Wavelength Routing Networks with Multiast Calls
Sridhar Ramesh George N. Rouskas Harry G. Perros
TR-01-02
January 7,2001
Abstrat
WepresentanapproximateanalytialmethodtoomputeeÆientlytheallbloking
prob-abilities in wavelength routingnetworks with bothuniast and multiast alls. The model is
fairly general and allows eah soure-destination pair to serve alls of dierent lasses, with
eah all oupying onewavelength per link. Ourapproah involvesthe followingsteps. We
rst onsider alinear,single-lass uniastnetworkwhere the arrivalproess of alls on
multi-hop routes is modied slightlyto obtain anapproximate network model with aprodut-form
solution. This method is used to solvesingle-lass uniast wavelength routingnetworks with
asmall numberof hops. Next, alllassesof alls onapartiular routeare aggregatedto give
an equivalent single-lassmodel. Wethen extendthe pathdeomposition algorithmsthat we
havedevelopedforsingle-lassnetworkstohandlemeshnetworkswithmultiplelassesofalls.
We showhow to use these path deomposition algorithms to deompose largenetworks with
multiast paths intosmaller sub-systemswith onlylinearpaths, whih, in turn, aresolved by
theprodut-formapproximationalgorithm. Wealsoonsiderastate-dependentPoissonarrival
proessformultiastallswhihismoreaurateinapturingthebehaviorofthesealls. Tothe
bestofourknowledge,thisistherstmodelthatanbeusedtoanalyzeeÆientlywavelength
routingnetworksof arbitrarymeshtopologies,withmulti-lassservieandmultiastalls.
Department of ComputerSiene
North Carolina StateUniversity
Raleigh,NC 27695-7534
A basipropertyof singlemode optialberis its enormouslow-lossbandwidthof several tensof
Terahertz. However, dueto dispersive eets and limitations in optial devie tehnology, single
hanneltransmissionislimitedtoonlyasmallfrationoftheberapaity. Totake fulladvantage
ofthepotentialofber,theuseofwavelengthdivisionmultiplexing(WDM)tehniqueshasbeome
the option of hoie, and WDM networks have been a subjet of researh both theoretially and
experimentally[3 ℄. Optialnetworkshavethepotentialofdeliveringanaggregatethroughputinthe
order of Terabits perseond, and they appearasa viable approah to satisfyingthe ever-growing
demandformore bandwidthperuseron a sustained,long-term basis.
ThewavelengthroutingmesharhitetureappearspromisingforWideAreaNetworks(WAN).
The arhitetureonsistsof wavelengthroutersinteronnetedbyberlinks. Awavelength router
isapableofswithingalightsignalatagivenwavelengthfromanyinputporttoanyoutputport.
A router may also be apableof enforing a shiftin wavelength [10 ℄, in whih ase a light signal
may emerge from the swith at a dierent wavelength than the one it arrived. By appropriately
onguringtherouters,all-optialpaths(lightpaths)maybeestablishedinthenetwork. Lightpaths
represent diret optial onnetions without any intermediate eletronis. Beause of the long
propagation delays, and thetimerequired to onguretherouters, wavelengthrouting WANs are
expeted to operate in iruit-swithedmode. This arhitetureis attrative for two reasons: the
same wavelengthanbeusedsimultaneouslyatdierentpartsofthenetwork,andthesignalpower
is hanneled to the reeiver and is not spread to the entire network. Hene, wavelength routing
WANs an behighly salable.
Given theinstalled baseof berand thematuringof optialomponent tehnology,itappears
that wavelength routing tehnology is poised to make the leap from promise to reality. Thus, it
is not surprising that a signiant amount of researh eort in reent years has been devoted to
understandingand addressingtheset ofproblemsand issuesthatariseinthedesignofwavelength
routing networks[3 ℄. One problemthat has attrated onsiderableattention isthat of omputing
all blokingprobabilitiesinsuhnetworks. Several variationsof thisproblem have beenstudied,
mainly diering in the underlyingassumptions regarding key elements of the wavelength routing
arhiteture, inluding: the availability, type, and loation of onverters, the routing algorithm
(whih nds a path for an inoming all), the wavelength alloation poliy (whih assigns a free
wavelengthtoanewall),thetraÆtype(uniastand/ormultiast),andthearrivalproess(mostly
Poisson, with a few exeptions). A major diÆulty is analyzing wavelength routing networks is
The problem of omputing all bloking probabilities under stati or xed alternate routing
with randomwavelength alloationand withorwithoutonvertershas been onsideredinseveral
studies, inluding[1, 8 , 2 , 5, 12 , 14 ℄. Eah work develops approximate tehniquesto analyze the
blokingperformaneof wavelength routingnetworks. Ingeneral, these approximationsarebased
on various independene assumptions (e.g., independene of link loads, link bloking events, or
wavelength usage), and also exploit the analytial tratability of the random alloation poliy.
Other wavelength alloation shemes, aswell asdynami routing are harder to analyze. First-t
wavelengthalloationwasstudiedusingsimulationin[8℄,anditwasshowntoperformbetter than
random alloation, whilean analytialoverow modelfor rst-t alloation was developed in [6℄.
A dynamirouting algorithm that selets the leastloaded path-wavelength pair was also studied
in[6℄,andin[9 ℄ anunonstraineddynamiroutingshemewithanumberofwavelengthalloation
poliieswasevaluated.
Exept in [12 , 13℄, all of the above studies assume that either all or none of the wavelength
routers have fullwavelengthonversion apabilities. The two studiesin[12,13 ℄onsidernetworks
with sparse onversion, whereby only a subset of the set of nodes are equipped with onverters.
Speially, [12 ℄ takes a probabilistiapproah in modelingwavelength onversion by introduing
theonverterdensity,whihrepresentstheprobabilitythatanodeisapableofonversion
indepen-dentlyofothernodesinthenetwork. Thefousof[13 ℄,ontheotherhand,isonapaityplanning,
and a dynami programming algorithm to determine the loation of onverters on a single path
that minimizes average or maximum bloking probability is developed under the assumption of
independentlinkloads. A dierenttype ofwavelength onversionis studiedin[16,15 ℄, whereitis
assumedthatallnodesinthenetworkareequippedwithdevieswhihanonverteahwavelength
to only a subset of the supported wavelengths (usually, to ones lose to the given wavelength in
the optial spetrum). This type of onversion is referred to as limited onversion, as opposed to
the full onversion (i.e., the apability of onverting a wavelength to any other supported
wave-length) assumed bypreviously disussedworks. Approximate analytialtehniquesare developed
in [16 ,15 ℄ to omputeallblokingprobabilitiesinnetworkswithlimited onversion.
All of the above studies desribe tehniques to analyze networks in whih alls are
point-to-pointandarrivalsarePoisson. Astudyofallblokingundernon-PoissoninputtraÆispresented
in [14 ℄, where the assumption is made that link loads are statistially independent. Finally, the
problemofroutingandwavelengthassignmentwhenthenetworkarriesmultiastallsisaddressed
in [11 ℄.
approah [4℄. Further, thedevelopment of some of thetehniquesis based on additional
assump-tions, for instane, thatstatistis of link loadsare mutuallyindependent, linkbloking events are
independent,or that wavelength use on eah link is haraterized by a xed probability
indepen-dent ofotherwavelengthsand links. Linkdeompositionhasbeenextensivelyusedinonventional
iruit-swithed networks where there is no requirement for the same wavelength to be used on
suessive links of the path taken by a all. The auray of these underlying approximations
also dependson thetraÆ load,the network topology,and the routingand wavelength alloation
shemesemployed. While link deompositiontehniquesmake it possibleto studythequalitative
behaviorofwavelengthroutingnetworks,moreaurateanalytialtoolsareneededtoevaluatethe
performane of these networks eÆiently, as well as to takle omplex network design problems,
suhasseleting thenodeswhereto employonverters.
The authors have developed an iterative path deomposition algorithm [21 , 19 ℄ for analyzing
arbitrary wavelength routingnetwork topologies withsparseonversion. Speially,we analyzea
givennetworkwithpoint-to-pointallsbydeomposingitintoanumberofpathsub-systems. These
sub-systems are analyzed in isolation usingour approximation algorithm for omputing bloking
probabilitiesinasinglepathofanetwork[18 ℄. Theindividualsolutionsareappropriatelyombined
to forma solutionfortheoverall network, and theproess repeats untiltheblokingprobabilities
onverge. Our approah aounts for the orrelation of bothlink loads and link bloking events,
giving aurate results for a wide range of loads and network topologies where only a xed but
arbitrarysubsetofnodesareapableofwavelengthonversion. Therefore,ouralgorithman bean
important tool inthe development and evaluation of onverter plaement strategies. Also,in [20 ℄
westudiedtherst-tandmost-usedwavelengthalloationpoliies,andweshowedthattheyhave
almost identialperformaneinterms of blokingprobabilityforall allsinthe network. We also
demonstrated thatthe blokingprobabilitiesunder therandomwavelength alloationpoliy with
no onverters and with onverters at all nodes provide upper and lower bounds for the values of
the blokingprobabilitiesunder therst-t and most-usedpoliies.
Inthispaperwedemonstratehowpowerfulthepathdeompositionapproahanbeby
extend-ing the deompositionalgorithm in[21 ℄ to evaluate the blokingperformane of optial networks
with multiplelasses of allsand multiast traÆ. While multiastalls where onsideredin[11 ℄,
all previous studies on all bloking probabilities have mostly foused on single-lass wavelength
routing networks. However, future networks will be utilizedby a wide range of appliations with
varying harateristis in terms of their arrival rates and all holding times. To the best of our
0
1
2
hop 1
hop 2
. . .
. . .
j
. . .
k-1
k
hop k
i-1
λ
12
λ
22
λ
ij
λ
j+1,k
λ
λ
11
1j
Figure 1: A k-hoppath
have been analyzed. Further, we analyze a state-dependent Poisson arrivalproess for multiast
alls whih ismore aurate thanaPoissonmodelinapturing thebehaviorof thesealls.
Thedevelopmentof ourapproximateiterativedeompositiontehniquesinvolvesthefollowing
steps. The wavelength routing network is rst deomposed into several smaller sub-systems. The
arrival rates of alls on eah route in the smaller sub-systems is adjusted to reet the arrival of
alls on longer routes in the original network, as well as the arrival of multiast alls using the
route. Eah sub-systemis thensolved asfollows. The arrivalproess ofalls on multi-hop routes
is modied slightly to obtain a modied multi-lass network model. Next, all lasses of alls on a
partiular routeareaggregatedto givean equivalentsingle-lass model. Thisequivalentmodelhas
thesame allblokingprobabilityonanygiven route asthemodiedmulti-lassnetwork,and an
be solved usingourpreviousalgorithms.
In Setion2, we desribe thenetwork under study. In Setion3, we explainhow themodied
multi-lassmodeland theequivalentsingle-lassmodelareobtainedforasinglepathofanetwork.
InSetion4,wedesribeadeompositionalgorithmformeshnetworks. InSetion5,weextendthis
deompositionalgorithm for networkswith multiast traÆ. Setion6 presentsnumerialresults,
and weonlude thepaperinSetion7.
2 The Multi-Class Wavelength Routing Network
We onsider a wavelength routing network with an arbitrary topology. Eah link in the network
supportsexatlyW wavelengths,and eahnode isapableoftransmittingand reeivingon anyof
these W wavelengths. Call requestsbetween a soureand a destination node arrive at the soure
aording to a Poisson proess with a rate that depends on the soure-destination pair. If the
request anbesatised,anoptialiruitisestablishedbetweenthesoureanddestinationforthe
durationoftheall. Inaddition,there aremultiastallsbetweena soureandasetofdestination
nodes. The arrival of multiast alls is modeled by a state dependent Poisson proess. Further,
alls(multiastallsoruniastallsbetweenanysoure-destinationpair)maybeofseverallasses.
of theall.
In our model, we allow some of the nodes in the network to employ wavelength onverters.
Thesenodesanswithaninomingwavelengthtoanarbitraryoutgoingwavelength. (Whenthere
areonvertersatallnodes,thesituationisidentialtothatinlassialiruit-swithingnetworks,a
speialaseofthemoregeneralsenariodisussedhere.) Ifnowavelengthonvertersareemployed
in the path between the soure and the destination, a uniast all an only be established if the
samewavelengthisfreeonallthelinksusedbytheall. Ifthereexistsnosuhwavelength,theall
isbloked. Similarly,amultiastallan onlybeestablishedinanetworkwithoutonverters ifthe
same wavelengthis freeon alllinksofthemultiasttree onnetingthesouretothedestinations.
This isknownasthewavelength ontinuityrequirement (see[21 ℄), and itinreases theprobability
of all bloking. On the other hand, if a allan be aommodated, it is randomly assigned one
of the wavelengths that are available on the links used by the all 1
. Thus, we onlyonsider the
random wavelengthassignment poliyin thispaper.
Sine our model is an extension of a model developed for a single path, we introdue some
relevant notation for a linear, uniast wavelength routing network. A k-hop path (see Figure 1)
onsists of k+1 nodes. Nodes(i 1) and i, 1 ik,are said to be onneted bylink (hop) i.
Callsoriginatingat nodei 1and terminatingat nodej usehopsithroughj;j i1,whihwe
shall denote bythepair (i;j). Callsbetweenthese two nodesmaybelongto one ofR lasses, and
these alls aresaid to useroute (i;j). We also denethefollowingparameters.
(r)
ij
;ji;1r R ,isthePoissonarrivalrateofallsoflassr thatoriginateatnode(i 1)
and terminate at nodej.
1=
(r)
ij
isthemeanoftheexponentiallydistributedservietimeofallsoflassrthatoriginate
at node (i 1) and terminateat node j. We alsolet (r)
ij =
(r)
ij =
(r)
ij .
N (r)
ij
(t)isthe numberof ative alls at timeton segment (i;j) belongingto lass r.
F
ij
(t) is the number of wavelengths that are free on all hops of segment (i;j) at time t. A
allthat arrivesat timetand uses route (i;j) is bloked ifF
ij
(t)=0.
1
Inanetworkwithwavelengthonverters,awavelengthisrandomlyassignedwithineahsegmentofapathwhose
3.1 The Single-Class Case
Inthissetionwebrieyreviewsome ofourpreviousresultsforapathofasingle-lasswavelength
routing network. These results were publishedearlier in [18, 21 ℄, and are inluded here for
om-pleteness. Consider the k-hop pathshown in Figure 1. Let the state of thissystem at time t be
desribed by thek 2
-dimensionalproess:
X
k
(t)= N
11 (t);N
12
(t);;N
k k (t);F
12
(t);;F
1k (t);F
23
(t);;F
(k 1)k (t)
(1)
A loser examination of the proess X
k
(t) reveals that it is not time-reversible (see [18 ℄). This
result is trueingeneral, when k2and W 2.
Sine thenumber ofstates of proess X
k
(t) grows very fastwith the number k of hopsin the
path and thenumberW ofwavelengths, itis notpossibleto obtaintheallblokingprobabilities
diretly from the above proess. Consequently, an approximate model wasonstruted in[18 ℄ to
analyze asingle-lass, k-hoppathofa wavelengthroutingnetwork. Theapproximationonsistsof
modifyingtheallarrivalproessto obtainatime-reversibleMarkovproessthathasalosed-form
solution. To illustrateour approah, let usonsiderthe Markov proess orresponding to a2-hop
path:
X
2
(t)=(N
11 (t);N 12 (t);N 22 (t);F 12 (t)) (2)
We nowmodifythe arrivalproessof alls that usebothhops(a Poisson proess withrate
12 in
the exat model) to astate-dependent Poisson proess withrate
12 given by: 12 ( n 11 ;n 12 ;n 22 ;f 12 )= 12 f 12 (W n 12 ) f 11 f 12 (3)
The arrival proess of other alls remain as inthe original model. As a result, we obtain a new
Markovproess X 0
2
(t)with thesame state spae and thesame state transitionsas proess X
2 (t),
butwhihdiersfrom thelatterin some ofthestate transitionrates.
We made the observation in [18 ℄ that under the new arrival proess (3) for alls using both
hops, theMarkovproess X 0
2
(t)is time-reversibleandthe stationary vetor isgiven by:
( n 11 ;n 12 ;n 22 ;f 12 )= 1 G 2 (W) n11 11 n 11 ! n12 12 n 12 ! n22 22 n 22 ! f 11 f 12 ! n 11 W n 12 n 22 f 12 ! W n 12 W n 12 n 22 ! (4) whereG k
11 12 22 ij ij
j 2. It an be veried[18 ℄ that
P(n 11 ;n 12 ;n 22 )= 1 G 2 (W) n 11 11 n 11 ! n 12 12 n 12 ! n 22 22 n 22 ! (5)
Likewise, for a k-hop path, k 2, withthe modied state-dependent Poissonarrivalproess, the
marginal distributionoverthestates forwhih N
ij
(t)=n
ij
,1ijk,isgiven by:
P(n
11 ;n
12 ;;n
kk )= 1 G k (W) Y f(i;j)j1ijkg n ij ij n ij ! (6)
It is easilyseen that thisdistribution is thesame asinthe ase of a network withwavelength
onverters at eah node. An interesting feature of having wavelength onverters at every node
is that the network has a produt-form solution even when there are multiple lasses of alls on
eah route, aslong as allarrivals arePoisson,and holding times are exponential [7, 4℄. Further,
when alls of all lasses are assigned wavelengths from the same set, we an aggregate lasses to
get an equivalent single-lass model with the same steady-state probability distribution over the
aggregated states, asweshownext.
3.2 The Multi-Class Case
Let us now onsidera k-hop path with wavelength onverters at all nodes, and with R lassesof
alls. If (r)
ij
, 1 i j k, 1 r R , is the arrival rate of alls of lass r on route (i;j), and
1= (r)
ij
, 1 ij k,1 r R , is the mean of the exponentialholding time of alls of lass r,
the probabilityof beinginstate n= n (1) 11 ;n (2) 11 ;;n
(R)
11 ;n
(1)
12
;;;n (R)
kk
is given by:
P(n)= 1 G k (W) 0 B Y f(i;j)j1ijkg R Y r=1 ( (r) ij ) n (r) ij n (r) ij ! 1 C A (7) Let ij = P r (r) ij ands ij = P r n (r) ij
. Asdened,s
ij
isthetotalnumberofallsofalllassesthat
usesegment(i;j)ofthepath,and
ij
isthetotaloeredloadofthese alls. Takingthesummation
of (7)over allstates suh that P r n (r) ij =s ij
,1ijr,weobtain:
P 0 (s 11 ;s 12 ;;s
kk )= X fnj P r n (r) ij =s ij g
P(n)= 1 G k (W) Y f(i;j)j1ijkg s ij ij s ij ! (8)
Observe thatthis is idential to the solution(6) forthe single-lassase obtained by substituting
pathwithonvertersatallnodes,weobtainasystemequivalenttoasingle-lasspathwith
onvert-ers. In Setion3.1, we showed that the modied single-lasswavelength routing network without
onverters has a steady-state marginal distribution similarto the exat single-lass network with
onverters. We nowshow that a modied multi-lass network withoutwavelength onverters an
also be subjeted to lass aggregation that results in an equivalent single-lassmodel. The
mod-iation applied to the arrival proess of alls is similar to the single-lass ase, and it is given
by: (r) ij (x)= (r) ij f ij P i l =1 s l i +f ii P i+1 l =1 s l (i+1) +f (i+1)(i+1) P j 1 l =1 s
l (j 1) +f
(m 1)(m 1) ) f ii f (i+1)(i+1) f jj (9)
Then, the probability that the equivalent single-lass network without onverters is in state
S x = s 11 ;s 12 ;;s
1k ;s
22 ;;s
kk ;f
12 ;f
13 ;;f
(k 1)k
isgiven by:
(S x )= 0 Y i;j s ij ! s ij ij 1 A 0 B B B B B B k Y l =2 0 f 1(l 1) f 1l 1 A 8 < : Q l 1 m=2 0 f m(l 1) f
(m 1)(l 1)
f ml f (m 1)l 1 A 9 = ; 0 f l l +n l l f
(l 1)(l 1)
f l l f (l 1)l 1 A 0 f l l +n l l f l l 1 A 1 C C C C C C A (10)
One again,theparameters ofthe single-lassmodelaregiven by:
s ij = R X r=1 n (r) ij
1i<jk;
ij = R X r=1 (r) ij
1i<j k (11)
3.3 Bloking Probabilities in the Multi-Class Case
Sine thearrivalrateof alls ofeah lass oneah route isPoisson,theblokingprobability,Q (r)
ij ,
of a allof lassr usingroute (i;j) isjust thefration of timethat there areno wavelengths that
are freeon all hopsalong route(i;j) (see thePASTAtheorem in[17 ℄). Thus, we have:
Q (r) ij = lim !1 R t=0 I fF ij (t)=0g dt
; where I
fF ij ()=0g = 8 < :
1; if F
ij
()=0
0; otherwise
(12)
As an be seen,theblokingprobabilityislass-independent.
Next,we fous on theallblokingprobabilitiesinthemodiedmodel. The arrivalproess of
alls of lassr on route (i;j) isa state-dependent Poisson proess whoserate at time,A (r)
A (r) ij () (r) ij
(X())=
ij F ij () Q j 1 k=i P k l =1 N l k
()+F
kk () F ii F i+1;i+1 F jj (13)
Notethat themodied arrivalproess satises theriterion:
(r 1 ) ij (x ) (r 2 ) ij (x ) = (r 1 ) ij (r 2 ) ij
1r
1 ;r
2
r (14)
By applying the PASTA theorem onditioned on being in state x, the onditional all bloking
probability,P (r)
ij
(x), ofallsoflassronroute(i;j)isgivenbythefrationoftimespentinstatex
inwhihthere are nowavelengths that arefreeon all hopsof route (i;j). Therefore:
P (r)
ij
(x )= lim
!1 R
t=0 I
fFij(t)=0;X(t)=x g dt R t=0 I fX(t)=x g dt = 8 < :
1; if f
ij =0 0; otherwise (15) Let P (r) ij
be the unonditionalprobability that a all of lass r, on route (i;j) gets bloked in
the modiedmulti-lassmodel. Thisis given by:
P (r) ij = P x (r) ij (x)(x )P (r) ij (x) P x (r) ij (x)(x ) = P fxjf ij =0g (r) ij (x )(x) P x (r) ij (x)(x ) (16)
and analsobeseento beindependentofthelassr. Thus,byomputingtheblokingprobability
on theequivalent single-lasspath, we an obtainthesolutionto themulti-lass path.
4 Bloking Probabilities in Mesh Topologies
The solutionto single-lassnetworkswitha meshtopology, wavelengthonverters at an arbitrary
subsetof nodes,and stati orxedalternaterouting,hasbeenpresentedin[21 , 19 ℄. Thissolution
involvesdeompositionof the network into shortpath segments withtwo orthree hops, and
ana-lyzing these approximatelyusingexpression (4), ortheorresponding expressionfora 3-hop path
whih an be found in [21 ℄. The solutions to individual segments are appropriately ombined to
obtainavaluefortheblokingprobabilityofallsthattraverse morethanonesegment. Theeet
of thewavelength ontinuityrequirement is aptured by an approximateontinuity fator that is
used to inrease the bloking probability of alls ontinuing to the next segment to aount for
the possiblelakof ommon free wavelengths in thetwo segments. The proess repeats until the
blokingprobabilitiesonverge. Byapplyingthetransformationsin(11),thesamealgorithmsmay
beusedto alulateblokingprobabilitiesformulti-lassnetworks. Speially,weusethese steps
sub-systemsusingthealgorithm in[19 ℄.
2. Time-reversible proess approximation: For eah single-path sub-system, modify the
arrivalproess as given by expression (9) to obtainan approximate time-reversible Markov
proessforthe path.
3. Class aggregation: For eah sub-system, apply the transformations in (11) to obtain an
equivalent single-lasspathsub-system.
4. Calulation of bloking probabilities: For eah path sub-system, obtain the bloking
probabilitiesasfollows. Ifthepath isat mostthree hopslong,useexpressions(16) and(10)
diretly. If thepathsub-system is longerthanthree hops, analyze itbydeomposingit into
2-or3-hoppathswhiharesolvedinisolation,andombinetheindividualsolutionstoobtain
the blokingprobabilitiesalong theoriginallongerpath (see[18 ℄).
5. Convergene: Repeat Steps 2to 4,after appropriatelymodifyingthe originalarrivalrates
to eahsingle-pathsub-systemtoaountforthenewvaluesoftheblokingprobabilities
ob-tainedinStep4(see[19 ℄),untiltheblokingprobabilitiesonvergewithinaertaintolerane.
5 Evaluating Call Bloking Probabilities with Multiast Calls
In this setion, we examine how the path deomposition tehnique an be used to solve for all
bloking probabilities in multi-lass wavelength routing networks with multiast alls. We rst
assume that multiast all arrivals are also Poisson. Then, we solve for state-dependent Poisson
arrivalsusingan approximation algorithm.
5.1 Multiast alls with Poisson arrivals
Considera single-lasswavelengthroutingnetworkwithonlyuniastalls. Tothisnetwork,let us
add asingle-lassofmultiast allswhiharerouted usingstati orxedalternaterouting. Under
stati routing,asingle multiasttree isavailableforaallbetweena givensoure anddestination
set. Ifthe allannot be establishedoverthetree due to lakof free wavelengths,then theallis
bloked. With xed alternate routing,a (typiallysmall) numberof multiast trees are available
foreah multiastall. Upona allarrival, theorresponding treesareonsideredina xedorder
for arrying the all. In thisase, the all is bloked ifno free wavelength is found in anyof the
1 2 v
There are two possibilities. First, the soure node i, and all the destination nodes may lie on a
single linear path. Let j
m and j
n
be the nodesat the extremities of thispath (j
m and j
n
an be
anyof the nodes infig[V). Forpurposesof evaluating allblokingprobabilities,thismultiast
session is equivalent to a uniast all between node j
m and j
n
. Let
j
m ;j
n
be the Poisson arrival
rateofuniastallsbetweennodesj
m andj
n ,
1
jm;jn
bethemeanoftheirexponentialholdingtime,
and
j
m ;j
n =
jm;jn
jm;jn
. Let
i;V
be the Poisson arrival rate of multiast alls from soure node i to
a set of destination nodes V, 1
i;V
bethe mean oftheir exponential holdingtime, and
i;V =
i;V
i;V .
Then, thegiven multiastnetwork anbe analyzedasauniast network withthefollowingsimple
modiation:
(new)
jm;jn =
jm;jn +
i;V
(17)
The resultingnetwork maybeanalyzed usingthepathdeompositionalgorithm in[21 ℄.
Ingeneral,however, thesoureanddestination nodesofamultiastallneednotlieonasingle
linearpath. Buteveninthisase,themultiasttreemaybebrokenupintoseverallinearsegments,
whereeah segment startsatthesoure(orabranh pointinthetree) andterminatesata branh
point (or a destination node). This network an be also analyzed using the path deomposition
algorithm in[21 ℄ byonsidering eah segment ofthe multiasttree asa sub-system. Inturn,eah
resulting sub-systemmay beanalyzed asauniast network after applyingthemodiationshown
in (17) for eah multiast group using the sub-system. The bloking probability of a multiast
all an then be expressed in terms of the bloking probabilityon the various segments to whih
its multiasttree has been deomposed. Note that, sinethedeompositionalgorithm in [21℄an
handlexed alternateroutingforuniast alls,itan be extendedin astraightforward mannerto
handlexed alternateroutingformultiast alls.
Toanalyzeamulti-lassnetwork,werstonstrutanequivalentsingle-lassnetwork(inluding
both uniast and multiast alls) in the mannerdesribed in Setion 3.2. Then, the modiation
in (17)is appliedto thisequivalentsingle-lass network to obtainasingle-lassnetwork withonly
uniast alls. Finally,thedeompositionalgorithmin[21 ℄ is appliedto theresultingnetwork.
5.2 Multiast Calls with State-Dependent Poisson Arrivals
Our disussionin theprevious setion assumed that multiast alls arrive aording to a Poisson
proess. Inpratie,however, itisunlikelythat therewillbeseveral onurrentlyativemultiast
sessions among the same set of soure and destination nodes. To aount for this fat, we now
dened bytheorderedpair G
i
=(i;V),where iisthe sourenodeand V isthe setof destination
nodes. The ordered pair G
i
may also be used to represent the route of the multiast group.
Likewise, the route of a uniast all from node i to node j is denoted by the ordered pair (i;j).
LetR representthetotal numberofroutesinthenetwork representing bothuniastandmultiast
alls.
We rst onsider the single-lass ase. Let
(i;j)
be the Poisson arrival rate of alls on a
uniast route. Let 1
(i;j)
be the mean of the exponential holding time. Let 1
G
i
be the mean of
the exponentiallydistributedtime untilthearrivalofa multiast allto groupG
i
,onditionedon
there being no ative sessions. The arrivalproess of multiastalls is,therefore, state-dependent
Poisson. Let 1
G
i
bethemeanof theexponentialholdingtime. We usea Poissonapproximationto
obtaintheblokingprobabilitiesof thistype ofnetwork,whihwe nowproeed to desribe.
Suppose thewavelength routingnetwork has an unlimitednumberof wavelengths. Then, the
state of the network may just be dened by the number of ative alls on eah route, inluding
multiast routes. Let N
r
(t) be the number of ative alls on route r at time t. The state of
the system at time t may, therefore, be represented by S(t)=(N
1 (t);N
2
(t);;N
R
(t). Let (n)
represent the probability P (S(t)=n) in steady state. When there are an unlimited number of
wavelengths, theMarkov hain S(t)hasa produt-formsolution,and (n) may be writtenas:
(n
1 ;n
2 ;::::;n
R )= R Y r=1 P r (n r ) where P r (n r
) isthemarginalprobabilitythatthenumberofative alls onroute r isn
r
insteady
state. Ifr isa uniastroute,P
r (n
r
)is given by:
P r (n r )= 1 n r ! r r ! n r exp r r !
Formultiast routes,we have:
P r (0)= r r + r P r (1)= r r + r P r (n r
)=0 if n
r >1
Thus, the mean arrival rate of alls on multiast route r is given by r r r + r
. We substitute this
to obtain an approximate state-independent Poisson arrival model for multiast alls. Thus, the
arrivalrateof allsto multiastroute r is assumedPoissonwithmean
r = r r r + r
With a multi-lass network, we rst apply the Poisson approximation to eah lass of multiast
alls with state-dependent Poisson arrivals. From this, an equivalent single-lass network may be
onstruted inthesame way asshown inSetion3.2. Then, themodiationin(17) isapplied to
thissingle-lassnetwork.
6 Numerial Results
In this setion, we validate the approximate method desribed in Setion 4 by omparing the
blokingprobabilitiesforeahrouteasobtainedfromtheapproximatemethodwiththoseobtained
throughsimulationof theexat model.
6.1 Multi-Class Networks with Uniast TraÆ
Werstprovideresultsfor3-hoppathswhiharethebasibloksofourdeompositionalgorithm.
In Table 1 we show the arrival and servie rates for alls on eah route (i;j);1 j 3, of a
3-hop path. There are R = 3 lasses of alls for eah route. In Figure 2, we plot the bloking
probability against the number W of wavelengths for three (out of the six) types of alls in this
path: alls usingRoute (1;1), i.e., the rst hop of the path, alls on Route (2;3), that is, those
using the last two hops, and alls on Route (1;3) using all three hops of the path. As we an
see, theblokingprobabilitydereases asW inreases, asexpeted. We also observe thatalls on
Route(1;3)(i.e.,allsusingallthreehopsofthepath)experienethehighestblokingprobability,
againasexpeted. Mostimportantly,however,weanseethatthereisgoodagreementbetweenthe
values ofthe blokingprobabilities obtainedthroughour analytialtehnique and those obtained
through simulation. Similar results have been obtained for dierent values for the arrival and
servie parameters and fordierent numberof lasses, indiatingthat ourapproximatemethodis
aurate overa widerangeof networkharateristis.
Next, we onsider a network with a topology similar to that of the NSF network, shown in
Figure 3. There are 16 nodes, and 240 uni-diretional routes. There are three lassesof alls on
eah route. The arrivaland servierates of allsof a partiular lass are thesame on eah route,
and areshownin Table 2. The bloking probabilitiesareplotted in Figure 4 forfour routes, asa
funtion of the number of wavelengths on eah link. Route A is a single-hop route from nodes 1
to 5. Route B has two hops, onneting node 1 to node 3 via node 2. Route C has three hops,
Route Class1 Class2 Class3
(i;j)
(1,1) 3.0 3.0 1.0 3.0 1.0 3.0
(1,2) 1.0 6.0 2.0 6.0 1.0 6.0
(1,3) 1.0 2.0 1.0 2.0 2.0 2.0
(2,2) 1.0 3.0 1.0 3.0 2.0 3.0
(2,3) 1.0 2.0 1.0 2.0 3.0 2.0
(3,3) 3.0 6.0 1.0 6.0 1.0 6.0
5
6
7
8
9
10
11
12
10
−4
10
−3
10
−2
10
−1
10
0
Number of wavelengths
Blocking probability
Simulation
Approximation
Route 11
Route 23
Route 13
From Figure 4 we an see that the length of the path used by a all onsiderably aets the
blokingprobabilityexperienedbytheall,an observationthatisonsistentwithallourprevious
resultsin thissetion. Speially,foragiven numberW ofwavelengths, theblokingprobability
inreases withthenumberofhops inaroute, suh thatallson Route A(a single-hop path)have
the lowest blokingprobabilitywhilealls onRouteD (a four-hoppath) thehighest. Further, the
results indiatethat our approximation method an be used to estimate aurately the bloking
probabilitiesforall allsin thenetwork.
6.2 Multi-lass Networks with Multiast TraÆ
In this setion, we ompare the results of theapproximate algorithm fornetworks withmultiast
alls to simulation results. First, we onsider a network with the simple \T" topology shown in
Figure 5. The network isomposed of ve nodesand two linear paths. In additionto eah of the
twenty possible uniast routes, we assume the following sets of links to arry multiast alls: (i)
Multiast treef1;2;3g and (ii)Multiast tree f1;2;3;4g.
Fortheresults presentedhere,we assumed thateah of theabove multiasttreesan arryat
most two onurrently ative multiast sessions. For instane, eah multiast tree is used by two
multiastgroups,eah of whihallows at mostone ative multiast sessionat anygiven time. We
also assumed three lasses of alls on eah (uniast or multiast) route. Table 3 shows thevalues
of thearrivalandservie rateparametersassumed. Aswean see, thismulti-lass network isvery
general sine the arrival and servie parameters depend not only on the lass of a all, but also
on the route taken (i.e., the soure and destination(s) of the all). This example illustrates the
potential of our path deomposition tehnique whih an be used to analyze wavelength routing
networks with multiple lasses of traÆ, and eah lass onsisting of alls with a wide range of
behavior.
In Figure 6, we plot the bloking probability values against the number of wavelength per
link for a few of the routes. Two sets of results are shown, one set obtained by simulation and
one obtained by approximate analysis. We again observe the familiar pattern of higher bloking
probabilityforroutesonsistingoflarger numberoflinks. In partiular,multiastallsexperiene
higher probability than uniast alls, as expeted. (Note: Figure 6 plots the probability of full
bloking[11 ℄,wherebyamultiastallisblokedwhenitisimpossibletoestablishtheallbetween
thesoureandallthedestinationnodes. It is,however, straightforwardtoomputetheprobability
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Figure3: The NSFnetwork
Table 2: Arrivaland servieparameters forthe NSFNetwork
Class1 Class2 Class3
0.1 0.2 0.15
3.0 6.0 4.0
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
10
−4
10
−3
10
−2
10
−1
10
0
Number of wavelengths per link
Call Blocking Probability
Simulation
Approximation
Route A
Route B
Route D
Route C
again observe the good agreement between analysis and simulation over a range of all bloking
probabilityvalueswhih spansfour ordersofmagnitude.
Next,weonsidera33torusnetworkwithmultiastalls. The network topologyisshownin
Figure7. Thereare18one-hopuniastroutesand18two-hopasshowninthegure. Therearesix
multiasttrees, eah of whih may be listedbythe set of linksthey onsist of: f1;5;6g, f2;4;7g,
f3;1;10g, f3;10;12g, f6;10;13;14g, and f8;13;15g. We assume these onstitute routes 37 to 42.
Eah multiast routeis assumedto arry at mosttwo onurrent multiast alls.
Therearethree lassesofalls(uniast ormultiast)on eahroute. The arrivalrate ofallsof
Class1oneahuniastroutewasassumedthesameandsetto0.4(allspertimeunit). Thearrival
rate ofClass1allson eahmultiastroute,onditionedonthere being noexistingmultiastalls
wasassumedequalto0.4. Themeanholdingtimewasassumedthesameonallroutesandequalto
1/3 timeunits. ThearrivalrateofallsofClass2oneahuniastroutewasassumedthesameand
setto0.6(allspertimeunit). ThearrivalrateofClass2allsoneahmultiastroute,onditioned
on there being no existing multiast alls wasassumed equalto 0.6. The mean holding timewas
assumed the same on all routes and equal to 1/5 time units. The arrival rate of alls of Class 3
on eah uniast route was assumedthe same and set to 0.5 (alls pertime unit). Thearrivalrate
of Class3allson eah multiastroute,onditioned onthere being noexisting multiastallswas
assumed equalto 0.5. Theholdingtimewasassumedthesameon allroutesand equalto 1/6 time
units.
InFigure8,weplottheallblokingprobabilitiesobtainedthroughtheapproximationmethod
and throughsimulation,assumingrandomwavelength alloation. We also plottwo additionalsets
ofsimulationresults: onesetassumingrst-talloationandnowavelengthonversion,andoneset
for anetwork with onverters at every node (in whih ase wavelength alloationisnot an issue).
It maybe observed that the resultsobtained by approximateanalysis are loser to theresults for
the rst t simulation forthe uniastroute 3 whih hasa single hop. However, formultiple hops
and for multiastroutes, the simulationand analytialresults withrandomwavelength alloation
are inloseragreement.
6.3 Disussion
Results similarto the onespresentedin Figures2,4,6, and 8 have been obtainedfor a widerange
of traÆloadsanddierentlassesofalls,and forothernetworktopologies. Ourmainonlusion
probabil-0
1
2
3
4
1
2
3
4
Figure 5: A simplemultiastnetwork topology
Table 3: Arrivaland servie parametersforthesimple multiastnetwork
Route (0;1) (0;2) (0;3) (0;4) (1;2) (1;3) (1;4) (2;3) (2;4) (3;4) f1;2;3g f1;2;3;4g
Class1 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0
Class2 0.5 1.5 0.5 1.5 0.5 1.5 0.5 1.5 0.5 1.5 0.5 1.5
7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0
Class3 0.5 0.5 1.0 0.5 0.5 0.5 1.0 0.5 0.5 0.5 1.0 0.5
9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Number of wavelengths per link
Call blocking probability
Approximate analysis
Simulation
Route (0,1)
Route (0,2)
Route (0,1,2,3)
Route (0,1,2,3,4)
1
2
3
4
5
6
7
8
9
4
3
7
8
10
11
12
16
17
18
6
15
19 -> 1 + 6
20 -> 2 + 8
21 -> 1 + 7
22 -> 2 + 15
23 -> 1 + 3
24 -> 5 + 8
25 -> 1 + 4
26 -> 5 + 9
27 -> 2 + 3
28 -> 5 + 6
29 -> 2 + 4
30 -> 5 + 7
31 -> 10 + 14
32 -> 11 + 15
33 -> 10 + 12
34 -> 13+15
35 -> 11 + 12
36 -> 13 + 14
Interconnection Pattern
2-hop routes
2
1
13
14
9
5
Figure 7: The33 torusnetwork
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Number of wavelengths per link
Call blocking probability
Simulation (random allocation)
Analysis (random alloc)
Sim (first fit)
Sim (with converters)
Unicast Route 3
Unicast Route 19
Multicast Route 39
tehniqueaordsasigniantredutioninthetimeforomputationof blokingprobabilities. For
instane, thesimulation program takes approximately 100 minutes whilerunning on a Sun Ultra
10 workstation, to ompute all blokingprobabilities for thenetwork with a topologysimilar to
theNSFnetwork. Theapproximationmethod,ontheother hand,takeslessthanaminuteforthe
same network. In addition, to obtainstatistially signiant results, thesimulationprogram runs
untilthereareatleast10 5
allarrivalsofeverylass. Forallblokingprobabilitieslessthan10 4
,
thesimulationprogrammustonsidermorethan10 6
allarrivalsofeahlass,resultinginaneven
longer omputationtime.
7 Conluding Remarks
We have onsidered the problemof omputingall bloking probabilitiesin multi-lass multiast
wavelengthroutingnetworkswhihemploytherandomwavelengthalloationpoliy. Ourapproah
onsists ofmodifyingtheallarrivalproess to obtainanapproximatemulti-lassnetwork model,
using lass aggregation to map this to an equivalent single-lass network, and employing path
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