Andrea Bobbio Miklos Telek
Dipartimentodi Elettronia Departmentof Teleommuniations
Universitadi Bresia TehnialUniversity of Budapest
25123 Bresia, Italy Budapest, Hungary
email: bobbioiil64.ilea.it email: telekplan.hit.bme.hu
Abstrat
The paper disusses a lass of Markov Regenera-
tive Stohasti Petri Nets(MRSPN) haraterized by
the fatthatthe stohastiproesssubordinatedtotwo
onseutiveregenerationtimepointsisasemi-Markov
reward proess. This lass of SPN's an aommo-
date transitions with generally distributedring time
andassoiatedmemorypoliyofbothenablingandage
type,thusgeneralizingandenompassingalltheprevi-
ous denitionsof MRSPN.An uniedanalytial pro-
edure is developed for the derivation of losed form
expressions for the transient and steady state proba-
bilities.
Key words: Stohasti Petri Nets, semi-Markov
RewardModels,Markov regenerativeproesses.
1 Introdution
In the usual denition of Stohasti Petri Nets
(SPN) all the timed transitions have assoiated an
exponential random variable, so that their modeling
poweris onnedto Markoviansystems. The analy-
sis ofstohastisystemswithnon-exponentialtiming
is of inreasinginterestin the literatureand requires
thedevelopmentofsuitablemodelingtools. Reently,
someeorthasbeendevotedtogeneralizetheonept
of SPN, by allowing the ring times to be generally
distributed.
An extensivedisussion ofthe semantisof SPN's
withgenerallydistributedringtimesisin[1℄,whereit
is shown that eah non-exponentialtransitionshould
be assigned a memory poliy hosen among three
proposed alternatives: resampling, enabling and age
memory. We refer to this model as Generally Dis-
tributed TransitionSPN (GDTSPN).Ingeneral,the
stohastiproessunderlyingaGDT SPNistooom-
plex to be analytially tratable, while a simulative
solutionhasbeeninvestigatedin[16℄.
With the aim of providing a modeler's represen-
representation [17℄, variousrestritionsofthegeneral
GDT SPN model have been disussed in the litera-
ture[5℄. AlassiationofSPNmodels,basedonthe
nature of the assoiated marking proess, has been
proposedbyCiardoet al. [9℄.
A partiular ase of non-Markovian SPN, is the
lass of Deterministi and SPN (DSPN) dened in
[3℄. A DSPNis anon-MarkovianSPN,where allthe
transitions are exponential, but in eah marking, at
mostonetransitionisallowedtohaveassoiatedade-
terministi ring time with enabling memory poliy.
Onlythe steady stateanalysis waselaborated in [3℄.
Animprovedsteadystatealgorithmwaspresentedin
[20℄,andsomestruturalextensionswereinvestigated
in[10℄. Choietal. [7℄havereognizedthatthemark-
ing proess underlying a DSPN is a Markov Regen-
erativeProess[11℄ forwhihalosedform transient
solutionis available. This observation has opened a
veryfertileline ofresearh aimedat thedenition of
solvable lassesof models whose underlying marking
proessisaMarkov RegenerativeProess(MRP),and
thereforereferredtoasMarkov RegenerativeStohas-
tiPetri Nets(MRSPN).
Following this line, Choi et al. [8℄ have investi-
gatedalassofmodelsinwhihonetransitionwitha
generallydistributedringtimeandenablingmemory
poliyisallowedto beenabledineahmarking. Ger-
man and Lindemann [15℄ haveproposed anumerial
solution of the same model based on the method of
supplementaryvariables[12℄.
In the mentioned referenes, the generally dis-
tributed (or deterministi) transitions must be as-
signedaringpoliyofenablingmemorytype 1
. The
enablingmemorypoliy means[1℄that wheneverthe
transition beomes enabled anew, its ring distribu-
1
Theenablingmemoryassumptionisrelaxedin[10℄forvan-
ishingmarkingsonly.Sinevanishingmarkingsaretransversed
inzerotime,thisassumptiondoesnot modifythebehaviorof
outringinpriorenablingperiodsislost. Inthelan-
guage of queueing systems the above mehanism is
referredto aspreemptive repeat dierent (prd)poliy
[14,19℄.
The possibility of inorporating non-exponential
transitions with assoiated age memory poliy has
been rst explored in [6℄. The age memory is able
to apture preemptive mehanisms of resume (prs)
type, where an interrupted ativity is reovered by
keepingmemory of theworkalready performed, and
upon restart, only the residual servie needs to be
ompleted. Thismodelingextensionisruialinon-
netionwithfaulttolerantanddependableomputing
systems, where aninterruptedtaskmust be resumed
from thepointitwasinterrupted.
The paper investigates the nature of GDT SPN
withombinedmemorypoliiessuhthattheunderly-
ingmarkingproessisaMRP.Thetimed transitions
oftheGDT SPNarepartitionedintotwosubsets: the
EXPtransitionshaveanexponentiallydistributedr-
ingtime,whilefortheGENtransitionstheringtime
is anyrandom variable(inluding thedeterministi).
TheativityyleofaGENtransitionistheintervalof
time in whih thetransition has a non-nullmemory.
Westudy the ase of MRSPN with non overlapping
ativityyles, suhthat the markingproesssubor-
dinatedtotwoonseutiveregenerationtimepointsis
a semi-Markovreward proess. The proposed model
generalizesandenompassesallthepreviousformula-
tionsofMRSPN.
InSetion2,theonditionsunderwhihthemark-
ing proess underlying aGDT SPN isa MarkovRe-
generativeproess are set in verygeneral terms. In
Setion 3, theinuene of thememory poliy onthe
ativity yle ofa transition is disussed. InSetion
4, the subordinated proess in a MRSPN with non-
overlappingativityylesisharaterized,andauni-
ed analytial solution for the transient and steady
statetransitionprobabilitymatrixisproposedinSe-
tion5.
2 Markov Regenerative Stohasti
Petri Nets
A marked Petri Net is a tuple PN = (P;T;I;O;
H;M); where: P = fp
1
;p
2
;:::;p
np
g is the set of
plaes,T =ft
1
;t
2
;:::;t
nt
gisthesetoftransitionsand
I,OandHaretheinput,theoutputandtheinhibitor
funtions,respetively. M=fm
1
;m
2
;:::;m
np gisthe
marking. Thegenerientrym
i
isthenumberoftokens
in plaep
i
,inmarkingM.
Input andoutputarshaveanarrowheadontheir
destination,inhibitorarshaveasmallirle. Atran-
inputplaes ontainsat least as manytokens asthe
multipliity of the input funtion I and eah of its
inhibitorinputplaes ontainsfewertokens thanthe
multipliity of theinhibitorfuntion H. An enabled
transition res by removing as many tokens as the
multipliity of the input funtion I from eah ordi-
naryinput plae, and adding asmany tokens asthe
multipliity ofthe output funtion O toeah output
plae. The number of tokens in an inhibitor input
plaeisnotaeted.
AmarkingM 0
is saidto beimmediately reahable
from M, when is generated from M by ring an en-
abledtransition. ThereahabilitysetR(M
0
)istheset
ofallthemarkingsthatanbegeneratedfromanini-
tialmarkingM
0
byrepeatedappliationoftheabove
rules. If the set T omprisesboth timed and imme-
diate transitions, R(M
0
) is partitioned into tangible
(noimmediatetransitionsareenabled)andvanishing
markings. Sinetheeet of vanishingmarkingsan
be inorporatedinto the tangible ones, aording to
[2℄, wedo notaountin this paperfor thepresene
ofimmediatetransitions. LetN betheardinalityof
thetangible subsetofR(M
0 ).
Denition 1 - A stohasti GDT SPN is amarked
SPNinwhih [1 ℄:
To any timed transition t
k
2 T is assoiated a
random variable
k
,with umulativedistribution
funtion G
k
(x), modeling the time neededby the
ativityrepresentedbyt
k
toomplete, whenon-
sideredin isolation.
Eah timed transition t
k
is attahed a memory
variable a
k
and a memory poliy; the memory
poliy speies the funtional dependene of the
memoryvariable onthe pastenablingtimeofthe
transition.
Ainitial probability isgiven onR(M
0 ).
Thememory variable a
k
, assoiated to transition t
k ,
isafuntionalthatdependsonthetimeduringwhih
t
k
hasbeenenabled. Thememoryvariables together
withtheir memorypoliy univoally speifyhowthe
underlying stohasti proessis onditioned upon its
pasthistory. Thesemantisofdierentmemorypoli-
ieshasbeendisussedin [1℄wherethreealternatives
havebeenproposedandexamined.
Resampling poliy - The memory variable a
k is
resetto zeroatanyhangeofmarking.
Enabling memory poliy - The memory variable
a aounts for the elapsed time sine the last
k
sitiont
k
isdisabled(evenwithoutring)theor-
respondingenablingmemoryvariableisreset.
Age memory poliy - The memory variable a
k
aountsfortheelapsedtimesinethelastepoh
in whiht
k
hasbeenenabledwithoutring. The
memory variableis resetonlywhen t
k
res(and
notwhenitissimplydisabled).
Attheentraneinanewtangiblemarking,theresid-
ual ring time is omputed for eah enabled timed
transition givenitsmemoryvariable,sothat thenext
marking isdeterminedby theminimal residualring
time among the enabled transitions (rae poliy [1℄).
Beause ofthememorylessproperty, thevalueofthe
memoryvariableisirrelevantindeterminingtheresid-
ualringtimeforexponentialtransitions, sothatthe
three mentionedpoliiesare ompletelyequivalentin
this ase. Hene,for anexponentialtransitiont
k , we
assume, onventionally,that theorrespondingmem-
ory variableis alwaysidentiallyzero. Weanthere-
forepartitionthesetofthetransitionsintoEXPtran-
sitionswithassoiatedanexponentialr.v. andidenti-
ally zeromemoryvariable,andGENtransition with
assoiatedany r.v. (inludingthedeterministiase)
andmemoryvariableinreasingintheenablingmark-
ings.
Denition 2 - The stohasti proess underlying a
GDT SPN is alled the marking proess M(x) (x
0). M(x) isthe markingof the GDT SPNattimex.
A singlerealizationofthemarkingproessM(x)an
bewrittenas:
R = f(
0
;M
0 );(
1
;M
1
);:::;(
i
;M
i );:::g
whereM
i+1
isamarkingimmediatelyreahablefrom
M
i ,and
i+1
i
isthesojourntimein markingM
i .
With theabovenotation, M(x)=M
i for
i
x <
i+1 .
Assertion 1 - If at time +
i
of entrane in a tan-
gible marking M
i
all the memory variables a
k (k =
1;2;:::;n
t
) are equal to zero,
i
is a regeneration
time pointfor the markingproess M(x).
Infat,ifallthememoryvariablesareequalto0,the
future ofthemarkingproessisnotonditionedupon
thepastanddependsonlyonthepresentstate;hene,
theMarkovpropertyholds.
Let us denote by
n
thesequene ofthe regenera-
(n)
point
n
is alled a regeneration marking. The se-
quene (
n
;M
(n)
)is aMarkovrenewalsequene and
the markingproess M(x) is aMarkov regenerative
proess[11,8, 9℄. FromAssertion1followsthat:
i) ifallthetransitionsareEXPallthememoryvari-
ablesare identially zero so that any instant of
timeisaregenerationtimepoint,andtheorre-
spondingproessisaCTMC;
ii) if at any ring all the memory variables of the
GENtransitionsarereset,theorrespondingpro-
essreduestoasemi-Markovproess.
iii) only GEN transitions are relevant to determine
theourreneofregenerationtimepoints.
Denition3 -AGDT SPN,for whih anembedded
Markovrenewalsequene(
n
;M
(n)
)exists,isalleda
Markov Regenerative Stohasti Petri Net (MRSPN)
[8℄.
Sine (
n
; M
(n)
) is a Markov renewal sequene, the
followingequalitieshold:
PrfM
(n+1)
=j;(
n+1
n )xj
M
(n)
=i;
n
;M
(n 1)
;
n 1
;:::; M
(0)
;
0 g =
PrfM
(n+1)
=j;(
n+1
n
)xjM
(n)
=i;
n g =
PrfM
(1)
=j;
1
xjM
(0)
=ig
(1)
TherstequalityexpressestheMarkovproperty(i.e.
in any regeneration time point the ondition on the
past is ondensed in the present state). The se-
ondequalityexpressesthetimehomogeneity(i.e. the
probabilitymeasuresareindependentofatranslation
alongthe time axis). Aording to [8, 11℄, we dene
thefollowingmatrixvaluedfuntionsV (x)=[V
ij (x)℄,
K(x)=[K
ij
(x)℄andE(x)=[E
ij
(x)℄(allofdimension
NN),suhthat:
V
ij
(x)=PrfM(x)=jjM(
0 )=ig
K
ij
(x)=PrfM
(1)
=j;
1
xjM(
0 )=ig
E
ij
(x)=PrfM(x)=j;
1
>xjM(
0 )=ig
(2)
V (x)isthetransitionprobabilitymatrixandprovides
theprobabilitythatthestohastiproessM(x)isin
markingj at timex givenit wasin i at x =0. The
matrixK(x)istheglobal kerneloftheMRPandpro-
time point is
1
and the next regeneration marking
is M
(1)
= j given markingi at
0
=0. Finally, the
matrixE(x)istheloal kernelsinedesribesthebe-
haviorofthemarkingproessM(x)insidetwoonse-
utive regeneration time points. The generielement
E
ij
(x)providestheprobabilitythattheproessstays
in statej at time x startingfrom i at
0
=0before
thenextregenerationtimepoint. Fromtheabovedef-
initions:
X
j [K
ij
(x) +E
ij
(x)℄ = 1
The transient behavior ofthe MRSPN anbeevalu-
ated bysolving the following generalizedMarkovre-
newalequation(in matrixform)[11,8℄:
V (x) = E(x) + KV (x) (3)
whereKV (x)isaonvolutionmatrix,whose(i;j)-
th entryis:
[K V (x)℄
ij
= X
k Z
x
0 dK
ik (y)V
k j
(x y) (4)
BydenotingtheLaplaeStieltjestransform(LST)of
afuntionF(x)byF
(s)= R
1
0 e
sx
dF(x),Equation
(3) beomesintheLST domain:
V
(s) = E
(s)+ K
(s)V
(s) (5)
whose solutionis:
V
(s) =[I K
(s)℄
1
E
(s) (6)
Ifthesteadystatesolutionexists,itanbeevaluated
as lim
s!0 V
(s).
As speiedby(2), K(x) andE(x) depend onthe
evolution of the marking proess between two on-
seutive regeneration time points. By virtue of the
time homogeneity property (1), we an always de-
ne thetwosuessiveregenerationtimepointstobe
x=
0
=0andx=
1 .
Denition 4 - The stohasti proess subordinated
tostatei (denotedbyM i
(x)) istherestrition ofthe
marking proessM(x) for x
1
given M(
0 )=i:
M i
(x) = [M(x) : x
1
;M(
0 )=i℄
AordingtoDenition4,M i
(x)desribestheevolu-
tionofthePNstartingattheregenerationtimepoint
x=0intheregenerationmarkingi,uptothenextre-
generation time point
1
. Therefore, M i
(x)inludes
allthemarkingsthatanbereahedfromstateibe-
fore thenextregenerationtime point. Theentries of
thei-throwofthematriesK(x)andE(x)aredeter-
minedbyM i
(x).
3 Non-Overlapping AtivityCyles
The analytial tratability of themarking proess
depends on the struture of the subordinated pro-
esses whih, in turns, is related to the topology of
thePN andtothememorypoliiesof theGENtran-
sitions.
Denition5 -AGENtransitionisdormantinthose
markingsinwhih theorresponding memoryvariable
is equal to zero and is ative in those markings in
whih the memory variable isgreater thanzero. The
ativityyleofaGENtransitionistheperiodoftime
in whih a transition is ative between two dormant
periods.
Let us onsider a single generi GEN transition t
g .
The ativity yle of t
g
is inuened by its memory
poliy,and anbeharaterizedin thefollowingway.
ResamplingMemory - Ift
g
isaresamplingmem-
ory transition, its ativity yle starts as soon ast
g
beomesenabled,andendsattherstsubsequentr-
ingof anytransition (inludingt
g
itself). Therefore,
duringtheativityyleofaresamplingmemorytran-
sitionnohange ofmarkingispossible.
Enabling Memory - If t
g
is an enabling memory
transition its ativity yle starts as soon as t
g be-
omes enabledwhen dormant, and ends either when
t
g
res,orwhenitbeomesdisabledbytheringofa
ompetitivetransition. Duringtheativityyle the
marking anhange inside the enabling subset of t
g
(where the enabling subset is dened as the subset
of onneted markings in whih t
g
is enabled). The
memoryvariable assoiated to t
g
growsontinuously
duringtheativityylestartingfrom0. Weassoiate
arewardvariableequalto1toallthestatesintheen-
ablingsubset,sothatthevalueofthememoryvariable
isrepresentedbythetotalaumulatedreward.
Age Memory - If t
g
is an age memory transition,
itsativityylestartsassoonast
g
beomesenabled
whendormant,andendsonlyattheringoft
g itself.
Duringtheativityyleofanagememorytransition
there is no restrition on the markings reahable by
the markingproess. The age memory poliy is the
onlypoliyinwhihatransitionanbeativeevenin
markingsinwhihitisnotenabled. Duringtheativ-
ityyle,thememoryvariableisnon-dereasinginthe
sensethatitinreasesontinuouslyin thosemarkings
inwhiht
g
isenabledandmaintainsitsonstantposi-
tivevalueinthosemarkingsinwhiht
g
isnotenabled.
Inorder to trakthe enabling/disablingondition of
t
g
duringitsativityyle,weintrodueareward(in-
diator)variablewhihisequalto1inthosemarkings
in whih t is enabledand equalto 0in those mark-
TableI -Charaterizationoftheativityyleofa
GENtransition t
g
Memory Resamp. Enabling Age
poliy
startof t
g
t
g
enabled t
g
enabled
ativity enabled when when
yle dormant dormant
endof ring ringor ring
ativity ofany disabling of
yle transition oft
g
t
g
reahable starting markingsin any
markings marking enabling reahable
only subset marking
memory inreasing inreasing inreasing
variable oronstant
ingsin whiht
g
isnotenabled. Thememoryvariable
orrespondstothetotalaumulatedreward.
The above features are summarized in Table 1. By
virtue of Assertion 1, a regeneration time point for
the marking proess ours when a ring auses all
theativeGENtransitions tobeomedormant.
Denition 6- Atransitionisdominant if itsativ-
ity ylestritlyontains the ativityylesof allthe
ative transitions.
Denition 7- AMRSPN withnon-overlapping a-
tivityylesisaMRSPNinwhih alltheregeneration
periodsare dominatedby asingletransition: any two
suessive regeneration time points orrespond to the
startandtotheendoftheativeyleofthedominant
transition.
Denition 7, inludes the possibility that the ative
yles of GEN transitions are ompletely ontained
intotheativeyleofthedominantone,heneallow-
ing thesimultaneous enablingofdierentGENtran-
sitionsinsidethesamesubordinatedproess. Inorder
further restrit the subordinated proess inside any
non-overlappingativityyletobesemi-Markov.
Assertion 2 - The subordinated proess underlying
any non-overlapping ativity yle is semi-Markov if
atanyringinside the ativityyleofthe dominant
transitionallthememoryvariablesoftheGENtransi-
tionarereset. Thisfathappens ifthetransitionsan
be partitioned into three lasses (exlusive, ompeti-
tiveandonurrent)andonlyexlusiveorompetitive
transitions areallowedtobeGEN[13 ℄.
Foraregenerationperiodwithoutinternalstatetran-
sitions (Markovian or semi-Markovian regeneration
period) any of theenabled transitions anbehosen
tobethedominantone.
4 The Subordinated Proess
Atx=
0
=0adominantGENtransitiont
g (with
memoryvariable a
g
and ring time
g
) startsits a-
tivityylein statei(a
g
=0). Thesuessiveregen-
erationtime point
1
is the end ofthe ativity yle
oft
g
aordingto therulessummarizedin TableI.
Let Z i
(x) (x 0) be theproess dened overthe
states reahable from i during the ativity yle of
t
g
, and r i
the orresponding binary reward vetor.
We assume in the following that Z i
(x) is a semi-
Markov proess aording to Assertion 2. The sub-
ordinatedproessM i
(x)(Denition4)oinideswith
Z i
(x)whentheinitialstateisstateiwithprobability
1(PrfZ i
(0) =ig=1). The memoryvariable a
g in-
reasesatarater i
j
(whihiseitherequalto0orto1)
whenM i
(x)=j.
Weonsiderseparatelythefollowingasesdepend-
ingwhetherthedominanttransitiont
g
isofenabling
oragememorytype.
4.1 Enabling type dominant transition
The dominant GEN transition t
g
is of enabling
type. The statespae of thesubordinated proess is
partitioned into two subsets: R i
ontains the states
inwhiht
g
isontinuouslyenabled,andR i
ontains
thestatesinwhiht
g
beomesdisabledbytheringof
aompetitive transition. The rewardvetor isequal
to 1 for j 2 R i
and 0elsewhere. The next regener-
ation time point ours beauseone of the following
twomutuallyexlusiveevents:
t
g
res: this event an be formulatedas aom-
pletion time problem [4℄ when the aumulated
reward(memoryvariable)a
g
reahesan absorb-
ingbarrier equaltotheringrequirement
g .
t is disabled: this eventanbeformulatedasa
r r
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t
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4
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Figure1-a)PNoftheperiodiallyselftestedM/M/1/k;
b)orrespondingreahabilitygraph.
rst passage time in the subset R i
, and there-
fore R i
is made absorbing in the subordinated
proess.
Wefurtherpartiularizethefollowingtwoases:
CASE A - no other GEN transitions are ativated
during the ativity yle of t
g
. The subordinated
proess Z i
(x) isaCTMC.
CaseAistheoneonsideredintheDSPNmodel
dened in [3,7,20℄, andin thesuessiveexten-
sionstogeneraldistributionselaboratedin[8,15℄.
All the examplesreported in the mentioned pa-
persbelongtothis ase.
CASEB-duringtheativityylesoft
g
,Assertion2
issatisedandthesubordinatedproessisasemi-
Markov proess.
The Markovian (semi-Markovian) regeneration
period belongs to Case A (Case B), where R i
ontainsonlythe initial state. Thesteady state
analysis of aMRSPN with semi-Markoviansub-
ordinated proesshasbeenonsideredin[9℄.
Example1-Aperiodially self-testedsystem.
A system is exeuting tasks aording to a
M/M/1/kqueue(Figure1a). Plaep
1
representsuser
thinking andp
2
isthequeueinludingthetaskunder
servie. t
1
is the exponential submitting time with
markingdependentratem
1
,andt
2
istheexponen-
tialservietimewithrate. p
3
representsthesystem
waiting forthetest andp
4
thesystemunder test. t
3
is thedeterministitesting interval, andt
4
theexpo-
nentiallydistributed testdurationwithrateÆ. When
t
3
restheexeutionoftheM/M/1/kqueueisfrozen
untilthetest isompleted (t res). Thestatespae
Æ
Æ
Æ
Æ
q
Æ
Æ Æ s
1
s
2
s
4 t
1 t
3
t
2 t
4
1010
1001
0101
a) b)
p
1
p
2 p
4 p
3
q
?
?
?
?
?
?
?
?
Æ
s
3
J
J
J
J
^ J J J
℄
J
J
J
J
^ J J J
℄
/ 0110
b
t
1
t
1 t
2 t
3
t
3 t
4 t
4
Figure2-PreemptiveM/G/1/2/2queuewithtwolasses
ofustomers.
of the PN of Figure 1a with k = 2 ustomers is in
Figure 1b. All thestates anbe regenerationstates,
but notall the transitions provideregeneration time
points. States s
4 , s
5 and s
6
are alwaysregeneration
statesfrom whih asingleEXP transitionis enabled
(CaseA).Statess
1 ors
2 ors
3
areregenerationstates
onlywhenenteredbyring t
4
,i.e. whenthe ativity
yleofthedominantGENtransitiont
3
starts. Dur-
ingthe ativity yleof t
3
, the subordinated proess
anmove among s
1 , s
2 and s
3
whih thereforeform
thesubordinated CTMC(CaseA).
Iftransitionst
1 andt
2
areGENwithenablingmem-
orypoliy,thefeatures ofstatess
4 ,s
5 ands
6 donot
hange,whilethesubordinatedproessduringthea-
tivityyleoft
3
beomessemi-Markovianthusrepre-
sentingaCaseBexample.
4.2 Age type dominant transition
Thesituation in whih thedominant GENtransi-
tiont
g
is ofagetypehasbeenaddressedfortherst
timein [6℄. Thestatespaeof thesubordinatedpro-
essR i
ontainsallthestatesreahableduringthea-
tivityyleoft
g
,andthedisablingsubsetR
isempty
(the onlyriterionfortheterminationofthe ativity
yleistheringoft
g
). Therewardvetorisequalto
1forthestatesj2R i
inwhiht
g
isenabledand0for
thestatesj2R i
inwhiht
g
isnotenabled. Thering
oft
g
anbeformulatedasaompletiontimeproblem
[4℄ when the aumulated reward(memory variable)
a
g
reahestheringrequirement
g
. Wefurtherpar-
tiularizethefollowingtwoases:
CASE C - During the ativity yle of t
g
no other
GEN transitions are ativated and the subordi-
natedproessisarewardCTMC.
CASED - duringtheativityylesoft
g
,Assertion
2issatisedandthesubordinatedproessisaRe-
ward semi-Markovproess.
Example 2 - Preemptive M/G/1/2/2 with dierent
ustomers
Inthisexample,CasesCandDaremixedinasin-
glePN[5℄. ThePNofFigure2amodelsaM/G/1/2/2
queueinwhihthejobssubmittedbyustomer2have
higherpriorityandpreemptthejobssubmittedbyus-
tomer1. Theserverhasaprsserviedisipline. Plae
p
1 (p
3
)representsustomer1(2)thinking,whileplae
p
2 (p
4
)representjob 1(2) under servie. Transitions
t
1 and t
3
are EXPandrepresentthe submissionofa
job of type 1or 2, respetively. t
2 and t
4
are GEN
transitions,andrepresenttheompletionofservieof
ajob oftype1or2,respetively. Aprs serviedisi-
plineismodeledbyassigningtot
2 andt
4
anagemem-
orypoliy. Theinhibitorarfromp
4 tot
2
modelsthe
desribedpreemptionmehanism: assoonasatype2
jobjoinsthequeuethetype1jobeventuallyunderser-
vie isinterrupted. Thereahabilitygraphof thePN
ofFigure2aisinFigure2b. Underaprsservie,after
ompletionofthetype2job,theinterruptedtype1job
isresumedontinuingthenewservieperiodfromthe
pointreahed just before thelast interruption. From
Figure 2b, it is easily reognized that s
1 , s
2 and s
3
anallberegenerationstates,whiles
4
anneverbea
regenerationstate(ins
4
atype2job isalwaysinex-
eution sothat itsorrespondingmemoryvariablea
2
is never0). Onlyexponentialtransitionsare enabled
in s
1
and the next regeneration states an be either
s
2 ors
3
depending whether t
1 ort
3
resrst. From
state s
3
thenext regenerationmarkinganbeeither
states
1 ors
2
dependingwhetherduringtheexeution
ofthetype2jobatype1jobdoesrequireservie(but
remainsblokeduntilompletionofthetype2job)or
doesnot. Thesubordinated proessis aCTMC, and
belongstoCaseC.Froms
2
thenextregenerationstate
anbeonlys
1
,butmultipleyles(s
2 -s
4
)anour
dependingwhethertype2jobsarrivetointerruptthe
exeutionofthetype1job. Thesubordinatedproess
isaSMP(t
4
isGEN),andbelongstoaseD.
5 Unied Transient Analysis
TheglobalandloalkernelsK(x)andE(x)anbe
evaluated rowbyrow. Inthis setion, weprovidean
uniedanalytialproedurefordeterminingin losed
form the entries of a generi row i, given that i is
a regeneration marking whose subordinated proess
is asemi-Markov rewardproess as desribed in the
previoussetion.
Let Q i
(x) = [Q i
k `
(x)℄ be the kernel of the subor-
dinatedsemi-Markovproess(Z i
(x)). Z i
(x)startsin
markingM
i
(Z (0)=i),sothattheinitialprobability
vetorisV i
0
=[0;0;:::;1
i
;:::; 0℄(avetorwithall
theentriesequalto0butentryiequalsto1). Forno-
tationalonvenienewedonotrenumberthestatesin
Z i
(x)sothatallthesubsequentmatrixfuntionshave
thedimensions(NN)(ardinalityof R(M
0 )),but
with the signiant entries loated in position (k;`)
only, withk;`2R i
[R i
. Wedenoteby H thetime
durationuntiltherstembeddedtimepointin Z i
(x)
fromtimex=0.
Letusx thevalueof theringrequirement
g
=
w, and let us dene the following matrix funtions
P i
(x;w),F i
(x;w),D i
(x;w)and i
:
P i
k `
(x;w) = PrfZ i
(x)=`2R i
;
1
>xj
Z i
(0)=k2R i
;
g
=wg
F i
k `
(x;w) = PrfZ i
(
1
)=`2R i
;
1 x;t
g resj
Z i
(0)=k2R i
;
g
=wg
D i
k `
(x;w) = PrfZ i
(
1
)=`2R i
;
1 xj
Z i
(0)=k2R i
;
g
=wg
i
k `
= Prfnexttangiblemarkingis`j
urrentmarkingisk; t
g resg
(7)
By the above denitions, the entries P i
k `
(x;w) and
F i
k `
(x;w) are signiant only for k;` 2 R i
and are
0 otherwise; the entries D i
k `
(x;w) are signiant for
k2R i
and`2R i
,andare0otherwise.
P i
k `
(x;w) is theprobabilityof beingin state `2
R i
attimex beforeabsorptioneitherat thebar-
rierworintheabsorbingsubsetR i
,startingin
statek2R i
atx=0.
F i
k `
(x;w)istheprobabilitythatt
g
resfromstate
`2R i
(hitting theabsorbingbarrier win `) be-
forex,startinginstatek2R i
at x=0.
D i
k `
(x;w)is theprobabilityofrstpassagefrom
astatek 2 R i
to a state` 2R i
before hitting
thebarrierw, startingin statek2R i
at x=0.
i
isthe branhing probabilitymatrixand rep-
resentsthe suessor tangible marking ` that is
reahedbyring t
g
in statek2R i
(the ringof
t
g
in the subordinated proess M i
(x), an only
ourinastatek inwhih r i
k
=1).
From(7),itfollowsforanyx:
X
i i [P
i
k `
(x;w)+ F i
k `
(x;w)+ D i
k `
(x;w)℄ = 1
g
funtion of the r.v.
g
assoiated to the transition
t
g
,theelementsofthei-throwofmatriesK(x) and
E(x) anbeexpressedasfollows,asafuntionofthe
matriesP i
(x;w),F i
(x;w)andD i
(x;w):
K
ij (x) =
Z
1
w=0 [
X
k 2R i
F i
ik
(x;w) i
k j +
D i
ij
(x;w)℄dG
g (w)
E
ij (x) =
Z
1
w=0 P
i
ij
(x;w)dG
g (w)
(8)
In order to avoidunneessarily umbersome nota-
tion in the following derivation, we neglet the ex-
pliitdependeneonthepartiularsubordinatedpro-
essZ i
(x),byeliminatingthesupersripti. Itishow-
evertaitlyintended, that allthequantities r, Q(x),
P(x;w), F(x;w), D(x;w), ,R andR
referto the
spei proess subordinated to the regenerationpe-
riod startingfrom statei.
5.1 Derivation of P(x;w), F(x;w) and
D(x;w)
The derivation of these matrix funtions is de-
sribedin moredetailin [21, 6℄ andfollowsthesame
pattern of theompletion time analysis presentedin
[19,4℄.
Theorem1- FortheringprobabilityF
k `
(x;w)the
following doubletransformequationholds:
F
k `
(s;v) = Æ
k ` r
k
[1 Q
k
(s +vr
k )℄
s+ vr
k
+
X
u2R Q
k u
(s +vr
k )F
u`
(s;v)
(9)
Proof - Conditioningon H =h and
g
=w, let us
dene:
F
k `
(x;wjH=h) =
8
>
>
>
>
>
>
<
>
>
>
>
>
>
: Æ
k ` U
x w
r
k
if:hr
k w
X
u2R dQ
k u (h)
dQ
k (h)
F
u`
(x h;w hr
k )
if:hr
k
<w
(10)
In (10), two mutually exlusive events are identi-
ed. Ifr 6=0andhr w,asojourntimeequalsto
ingtime(nextregenerationtimepoint)is
1
=w=r
k .
If hr
k
< w then atransition ours to stateu with
probability dQ
k u (h)=dQ
k
(h) and theresidual servie
(w hr
k
)shouldbeaomplishedstartingfromstate
uattime(x h). TakingtheLSTtransformwithre-
spettox(denotingthetransformvariablebys),the
LTtransform withrespetto w (denoting the trans-
form variable by v) of (10) and unonditioning with
respettoH,(10)beomes(9). 2
Theorem2- ThestateprobabilityP
k `
(x;w)satises
the following doubletransformequation:
P
k `
(s;v) = Æ
k `
s[1 Q
k
(s +vr
k )℄
v(s +vr
k )
+
X
u2R Q
k u
(s +vr
k )P
u`
(s;v)
(11)
Proof - Conditioningon H =h, and
g
=w letus
dene:
P
k `
(x;wjH =h) =
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
: Æ
k `
U(x) U
x w
r
k
if:hr
k w
Æ
k `
[U(x) U(x h)℄+
X
u2R dQ
k u (h)
dQ
k (h)
P
u`
(x h;w hr
k )
if:hr
k
< w
(12)
ThederivationofthematrixfuntionP(x;w)based
on (12)follows the samepattern asfor the funtion
F(x;w)[21℄. 2
Theorem3- Theprobability D
k `
(x;w)of rstpas-
sage into R
satises the following double transform
equation:
D
k `
(s;v) = 1
v Q
k l
(s+ vr
k )+
X
u2R Q
k u
(s + vr
k )D
u`
(s;v)
(13)
Proof - Conditioningon H =h, and
g
=w letus
dene:
D
k `
(x;wjH=h) =
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
0 if:hr
k w
dQ
k ` (h)
dQ
k (h)
U(x h)+
X
u2R dQ
k u (h)
dQ
k (h)
D
u`
(x h;w hr
k )
if:hr
k
< w
(14)
The derivation of the matrix funtion D(x;w)
basedon(14)followsthesamepatternasforthefun-
tionF(x;w)[21℄. 2
5.2 ThesubordinatedproessisaReward
CTMC
Let us onsider the partiular ase in whih the
subordinated proess Z(x) is a reward CTMC with
innitesimal generator A = fa
k `
g. Let us suppose
that the states numbered 1;2;:::;m belong to R
(1;2;:::;m2R )andthestatesnumberedm+1;m+
2;:::;nbelongto R
(m+1;m+2;:::;n2R
). By
this orderingof statesA an be partitioned into the
followingsubmatriesA=
B C
U
1 U
2
whereBon-
tains theintensityof thetransitions inside R ,andC
ontainstheintensityofthetransitionsfromRtoR
.
U
1 andU
2
refertotheportionofthestatespaenot
involvedintheurrentsubordinatedmarkingproess,
andare,thus,notinuentialfortheproblemathand.
Forthisreason,theirentriesanbeassumedequalto
zero.
Corollary 4 - The entries of the matrix funtions
P
k `
(x;w), F
k `
(x;w) and D
k `
(x;w), in double trans-
formdomain, takethe followingexpression:
(s+vr
k )F
k `
(s;v) = Æ
k ` r
k +
X
u2R a
k u F
u`
(s;v)
(s+vr
k )P
k `
(s;v) = Æ
k ` s
v +
X
u2R a
k u P
u`
(s;v)
(s+vr
k )D
k `
(s;v) = a
k `
v +
X
u2R a
k u D
u`
(s;v)
(15)
Proof - Thekernel(transitionprobabilitymatrix) of
thegivenCTMC anbewritten as:
Q
k ` (x) =
>
<
>
: a
k `
a
k k (1 e
ak kx
) if:k 6= `
0 if:k = `
(16)
andinLST domain:
Q
k ` (s) =
8
>
<
>
: a
k `
s a
k k
if:k 6= `
0 if:k = `
(17)
witha
k k
= P
`2R i
[R i
;`6=k a
k `
By substituting (17) into (11), (9) and (13), the
orollaryisproved.2
Equations(15)anberewrittenin matrixform:
F
(s;v) = (sI+vR B) 1
R
P
(s;v) = s
v
(sI+vR B) 1
D
(s;v) = 1
v
(sI+vR B) 1
C
where I is theidentity matrixand R is thediagonal
matrixof therewardrates (r
k
); thedimensionsof I,
R,B,FandPare(mm),andthedimensionsofC
andDare(m(n m)).
6 Numerial Results
Anumerialderivationofthetransientstateprob-
abilities of the M/D/1/2/2 system desribed in Ex-
ample 2 of Setion 4.2 is provided. We onsider in
details the partiular ase in whih the GEN transi-
tionst
2 and t
4
areassumed to bedeterministi with
duration,while t
1 andt
3
are EXPwithparameter
[6℄. The reahabilitygraphin Figure 2b omprises
4states. Let us build upthe K
(s)and E
(s) ma-
triesrowbyrow,takingintoonsiderationthatstate
s
4
anneverbearegenerationmarkingsineatype2
jobwithnonzeroagememoryisalwaysative.
i)- The startingregenerationstate iss
1
- Nodeter-
ministi transitions areenabled: the state is Marko-
vian and the next regeneration state an be either
states
2 ors
3
. Thenonzeroelementsof the1-st row
ofmatriesK
(s)andE
(s)taketheform:
K
12 (s) =
s +2
; K
13 (s) =
s +2
E
11 (s) =
s
;
ii) - Thestartingregenerationstateiss
2
-Transition
t
2
isdeterministisothat thenextregenerationtime
point is the epoh of ring of t
2
. The subordinated
proess M 2
(x) omprises states s
2 and s
4
and is a
semi-Markovproess(CaseD) sinet
4
is determinis-
ti. Thekernelofthesemi-Markovproessis:
Q
(s)=
0 0 0 0
0 0 0
s +
0 0 0 0
0 e s
0 0
The rewardvetorisr (2)
= [0;1;0;0℄, andtheonly
nonzero entryof the branhing probability matrix is
(2)
21
=1. ApplyingEquations(9)and(11)weobtain
thefollowingresultsforthenonzeroentries:
F
22
(s;w) =
1
s+w+ e s
P
22
(s;w) =
s=w
s+w+ e s
P
24
(s;w) =
(1 e s
)=w
s+w+ e s
Applying (8), and after inverting the LT transform
with respet to w, the LST matrix funtions K
(s)
andE
(s)beome:
K
21
(s) = e
(s+ e s
)
E
22 (s) =
s[1 e
(s+ e s
)
℄
s+ e s
E
24 (s) =
(1 e s
)[1 e
(s+ e s
)
℄
s+ e s
iii)- Thestartingregenerationstateiss
3
- Thesub-
ordinated proessM 3
(x)isaCTMC(CaseC),hene
theresultsofSetion5.2apply. Theinnitesimalgen-
erator oftheCTMC is:
A=
0 0 0 0
0 0 0 0
0 0
0 0 0 0
and the reward vetor is r (3)
= [0;0;1;1℄. The
branhing probabilities arising from the ring of t
are
31
= 1 and
42
= 1. Applying the rst and
seondequation in (15),the nonzeroentries takethe
form:
F
33
(s;w) =
1
s++w
F
34
(s;w) =
(s+w)(s++w)
P
33
(s;w) =
s
w(s++w)
P
34
(s;w) =
s
w(s+w)(s++w)
Inverting the above equations with respet to w,
takingintoaountthebranhingprobabilities,yields:
K
31
(s) = e (s+)
K
32
(s) = e s
(1 e
)
E
33 (s) =
s
s+ (1 e
(s+)
)
E
34 (s) =
s+ (1
s
s+ e
)e s
The time domain probabilities are alulated by
rstderivingmatrixV
(s)from(6)usingastandard
pakage for symboli analysis (e.g. MATHEMAT-
ICA), and then numerially inverting the resulting
LST expressionsresortingto the Jagerman'smethod
[18℄. Theplotofthestateprobabilitiesversustimefor
statess
1 and s
4
is reported in Figure 3,for a deter-
ministi servieduration =1 andfor twodierent
valuesofthesubmittingrate=0:5and=2.
7 Conlusion
The GDT SPN model, whose semantishas been
disussedin[1℄,providesanaturalenvironmentforthe
denitionofalassofanalytiallytratableMRSPN's.
ThepaperhasonsideredtheaseofGDT SPNwith
non-overlappingativity yles, suh that the mark-
ing proess subordinated to theativity yle of the
dominanttransitionisarewardsemi-Markovproess.
Theinlusionofarewardvariableinthedesriptionof
thesubordinated proesshasprovento beveryee-
tivetehniquefor extendingthe desriptivepowerof
themodel to age memorypoliies, and forproviding
auniedproedure fortheanalytialsolution.
0.5 1 1.5 2 2.5 3 3.5 t 0
0.2 0.4 0.6 0.8 1 st. prob.
λ = 0.5 λ = 2 p (t) 1
p (t)
4
Figure 3 - Transient behavior of the state probabilities
for thepreemptiveM/D/1/2/2 systemwithdierentus-
tomers.
Aknowledgements
ThisworkhasbeenpartiallysupportedbyPHARE-
ACCORD under grant No. H-9112-0353 and by
OTKA undergrantNo. W-015859.
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