviearerepresentedbytokensinp
2
. Atokeninp
3
meansthattheserverisbusywhilea
tokeninp
4
meansthattheserverisidle. Transitiont
g
isthejobservietime;whenthejobisompleted
theustomerreturnsin histhinkingstate. Transitiont
i
isanimmediatetransition modellingthestart
ofserviei.e. thetransferofthejob fromthequeuetotheserver.
In[3,13℄,thefollowingassumptionsweremade. t
a
isexponentiallydistributedwithratem
1
being
m
1
is thenumber oftokens in p
1
and =0:5 job/hour. t
g
is aDET transition modeling aonstant
servietimeofdurationd=1:0hour.
ThereduedreahabilitygraphofthePN(aftereliminatingthevanishingmarkingsarisingfromthe
immediate transitiont
i
[2℄) isomposedof three states,alled s
1
;s
2 and s
3
in Figure 1b. The PNof
r
r r
? 6
? 6 s
1
s
2
s
3 t
1 t
2
t
2 t
1 p
2
p
3
20
11
02
a) b)
p
4
?
?
?
?
?
?
6 -
r r
) A
A
A U
A A A K
2001
1010
0110 t
1
t
3 p
1
t
2
p
1
p
2 t
2 t
1
Figure1-a) -PNmodellingtheatomioperation ofa M/D/1/2/2(after
[3 ℄);b)-orrespondingreduedreahabilitygraph;)-simplied
PN.
Figure1isintendedtoshowindetailstheatomistepsbywhihaustomersubmitsajobandthejob
isservied. Figure2shows,however,asimplerPNisomorphitotheoneofFigure1.
Tokensinplae p
1
ofFigure2representustomersin thethinkingstate,whilep
2
ontainsthejobs
inthequeue(inluded theoneunder servie). t
1
isthesubmittingtimeandt
2
istheservietime. Itis
easyto verifythat theabovePN generatesthesamemarkingproessX
T
() of Figure1b)when t
1 is
exponentialwithrate m
1
and t
2
is DET. Theprobabilitiesversustimeof thetwostatess
1 and s
3
arereportedin Figure3in solidline.
ApproximatingtheDSPNofFigure2bymeansofthePHSPNmodelisstraightforward. Transition
t
2
isassignedaPHdistributionandanenablingmemorypoliy,inonformitywithpoint3)ofDenition
5. SinetheErlangdistributionisthePHwiththeminimumoeÆientofvariation[4℄itisappropriate
to approximate theDSPN by assigning t
2
anErlang distribution of inreasing order. In Figure 3we
omparetheresultsobtainedfromthePHSPNmodel,byreportingthebehaviorofthestateprobabilities
versustimeintwoases: whena)therandomringtimeassignedtot
2
isErlang(5)(dashedline),and
b)whenisErlang(100)(dottedline). InbothasestheexpetedvalueoftheErlangmatheswiththe
value d = 1:0hours of the DET model, being all the other parameters unhanged. It is interesting
to observethat withtheErlang(5)theloalmaxima andminimain theprobabilitybehaviordoesnot
appear, whilethevisualagreementisverysatisfatoryintheaseoftheErlang(100).
As a further omparison, Table I shows the values for the steady state probabilities alulated
from the DSPN model and from the PHSPN model when t
2
is assumed to be Erlang(5),Erlang(10),
Erlang(100)andErlang(1000),respetively. Itshouldbestressedthatthepresentaseanbeonsidered
asaworstase examplesineaDET type variablean be loselyapproximatedby aPH only asthe
numberofstages growsto1[19,9℄.
5.2 Preemptive Queue
Let us assume aM/G/1/2/2 with a preemptive servie and the samekind of ustomers. The job in
exeutionis preemptedassoonasanewjob joinsthequeue. Twoasesanbeonsidered depending
whetherthejobrestartedafterpreemptionisresampledfromthesamedistributionfuntion(preemptive
repeatdierent poliy- prd),orisresumed(preemptive resume poliy- prs).
5.2.1 prdpoliy
WithreferenetoFigure2,eahtimetransitiont
1
res(athinkingustomersubmitsajob)whilep
2 is
marked (a jobis urrently under servie)transitiont
2
should beresetand resampled. Inthe PHSPN
modelthismehanismanbesimplyrealizedbyassigningtot
2
aresamplingpoliy. Itiseasytoprove
PHSPN
State DSPN
Erl(5) Erl(10) Erl(100) Erl(1000)
TABLEI- Non preemptivepoliy
s
1
0.37754 0.38307 0.38039 0.37783 0.37757
s
2
0.48984 0.46773 0.47845 0.48867 0.48972
s
3
0.13262 0.14920 0.14116 0.13350 0.13271
TABLEII- Preemptiveprdpoliy
s
1
0.33942 0.35317 0.34642 0.34014 0.33950
s
2
0.44038 0.43122 0.43572 0.43991 0.44034
s
3
0.22019 0.21561 0.21786 0.21995 0.22017
TABLE III- Preemptiveprs poliy
s
1
0.35015 0.36194 0.35618 0.35076 0.35021
s
2 +s
3
0.45429 0.44193 0.44801 0.45365 0.45423
s
4
0.19556 0.19613 0.19582 0.19558 0.19556
thattheunderlyingproessX
T
()isasemi-Markovproess,sineeahtimethe(generallydistributed)
transitiont
2
isentered,aregenerationpointisprodued sineanewjob starts.
Evenifthelassofsemi-markovproessesisapropersublassoftheMarkovregenerativeproesses,
theabovepreemptivemehanismannot be naturallygeneratedfrom theurrentdenition ofDSPN.
Infat,sinet
1
isnotompetitivewithrespettot
2
,theringoftheformerdoesnotdisablethelatter,
thatindeedisnotresampled. ThePNinFigure4desribesthepreemptionwithoutthisanomaly. Plae
p
1
in Figure4ontainstheustomers thinking, whileplae p
2
ontainsthenumberof submittedjobs
(inluded theoneunderservie). Plaep
3
representsasinglejobgettingservie: servieisinterrupted
(t
2
is disabled)ifanew jobjoins thequeue(if transitiont
3
resbefore t
2 ). t
1 and t
3
areassigned the
exponentialsubmittingtimeandtransitionst
2 andt
4
thegenerallydistributedservietime. Assigning
anenablingmemorypoliyto t
2 andt
4
theM/G/1/2/2systemwithprdpreemptionisgenerated.
Table II ompares the steady state probabilitiesassuming thesubmitting and servie time distri-
butions idential to the non preemptivease. The transient behavior is ompared in Figure 5where
theresultsfrom theDSPNmodelaredrawnin solidlinewhiletheresultsfrom thePHSPNmodeland
withtheservietimegivenbyanErlang(5)andanErlang(100)aredrawnindashedandindottedline,
respetively.
5.2.2 prs poliy
Theprspoliymeansthatwhenanewjobjoins thequeuethejobunderservieispreempteduntilthe
newlyarrivedjob ompleteshis servie. Thepreemptedjob isresumedand putto exeutionfrom the
pointofpreemptionwithoutlossoftheworkpreviouslyperformed.
TheprsmehanismfortheM/G/1/2/2queueorrespondsto thePN ofFigure4whent
2 andt
4 is
assigned anagememorypoliy. Thepreemption mehanismdoesnott therulesofDenition 5and
thusannotbemodeled in theframeworkof theatualimplementation oftheDSPNmodel. Whent
2
tiveM/D/1/2/2
andt
4
arebothErlang(100)thenumerialresultsforthesteadystateprobabilitiesares
1
=0.4, s
2
=
0.4,s
3
=0.2. ThetransientbehaviourisdepitedinFigure 9asCase III.
5.3 Preemptive Queue with Dierent Classes of Customers
Ainterestingaseariseswhenthetwoustomersareofdierentlasses,andustomeroflass2preempts
ustomer of lass 1 but not vie versa. A PN illustrating the M/G/1/2/2 queue in whih the jobs
submitted by ustomer 2 have higher priority over the jobs submitted by ustomer 1 is reported in
Figure 6. Plae p
1 (p
3
) represents ustomer 1 (2) thinking, while plae p
2 (p
4
) represent job 1 (2)
under servie. Transitiont
1 (t
3
) is the submission of a job of type 1 (2), while transition t
2 (t
4 ) is
theompletion of servieof ajob oftype1(2). Theinhibitor arfrom p
4 tot
2
models thedesribed
preemptionmehanism: as soonasatype2jobjoins thequeuethetype1jobeventuallyunderservie
isinterrupted.
Ifweassumethattheservietimeisnotexponentiallydistributed,twopossiblepreemptionpoliies
anbeonsidered dependingwhether the job oftype 1,restarted after preemption, is resampled(prd
ase) or is resumed (prs ase). In the PHSPN model, the two poliies an be naturally realized by
assigning to the servie transitions t
2 and t
4
an enabling memory poliy in the prd ase and a age
memorypoliy intheprsase.
SineintheDSPNonlytheprdpoliyissupportedthetransientresultsforprdpoliybythedierent
methodsarereportedinFigure7andinTableIIIforwhatonernsthesteadystateprobabilityvalues.
The eet of the dierent kinds of preemptions are ompared for the DSPN in Figure 8and for the
PHSPNinFigure9. (CaseIreferstothenonpreemptivesystem,CaseIItothepreemptivesystemwith
identialustomersandprdpoliy,CaseIIItothepreemptivesystemwithidentialustomersandprs
poliy, CaseIV to thepreemptivesystemwith dierent ustomers andprd poliy, andCase V to the
preemptivesystemwith dierent ustomersand prs poliy). Inthe DSPNmodel onlyasesI,II and
IVanbeomputed.
6 Computational omplexity
Let us briey summarize the elementary omputational steps for the two onsidered methodologies
(DSPN and PHSPN), taking into aount that the DSPN solution requires manual and automati
manipulation,whilethePHSPNsolutionisfullysupportedbyatool.
r r r
A
A
A U
A
A
A U
A
A
A U
A
A
A U
A
A
A U
6 6
-
?
?
? 6
? 6 s
1
s
2
s
3 t
1 t
2
t
4 t
3 t
1 t
3
t
2 t
4
2001
1110
0201
a) b)
p
1
p
2
p
3
p
4
Figure3-PreemptiveM/D/1/2/2ithidentialustomers
6.1 Evaluation of DSPN model
Aordingto[13℄, weandividetheomputationalmethodin thefollowingsteps:
1. generation ofthereahabilitytree;
2. manualderivationoftheentries oftheK(t) andE(t)matriessymboliallyin Laplaetransform
domain;
3. symbolialmatrixinversionand matrixmultipliationbyusingastandardpakage(e.g. MATH-
EMATICA)in ordertoobtaintheV (t)(Equation4)matrixintheLTdomain;
4. time domain solution obtained by a numerial inversion of the entries of the V (t), resorting
to the Jagerman's method [27℄. Forthe sake of uniformity, this step has been implemented in
MATHEMATICAlanguage.
Step1)anbeperformedwithanyPNpakage. Step2)isdonemanually,anditsdiÆultydepends
on the non-zeroentries of the involved matries, and on the omplexity of the CTMCs subordinated
to the dierent deterministi transitions. The omputational omplexity of step 3) depends on the
dimensionof the matries(i.e. the numberof tangible markings)and theomplexityof theelements
of the kernels (whih is similar to the diÆulty of the rst step). The omplexity of the numerial
inversion at step 4) also depends on two fators; the omplexity of the funtion to invert, and the
presribedauray.
Fortheexampledesribedintheprevioussetion,theomputationaltimeforthesymboliinversion
wasnotsigniant,while thenumerialinversionrequiredabout30sonanIBMRISC 6000mahine,
foreah pointofthegraph.
6.2 Evaluation of PHSPN model
Forthe evaluation of this model weused theESPtool([20℄). Theproedure anbedevided into the
followingsteps:
1. generation ofthereahabilitytree;
2. generation oftheexpandedCTMC;
3. solutionoftheresultingCTMC.
Step 1) is standard. The omputational omplexity of steps 2) and 3) depends on the number
of tangible states and on the order of the PH distribution assoiated to eah transition. With PH
M/D/1/2/2withidentialustomers.
transitions of order n theardinality of the expanded CTMC is 2n+1in Case I,2n+1 in Case II,
n 2
+n+1in CaseIII,3n+1in CaseIV,n 2
+2n+1inCaseV.Inthistrivialexample,withn=100
(Erl100) thegenerationoftheCTMCtakes2m forCaesII andV,andthewhole analysistwofurther
minutesonthesameIBMRISC6000omputer.
7 Conlusion
Thedevelopmentofmethodologiesabletoaommodatenonexponentialrandomvariablesisofinreas-
ing interest in the analysis of stohasti systems. The paper has examined and ompared PN based
modelswhose denitionallowsthemodelertoassoiate,tosomeextent, nonexponentialdistributions
totimed PNtransitions.
Themodelingpowerandthenumerialapabilitiesareinvestigated,withpartiularreferenetothe
DSPNmodel, inwhihasingledeterministitransition anbeassigned ineahmarking(beingallthe
other transitions exponential), andthe PHSPN model in whih eah transition anbeassigned aPH
distributed ringtime.
Asimplequeueingsystemisompletelyanalysed. Evenifthedeterministidistribution istypially
non PH, anapproximationerror for the steady stateprobabilities of the order of 10 2
is reahed by
modelingthedeterministitransitionwithanErlang(5)andanerroroftheorderof10 4
bymodeling
the deterministi transition with an Erlang(1000). However, the use of PH distribution and of the
PHSPNmodeloersthemodeleramoreexibletoolfordeningamoreextendedinterationsbetween
theserverandthejobin progress.
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s
1
s
2
s
4 t
1
t
3
t
2
t
4
1010
1001
0101
a) b)
p
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p
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p
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r
?
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?
?
?
s
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J
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^ J J J J
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^ J J J J
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