WDM Network Design by ILP Models Based on Flow Aggregation

44 Pages·2013·2.4 MB·English

IEEE/ACMTRANSACTIONSONNETWORKING,VOL.15,NO.3,JUNE2007 709
WDM Network Design by ILP Models
Based on Flow Aggregation
Massimo Tornatore,Member,IEEE, GuidoMaier,Member,IEEE,and AchillePattavina,SeniorMember,IEEE
Abstract—Planning and optimization of WDM networks has planning tools are needed for the future which can determine
raised much interest among the research community in the last howtoutilizeefficientlythenetworkresourcesinareasonable
years.IntegerLinearProgramming(ILP)isthemostusedexact
computationaltime.
methodtoperformthistaskandmanystudieshavebeenpublished
Sincesomeyearsago,researchonopticalnetworkshasbeen
concerning this issue. Unfortunately, many works have shown
that, even for small networks, the ILP formulations can easily investigating design and optimization techniques. The various
overwhelm the capabilities of today state-of-the-art computing proposed solutions can be classified into two main groups:
facilities. So in this paper we focus our attention on ILP model heuristic methods and exact methods. The former return
computationalefficiencyinordertoprovideamoreeffectivetool
sub-optimal solutions that in many cases are acceptable and
in view of direct planning or other benchmarking applications.
havetheadvantageofrequiringalimitedcomputationaleffort.
Our formulation exploits flow aggregation and consists in a new
ILP formulation that allows us to reach optimal solutions with The latter are much more computationally intensive and do
less computational effort compared to other ILP approaches. notscalewellwiththenetworksize,beingevennotapplicable
This formulation applies to multifiber mesh networks with or in some cases. However, since the exact methods are able to
withoutwavelengthconversion.Afterpresentingtheformulation
identifytheabsoluteoptimalsolution,theyplayafundamental
we discuss the results obtained in the optimization of case-study
roleeitherasdirectplanningtoolsorasbenchmarkstovalidate
networks.
andtestheuristicmethods.
Index Terms—Design methodology, integer programming, op-
Theworkwearepresentingconcernsexactmethodstoplan
ticalfibercommunication,wavelengthdivisionmultiplexing.
andoptimizemultifiberWDMnetworks.Inparticularwefocus
onIntegerLinearProgramming(ILP),awidespreadtechnique
I. INTRODUCTION to solve exact optimization: we propose a new formulation of
the optimization problem that we call source formulation, in
INRECENTopticalnetworks,theintroductionofthewave- that it exploits the aggregation of all the flows generated in a
singlesourcenode[1].Oursourceformulationisequivalentto
lengthdivisionmultiplexing(WDM)techniquehasopened
thewellknownflowformulation,butitallowsarelevantreduc-
the road to a new paradigm of transport infrastructure evolu-
tionofthenumberofvariablesandofconstraints,thussensibly
tioncharacterizedbyhighcapacityandhighreliability.Onthe
diminishing computation time and memory occupancy during
switching equipment side, optical cross connects (OXC) sys-
optimizationruns.
tems have become available, beside the more mature optical
Thepapersummaryisasfollows.InSectionII,weintroduce
add-dropmultiplexers.Thisopeneduptheroadtothepossibility
our solution by presenting a short review of the literature re-
of deploying complex WDM networks based on mesh topolo-
gardingILPapplicationstoWDMoptimization.InSectionIII,
gies,whileinthepastsingleringoroverlaidmulti-ringwerethe
thesourceformulationispresentedandexplainedintodetailsin
mostusedarchitecturesforWDM.Inordertotransferdatabe-
the two versionsfor network withor without wavelengthcon-
tweentwonodes,anopticalconnectionneedstobesetupand
version.Finally,inSectionIV,resultsobtainedbyapplyingthe
routedattheopticallayerasinacircuit-switchednetwork.
source formulation to case-study networks are shown and the
TheincreaseinWDMcomplexitybroughttheneedforsuit-
new formulation is compared to the traditional flow and route
ablenetworkplanningstrategiesintotheforeground.Problems
formulationstopointouttheadvantagesofthemethodweare
such asoptimal dimensioning, routing and resource allocation
proposing.AnAppendixisincludedtoshowtheequivalenceof
foropticalconnectionsmustbecontinuouslysolvedbynewand
flowandsourceformulation.
oldoperators,toplannewinstallationsortoupdateandexpand
the existing ones. These problems can no longer be manually
solvedincomplexnetworkarchitectures,asitusuallyhappened II. WDM NETWORK OPTIMIZATION BY INTEGER
intheearlierexperimentalWDMinstallations.Computer-aided LINEARPROGRAMMING
Network design and planning is carried out with different
techniques according to the type of traffic the network has to
ManuscriptreceivedMarch28,2003;revisedApril30,2004,andNovember support.Weinvestigatethestatictrafficcaseinwhichaknown
17,2005.ThisworkwassupportedinpartbytheEUNetworkofExcellence
setofpermanentconnectionrequestsisassignedaprioritothe
"E-Photon/ONe+.ApreliminaryversionofthispaperwaspresentedattheIEEE
INFOCOM2002,NewYork,NY. network.Theconnectionsrequestedbythenodesatagiventime
The authors are with the Department of Electronics and Information, toaWDMnetworkalltogetherformtheofferedtrafficmatrix
Politecnico di Milano, 20133 Milan, Italy (e-mail: [email protected];
virtualtopology(aliasvirtualtopology).Eachrequestisforone
[email protected];[email protected]).
DigitalObjectIdentifier10.1109/TNET.2007.893158 ormorepoint-to-pointopticalcircuits(lightpaths)abletocarry
1063-6692/$25.00©2007IEEE
710 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.15,NO.3,JUNE2007
agivencapacityfromthesourceterminationtothedestination aspectsofWDMnetworks.Forexample,in[9]theoptimization
termination. We assume that all the WDM channels carry the iscarriedoutinordertoemphasizethedifferencebetweenWP
samecapacity.LightpathsareroutedandswitchedbytheOXCs and VWP scenarios. Ref. [6] studies the effects of imposing a
ofthenetworkandthetwolightpathterminationsarelocatedin constraint on the average delay seen by a source-destination
thesourceanddestinationOXCs. pairandtheamountofprocessingrequiredatthenodes,while
Weassumethatthechannelscomposingthelightpath(onefor in [10] possible utilization of bounds derived from the two
eachfiberitcrosses)mayhavedifferentwavelengthsormaybe formulationsbyrelaxationoftheintegerconstraintsarestudied
allatthesamewavelength,accordingtotheavailabilityofthe and compared. In other works, the authors have selected as
wavelength conversion function in the transit OXCs. To sim- costfunctionsthenumberofwavelengths[9],[11]orthetotal
plify, we have considered two extreme cases referring to def- number of WDM channels in the network [12], [13]. In [14]
initions introduced in [2]: the virtual wavelength path (VWP) authorsproposenewILPformulations,whichtend tohavein-
network case, in which all the OXCs are able to perform full tegeroptimalsolutionsevenwhentheintegralityconstraintsare
wavelengthconversion,andthewavelengthpath(WP)network relaxed, thereby allowing the problem to be solved optimally
case,inwhichnowavelengthconversionisallowedinthewhole by fast and highly efficient linear (not integer) programming
network and lightpaths are subject to the “wavelength conti- methods. In [15] an exact linear formulation was presented
nuity” constraint, that is absent in the VWP case. It is impor- for the logical topology design problem with no wavelength
tanttonotethatwavelengthassignmenttolightpathsinWPcase converters. In [16] the authors have investigated the so called
isanNP-completeproblem(itisequivalenttothewell-known RWA-P,i.e.,theRWAproblemwhileallowingfordegradation
graph-coloringproblem)[3]. ofroutedsignalsbyopticalcomponents.
Today,WDMnetworksareoftendesignedinordertobere- In optimization of multifiber WDM networks optimal allo-
silient to failures that may occur to switching or transmission cation of fibers has also to be solved, thus complicating the
equipment. Though automatic lightpath protection is very im- problem of lightpath set up into routing, fiber and wavelength
portanttoday(giventhehighbit-ratesthataWDMchannelusu- assignment (RFWA). Solving RFWA becomes really chal-
allycarries,e.g.2.5to40Gb/s),thisfeaturewillnotbecovered lengingevenwithrelativelysmallnetworks,especiallybecause
inthiswork,forthereasonsthatwillbeexplainedlateron. routingandwavelengthassignmentiscoupledtodimensioning.
StaticoptimizationofaWDMnetworkcanbesummarizedas In this case, a new set of variables representing the number
follows:givenastatictrafficmatrix,findtheoptimumvaluesof of fibers of each physical link must be considered in addition
asetofnetworkvariablesthatminimizesagivencost(orobjec- to the flow or the route variables defined above for the two
tive)function,underasetofconstraints.Thechoiceofvariables, formulations. This implies that RFWA has also to include the
costfunctionandconstraintsgreatlyvariesfromcasetocase.In highly complex localization problem. The choice of complex
thepastmostofstudiesregardingWDMnetworkplanningwere costfunctionssuchasthosecomprisingnodeorductcostmakes
aimedatvirtualtopologyoptimizationwithsingle-fiberWDM theachievementofILPoptimalsolutionverychallengingeven
links[4],[5].Thecostfunctiontobeoptimizedwaseitherthe for very small networks [17] (this is even worse in the case
numberofwavelengthsnecessarytoroutethestatictrafficorthe of nonlinear objective function that require integer nonlinear
networkload(thenumberofchannelsroutedonthemostloaded programming[18]).
link of the network) [6]. In [6] the authors introduce an ILP When the problem becomes computationally impractical,
modelbasedonaggregatedflowsappliedtovirtualtopologyop- routeformulationbecomesmoreusefulthanflowformulation.
timization.Intheworkweareproposingthevirtualtopologyop- IfitisacceptablethatRFWAisperformedinaconstrainedway,
timizationisaccompaniedbycostminimizationofamulti-fiber then the solution complexity of the route formulation can be
physicalnetwork:thenumberoffibersperlinkneededtosup- controlled.Forexample,allthelightpathscanbeconstrainedto
portapre-assignedtrafficisavariableoftheproblemtobemin- beroutedalongthefirst shortestpathsconnectingthesource
imized,whiletheamountofwavelengthsperfiberispreset[7]. to the destination. Differently from the flow formulation, the
WDMnetworkoptimizationbyILPhasbeenwidelystudied complexity of which is strictly dependent on physical and
inliterature.Wecansubdivideresearchcontributionsintotwo virtual topologies, the complexity of the route formulation
groups according to the type of networks to which they are decreases with the number of paths that can be employed to
applied: routethelightpaths.Multifibernetworkoptimizationwithroute
• WDMnetworkswithsingle-fiberlinks; formulationandconstrainedroutinghasbeenstudiedin[9]and
• multifiberWDMnetworks. [19]–[23].
In the first group the problem consists in optimal routing and Beside route formulation with constrained routing, other
wavelength assignment (RWA) of the lightpaths. This is a methods to control complexity have been proposed. A possi-
NP-completeproblem,asitwasdemonstratedin[3],[8].Two bilityistostopthebranch-and-boundalgorithm(typicallyused
basic methods have been defined to model the RWA problem: to solve ILP problems) after finding the first or a pre-definite
flow formulation(FF) and route formulation (RF) [9]. In the number of integer solutions. Ref. [17] shows that acceptable
former the basic variables are the flows on each link relative results (though quite far from the optimal solution) can be
to each source-destination OXC pair; in the latter the basic obtained when the branch-and-bound duration is fixed to 10
variables are the paths connecting each source-destination minutes.Ref.[24]proposedthatthewholeRFWAproblemcan
terminationpair.Boththeseformulationshavebeenemployed besolvedasasequenceofsimplerproblems(e.g.firstrouting,
to solve various sorts of problems and to investigate different then fiber assignment, and so on). Other possible approaches
TORNATOREetal.:WDMNETWORKDESIGNBYILPMODELSBASEDONFLOWAGGREGATION 711
twoformulationsbydescribinghowtoobtainanequivalentFF
(SF) solution, given a SF (FF) solution. In other words, if the
objective is to evaluate the number and the distribution of the
fibersinthenetwork,wecansimplyapplySFinordertoachieve
thesolution.Then,ifweareinterestedalsointhedetailsofthe
RWA(i.e.,theroutingandwavelengthassignmentofeachcon-
nectionrequest),wehavetotransformtheSFsolutioninaFF
(or equivalent solution). This second step is absolutely negli-
gible from a complexity point of view: if the first SF step is a
localizationproblem,thesecondstep(neededtotransformthe
Fig. 1. Example of (a) three distinct source-destination commodities and
SFsolution indetailed RFWA description)has the samecom-
(b)the corresponding single source commodity, which will be exploited in
sourceformulation. plexityof a mere max-flow algorithm (for further details refer
totheAppendix).Soallthecomputationaltimesreportedinthe
following are related to the SF step, disregarding the possible
are:exploitationoflagrangeanrelaxation[23],[25],relaxation followingtransformation.
ofintegerconstraints[19]andrandomizedrouting[12]. We explain now the details of the source formulation, for
Undoubtedly, the massiveneed for computational resources which two different versions are reported related to networks
(i.e., processing time and memory occupation) represents the withorwithoutwavelengthconversioncapability.
main obstacle to an efficient application of ILP in optical net-
work design. Constrained routing and the other simplification A. SourceFormulationforVWPNetworks
techniquesareabletoovercomethislimitation,butthesolution
First we consider a VWP network, provided with full
theyproduceisonlyanapproximationoftheactualoptimalnet-
wavelength conversion as defined in Section II. The physical
workdesign.ThegreatadvantageofILPoverheuristicmethods
topologyismodeledbythegraph .Physicallinks
istheabilitytoguaranteethattheobtainedsolutionistheabso-
are represented bythe undirected edges with ,
lute optimum value.Anyof the abovetechniquesaimed at re-
while the nodes , with ,
ducingthecomputationalburdenimpliesthattheILPapproach
represent the OXCs. Each link is equipped with a certain
loses its added value, even if the approximated solutions may
amount of unidirectional fibers in each of the two directions;
beclosetotheexactone.Ourworkdevelopsandappliesanew
fiberdirectionisidentifiedbythebinaryvariable .Finally,the
formulationofRFWAproblemwhichisabletoprunevariable
virtualtopologyisrepresentedbythesetofknownterms ,
multiplicity without introducing any approximation, thus pre-
each one expressing the number of connections that must be
servingtheaddedvalueofmathematicalprogramming.
established from the source node to the destination node .
Unidirectionalpoint-to-pointconnectionsareconsidered(thus,
III. SOURCEFORMULATIONOFTHERFWAPROBLEM inthegeneralcase, ).
Thevariablesinthesourceformulationarethefollowing:
Let us consider a multifiber WDM network environment
• is the number of WDM channels on link on fibers
under static traffic, in which the number of wavelengths per
havingdirection whichhavebeenallocatedtolightpaths
fiber is given a priori, while the fiber numbers of each
generatedatnode ;
physical link are variables of the problem. Traditional ILP
• isthenumberoffibersonlink withdirection .
formulationsbasedonfloworrouteparadigm1solvetheRFWA
Itshouldbenotedthattheflowvariables aredefinedin
problem managing source-destination commodity, that is to
suchawaythatallthetrafficoriginatingfromthesamenodeand
say that these formulations route static connection requests
travelingonthesamelinkinthesamedirectionisrepresented
identified by a source and a destination node on the graph
inanaggregatedform,regardlessofthedestination.Thisisthe
representingtheWDMnetwork[seeFig.1(a)].
mainaspectthatdifferentiatessourcefromflowformulation.
Inourproposal,theILPformulationwillconsiderallthecon-
Thefollowingadditionalsymbolsaredefined:
nectionsoriginatingfromasinglesourceOXCasasinglecom-
• identifies the set of fibers of link that are directed
modity [see Fig.1(b)]. Let us observethat single source com-
asindicated by ;for sake ofclarity, in the following we
modity on link assumes value equal to 2, because on
name a“unidirectionallink”;
that link there are two source-destination commodities having
• is the set of “unidirectional links” having the node
origininnode .Thankstothisnewmodel(thatfromnowon
as one extreme and leaving the node; analogously, is
wewillcallsourceformulationorSF),weareabletoprunethe
thesetof“unidirectionallinks”havingthenode asaone
number of variables associated to traffic flows, thus reducing
extremeandpointingtowardsthenode;
computational time and memory occupation compared to the
• isthetotalnumberofrequestedconnections
flowformulation.
havingnode assource.
Inordertominimizethenumberoffibersneededtosupport
Nowwecandetailthesourceformulation.Thecostfunction
a certain amount of traffic, source and flow formulations are
tobeminimizedisthetotalfibernumber
equivalent.IntheAppendix,weshowtheequivalenceofthese
1Fromnowon,theflowformulationcasewillbeconsideredthemaintermof
comparison.
712 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.15,NO.3,JUNE2007
Actually,thesourceformulationcanbeveryeasilyextended
tosolveoptimizationproblemsbasedonthelengthmetric.The
onlychangethatmustbemaderegardsthecostfunction,which
becomes
where isthegeographicallengthoflink .
Thesetofconstraintsisthefollowing:
(1)
(2)
(3)
integer (4)
integer (5)
Constraint(1)isasolenoidalityconstraintwhichimposesthat
thetotalflow(numberoflightpaths)generatedbynode andex-
itingfromitmustbeequaltothetotalnumberofconnectionre-
questshavingnode assource.Notethatthesolenoidalitycon-
straintisnotappliedoneachnode-pair(bywhichaconnection
is requested) but on the aggregated traffic relative to a source
node:thereforeitisnotdependentondestinations.
Constraint (2) is again a solenoidality constraint. It corre-
sponds to the following sequence. Let us take a node . We
Fig.2. Thesolenoidalityconstraintin(a)flowformulationand(b)sourcefor-
expresstheflowconservationconditionforeachothernodeof mulation.In(c),twoadmissiblesolutionsderivablefromtheprevioussource
the network , considering only traffic having as source formulationoutcome.
node.Thisconditionstatesthatthetotalflowgeneratedby and
leaving isgivenbythetotalflowgeneratedby andincidenton
minusthenumberofrequestedconnectionshaving assource flowsandjustoneleavingduetotheflowwhichisdroppedat
and asdestination . thatnode).
In Fig. 2, we show the different application of the It is worth noting that the source formulation does not re-
solenoidality constraint in the flow and source formulation turn a detailed mapping of routing (i.e., a path for each single
cases using two simple examples. The first example shown connectionrequest),evenifitoptimallyassignsthenumberof
in Fig. 2(a) refers to solenoidality constraint in the classical fibersneededtosupportthetraffic;asecondstepmustbeused
flowformulation.Asingleconnectionrequesthasbeenrouted toidentifytheroutingoftheconnections(seetheAppendix).In
betweensourcenode anddestinationnode throughnodes otherwords,thesourceformulationlosestheinformationofthe
and (dottedline).Theflowsassociatedtothisconnectionare routingofeachsingleconnectionduetotheaggregationofflows
representedbyasolidarrowintheroundwindowsthatmagnify onthebasicvariable.LetusrefertoFig.2(c):overthesource
thesituationinnodes , ,and :inthesource(destination)the formulationoutcomeshowninFig.2(b),wecanmaptwodis-
nodeleaving(entering)flowisequaltotheofferedtraffic(i.e., tinct (yet admissible) routing assignments (RA): in a first RA
atrafficunit),whileintheintermediatenodestheleavingflows thetwoconnectionsareroutedonthetwopaths
equaltheenteringflows(e.g.,innode theleavingarrowhasa and , while a second admissible RA could be
correspondententeringarrow). and .
Fig.2(b)referstosolenoidalityconstraintworkinginsource Thecapacityconstraint(3)allowsustodimensionthephys-
formulation case: a simple network case with two connection icalnetworkcapacity.Inordertoensureafeasibleresourceal-
requests(between and and and )isshown.Atthesource locationitimposesthatoneachlinkthesumofflowsgenerated
node ,thesumoftheleavingflowsisenforcedtobeequaltothe by all the nodes is smaller than the product of the number of
sumofthetrafficstoberoutedtowardsallthedestinations(in fibers by the number of wavelengths per fiber. The remaining
thisexampletwotrafficunits,onedestinedtonode ,theother constraints[(4)and(5)]enforcevariableintegrity.
destinedtonode ).Intheothernodesthesumofenteringflows Let us now discuss the source formulation complexityfor a
equalsthesumofleavingflowsplusthetrafficthatisdropped VWPnetwork.TableIshowstherelationsexpressingthetotal
at that node (e.g. in node in Fig. 2(b) we have two entering numberofvariablesandconstraintsasfunctionsofthephysical
TORNATOREetal.:WDMNETWORKDESIGNBYILPMODELSBASEDONFLOWAGGREGATION 713
TABLEI
COMPARISONONCONSTRAINTANDVARIABLENUMBERSBETWEENSOURCEANDFLOWFORMULATIONS
topology size and the number of node pairs requiring connec- have to be split, originating the new variables and
tions.Thecorrespondingrelationsfortheflowandrouteformu- .Thesetofconstraintsismodifiedasfollows:
lationarereportedforcomparison(symbolsreportedinTableI
(6)
havebeenpreviouslydescribed,exceptfor thatrepresentsthe
meannumberofpossiblealternativeroutesbetweentwonodes
inthenetwork).Inthetable, isthenumberofsource-destina- (7)
tionnode-pairsrequiringconnections,thatisupper-boundedby
thenumberofnodepairsofthevirtualtopology .
(8)
Thenumberofvariablesofthesourceformulationgrowswith
theproductofthenumberoflinksbythenumberofnodes.Inthe
flowformulationitgrowsinsteadwiththeproductofthenumber (9)
of links by the number of node pairs requesting connections.
So,fromthevariablenumberpointofview,sourceformulation (10)
shouldbemoreefficientthanflowformulationunderthecondi-
tion ,thatispresumablyacommonsituationinrealnet- integer
works.Ifthereisatleastonelightpathrequestedbyeachnode
integer
pair,thenwecouldset (themaximalvaluethat
integer
couldbeachievedby ).Sourceformulationinthiscaseallows
areductionofthenumberofvariablesbyafactor compared integer
to flow formulation. The same order of reduction is obtained
The solenoidality constraints are split into the sets (6) and
onthenumberofconstraints,whosecomplexitydecreasesfrom
(8)inordertoimposeflowconservationindependentlyforeach
to . The previous comparison is focused on
wavelength.Also,thecapacityconstraint(10)ismodified.The
thedifferencebetweenflowandsourceformulations.Asfarthe
newconstraints(7)and(9)expressthedistributionofthetotal
routeformulationwithoutconstrainedroutingisconcerned,we
number of connections among the different wavelengths for a
caneasilynoticethatthevariablesnumberisdependentonthe
sourcenodeandforasource-destinationpairrespectively.
term ,i.e.,thetotalnumberofpossiblealternativepathsfor
Table Icomparesthe complexityof sourceand flowformu-
eachnodepairrequiringconnectionsinanetwork.Thismeans
lationsalsointheWPcase.Inbothformulationsthenumberof
that the number of variables tends to grow very quickly with
constraintsandvariablesincreaseslinearlywith .Itisimpor-
networkconnectivityanddimension,sothat,forexampleinour
tanttonoticethattheincreaseof intheWPscenarioisac-
casestudy-networks,flowformulationismodelizedbyalower
companiednotonlybyagrowthofvariableandconstraintnum-
numberofvariablesthanrouteformulation.
bers,butalsobytheextensionofrangeofpossiblevaluesthat
the variable can take. Although not directly arguable from the
B. SourceFormulationforWPNetworks table,thishasagreatimpactoncomputationaltimeandmemory
requirement. The advantage of the source formulation can be
Thesourceformulationcanbeextendedtonetworkswithout evaluatedinasimplewaybyconsideringafully-connectedvir-
wavelength conversion.ILPcomplexity inthe WP case grows tualtopologyinwhich .Undersuchassumption
with the number of wavelengths per fiber and constraints thedominanttermofthenumberofvariablesis and
become more complicated because wavelength continuity has forsourceandflowformulation,respectively.Asfor
to be imposed on the lightpaths. Nevertheless, the advantages thenumberofconstraints,thetwodominanttermsare
ofthesourceoverflowformulationarestillrelevant. and ,respectively.
The cost function is the same as in the VWP case Finally,weshallmentionalimitationofthesourceformula-
Section III-A. A new index must be tion.Unfortunately,thisformulationcannotbeextendedtoop-
added to identify the wavelength of the WDM channels, in timize path-protected WDM networks. In fact path protection
order to impose the wavelength continuity constraint along a requirestoroutelightpathsunderthelink-disjointconstraint,so
lightpath. Flow variables defined in the VWP case are trans- thataworkinglightpathcannotshareanyphysicallinkwithits
formed: now indicates the number of WDM channels protectionlightpath.Thebasicvariables containsinforma-
having wavelength which on the “unidirectional link” tionconcerningalltheconnectionshavingthesamesourcenode
carry lightpaths generated at node . The known terms and aggregated together. No explicit reference can be inferred
714 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.15,NO.3,JUNE2007
Fig. 3. Physical topologies of two case-study networks: (a) NSFNET and
(b)EON.
TABLEII
ILPVARIABLESANDCONSTRAINTSFORNSFNETANDEON
INTHEVWPCASE
regardinglightpathshavingthesamesourceandthesamedes-
tination,sothatthelink-disjointconstraintcannotbeenforced.
Anyway, other protection techniques, such as link protection,
couldbeplannedusingsourceformulation.Infactanapproach
tolinkprotectionconsistsinprovidingforeachlink(i.e.,forall
itsfibers)analternativerouteinordertofacelinkfailure:such
afeaturedoesnotneedinformationrelatedtotrafficdestination Fig.4. ILPvariablesandconstraintsforNSFNETintheWPcase.
node.Asourceformulationbasedmodelforlinkprotectionin
bothdedicatedandsharedcasesiscurrentlyunderstudy.
Table II shows the number of variables and constraints that
IV. CASESTUDIESANDRESULTCOMPARISON areinvolvedintheILPproblemappliedtothetwonetworksin
theVWPcase.Theyclearlyshowtheadvantageachievedusing
In this section we present and discuss the results obtained
theaggregationofflows.Dataarecomputedusingtherelations
byILPoptimizationexploitingsourceformulationontwocase-
reportedinTableIexceptthenumberofvariablesinroutefor-
studynetworktypesincomparisonwithresultsobtainedusing
mulation,whichneedsasinputvariablealladmissiblepathsin
traditional flow or route formulations. Well-known mesh net-
thenetworkbetweennodesrequiringconnections.So,inorder
works are considered first, that is the National Science Foun-
to run an optimization based on non-constrained route formu-
dationNetwork(NSFNET)andtheEuropeanOpticalNetwork
lation we have precomputed all the possible alternative paths
(EON). Then a class of networks called “wheel networks” are
usingagreedyroutine:ouralgorithmtakesaboutfivehoursto-
considered,inwhichthevariationoftheconnectivityindex[2],
computethe14604pathsintheNSFNETnetwork,whileinthe
[11] defines a set of topologies ranging from the ring to the
EON,duetoitsgreaterdimension,ouralgorithmtakesaboutten
full-meshnetwork.
hours to compute about 1.5710 paths between a single node
andalltheothernodestakenasdestinations.Thehugenumber
A. NSFNETandEON
ofvariablesinthislastcaseinducedusnottoproceedonEON
DataregardingthephysicaltopologyofNSFNETandEON, optimizationbasedonunconstrainedrouteformulation.
representedinFig.3(a)and(b),havebeentakenfrom[21]and For the WP case, a comparison can be done taking into ac-
[26], respectively. NSFNET has 14 nodes and 22 links, while countrouteformulationcomplexity.Inthislattercase,aswecan
EON has 19 nodes and 39 links. The virtual topologies are arguefromFig.4,thenumberofconstraintsissmallerthanin
based on the static (symmetric) traffic matrices derived from sourceformulation,butthenumberofvariablesgrowsrapidly.
realtrafficmeasurementswhicharereportedinthesamerefer- In fact it is associated to the number of all the possible routes
ences.Thetwotrafficmatricescomprise360and1380unidirec- connectingeachnodecouple,thatincreasesexponentiallywith
tionalconnectionrequestsforNSFNETandEON,respectively, networkdimensionandinparticularitisinfluencedbythecon-
whilethe distinctnodepairsrequiringconnectionsarerespec- nectivityindexofthenetwork.Inthefollowingwewillseehow
tively 108 and 342 (each node pair can require more than one theincreaseinnetworkdimensionandconnectivitywillaffect
connection).BothVWPandWPcaseshavebeenanalyzed. ILP model performance. To solve the ILP problems we used
TORNATOREetal.:WDMNETWORKDESIGNBYILPMODELSBASEDONFLOWAGGREGATION 715
TABLEIII TABLEV
VWPNSFNETOPTIMIZATION:COMPUTATIONALTIME VWPEONOPTIMIZATION:COMPUTATIONALTIMEANDgap
BETWEENINTEGERSOLUTIONFOUNDANDLOWERBOUND
RETURNEDBYBRANCHANDBOUNDALGORITHM
TABLEIV
VWPNSFNETOPTIMIZATION:MEMORYOCCUPATION
TABLEVI
WPNSFNETOPTIMIZATION:TIMEREQUIREDTOFILLUP
100MBOFMEMORY
the software tool CPLEX 6.5 based on the branch-and-bound
method[27].Ashardwareplatformaworkstationwitha1GHz the source formulation than with flow formulation. Moreover
processor was used. The available memory (physical RAM + source formulation always requires a smaller memory amount
swap)amountedto900MByte. andashorterrundurationthantheothertwoformulations.The
Beforethepresentationofnumericalresults,itiscrucialtore- gap in run duration between source and the other two tends
memberthatthethesourceandtheflowformulationareequiva- to increase with the parameter. This is probably due to the
lent(seetheAppendix).Thisequivalenceisconfirmedinallthe extension of the range of the possible values that the variable
networkcasesinwhichbothformulationssucceedinfindingthe can take.
optimum value: this value in fact results to be the same in the Table V shows resource occupation comparison between
twoformulations. sourceandflowformulationinEONnetwork(therouteformu-
We have already shown the advantage of source formula- lationisnotfeasibleduetothehugenumberofvariables);again
computational times in bold are associated to runs succeeding
tionversusflowandrouteformulationintermsofvariableand
infindingoptimalvalues.Wewillshowherethegapparameter
constraint numbers. It is important to see how much this ad-
tocomparethequalityofintegersolutionfound:neithersource
vantage affects the actual computational performance of ILP.
formulationnorflowformulationsucceedsindemonstratingthe
TablesIIIandIVdisplaycomputationaltimeandmemoryoc-
optimality of the returned integer solution except for ,
cupation measurements of NSFNET optimization in the VWP
but quality of source formulation solutions is evidently better
case(s,m,h,anddstandforseconds,minutes,hoursanddays,
than quality of flow solutions. So O.O.M event happens in all
respectively, while MB stands for megabyte). Computational
the optimizations, while for the amount of occupied
times in bold are associated to runs succeeding in finding op-
memoryisabout10MBinbothcases.
timalvalues.
NSFNEToptimizationintheWPscenariotakesaverylong
To clearly understand the reported data, a particular aspect
timewiththehardwareweemployed.Insomecasesitwastoo
of ILP must be clarified. The branch-and-bound algorithm
longtowaiteitherfortheoptimalresultorforanout-of-memory
progressively occupies memory with its data structure while
event.Thus,inTableVIwehavereportedthetimenecessaryto
it is running. When the optimal solution is found, the algo-
fillupofthefirst100MBofmemory.Thecasewith
rithm stops and the computational time and the final memory
hasprovedtobetoocomplextobesolvedinareasonabletime
occupation can be measured. In some cases, however, all the
and therefore it has been omitted (except for in the
available memory is filled up before the optimal solution can
sourceformulation).Thespeedofthebranch-and-boundalgo-
be found. In these cases, CPLEX returns the best but non-op-
rithmappliedtotheflowformulationdecreasesdramaticallyfor
timal solution that branch-and-bound has been able to find
highvaluesof .Althoughinthesourceformulationthespeed
and forces the execution to quit. These cases are identified
does not decrease so much, the model simplification allows a
by the out-of-memory tag (O.O.M.) and the computational significantcomputationaltimedecrementcomparedtoflowfor-
time measures how long it has taken to fill up memory. This mulation.Therouteformulation,despitethegreatdifferenceon
integer solution, forced to be returned because of the limited thenumberofvariables,hascomparableresultsregardingcom-
amountofmemory,isassociatedtotheso-calledgapparameter putationalperformance;thisisprobablyduetothestructureof
that expresses the percentage difference between the integer route variables that make simpler to set the wavelength conti-
solution found and the minimal possible value the solution nuityconstraints.
couldreach(i.e.,alowerboundreturnedbybranchandbound Nowwearegoingtocomparethesourceandtheflowformu-
algorithm).Thisparameterreturnsanestimationofthequality lationsonthebasisofthefinalvalueofthecostfunction.Inall
ofthenon-optimalintegersolutionfoundintermsofmaximal thecasesinwhich,forbothformulations,thebranch-and-bound
possible distance from the optimum. From Tables III and IV ends up before an out-of-memory event, the final values ob-
we can see that the out-of-memory event is less frequent with tainedare coincident, thus proving the equivalence of source
716 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.15,NO.3,JUNE2007
Fig.5. Source-flowcomparisononthefinalnumberoffibersintheVWPcase,as(a)absoluteand(b)percentrelativedifference.
andflowformulation.Inalltheothercases,thebestintegerso-
lutionreturnedbytheSFissmallerthanorequaltotheFFbest
integersolution.Furthermore,thegapparameterassociatedin-
teger solutions is always smaller for SF than for FF, showing
thatSFoptimizationrunsareabletogetclosertolowerbound
oftheproblem.
We focus our analysis on the performance comparison be-
tween flow and source formulation. The following parameters
areintroduced:
• :totalfibernumberreturnedbyILPbased
onthesource(flow)formulation;
• :difference ;
• :percentrelativedifference
.
Wewillshowresultsconcerningthetwocase-studynetworks.
Fig.6. Source-flowcomparisononthefinalnumberoffibersintheWPcase
Thedifferencebetweenthetwoformulationsobtainedinthe forNSFNET.
VWPscenarioarerepresentedinFig.5(a)(absolutevalues)and
inFig.5(b)(percent)asfunctionsoftheparameter .Theab-
solute difference is on averagegreater for the EON which has
a larger number of nodes and links. Convergence between the
twoformulationsoccurs,forexample,intheNSFNETcasefor
and ,inaccordancewithTableIV.2Itshouldbe
notedthatsub-optimalsolutionswiththeflowformulationcan
beupto18%worsethanthecorrespondingsolutionsproduced
bythesourceformulation.
InFig.6, and aredisplayedforNSFNETintheWP
case.Inthiscase,thestrongincreaseofvariableandconstraint
number with causes a relevant increase of the differences
between the two formulations. It is worth noting that both SF
orFFareunabletoreachtheendofoptimizationruns(except
for the case). We evaluate the quality of the integer
solution not only comparing its numerical value (as shown in
Fig.7. Source-flowcomparisononthefinaltotalfiberlengthintheVWPcase,
Fig.6),butalsocomparingthegapparameterassociatedtothe aspercentrelativedifference.
integer solution: for 2, 4, 8, 16 the value of the gap is
equalto0%,1.9%,5.5%and12%forSFand0.3%,2.4%,8%, multifiberWDMnetworkoptimizationhavingthetotalnumber
and 30% for FF. In conclusion, integer solutions provided by of fibers as cost function. This interpretation of network cost
SFoutperformsolutionsprovidedbyFFasfarasthenumerical is called hop metric and it models a situation in which all the
values,computationaltimes andproximitytothe lowerbound fibersofthenetworkhavethesamecost.However,inrealnet-
areconcerned.Uptothispointofthepaperwehavedescribed worksthecostofalinkalsodependsonitsgeographicallength,
whichforexampledeterminesthenumberofopticallineampli-
2InthecaseW = 4,SFandFFreturnthesameintegersolution,whichis
fiersthatmustbeinstalled.Measuringthecostofafiberinthis
verylikelytobetheoptimal,butthebothofthemfailinprovingtheoptimality
ofintegersolutionfound. situationbecomesmuchmorecomplicatedandthe hopmetric
TORNATOREetal.:WDMNETWORKDESIGNBYILPMODELSBASEDONFLOWAGGREGATION 717
Fig.8. NSFNETtotalfibernumberoptimizedbyILPsourceformulationandbyadeterministicheuristic,inthe(a)VWPand(b)WPcases.
is not appropriate anymore. Another simple alternative is the
TABLEVII
length metric, which assigns a cost to each fiber proportional WHEELNETWORKSOPTIMIZATION:RESULTSONFIBERNUMBER
to the geographical length of the link it belongs to. Although
stillnotcompletelyrealistic(e.g.,itdoesnottakeintoaccount
thatthecostoftheductshouldbesharedbyallthefibersofa
link),itcouldbeusefulinmanysituations(e.g.,whenthecost
ofopticallineamplifiersisanimportantissue).Clearly,thehop
metric can be regarded as a particularcase oflength metricin
which all the links have unity length.3 We have tested source
formulation based on length metric on NSFNET and EON in
comparisonshowsthattheresultsofthetwotechniquesarequite
theVWPcase.Inafashionsimilartohopmetric,thefollowing
close:theheuristicapproachisabletoprovidegoodsub-optimal
parametershavebeendefined:
results,butonlytheexactapproachallowstoreachtheabsolute
• :totalfiberlengthreturnedbyILPbasedon
optimum(ortocomeclosertoitwhenlimitationsonmemory
thesource(flow)formulation;
orcomputationaltimepreventsbranch-and-boundtoconverge).
• :difference ;
Asfarasthecomputationaltimeisconcerned,wehavenoticed
• : percent relative difference
thatheuristic and source-formulation ILP behavesimilarly for
.
VWPnetworks.IntheWPcase,however,heuristicmethodsare
Link lengths were assigned for the two networks according
muchfasterthanILP,evenwhensourceformulationisadopted.
to[21]and[26].
It can be noticed in Fig. 8(b) that heuristic has been the only
Fig. 7 displays the percent relative difference between
possible approach to obtain a result with , given the
the total fiber lengths obtained applying the source and the
hardware limitations of our workstation. Concluding, SF has
flow formulation. The same conclusions drawn for the hop
provideda useful benchmarkto evaluatethe performanceofa
metriccanbeextendedtothesenewoptimizationexperiments.
heuristicstrategyonNSFNETandEON,thatcanbeconsidered
Source formulation performs better in all the cases in which
twosignificanttest-casenetworks.
an out-of-memory event occurs; otherwise, the results are
coincident, but source formulation converges more rapidly
B. WheelNetworks
(computationaltimesareomittedforbrevity).Itisworthnoting
that the length metric results in more solutions found for the Inordertoshowtheeffectivenessofoursourceformulation,
considered set of with respect to hop metric case: this is wehaveperformedoptimizationexperimentsalsoonthesetof
becausetherearelesstie-breaksinB&Balgorithmthaninthe 8-node “wheel networks” shown in Fig. 9. This network class
hopmetriccase,wherethe weightassigned toeachlinkis the definestopologieswithincreasingconnectivitydegree,starting
same.Inconfirmationofthisobservation,thevaluesofthegap fromtheringnetworkandendingwithafullmeshnetwork.This
parameter for the length metric are always smaller than those is obtained by increasing the number of edges with respect to
reportedforhopmetricinTableV. initial ring topology, so that the connectivity index (i.e., the
Finally,weshowacomparisonbetweentheILPoptimization ratiobetweenthenumberoflinksintheconsiderednetworkand
carried out by source formulation and the optimization by the thenumberoflinkinthefull-meshnetworkcase)assumesthe
heuristicapproachdescribedin[28].LetusconsiderNSFNET values 0.29 (ring), 0.43, 0.57, 0.71, 1 (full-mesh). Again, we
andthehopmetric.InFig.8ILPandheuristicfinalresultsare haveassumeddifferentvaluesof ,thatis {2,4,8,16,32}.
displayedintheVWP[Fig.8(a)]andWPcase[Fig.8(b)].The Thisnewclassofnetworktopologieswillallowustobetterap-
preciate the behavior of SF with respect to FF and RF, while
3Neitherhopnorlengthmetrictakethenodecostintoaccount.Node-cost
optimizationissuesarenotcoveredbythispaper. varyingonecrucialnetworkparameter,theconnectivityindex.
718 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.15,NO.3,JUNE2007
Fig.9. Wheelnetworkswithdifferentconnectivitydegrees.
Fig.10. ComputationaltimefoSF,FF,RFfordifferent(cid:11)andW values.
Weassumethatofferedtrafficisuniformandequaltoonecon- MoreoverSFallowsustoobtainoptimalvaluesalsoinsome
nectionrequestforeachnodecouple. networkcasesinwhichFFandRFfail;infactfor 0.71with
ThenumericalresultsobtainedbySFforthesenetworksare 4,8and with 16SFsucceedsinfindingop-
summarizedinTableVII.Letusnowanalyzeindetailthesere- timalvaluesthatarenotreachedbyFFandRF(thesevaluesare
sults.SF,FFand RFallleadtotheoptimum solutionfor reportedinitalicstyle).Forthesethreecases,Fig.10compares
0.29,0.43,0.57andallvaluesof ,for 0.71with thecomputationaltimesSFtakestoreachtheoptimalvaluewith
2, 16,32and for with 16,32(the corresponding the computational times required by FF and RF to reach their
numericalvaluesareinboldinTableVII.Whenallthethreefor- bestintegervaluesunderO.O.M.condition.
mulationssucceedinfindingoptimalvalues,SFtakessensibly Finally,inthecases with 2,4,8,thethreeformu-
lowercomputationaltimesthantheothertwoformulations(see lationsfailinprovingtheoptimalityofthebestintegersolution
thecorrespondingvaluesinFig.10,wherewehavedrawnthe returned, so the computational times refer all to O.O.M cases
computationaltimesforeachvalueofconnectivityindex). (timeneededtooccupythewholememory).So,tobetterappre-