OptimalWavebandSwitchinginWDMNetworksLi-WeiChen,PoompatSaengudomlert,E - PDF document

OptimalWavebandSwitchinginWDMNetworksLi-WeiChen,PoompatSaengudomlert,EytanModiano—Switchingtrafctogetherinbundlesofwavelengthswavebandscangreatlyreducetheswitchingcostsofthenet-work.Weconsidertheproblemofpartitioningwavelengthsintowavebandsforstarnetworksusingtheminimumnumberoftotalwavebands.Weprovideagreedyalgorithmforwavebandparti-tioningandshowthatitisoptimalinthatitrequirestheminimumnumberofwavebandssubjecttousingtheminimumpossiblenum-berofwavelengths.Wealsogiveanalgorithmforallocatingcallsfromanyadmissibletrafcsettothewavebandsinanon-blockingmanner.Finally,weshowthattheincreaseinthenumberofwave-bandsrequiredislogarithmicinthenumberofcallsandpolyno-mialinthesizeofthenetwork.I.INTRODUCTIONWithtrafÞcdemandscontinuingtoincreaserapidlyeachyear,wavelength-divisionmultiplexing(WDM)hasemergedasanattractivesolutionforincreasingcapacityinopticalnet-works.WDMallowsmultipledatastreamstobecarriedusingthesameÞberlink,aslongaseachdatastreamoccupiesadif-ferentwavelength[1].MuchoftheworkcurrentlyintheRWAliteratureconsidersroutingatthewavelengthlevel([2],[3],[4],[5],[6],[7],[8],[9]).Asthenumberofwavelengthsincreases,thecostofac-cessingandmanagingallthesewavelengthsateachnodegrowsrapidly,makingthisapproachdifÞculttoscale.Inparticular,eachnodeneedstobeabletoswitcheachwavelengthindepen-dently,increasingtheswitchingcomplexityrapidlyasthesize fiber 1 fiber 2 fiber N . . . . . . . . .. . .. . .. . .. . . . . . . . .. . .. . . . . .. . .. . . Nx NNx NNx N Fig.1.Switchingrequirementsforanodewithdegree.ThenodehasinputÞbersandoutputÞbers,andneedstobeabletoswitchanyofthewavelengthsoneachinputtoanyoutput.ThisrequiresWeconsidertheproblemofmultiplesourceanddestinationnodesinastartopology.Weexaminetheclassofwavelength-efÞcientbandingalgorithmsthatusenomorethantheminimumnumberofwavelengthsrequired,andprovideanoptimalband-ingalgorithmthatminimizesthetotalnumberofwavebandsrequiredsubjecttothisminimum-wavelengthconstraint.Wealsoderivethenumberofwavebandsrequiredasafunctionofthenumberofnodesandthenumberofcalls,andanalyzetheincreaseintherequirementsasafunctionofthetwoparame-A.SystemModelWeconsideranetworkconsistingofnodeswhereeach 1604 canconsideronlymaximaltrafÞc,sinceforanynon-maximalset,wecaninsertÞctitiouscallstoobtainamaximalset.Thesesetsarethecasesofinterestsincetheysendthemaximumpos-sibletrafÞcintothenetwork.[8]showedthatforsucha-nodetopologywithnowavebanding(or,alternatively,whereallwavebandsareofsizewavelengthsarebothnecessaryandsufÞcienttosupportanymaximaltrafÞcset.Thereforeaminimumofwave-lengthsarealsorequiredbyanywavebandpartitioningscheme.Wesaythatawavebandcanbefullyutilizedifitispossibletoassignenoughcallstothatwavebandsuchthateverywave-lengthinthewavebandisutilizedoneverylinkinthenetwork.NotethatforamaximaltrafÞcset,iftheminimumnumberofwavelengthsaretobeused,theneverywavebandmustbefullyutilized;otherwise,therewillexistalinkthatdoesnothavesufÞcientlymanywavelengths.ThisfollowsfromthefactthatmaximaltrafÞcsetshavecallsoneachlink,andonlyatotalwavelengthsareavailable.InSectionII,weaskthequestionofhowlargethelargestwavebandcanbebeforewecannotguaranteethatitcanbefullyutilizedbyanyadmissibletrafÞcset.ThiswillbeusefulinSec-tionIII,whereweshowthatoncewehaveanexpressionforthelargestwavebandsizethatcanbefullyutilized,agreedyalgo-rithmcanbeusedinconjunctionwiththisexpressiontorecur-sivelypartitionthewavelengthsintowavebands.Thegreedyapproachwillbeshowntobeoptimalinminimizingtheresult-ingnumberofwavebandssubjecttousingtheminimumnum-oftotalwavelengths.SectionIVdescribeshowtoassigncallsfromatrafÞcsettowavebandsgivenawavebandparti-tion,andprovidesanupperboundonthenumberofwavebandsrequiredbythegreedyalgorithm.TheboundwillshowthatthisnumbergrowsonlylogarithmicallywiththetotalnumberofwavelengthsII.MAVEBANDRecallthatformaximaltrafÞcsets,everywavebandmustbefullyutilizedifwearetouseonlytheminimumnumberofwavelengths.Inthissection,weconsidertheproblemofde-N,P,thelargestwavebandthatcanbefullyutilizedbyallmaximaltrafÞcsets,forgeneral-porttrafÞcin-nodestar.Theapproachwewilluseistoshowanequiv-alencebetweenamaximallyutilizedwavebandandamaximalbipartitegraphmatching,andthenrelyongraphtheoreticre-sultstoprovidenecessaryandsufÞcientconditionsforobtain-ingmaximalmatchings.WecanrepresentanytrafÞcsetonastartopologybyabi-partitegraph.Thegraphhastwosetsofnodeseach,denoted.Nodesinrepresentsourcesofcalls,whilenodesrepresentdestinations.Anedgeexistsbetweenanodeandanodeifthereisacallfromnodeinthestar.Theedgeisgivenaweightbasedonthenumberofcallsbetweenthatsource-destinationpair.DenotethesetofalledgesbyDeÞneamaximalmatchingtobeasetofedgessuchthatexactlyoneedgeisincidentoneverynodeinthebipartitegraph.Allcallsinamaximalmatchingcanusethesamewaveband,sinceeverycallleavingagivensourcenodegoestothesamedestination,andviceversa.Callsfromthemaximalmatching 1 4 2 3 1 2 3 4 1 2 3 4 21322123 Fig.2.A4-nodestarwithatotalof8calls,asdenotedbytheweightedarrows.Thecallscorrespondtoamaximalmatchingonthebipartitegraph,andcouldallÞtonasinglewaveband.Thewavebandcouldconsistofatmostasinglewavelength,sincemorecallsfromnode1to3cannotbefoundtofullyutilizealargerwaveband. 1 2 3 4 1 2 3 4 vN(v) 1 2 3 4 1 2 3 4 vN(v) Fig.3.ToapplythetestgivenbyHallÕsTheorem,asubsetofthesourceisÞrstchosen.Inthiscase,consistsof2nodes,andtheneighbor-contains4nodes.ThereforethetestispassedforthischoiceofThistestmustberepeatedforallpossiblechoicesofcanbeassignedtofullyutilizeawavebandofsizeatmostequaltothesmallestedgeweightinthematching.(Ifalledgesarenotofthesameweight,somecallsinthematchingwillnotbeassigned,andmustbeallocatedtosubsequentwavebands.)ThisisillustratedinFigure2.Therefore,theproblemofÞndingN,PcanbereducedtoÞndingthelargestwavebandsizeforwhichwecanstillbeguaranteedtoÞndamaximalmatchingwherethesmallestedgeisatleastequaltothatsize.Conveniently,atheoremexistswhichprovidesnecessaryandsufÞcientconditionsfortheexistenceofmaximalmatchings.Hall’sTheoremem:Inabipartitegraph,deÞnetheneighborhoodofasubsettobethosenodesinwhichareconnectedviasomeedgeintosomenodeinThenthereexistsamaximalmatchingifandonlyif,forevery,itsneighborhoodhassizeHallÕsTheoremthereforeprovidesthebasisfordeterminingtheexistenceofmaximalmatchings.Thefollowingtestisap-plied:asubsetofthesourcenodesischosen.Iftheneigh-borhoodofthesubsetisofsizegreaterthanorequaltothesizeofthesubsetitself,thetestispassed.ThisisshowninFigure3.Thetestisthenrepeatedforallpossiblesubsets.Ifthetestispassedforallsubsets,thenamaximalmatchingexists.Ifatleastonetestisfailed,thennomaximalmatchingexists.WecandetermineifawavebandofagivensizecanbefullyutilizedbyagiventrafÞcsetasfollows.DeterminethebipartitegraphcorrespondingtothetrafÞcset,anddeleteanyedgeswithweightlessthan.(Thisguaranteesthatanymaximalmatch-ingfoundwillhaveminimumedgeweightatleast.)ThetestsgivenbyHallÕsTheoremcanthenbeappliedtothisgraphtodetermineifamaximalmatchingexiststhatissufÞcientlylargetofullyutilizethewaveband.Ifthetestfails,thenawavebandofsizeistoolargetobesufÞcientlyutilized.ThistestshouldbeappliedtoallmaximaltrafÞcsetstoguaranteethatimalsetcanfullyutilizethewaveband. 1605 nm mP nm mP N-m (n-m)P (b)(a) Fig.4.Atmostcallscanbesenttonodesin,andhencecallsmustgotonon-neighborhoodnodes.Inprinciple,theprecedingapproachcouldbeusedtode-N,Pnumericallybybruteforce.However,wewillseethataclosed-formsolutioncanbeobtained.Themethodforobtainingtheclosed-formexpressionforN,PreliesonattemptingtoconstructabipartitegraphwhichcausesthetestgivenbyHallÕsTheoremtobefailed.(Inaslightabuseofnotation,wecallsuchabipartitematchingaÒcounterexampleÓ.)N,Pisthenthelargestwavebandsizeforwhichnocounterexampleexists.Inorderforthetesttobefailed,amaximaltrafÞcsetmustbefoundforwhichwecanchooseasuchthatthesizeoftheissmallerthanthesizeof.Wethereforewishtoconstructacounterexamplewhere,if,then,whereUnderthe-portmodel,thenodesincansendatmostcallstonodesin.Theremainingresidualtrafcthereforeatleast.Thesecallsaresenttonodesoutsidetheneighborhood,andhencemustbelongtoedgesadjacenttoanon-neighborhoodnode.Calltheseedgesneighborhoodedges.Non-neighborhoodedgeshaveweightlessthanN,Pandareremovedfromthegraphbeforethesearchformaximalmatchingsbegins,sincetheydonotcontainenoughcallstofullyutilizethewaveband.Thisisil-lustratedinFigure4.Thereareatmostneighborhoodedges.WecanthereforeconstructacounterexampleifandonlyiftheresidualtrafÞccanbedividedamongthenon-neighborhoodedgessuchthatnonon-neighborhoodedgehasweightgreaterthanorequaltoN,P.Sincetherearecallsandnon-neighborhoodedges,thereisatleastonenon-neighborhoodedgewithweightatleast N,Pischosentobeatmostthisnumber,thennocounterexampleexistsforthegivenvaluesof.WecanN,PtoguaranteethatcounterexampleexistsbyminimizingoverandchoosingN,Patmostthisminimum: =min nP WeÞxforthemomentandconsidertheminimizationover.Since,theminimizationissubjecttotheconstraint.Equation1isminimizedbychoosingsmallaspossible.Sincearebothintegerquantities,weshouldchoose;conversely,Makingthissubstitution,theminimizationbecomes: NŠ(nŠ1)](2)Sincetheceilingfunctionismonotonic, n[NŠ(nŠ1)]=
minnP NŠ(nŠ1)]Ignoringintegralityconstraints,theright-handsizeiseasilyshowntobeminimizedat .Ifisodd,then anintegerandhenceisavalidchoicefor.Wesubsequentlyobtainavaluefor N+1 2N+1 2=
4P +1)oddiseven,thensinceP/n isconvex,theminimizingvalueofmustbeoneoftheintegersadjacentto ,namely 2=N 2orN+1 2=N .Itiseasytoverifythateithercaseresultsinthesamevalueof N 2N 2+1=
4P +2)evenInsummary, even III.OAVEBANDARTITIONINGThegoalofwavebandpartitioningistoassigneachwave-lengthtoawavebandsuchthat(1)foranyadmissibletrafÞcset,allwavelengthsinagivenwavebandcanberoutedtogetherwithoutbeingdemultiplexed,and(2)thetotalnumberofwave-bands(subjecttotheminimum-wavelengthconstraint)ismini-mized.Wewillshowinthissectionthattheoptimalmethodofaccomplishingthesegoalsistouseagreedyalgorithm.Thegreedyalgorithmforwavebandassignmentlooksatthewavelengthsrequiredtosupport-porttrafÞcanddeter-minesthelargestwavebandsizethatcanalwaysbefullyuti-lizedbyanyadmissibletrafÞcset.Itthencreatesawavebandofthislargestsize,andrepeatsthisprocessrecursivelyusingtheremainingwavelengths.Supposethesizeofthelargestwavebandis.Sincethelargestwavebandcanalwaysbeallocatedwithoutwaste,ithan-callstoandfromeachnode.TheremainingtrafÞc 1606 20 wavelengths14 wavebandswaveband partitioning Fig.5.Thewavebandpartitioningfora20-port5-nodestar.thereforeformsa-porttrafÞcset.Thegreedyalgo-rithmdeterminesthesizeofthelargestwavebandforthisresid-ualtrafÞcsetbyrepeatingtheaboveprocess,proceedingrecur-sivelyuntilallwavelengthshavebeenpartitionedintowave-Aformalstatementofthealgorithmisasfollows:1)Initializetobethenumberofportsandtobethenumberofnodes.2)Using(3),determine,themaximumsizeofwave-bandfora-nodestarsuchthatnowavelengthsinthebandarewastedregardlessofthetrafÞcset.3)Let4)Ifgoto1.NotethatthenumberofwavebandsN,Gusedbythegreedyalgorithmisnondecreasingin;thatis,N,GN,G.Thispropertywillproveusefullater.Example1:Considera5-nodestarwith=20.Usingthegreedyalgorithm,wewoulddeterminethatthelargestwave-bandshouldbe (N+1)2\f=(4)(20) Afterthisstep,17wavelengthsremaintobepartitioned.Repeating,wedeterminethenext-largestwavebandtobe .Thisleaves15wavelengthstobepartitioned.Iteratingthroughthisprocedureproducesawavebandparti-tioningof.Noticethatthere wavebandsconsistingofasinglewavelength.Therearealways single-wavelengthbandsforlarge .TheÞnalwavebandpartition-ingisshowninFigure5.Wewillnextshowthattheapproachofthegreedyalgo-rithmisoptimal,inthatitresultsinthecreationofthemini-malnumberofwavebandssubjecttotheminimum-wavelengthconstraint.Wedosoviaaninductionproof.Foranodestar,lettheminimumnumberofwavebandsrequiredbedenotedbyStep1:BaseCase Considerthecaseof=1.Thenthereisonlyasinglewave-length,andthemaximumbandsizeistriviallyequalto1.Thereforethegreedyalgorithm,whichwouldproduceasinglebandofsize1inthiscase,isoptimal.Step2:InductionStep Intheinductionstep,weconsidera-portstarandassumethatforanyvalueof,wearegiventhatthegreedyalgorithmisoptimal.WethenwishtoprovethatthegreedyalgorithmisoptimalforLetthenumberofwavelengthsassignedtotheÞrstwave-bandbythegreedyalgorithmbe,andletthenumberofwavelengthsassignedbyanyotheralgorithmbe.ThenwecanexpressthetotalnumberofwavebandscreatedbythetwoalgorithmsasN,GN,O,respectively.TheÞrstquantitycanbewrittenas:N,G)=1+N,G=1+withthesecondequalityresultingbecausebytheinductionhy-pothesisthegreedyalgorithmisoptimalfor.Thesecondquantityis:N,O)=1+N,OwherethesecondinequalityresultsfromthefactthatBytheinductionhypothesis,sincethegreedyalgorithmisoptimalfor,andthegreedyalgorithmisnon-decreasinginisalsonon-decreasingforComparingequations4and5andnotingthat,weconcludethatsinceisnon-decreasing,N,GN,Oandhencethegreedyalgorithmisoptimalforaswell,concludingtheproof.IV.RWAUAVEBANDSA.WavebandRWAAlgorithmThepartitioningofwavelengthsintowavebandsisaprocessthatisdoneonlyonceforanygivennetworkwithÞxed.ThewavebandsarereusedforanyadmissibletrafÞcsetsimplybyreconÞguringtheswitchatthehub.Whatnowre-mainsistheproblemofwavelengthassignmentandthesub-sequentswitchingofthewavebandsforeachadmissibletrafÞcset.Oncethisproblemissolved,whenthereisanewcallar-rival,thewavelengthassignmentandswitchsettingsaresimplyreconÞguredtosupporttheresultingnewtrafÞcset.Wecanusethesameapproachasinthedevelopmentandproofofthegreedyalgorithmforwavelengthassignment.Giventhewavebandpartition,wechoosethelargestwavebandandÞndasetofcallsthatfullyutilizethatwaveband.WeareguaranteedtoÞndsuchasetbyourchoiceof.Removingthosecallsfromconsideration,weproceedtothenext-largestwaveband,andrepeatrecursivelyuntilallcallsareassigned.Formally,giventhewavebandpartition,theprocedureforas-signingcallsfromanyadmissibletrafÞcsettowavelengthsinthosebandsisasfollows:ObtainasetofcallswhichfullyutilizesthelargestfreeThiscanbeaccomplishedbydrawingthebi-partitegraphcorrespondingtoalltrafÞccarryingmorecalls.FromSectionII,weknowissuf-Þcientlysmallthatamaximalmatchingwithminimumedgeweightatleastcanalwaysbefound.Assigncallsfromthatmatchingtofullyutilizethewaveband.Removeassignedcalls,andrepeat.RemoveallcallswhichhavebeenassignedwavelengthsfromthetrafÞcset.IfthetrafÞcsetisnowempty,stop.Otherwise,re-turntoStep1.Weemphasizethattheproblemofpartitioningthewave-bandsisdisjointfromthewavelengthassignmentforcallsinthetrafÞcset.ThewavebandpartitionisforagivenandanyadmissibletrafÞcsetusesthesamewavebandpartition;onlythewavelengthassignmentandswitchingchanges. 1607 B.UpperBoundonNumberofWavebandsHavingobtainedanoptimalalgorithmforallocatingwave-bands,itisnaturaltoaskhowmanywavebandsarerequiredforgivenvaluesof.Thequestionisrelevantbe-causeifroutingisdoneatthewavebandlevel,thenthenumberofwavebandsdeterminestheswitchingcost.Inprinciple,thenumberofwavebandscouldbedeterminednumericallybyit-eratingthroughthegreedyalgorithm(aprocedurethatisnotcomputationallydifÞcult)foreachsetof.Inthissection,analternateapproachbasedonrelaxingtheintegralityconstraintsassociatedwithisusedtoobtainaclosed-formupperboundontheoptimalnumberofwavebands.Considerthecaseofchoosingthelargestwavebandtobe ,wheretheintegralityconstraintsonhavebeenrelaxed.Wethenusethistoobtainanupperboundonthevalueofthelargestwavebandinthegreedyalgorithm,andtrackthevalueofthrougheachiteration.Letbethevalueofafterrunningtheiterationofthegreedyalgorithm.Itcanbeshownthattheseriesprogressedaccordingtotherelation: +1) ,thenthenumberofbandsissimplyequal sinceeachbandiscomposedofonlyasinglewave-length.Thereforeconsider anddeterminethenumberofbandsrequiredtoreducethenumberofunassignedwavelengthsto .Thenthetotalnumberofwavebandswouldbe .WeÞrstsolveforasfollows:+1) 4\r1Š4 +1)+1) 4k=log(N+1)2 4P log1Š4 Thisgivesanupperboundonthenumberofwavebands 4+log(N+1)2 4P log1Š4 (N+1)2�,P(N+1)2 P,P Toobtainasenseofhowquicklythenumberofwavebandsincreaseswith,weusetheapproximationthat,forlarge log1Š4 +1) whichgivesus+1) 1+log +1) 0 100 200 300 400 500 600 700 800 900 1000 0 20 40 60 80 100 120 140 PbandsNumber of bands for a 10node star actualbound Fig.6.Acomparisonoftheactualnumberofwavebandsrequiredfora10-nodestartotheupperbound.Fromthis,wecanseethatthenumberofwavebandsgrowswithorderlog(P/N.Figure6plotstheboundcom-paredtotheexactnumberofwavebandsrequired.V.CWeconsideredtheproblemofpartitioningwavelengthsintowavebandsforstarnetworks.Weprovideagreedyalgorithmforwavebandpartitioningandshowthatitisoptimalinthatitrequirestheminimumnumberofwavebandssubjecttousingtheminimumpossiblenumberofwavelengths.WealsogiveanalgorithmforallocatingcallsfromanyadmissibletrafÞcsettothewavebandsinanon-blockingmanner.Finally,weshowthatnumberofwavebandsrequiredgrowsaslog(P/Nisthenumberofportspernodeandisthenumberofnodes.[1]R.RamaswamiandK.N.Sivarajan,OpticalNetworks:APracticalPer-.MorganKaufmann,1998.[2]ÑÑ,ÒRoutingandwavelengthassignmentinall-opticalnetworks,ÓIEEE/ACMTrans.Networking,vol.3,pp.489Ð500,October1995.[3]L.LiandA.K.Somani,ÒDynamicwavelengthroutingusingcongestionandneighborhoodinformation,ÓIEEE/ACMTrans.Networking,vol.7,pp.779Ð786,October1999.[4]D.BanerjeeandB.Mukherjee,ÒWavelength-routedopticalnetworks:linearformulation,resourcebudgetingtradeoffs,andareconÞgurationstudy,ÓIEEE/ACMTrans.Networking,vol.8,pp.598Ð607,October2000.[5]A.Narula-Tam,P.J.Lin,andE.Modiano,ÒEfÞcientroutingandwave-lengthassignmentforreconÞgurableWDMnetworks,ÓIEEEJ.Select.AreasCommun.,vol.20,pp.75Ð88,January2002.[6]O.Gerstel,G.Sasaki,S.Kutten,andR.Ramaswami,ÒWorst-caseanal-ysisofdyanmicwavelengthallocationinopticalnetworks,ÓIEEE/ACMTrans.Networking,vol.7,pp.833Ð845,December1999.[7]P.Saengudomlert,E.Modiano,andR.G.Gallager,ÒOn-lineroutingandwavelengthassignmentfordynamictrafÞcinWDMringandtorusnet-works,ÓinProc.IEEEINFOCOM,April2003.[8]ÑÑ,ÒDynamicwavelengthassignmentforwdmall-opticaltreenet-works,ÓinProc.Allerton,September2003.[9]L.-W.ChenandE.Modiano,ÒEfÞcientroutingandwavelengthassign-mentforreconÞgurableWDMnetworkswithwavelengthconverters,ÓinProc.IEEEINFOCOM,April2003.[10]X.Cao,V.Anand,Y.Xiong,andC.Qiao,ÒPerformanceevaluationofwavelengthbandswitchinginmulti-Þberall-opticalnetworks,ÓinProc.IEEEINFOCOM,April2003.[11]R.Izmailov,S.Ganguly,V.Kleptsyn,andA.C.Varsou,ÒNon-uniformwavebandhierarchyinhybridopticalnetworks,ÓinProc.IEEEINFO-,April2003.[12]M.N.S.SwamyandK.Thulasiraman,Graphs,Networks,andAlgo-.NewYork:Wiley,1981. 1608 s.Inthisregion95%ofallrealizationsdonotreachspeedupsgreaterthan0:8s,andsomemayevendropbelow0:6s.Asincreases,however,the5%quantilerapidlyimprovestovaluesabove0:8s,andthemedianseemstogrowmonotonously.Ne where is the channel coefficient between the th transmit antenna and the th receive antenna. i,j’s are assumed to be In the literature, many research studies on survivability in WDM optical networks have focused on the recovery of point-to-point (unicast) traffic [4]-[9]. Yet, few research studies have focused on multicast traffic using concepts such as dedic x x2 x3 x4 x5 x6 x7 x8 x9 x First, the RF signal is downconverted to IF. Then a bandpass filter with a center frequency of i is employed to select the channel and suppress high frequency components. At this point, A/D conversion is performed, followed by quadrature downconversion of the IF signals to baseband and lowpass filtering. After some manipulations, the baseband signal observations are obtained as: X(n) = =AS(n). (4) INPUT STAGEOUTPUT STAGEnxmrxrmxn Fig.1.Schematicofathree-stageClosswitchwithinput/outputports.asabackuppathformostcircuitssothatbercutsandcomponentfailuresdonotleadtodisruptionofservicefortheir Wireless Broadband System