lients/servers intera tion SusannaDonatelli 1? ,SergeHaddad 2 ,Patri eMoreaux 2;3
, andMbayeSene 2
1
UniversitàdiTorino,DipartimentodiInformati asusidi.unito.it
2
LAMSADE{haddad, moreaux, sene}lamsade.dauphine.fr
3
RESYCOM,UniversitédeReimsChampagne-Ardenne
Abstra t. This paper presents a new methodto ompute bounds of
performan e parametersofMarkov hainsexhibitingapartition ofthe
statespa ewithsomefamilyofsubsetsvisitedinasequentialorder.We
use this stru ture to ompute bounds of steady state reward rates on
thesesubsetswithout omputingtheglobalsteadystateprobabilitiesof
thewhole hain.Themethodpresentedisbasedona ombinationofan
aggregation pro edure onthe subsets anda strongsto hasti ordering
ontheresultingaggregatedspa e.
1 Introdu tion
Performan emodelsbasedon ContinuousTime MarkovChains(CTMC)have
proven theirusefulness formany years.In this ontext, performan e measures
anbefrequentlyexpressedas(steadystate)expe tedrewardrates(or
instanta-neousrewardinsteadystate)R= P
s2S G
r(s)[s℄wherer(s)istherewardrate
asso iatedwiththestatesand the(row)ve torofthesteadystate
probabili-tiesoftheCTMCM=(S G
;Q;
0
)withstatespa eS G
,generatorQandinitial
probabilities
0
.Unfortunately,itisoftenimpossibleto ompute withan
an-alyti almethod duetothe omplexityoftheintera tionsamong theentities of
the system (there is no " losed" form for ). In these situations, we are lead
to numeri allysolvethelinearsystemQ=0.Thereare,however,well known
di ultiesforsolvingthisequation,amongthesethesizeofthestatespa e,and
formanysystems,thestinessofthelinearsystemwhentherearerareeventsin
themodelledsystem.To opewiththeseproblems,statespa eredu tion
meth-ods havebeen provenverypowerful.To omputetheexa tsolution,redu tion
methodsusuallyinvolvethewhole statespa e.Thesemethodsarebasedonthe
stru tureofthebehaviourofthesystem,liketensorbasedmethods,oronexa t
Markovian aggregation (lumping methods). Another large lass of methods is
basedonapproximatesolutions.Thepri etopayforthee ien yofthe
om-putation is then the di ulty to appre iate the quality of the result. Finally,
boundingmethods,oftenbasedonpartialredu tionofthe hain,areatradeo
?
Thisworkhasbeendevelopedwithintheproje tHCMCT94-0452(MATCH)ofthe
quanti ableestimation ofthequalityoftheresults.
Whateverthebounding method,it isalwaysguidedby thestru ture ofthe
model, dedu edfromthepropertiesofthestudiedsystem.An important
stru -turalpropertyofMisthepossibilitytofa torizethestatespa eintoapartition
of N subspa esS G
k
, allowingthede omposition R= P N k =1 R k .Depending on
thepropertiesofMwithrespe ttotheS G
k
,severalmethodshavebeendevised,
mainlybasedontheCourtoisresults[2,3℄and/orsto hasti orderings[10℄.
In this paper, we are on erned with systems exhibiting sequen es of
ele-mentary a tionssu hthat,in ea h sequen e,there is anonnull probability to
rea h the next a tion. Examples of su h systems are those with sequen es of
lient/serverlikeintera tions,repairablesystems,databasesystems.Many
au-thors,forinstan e [8,6,5,1℄,havestudied su h systemsandit hasbeen shown
that beside bounding the total probability of states orresponding to a given
sequen e(i.e.(S G
k
)),oneofthemainproblemsistoobtainboundsofthe
prob-ability distributioninside aS G
k
. Wepresenthere anew method to bound the
onditional steadystaterewardratesonthe subsetsofstates orresponding to
thesesequen esofa tions.Towardsthisgoal,weextendthestrongsto hasti
or-deringbetweenMarkovianpro essestoaggregatedversionsofthepro essesfrom
stru turalinformationabouttheMarkov hain,without omputingthe bounded
aggregated hain.
The organization of the paper is as follows: in Se tion 2 we present the
ontextoftheworkandourapproa h.InSe tion3,weintrodu ea rstmethod
to ompute lower bounds of rewardsrates on subsets of states, together with
resultsaboutstrongsto hasti orderingandaggregation.Wealsode neageneral
appli ationframeworkandwestudyadetailedexamplewithnumeri alresults.
InSe tion4,ase ondmethodisproposedwhi hextendsresultstoother lasses
of systems and we explain in details omputations of the bounds through an
example.Se tion5 on ludesthepaper.Anextendedversionofthispaper,with
detailed numeri alresultsisavailable[4℄.
2 Approa h
Weareinterestedinsystemswith lientandserverentitiesanda tivitieswhi h
maybe lassi edasautonomous lient'sa tivitiesandintera tiona tivities
be-tween lientsandservers.Whateverthe(highlevel)modellingformalismofsu h
systems(QN,SPN,:::),thisleadstoCTMCswhereitispossibletoidentifysets
S G
k
orrespondingtotheseintera tions.Inthepresentstudy,weindeedworkat
the Markov hain level. In many important ases, although the state spa e of
intera tions islargedueto their omplexity,the orrespondingrewardremains
"small"withregardtothe ontributionoftheautonomousa tivities.Thisisalso
the asefordependabilitymodels, where" lient/serverintera tions"statesare
Ifthe whole statespa e of the hain is S G = N k =1 S G k
, then the rewardis
R= P N k =1 R k andR k =(S G k ) P s2S G k r(s)(sjS G k
).We anregardthequantity
R =k = P s2S G k r(s)(sjS G k
) asthe (steadystate) onditionalrewardrate ofthe
measureonS G
k
.Toobtainbounds forRwemaythen omputeboundsforR
=k
andfor(S G
k
).Note howeverthatforsomeimportantmodels[8℄,onlyR
=k are
needed. Inthispaperwe on entrateourselvesonthe rst problem(obviously,
these bounds mustbe omputed without omputing the distribution on the
S G
k
).Morespe i ally,asshowedbymanyauthors[8,6,7,1℄,themostdi ult
problemisfrequentlytoderivealowerboundofR
=k
so thatweonlystudythis
problemin thefollowing.
Intherestofthepaper,werename,foreaseofreading,R
=k
asRandS G
k asS,
sothatwestudythe omputationofalowerboundR ofR= P
s2S
r(s)(sjS).
Notethat,in this paper, weassumethat there isa singleentrypointin S,
orrespondingtothebeginningofservi e.Infa t,ifthisisnotthe ase,Courtois
has shown that there is a family(
s 0
) of positivereals with P s 0 2in(S) s 0 =1 su hthat (sjS)= P s 0 2in(S) s 0 s 0
(sjS)where in(S)istheset of entrypoints
of S and s
0
thesteady stateprobabilitiesofthemodi ed hain withallinput
transitions of S "redire ted" to s 0
. Hen e, we may derive a bound R from
boundsR s
0
.
Evidently, a rst lowerbound ofR is min
s2S
fr(s)g(S). This is obviously
verypoor,sin eif9s:r(s)=0thenthelowerboundiszero.Togofurther, we
supposethat,on eagain,thespa eSitself,possessesastru turewhi hwillhelp
to de ne tighter bounds. If intera tionsare sequen esof elementary a tions,S
maybepartitionedinto subsets(S
i )
1in
su hthat ris onstantin ea h S
i .If r(s)=r i fors2S i ,thenR= P n i=1 r i (S i jS)(with(AjB)= P s2A (sjB)).
We propose two methods to nd a lowerbound of R , whi h are based on
an extension of thestrong sto hasti ordering of Markov hains to aggregated
versionsofthese hains(seeProposition1)without omputingthebounded
ag-gregated hain.We all"adaptedboundingmatri es"(seeDe nition1)matri es
whi hallowsu hanextension.The rstmethod bounds the onditional
proba-bilities (:jS)themselves.Asweshallsee inSe tion 3weobtainR >0under
onditionsdis ussed at theend of these tion. To opewith systemswhi h do
notsatisfytherequired onditionsofmethod1,weproposeinSe tion4ase ond
method, whi hrelaxesthese onditionsbutneedsmorestru turalproperties.
3 First method
Letusre allthatwewantto omputealowerboundof R
S =
P
s2S
r(s)(sjS)
where S isasubspa eof thewholestatespa e S of aMarkov hain.Moreover
we assumethat S may be partitioned into n subsets S
i su h that S i = fs 2 S j r(s) =r i g, sothat R S = P n i=1 r i (S i
jS). Inthis se tion we shalldire tly
bounds b [i℄,we anderivethelowerbound R S = n i=1 r i b [i℄ withR S >0
under onditionsexplainedattheendofthese tion.
Wethenpresentageneral lassofsystemsforwhi hourmethodisappli able,
followedbyanee tiveexampleofsu hasystem.
3.1 Idea of the method
Ourmain obje tiveisto obtainbounds b [i℄ using onlytheparametersofthe
model (for instan e the generator Q of the hain) and not the steady state
probabilities on S whi h will not be omputed. To this end, we rst restate
the problem in the framework of DTMCs. As usual with the uniformization
method,wetakeP =I jSj + 1 M Q jS whereM max s2S f Q[s;s℄gandQ jS isthe restri tionofQto S.
Sin e,forDTMCs,we anestablishalinkbetweenthesteadystate
probabil-itiesandthematrixP viatheso alledvisit-ratios,wede neboundsfor(S
i jS)
fromboundsofmatri esderivedfromthematrixP.
Denotingbys
0
theuniqueentrystatein S fromtherest ofthewholestate
spa e,itiswellknownthat
(sjS)= V(s 0 ;s) P s 0 2S V(s 0 ;s 0 ) (1) whereV(s 0
;s)denotesthemeannumberofvisitstosfroms
0
beforeleavingS.
Togetalowerboundof(S
i
jS),itisthensu ientto omputealowerboundof
P
s2Si V(s
0
;s)andanupperboundof P s 0 2S V(s 0 ;s 0
).Thelinkwiththematrix
P liesintherelationV(s
0 ;s)= P k 0 P k [s 0 ;s℄ A majorization of P s 0 V(s 0 ;s 0
) by a dire t ( omponentwise) majorization
of P may, in the general ase, provide a non stri tly substo hasti jSjjSj
matrix P +
so that the power sum of P +
would have no meaning. Hen e, we
ompute a majorization of (S
i
jS) by an upper bounding aggregation on the
(S i ),withannmatrix b P +
,theboundingbeing" ompatible"withthepower
sum operator.Onthe other hand,we omputeaminorization of (S
i
jS) bya
lowerbounding (againannmatrix) ofP[s
i ;S j ℄= P s j 2S j P[s i ;s j ℄8s i 2S i .
Anotheradvantageoftheintrodu tionofthesetwonnmatri esisthat,sin e
we ompute powers of matri es, smaller matri es will allow faster and more
a urate omputations.
Note that, in fa t, we may hoose dierent partitions (S
i
) for the upper
boundingof P s 0 V(s 0 ;s 0
)andthelowerboundingofV(s
0
;s).However,forease
ofreading,wekeepthesamepartitionfortherestofthisse tion.
3.2 Boundingsubsto hasti matri es
Inthis se tion,weprovideamean tode ne bounds forasubsto hasti matrix
and its powerseries. The following lemma sets out su ient onditions for a
nnmatrixtolowerboundV(s
0 ;S i )= P si2Si V(s 0 ;s i ).
Lemma1. Let P be a nn matrix su h that 8i;j, 8s i 2 S i ; P [i;j℄ P[s i ;S j ℄.Then 0 X k 0 ( b P ) k 1 A [i;j℄min s i 2S i fV(s i ;S j )g (2) Proof. Sin eV(s i ;S j )= P k 0 P k [s i ;S j
℄, letusprovebyindu tion that for
anyk:( b P ) k [i;j℄min s i 2S i fP k [s i ;S j ℄g.
Thepropertyis learlytrue from the hypothesis for k= 1.Assuming that
thepropertyistruefork,wehave
P k +1 [s i ;S j ℄ def = X s j 2S j n X l=1 X s l 2S l P k [s i ;s l ℄:P[s l ;s j ℄= n X l=1 X s l 2S l P k [s i ;s l ℄:P[s l ;S j ℄ n X l=1 ( b P ) k [i;l℄: b P [l;j℄
fromthehypothesisandthepropertyfork.
In ontrastwiththeminorization,themajorizationofV(s
0 ;S
i
)imposessome
onstraints on thebounding matrix. Adaptation is an extension of the strong
sto hasti ordering [10℄ on ept between Markovian pro esses to b P + and an aggregationofPonthe(S i
).Monotoni ity,likeforMarkovianpro esses,ensures
theexpansionofaninitialstrongsto hasti orderingto any(dis rete)time,by
transitionsofthe hain.
De nition1. AnnmatrixX is monotoni i 81i<jn; 81m
n; P m l=1 X[i;l℄ P m l=1 X[j;l℄ A nn matrix b P +
is an adapted (upper) bound of P with respe t tothe
partition(S
i )
1in
i(i)itismonotoni and(ii) 81in; 8s2S
i ; 81 mn; P m j=1 P s 0 2Sj P[s;s 0 ℄ P m j=1 b P + [i;j℄
Adaptation is asu ient ondition to upper bound the visit-ratios, as stated
in thenextpropositionwhi histheequivalenttotheresultforMarkovian
pro- esses.
Proposition1. If b
P +
isanadapted upper boundofP,thenfor any integerk
81in; 8s2S i ; 81mn; m X j=1 X s 0 2S j P k [s;s 0 ℄ m X j=1 ( b P + ) k [i;j℄ (3)
Proof. Theproofisdonebyindu tion.
A= m X j=1 ( b P + ) k +1 [i;j℄= m X j=1 n X l=1 ( b P + ) k [i;l℄ b P + [l;j℄= n X l=1 ( b P + ) k [i;l℄ m X j=1 b P + [l;j℄
Usinga lassi altransformation
A= " n X l=1 ( b P + ) k [i;l℄ # 2 4 m X j=1 b P + [n;j℄ 3 5 . . . + " t X l=1 ( b P + ) k [i;l℄ # 2 4 m X j=1 b P + [t;j℄ m X j=1 b P + [t+1;j℄ 3 5 . . . + " 1 X l=1 ( b P + ) k [i;l℄ # 2 4 m X j=1 b P + [1;j℄ m X j=1 b P + [2;j℄ 3 5 Bymonotoni ityof b P +
,ea hse ond termoftheprodu tsispositive,hen ewe
mayapplytheindu tiveinequalityonthe rsttermsfors2S
i AB= " n X l=1 X s 0 2S l P k [s;s 0 ℄ # 2 4 m X j=1 b P + [n;j℄ 3 5 . . . + " t X l=1 X s 0 2Sl P k [s;s 0 ℄ # 2 4 m X j=1 b P + [t;j℄ m X j=1 b P + [t+1;j℄ 3 5 . . . + " 1 X l=1 X s 0 2Sl P k [s;s 0 ℄ # 2 4 m X j=1 b P + [1;j℄ m X j=1 b P + [2;j℄ 3 5
We now apply the inverse transformation to B: B = P n l=1 P s 0 2S l P k [s;s 0 ℄ h P m j=1 b P + [l;j℄ i and sin e b P +
is an adapted upper bound of P, B C =
P n l=1 P s 0 2Sl P k [s;s 0 ℄ h P m j=1 P s 00 2Sj P[s 0 ;s 00 ℄ i
.Rearrangingtheorderof
sum-mation,wegetC= P m j=1 P s 00 2S j P n l=1 P s 0 2S l P k [s;s 0 ℄P[s 0 ;s 00 ℄= P m j=1 P s 00 2S j P k +1 [s;s 00 ℄
3.3 Bounds for onditionalsteady state probabilities
Gatheringpreviousresults,we anstatethefollowingtheoremwhi hprovidesa
lowerboundof(S
i
jS).Thenweexplainhowtoobtainanupperbound(S
i jS)
Theorem1. Let b
P and b
P be the matri es of lemma 1 and proposition 1,
andlet b
P +
isstri tlysubsto hasti . Then,for any i
(S i jS) P k 0 ( b P ) k [1;i℄ P n j=1 P k 0 ( b P + ) k [1;j℄ (4)
Proof. From(1), and hoosing i=1ands
i =s
0
weapplylemma1and
propo-sition1withm=nandtakingthelimitinthesumoverk.
Fromthepreviousbounds,we annow omputeanupperboundof(S
i jS).
The derivation of this bound is similar to the omplement property used for
probabilitydistributions:Pr(SnA)=1 Pr(A))Pr(A)1 Pr (SnA) with
Pr (SnA)Pr(A). LetV = P s2S V(s 0
;s),wehavefrom proposition 1,(with b P + stri tly sub-sto hasti ): P i j=1 V(s 0 ;S j ) P i j=1 P k 0 ( b P + ) k [1;j℄ def = V + i .Fromlemma1, V P n j=1 P k 0 ( b P ) k [1;j℄ def = V .Sin e(S i jS)= P s2S i V(s 0 ;s) V , P i j=1 (S j jS) = P i j=1 V(s0;Sj) V V + i V .Hen e (S i jS)maxf V + i V i 1 X j=1 (S j jS);1 X j6=i (S j jS)g (5) where (S j
jS)isthelowerbound givenbythetheorem1.
Theee tive omputation of theabovebounds of (S
i
jS) involvesthe
nu-meri al al ulation of two power series of matri es whi h are performed with
standardmethods[9℄.
3.4 A general appli ationframework
S
1
S
2
S
n-1
S
n
S
s
0
Inthisse tionweshowhowtobuildmatri es b
P and b
P foralarge lassof
systems.Thestru tureofthesesystemsmustbesu hthat weshouldbeableto
derivethe b P and b P +
matri esdire tlyfromthemodel.Moreover,thememory
requirementsforthe omputation of b
P and b
P +
must be polynomial, oreven
linear,withrespe ttothesizeofthemodel.Weassumethatthestru tureofthe
statespa e S is asdepi tedin Figure 1and that there is onlyoneentrystate
inS,whi hbelongstoS
1
.Finally,weimposethatthesystemmayleaveSfrom
ea hstateofS
n .
Thede nition of b
P is theeasiestone.Wesimplytake
b P = 0 B B B B B B . . . . . . . . . 1 C C C C C C A with b P [i;j℄ 8 < : min s2S i P s 0 2S j P[s;s 0 ℄ ifj=i orj =i+1 =0 otherwise Itis learthat b
P ful lls thehypothesisoflemma1.
Theideabehindthede nition of an upperbound b
P +
is tolowerthe
tran-sition probabilities of leaving S and to lower the "advan e" from S
i to S
i+1 .
Let us suppose that the stru ture of the system allows us to derive to
se-quen es(L
i
)and(N
i
)withthefollowingproperties.LetusdenotebyLeave(s)=
1 P s 0 2Snfsg P[s;s 0
℄thetransition probability ofleavingS from s,then 81
in; 0L i min s2S l ;li fLeave(s)g(L n
>0sin ethe hainmayleaveSwith
anonnullprobabilityfromea hstateofS
n ).L
n 1
lowerboundsthetransition
probabilityforthe hain to leaveS from S
n 1 orS
n and L
1
lowerbounds the
transition probability for the hain to leave S. (L
i
) is obviously an in reasing
sequen e.
TheN
i
(Next),instead,hastolowerboundthetransitionprobabilitiesfrom
anystateofS i tothesetS i+1 :0<N i min s2S i P s 0 2Si+1 P[s;s 0 ℄.Notethatwe
proposethe ondition 0<N
i
rather than0 N
i
, whi h seems quitenatural,
and provides asu ient ondition for b
P +
to bestri tly substo hasti (see the
proofoflemma2). Withthese onditions,wede nethematrix
b P + = 0 B B B B B B 1 L 1 N 1 N 1 1 L 2 N 2 N 2 . . . . . . . . . N n 1 1 L n 1 C C C C C C A with b P + [i;j℄=0if j6=iorj6=i+1 Lemma2. b P +
isstri tlysubsto hasti andisanadaptedupperboundofPwith
respe ttothe partition (S
i ) 1in . Proof. b P +
isstri tlysubsto hasti :for1in 1: b P + [i;i℄=1 L i N i 1 min s2S l ;li f1 P s 0 2S P[s;s 0 ℄g min s2S i P s 0 2Si+1 P[s;s 0 ℄=max s2S l ;li P s 0 2S P[s;s 0 ℄
min s2Si s 0 2S i+1 P[s;s 0 ℄max s2Si s 0 2S P[s;s 0 ℄ min s2Si s 0 2S i+1 P[s;s 0 ℄ 0
Forallothervaluesofj, learly b P + [i;j℄0.Moreover P j b P + [i;j℄=1 L i 1.Sin e1 L n <1,and8in 1; N i >0, b P + isstri tly substo hasti . b P +
ismonotoni :wehavetoshowthatfori<jandmgiven:A= P m l=1 b P + [i;l℄ P m l=1 b P +
[j;l℄=B andby transitivity, it issu ient to provethepropertyfor
j=i+1.Wehavethefollowingsituations:
m A B m<i 0 0 m=i b P + [i;i℄0 m=i+11 L i 1 L i+1 N i+1 1 L i+1 andAB sin e (L i )is in reasing m>i+11 L i 1 L i+1 (sameasabove) b P +
isanadaptedupperboundofP:giveni;j;mands2S
i
,letusprovethat
A= P m j=1 P s 0 2Sj P[s;s 0 ℄ P m j=1 b P + [i;j℄=B. Form<i: A=B=0. Form=i: A= P s 0 2Si P[s;s 0 ℄=1 P s 0 = 2S P[s;s 0 ℄ P s 0 2Si+1 P[s;s 0 ℄from
the properties of the hain in S, so that A 1 (1 P s 0 2S P[s;s 0 ℄) N i 1 L i N i =B Form>i:A= P s 0 2Si P[s;s 0 ℄ + P s 0 2Si+1 P[s;s 0 ℄=1 Leave(s)1 L i =B
againfromthepropertiesofthe hain inS.
Above results were establishedwith the hypothesis that we haveonly one
entrypoint belonging to S
1
. Courtois and Semal [2℄showed that if we havea
set S (0)
1
of entrypointsin S
1
, there are positivereals (a
s )
s2S (0)
1
su h that the
steadystatedistributionisalinear ombinationofthesteadystatedistributions
(s)
orresponding to the modi ed Markov hain where all inputs in S
1 are "redire ted" to s 2 S (0) 1 : = P s2S (0) 1 a s (s) , with P s2S (0) 1 a s = 1. Sin e our
boundsdonotdependonthea tualentrypointinS
1
,weseethattheyarestill
validforallthedistributions (s)
,thereforeforanydistribution orresponding
toa hainwithoutauniqueentrypointinS
1 .
3.5 Anexample
We illustrate the general framework exposed above and we present numeri al
results foradetailed exampleof asystem. Inthe framework of lients/servers
systems, let us assume that lients request a servi e that is omposed of two
phases: omputation of therequested results and transmissionof these results
from the server to the lient. The omputation may be done by n distin t
unitswithexponentialservi edistributionwithdierentrates
l
,andthe
trans-mission may be arried out in nt dierent ways, also with exponentialservi e
distributionwithrates
m
aset ofservers,S
i
isthesetofstateswheretheresultsofi 1 lient'srequests
ofthen 1arebeingtransmitted, omputedbutnotyetre eivedbythe lient,
andn irequestsforwhi htheserverisstillinthe omputationphase.Froma
stateinS
i
,aservermaybeginitstransmissionofresults,whi hmeansthatthe
hain leavesS
i
and entersS
i+1
, oranewrequest enters the servi esubsystem
orone lienthasre eiveditsresults,whi hmeansthatthe hainleavesS,oran
a tionindependentoftheservi eso urs,whi histranslatedbyatransitionof
the hain insideS
i
.Wesupposethat thesystemmaya tuallyleavetheservi e
area(i.e.S
n
)whenallrequestsareinthere eptionofresultsphase.It ouldnot
bethe ase:forexampleanErlangdistribution (withmorethanone stage)for
transmission, generates (Markovian)entrystates in S
n
that do not ful ll this
ondition.
Computation of the bounding matri es First, let us note that expressions (4)
and (5) for bounds do not depend on the a tual value of M involved in the
uniformization pro edure.Sin e it iseasy to omputeonesu h valuefrom the
model ofthesystem,wesimplyassumeherethatM is xed.
Wenow show howto omputethe matri es b P and b P + . Le us denote by min ; max ; min and max
therespe tiveminimumandmaximumofthevarious
ratesof omputationandtransmission.We hoose b P [i;i+1℄= (n i) min M and sin e b
P [i;i℄mustful ll b
P [i;i℄min
s2S i f1 P s 0 = 2S i q s;s 0 M g,weset b P [i;i℄=1 (n i) max+(i 1) max+xmax M wherex max
denotesanupperboundofthearrivalrate
ofnewrequests.ApossiblevalueforM isthenM=(n 1)maxf
max ; max g+ x max +1. Forwhat on erns b P +
,wede nethequantitiesL
i andN
i
.Sin e ontributions
to Leave(s) omefrom internal(endof servi eofa lient'request)andexternal
(newrequest)eventsofS,we anminorizeLeave(s)usingaminorizationofthe
arrivalrates(x min ).SowetakeL i = (i 1) min+xmin M
.Ontheotherside,we hoose
N i = (n i) min M .
Complexity redu tion Letus evaluatetheredu tionofthestatespa eusingthe
aggregatedstates orrespondingtotheS
i
insteadof thesubsetsS
i
themselves.
As we an hooseone omputationamong n andone transmissionamong nt,
jS i j= n i+n 1 n 1 i 1+nt 1 i 1 .Sin emin fn i;i 1gb n 1 2 ,wehave,with in reasingvaluesofn, n i+n 1 n 1 =((n 1) n 1 )=(n n 1 )or i 1+nt 1 i 1 = ((i 1) nt 1 )=(n nt 1 ),thatistosay,jS i j(n minfnt;n g 1 ).Forexample,
withn=100;n =5andnt=10wesubstituteoneelementforasubsetS
i with
about10 5
elements.
Numeri al results Wehave omputedboundingve torsb andb +
fora
simpli- edversionofourexample:weassumeonlytwostagesinthehyperexponential
distribution for omputation, and also two stages in the one for transmission.
min max min max
As quality riterion, we ompare thevalue = P
n
i=1 b
[i℄ to 1: themore is
nearfrom 1,thebetterwillbethelowerbound.Toobtainanindi ationonthe
variationof withrespe ttotherates and ,wehave omputed fordierent
ratios = max min min and = max min min
(0.1,0.5, 1) aswellasfor various
ratios
min
min
(0.1, 1, 10). Results are reported in table 1.It learly appears, as
ase min min P n i=1 b [i℄ 10.01 0.1 0.1 0.9023 20.1 0.1 0.1 0.8660 30.1 0.1 0.5 0.7995 40.1 0.5 0.1 0.5970 50.1 0.5 0.5 0.5585 60.1 1 0.1 0.4224 71 0.1 0.1 0.7596 81 0.5 0.1 0.3846 910 0.1 0.1 0.5660 1010 0.5 0.5 0.1453
Table 1.Summaryofnumeri alresults
expe ted, thattheloweris min
min
,thebetteristhebound.
i b min s0 b (s 0 ) max s0 b (s 0 ) b + + 10.07660250.54074740.5856063 0.6039496 0.67475010.1172291 20.16404490.24579430.2940281 0.3005905 0.37979690.2635029 30.21277730.06620720.0841023 0.0922648 0.20020991.1699492 40.25554540.01170330.0157206 0.0186712 0.14570606.8037702 50.29423710.00141860.0020100 0.0025982 0.135421251.121 60.32982000.00011940.0001782 0.0002516 0.1341221532.16846 70.36287010.00000690.0000108 0.0000167 0.13400968012.1245 80.39376730. 4.306E-07 7.304E-07 0.1340029183468.89 90.42278130. 0. 0. 0.13400277083390.2 100.45011530. 0. 0. 0.13400276.073E+08
Table 2.Comparisonbetweenboundsandexa tvaluesforb( ase2)
Given these results, we have omputed the possibleexa t valuesb (s
0 )
or-responding to modi ed hain with entry point in s
0 2 S 1 . We obtain b (s 0 )
by rst omputing the matrix P and then visit-ratios matrix V (V[s;s 0 ℄ = P k 0 P k [s;s 0
℄)(allmatri esarejSjjSj;inourexample,jSj=220).Wehave
i b mins 0 b 0 maxs 0 b 0 b 10.12016050.54074740.6145978 0.6326504 0.74566260.1786329 20.27472930.20448430.2819420 0.2896483 0.40939950.4134366 30.39780160.04339190.0720558 0.0795974 0.24830712.1195382 40.50553820.00588650.0119049 0.0142554 0.210801713.787473 50.59680590.00053920.0013374 0.0017443 0.2054544116.78769 60.67279500.00003400.0001038 0.0001479 0.20494921384.7932 70.73539480.00000150.0000055 0.0000086 0.204916723863.654 80.78659700. 0. 0. 0.2049153626920.53 90.82826210. 0. 0. 0.204915227810027. 100.86203580. 0. 0. 0.20491522.743E+09
Table 3.Comparisonbetweenboundsandexa tvaluesforb( ase3)
sultsarepresentedinTables2and3.Inthesetables, (i)= b [i℄ mins 0 b (s 0 ) [i℄ min s 0 b (s 0 ) [i℄
gives therelativeerror betweenmin
s02S1 b (s 0 )
[i℄ and b [i℄.Likewise, + (i)= b + [i℄ maxs 0 b (s 0 ) [i℄ maxs 0 b (s 0 ) [i℄
. We an observe that the relative error for the lower bound
in reaseswithi,andlessqui klyinthe ase2thanin the ase3.
Finally,letusnotethat,sin eweshouldobtainanonnulllowerbound,itis
ne essarythat,either r
1
6=0orN
1
6=0if r
1
=0,and moregenerally,N
i 6=0.
The ondition N
1
6= 0, also pointed out by other authors [7℄, means that, if
the reward rate is null in S
1
, the system must be able (i.e. with a non null
probability)toenterS
2
(beginningofthelast phaseofservi eforonerequest)
from ea h state of S
1
. Likefor S
n
(andL
n
) this maynot bealwaysthe ase.
This justi estheintrodu tionofanothermethod whi hrelaxesthis onstraint.
We make up for the weakening of the onditions on S
1
with more stru tural
informationaboutthestatespa eofthe hain.
4 A more elaborated method
Wederiveinthis se tionanotherlowerbound ofR
S =
P
s2S
r(s)(sjS) where
Sisasubspa eofthewholestatespa eS,partitionedinnsubspa esS
i ,sothat R S = P n i=1 r i (S i jS).
Inmanysituations,weareinterestedinsystemsforwhi hthesequen e(r
i )
is in reasing: the moretherequest isin progress,thegreater isthe asso iated
reward.Hen e,in ontrastwiththe rstmethod, wedonot omputea
ompo-nentwiseminorizationofb =S (b =S [i℄=(S i
jS)).Instead,wede neaprobability
ve torb su hthat b <
st b =S where< st
denotesthestrongsto hasti
order-ing.Thuswegetfrom<
st properties R S def = n X r i b [i℄ n X r i b =S [i℄=R S
ea hS
i
isnowitselfpartitioned into (S
i;j )
1jni
.Using theforwardequations
ofthe hainonS,andfromresultsofthepreviousse tion,wede neindu tively
b
[i℄ startingwithi=1.
Herealso,we ouldintrodu edierentpartitions(S
i;j
)ofagivenS
i
forupper
andlowerboundingofprobabilities.Wekeeponlyonepartitionforbothbounds
forreadability.
4.1 Dis rete strong sto hasti ordering
We restate in ourspe i ontext, the equivalen e of two of the de nitions of
strongsto hasti orderingbetweentwoprobabilitydistributionsoverf1;:::;ng
(proofsofthelemmasareomittedduetola kofspa eandmaybefoundin[4℄).
Lemma3. Letpandqbetwoprobabilitydistributionsoverf1;:::;ngandm
n, su h that 81 i m; P i j=1 p j P i j=1 q j and P m j=1 p j = P m j=1 q j .
Then, for any in reasing sequen e (r
i ) 1im of positive numbers: P m i=1 p i r i P m i=1 q i r i
Asu ient onditiontoensurethehypothesisofthepreviouslemmaisgiven
bythe lassi al omparison riterionforsumsofpositivenumbers.
Lemma4. Letpandqbetwoprobabilitydistributionsoverf1;:::;ngsu hthat
P m j=1 p j = P m j=1 q j (m n). If, 81<j m, pj pj 1 qj qj 1 then 81i m, P i j=1 p j P i j=1 q j .
Applyingtheselemmastoourproblem,weonlyneedto ndaprobability
distri-butionb su hthat b [i+1℄ b [i℄ (S i+1 ) (Si) = (S i+1 jS) (SijS) .Infa t,to omputeb ,itis
su ienttode neasequen e(
i )su hthat 1 =1and81in 1; i+1 (S i+1 ) (Si)
,sin ewe anthentakeb [i℄= i i 1 1 P n i=1 ii 11 .
4.2 Computation of theminorization ratios
In this se tion we provide an algorithm to ompute the ratios
i
. Let us rst
set some notations:
i
(row ve tor with jS
i
j omponents) is the restri tion of
toS
i ;
=i
(rowve torwith jS
i
j omponents) is the onditionalversionof
i : =i [s℄ = [s℄ (Si) ; b =i
(row ve tor with dimension n
i
) is the onditional steady
stateprobabilityve torontheaggregatedstatesS
i;j ofS i :b =i [j℄=(S i;j jS i )= P s2S i;j [s℄ (S i )
;Qis thegeneratoroftheMarkov hain. Sin ethestatespa e S is
partitionedintodisjointsubsetsS
i
,sequentiallyvisited,atransitiono ursonly
insideS i orfromS i toS i+1
.WemaythenviewQasablo kmatrixQ=[Q
i;j ℄.
Q
i;j
givesthetransitionratesfromstatesofS
i
to statesofS
j
1 Q 12 + 2 Q 22 =0 . . . i 1 Q i 1;i + i Q i;i =0 (6) . . . n 1 Q n 1;n + n Q n;n =0
Equation(6)mayberewritten(Q
i;i
isregularsin eQisagenerator):
i = i 1 :Q i 1;i :Q 1 i;i (7)
whi histhebasisfortheiterative omputationofthe
i .
Theith stepof the algorithm omputestwove torsb
=i and b + =i su h that b =i b =i b + =i
andthe number
i
(Si)
(Si 1)
from theset ofvaluespreviously
omputed: 81ji 1 ( b =j and b + =j with b =j b =j b + =j j with j (Sj) (Sj 1)
Notethat,although weonlywantto omputethe
i , weneed tointrodu e theboundsb =j andb + =j :asweshallsee,b =i 1 isrequiredtode ne i andb =i 1 andb + =i 1
arerequiredto omputeb
=i .
Initial step Let P
1
bea"pseudouniformized"(sin e Q
1;1
is notagenerator)
substo hasti matrixasso iatedwithQ
1;1 andMmax s2S f q s;s g:P 1 =I jS1j + 1 M Q 1;1 ,withI jS 1 j thejS 1
jIdentitymatrix.We hoose b P 1 (n 1 n 1 matrix)su h that b P 1 [x;y℄min s2S1;x f P s 0 2S 1;y P 1 [s;s 0
℄gandwe omputeanupperadapted
(toP)matrix b
P +
1
.Likewiththe rstmethod, usingthevisit-ratios( b
P
1 ful lls
thehypothesisoflemma1and b
P +
1
ful llsthoseofproposition1withrespe tto
thepartition(S 1;j )ofS 1 ),weset b =1 [j℄= P k 0 ( b P 1 ) k [1;j℄ P n1 l=1 P k 0 ( b P + 1 ) k [1;l℄ and b + =1 [j℄=1 X l6=j b =1 [l℄ sin eb =1 [j℄=1 P l6=j b =1 [l℄.
Computations fora step i Letus rstgiveanotherexpressionof Q 1
If Q i;i =[q s;s 0 ℄wehave:q s;s = q s;s >0and q s;s 0 = q s;s 0 <0fors6=s, andq s;s P s 0 6=s q s;s 0
,sin eQisagenerator.LetM
i max s2Si f q s;s gandP i thesubsto hasti (jS i jjS i
j)matrixsu hthat Q
i;i =M i ( I P i )sothat P i [s;s 0 ℄= qs;s M i if s=s 0 1+ q s;s 0 M i otherwise Sin eQ i;i isregular, Q 1 i;i = 1 Mi : P k 0 P k i
So,from equation(7)
i = 1 M i : i 1 :Q i 1;i : X k 0 P k i = (S i 1 ) M i : =i 1 :Q i 1;i : X k 0 P k i (8) Computationofb =i Sin e b =i [j℄ = 1 (Si) (S i;j
), we ompute a lower bound of (S
i;j ) and an upperboundof(S i ). Lowerbound of(S i;j ) From(8) (S i;j )= 1 M i : X s 00 2Si;j X s 0 2Si X s2Si 1 i 1 [s℄:Q i 1;i [s;s 0 ℄: 0 X k 0 P k i 1 A [s 0 ;s 00 ℄
Movingthesummation overs 00 (S i;j )= 1 M i : X s 0 2Si X s2Si 1 i 1 [s℄:Q i 1;i [s;s 0 ℄: X s 00 2Si;j 0 X k 0 P k i 1 A [s 0 ;s 00 ℄
Splitting thesummation overs 0
andmovingitpartially
(S i;j )= 1 M i : n i X v=1 X s2S i 1 i 1 [s℄: X s 0 2S i ;v Q i 1;i [s;s 0 ℄: X s 00 2S i;j 0 X k 0 P k i 1 A [s 0 ;s 00 ℄ Ifwehaveamatrix b P i (n i n i
,tobe omputed)su hthat b P i [x;y℄min s2Si;x f P s 0 2S i;y P i [s;s 0
℄gthen,fromlemma1
(S i;j ) 1 M i : n i X v=1 X s2S i 1 i 1 [s℄: X s 0 2S i ;v Q i 1;i [s;s 0 ℄: 0 X k 0 ( b P i ) k 1 A [v;j℄
Wenowsplitthesummationovers
(S i;j ) 1 M i : ni X v=1 ni 1 X u=1 X s2S i 1 [s℄: X s 0 2S ;v Q i 1;i [s;s 0 ℄: 0 X k 0 ( b P i ) k 1 A [v;j℄
andweintrodu eamatrixQ i 1;i (n i 1 n i
,tobe omputed)su hthatQ
i 1;i [x;y℄ min s2Si 1;x f P s 0 2S i;y Q i 1;i [s;s 0 ℄g.Hen e (S i;j ) 1 M i : ni X v=1 n i 1 X u=1 X s2S i 1;u i 1 [s℄: b Q i 1;i [u;v℄: 0 X k 0 ( b P i ) k 1 A [v;j℄
fromthepropertyoftheve torb
=i 1 weobtain (S i;j ) 1 M i : ni X v=1 n i 1 X u=1 (S i 1 )b =i 1 [u℄: b Q i 1;i [u;v℄: 0 X k 0 ( b P i ) k 1 A [v;j℄ that isto say (S i;j ) (S i 1 ) M i :b =i 1 : b Q i 1;i : 0 X k 0 ( b P i ) k 1 A [j℄ (9) Upperboundof(S i )
From(8), andwiththesamekindof derivation
(S i )= i (S i ) (S i 1 ) M i : X s2S i X s 0 2S i 1 =i 1 [s 0 ℄: X s 00 2S i Q i 1;i [s 0 ;s 00 ℄: 0 X k 0 P k i 1 A [s 00 ;s℄ (S i )= (S i 1 ) M i : n i X j=1 X s 0 2Si 1 =i 1 [s 0 ℄: X s 00 2Si Q i 1;i [s 0 ;s 00 ℄: X s2Si;j 0 X k 0 P k i 1 A [s 00 ;s℄ (S i )= (S i 1 ) M i : ni X j=1 ni X v=1 X s 0 2Si 1 =i 1 [s 0 ℄: X s 00 2Si;v Q i 1;i [s 0 ;s 00 ℄: X s2Si;j 0 X k 0 P k i 1 A [s 00 ;s℄ Let b P + i
(tobe omputed)beastri tlysubsto hasti andupperboundingn
i n
i
matrixwithrespe ttothepartition(S
i;j )ofS
i
,then, fromProposition1,
(S i ) (S i 1 ) M i : ni X j=1 ni X v=1 X s 0 2Si 1 =i 1 [s 0 ℄: X s 00 2Si;v Q i 1;i [s 0 ;s 00 ℄: 0 X k 0 ( b P + i ) k 1 A [v;j℄ (S i ) (S i 1 ) M i : ni X j=1 ni X v=1 n i 1 X u=1 X s 0 2S i 1;u =i 1 [s 0 ℄: X s 00 2S i;v Q i 1;i [s 0 ;s 00 ℄: 0 X k 0 ( b P + i ) k 1 A [v;j℄ Let b Q + i 1;i (n i 1 n i
,tobe omputed)besu hthat b Q + i 1;i [x;y℄max s2Si 1;x f P s 0 2S i;y Q i 1;i [s;s 0 ℄g.Then, (S i ) (S i 1 ) M i : ni X j=1 ni X v=1 ni 1 X u=1 X s 0 2S =i 1 [s 0 ℄: b Q + i 1;i [u;v℄: 0 X k 0 ( b P + i ) k 1 A [v;j℄
(S i ) (S i 1 ) M i : ni X j=1 ni X v=1 n i 1 X u=1 b + =i 1 [u℄: b Q + i 1;i [u;v℄: X k 0 ( b P + i ) k A [v;j℄
fromthepropertyoftheve torb +
=i 1
,whi h mayrewrittenas
(S i ) (S i 1 ) M i :b + =i 1 : b Q + i 1;i : 0 X k 0 ( b P + i ) k 1 A :1 T n i (10) with1 T ni
the olumnve torwithn
i
omponents,allequalto1.
Finally,by(9)and(10),weset
b =i [j℄ def = b =i 1 : b Q i 1;i : P k 0 ( b P i ) k [j℄ b + =i 1 : b Q + i 1;i : P k 0 ( b P + i ) k :1 T n i (11) Computationofb + =i Sin eb =i [j℄=1 P l6=j b =i [l℄, wesimplyde ne b + =i [j℄ def = 1 P l6=j b =i [l℄. Computationof i From (9), wehave (S i ) (Si 1) Mi :b =i 1 : b Q i 1;i : P k 0 ( b P i ) k :1 T ni and we ande ne i def = 1 M i :b =i 1 : b Q i 1;i : 0 X k 0 ( b P i ) k 1 A :1 T ni (12)
Note that we must have
2
6= 0to obtain non trivial lower bounds b [i℄.
From12,thisisequivalentto theexisten eofj and ksu h thatb
=1
[j℄>0and
b
Q
1;2
[j;k℄ >0.This meansthat there mustbeasubsetS
1;j ofS 1 andasubset S 2;k ofS 2
su hthatfromea hstateofS
1;j
(andnotofS
1
asinthe rstmethod),
thesystemmustbeableto jumpintoS
2;k .
4.3 Example
Thegoal ofthis se tionis toexplain howthe variousmatri es requiredbythe
algorithmjustpresentedmaybederivedfromthemodelofthesystem.
Numer-i al results are presented in the extended version [4℄ of the paper. We study
amodi edversionof the exampleof Se tion 3.5 and we on entrate ourselves
on the omputation phase of the request servi e in the sub-state spa e S
i of
S (n irequests in omputation phase,i 1requests in transmissionphase).
This omputation phaseis nowmade upof four steps,the transmissionphase
beingun hanged.An initialstage(rate
1
)isfollowedbyaforkprodu ingtwo
"sub-requests",withrate
2 and
3
.Thejoinofthesub-requests(rate
4 )ends
µ
2
/(µ
2
+µ
3
)
µ
1
µ
2
+
µ
3
µ
3
µ
2
µ
4
µ
3
/(µ
2
+µ
3
)
Fig.2.Distributionofthe omputationphaseservi e(method2)
The riti al hoi e of themethod is the de nition of thepartitions S
i;j for
theminorizationandmajorizationinthealgorithm(thesamepartitionwasused
inthepresentationofthealgorithm).This hoi eisatradeobetweenthe
om-plexityofthe omputations, theful llmentof thehypothesisof thetheorem1,
for thematri es b P i and b P + i
, thetightness ofthe bounds andthe information
dire tlyavailablefromthemodel.Alsonotethat thepartition orrespondingto
the minorization must be a re nement of the one for themajorization due to
theboundexpressionsofSe tion3.3andthe omputationofb +
=i .
Inourexample,wemay hooseS
i;j
asthesetofstatesforwhi hj stepshave
beendonebythewholen irequestsandwe an omputequantitiesL
i;j andN i;j (theL i andN i
valuesofthemethod1)tode neab +
=i
asinmethod1.Forwhat
on ernstheminorization,theseS
i;j
areatoo oarsepartition,providingaweak
boundingve torb
=i
:weneedmoreinformationabouttheadvan eofrequestsin
thedierentstepstode neavaluablelowerbound.Hen e weshalltakeS
i;| as
thepartitionfortheminorization,where|=(j
1 ;j 2 ;j 3 ;j 4 ), P 4 k =1 j k =n iand j k
isthenumberofrequestsinkthstep.Itisstraightforwardtoverifythatthis
partitionisare nementofthemajorizationpartition.Whateverthematrix,we
may hooseanyvalueM
i su hthatM i max s2S i f q s;s g.Forinstan e,wemay set M i =(n i)maxf 1 ; 2 + 3 ; 2 ; 3 ; 4 g+(i 1) max +x max +1. Minorization Here,S i;| =fsjj 1
requestsininitialstep,j
2
requestswithonestep
done,:::;j
4
requestswiththreestepsdoneg.
Re allingthatwemusthave b P i [x;y℄min s2S i;x P s 0 2Si;y P i [s;s 0 ℄,weset b P i [|;| 0 ℄= 8 > > > > > < > > > > > : j11 Mi if | 0 =(j 1 1;j 2 +1;j 3 ;j 4 ) j2(2+3) M i if | 0 =(j 1 ;j 2 1;j 3 +1;j 4 ) j3minf2;3g M i if | 0 =(j 1 ;j 2 ;j 3 1;j 4 +1) 1 r(|;| 0 ) Mi if | 0 =| 0 otherwise withr(| ;| 0 )=j +j ( + )+j maxf ; g+j +(i 1) +x .
Wemustalso have b Q i 1;i [x;y℄min s2Si 1;x s 0 2S i;y Q i 1;i [s;s 0 ℄sothat we de ne b Q i 1;i [|;| 0 ℄=j 4 4 if| 0 =(j 1 ;j 2 ;j 3 ;j 4 1)and0otherwise. Majorization Here,S i;j
=fsjjstepshavebedoneg(0j3(n i)).
We de ne b
P +
i
as with method 1 (but applied to the S
i;j ), whi h ensures that b P + i
is astri tly substo hasti and upperbounding matrix. Sin eweneed
L i;j min s2S i;l ;lj fLeave(s)g, we take L i;j = 4lnls(j) Mi , where lnls(j) is a
lower bound of the number of requests being in their last step of
omputa-tion: lnls(j)=maxf0;j 2(n i)g.ThenumbersN
i;j mustverify0<N i;j min s2Si;j P s 0 2S i;j+1 P[s;s 0
℄. An a urate hoi e, whi h anbe omputed
with-outoverheadduring the omputation of b P i is: N i;j =min |; P 4 k =1 (k 1)j k =j 1j1+(2+3)j2+minf2;3gj3 M i . b Q + i 1;i must ful lls b Q + i 1;i [j;j 0 ℄max s2Si 1;j P s 0 2S i;j 0 Q i 1;i [s;s 0 ℄, hen ewe de ne b Q + i 1;i [j;j 0 ℄ = 4 unls(j) if j 0
= j 3 and 0 otherwise. unls(j) is an
upperboundofthenumberofrequestsbeingintheirlast stepof omputation:
unls(j)=jdiv3.
5 Con lusion
Inthis paperwehavepresentedanewapproa hto omputeboundsof
perfor-man emeasuresof Markov hainsfrequentlyen ountered insystemmodelling.
Thisapproa hprovidesimportantsavingsin omputationandmemory
require-ments with respe t to an exa t omputation of the steady state distribution.
Whenasubsetofthestatespa e ofthe hain may bepartitioned in subspa es
sequentiallyvisited,wehavede ned bounds of onditionalsteady statereward
ratesonthesessubspa esoronthewhole subset.Twomethodshavebeen
pro-posed, orrespondingtodierentpropertiesofthemodel.Ourmethodsarebased
on a ombination of strong sto hasti ordering and aggregation. We have
ex-plained how to ompute these bounds and reported rst experiments showing
theinterestoftheapproa hwhenthesystemful llssuited hypothesis.Workis
in progressto evaluate a urately the behaviourof the results for the se ond,
moreelaborated,method.Wearealsostudyingtheappli abilityofourmethod
todomainsusuallyinvolvingboundingmethods,likeperforman eevaluationof
fault tolerant and repairable systems, for whi h the reparation rates may be
seen,themselves,asrewards.Inparti ular,weareworkingonappli ationofour
resultstosystemswithmultipleentrypointsinthesubsetsvisited.
Referen es
tivematri es and for their approximationsby de omposition. Journal of ACM,
31(4):804825,O tober1984.
3. P.J.CourtoisandP.Semal.Computableboundson onditionalsteady-state
prob-abilities inlarge markov hainsand queueingmodels. IEEE Journal onSele ted
AreasinCommuni ation(SAC),4(6):926937,September1986.
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