anovelroutingschemecalledSprayandWaitSprayandWaitboundsthetotalnumberofcopiesandtransmissionspermessagewithoutcompromisingperformanceUsingtheoryandsimulationsweshowthatiunderlowloadSprayandWait ID: 364307
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SprayandWait:AnEfÞcientRoutingSchemeforIntermittentlyConnectedMobileNetworksThrasyvoulosSpyropoulosDepartmentofElectricalEngineering,USCspyropou@usc.eduKonstantinosPsounisDepartmentofElectricalEngineering,USCCauligiS.RaghavendraDepartmentofElectricalEngineering,USCraghu@usc.eduABSTRACTIntermittentlyconnectedmobilenetworksaresparsewire-lessnetworkswheremostofthetimetheredoesnotexistacompletepathfromthesourcetothedestination.ThesenetworksfallintothegeneralcategoryofDelayTolerant anovelroutingschemecalledSprayandWait.SprayandWaitboundsthetotalnumberofcopiesandtransmissionspermessagewithoutcompromisingperformance.Usingthe-oryandsimulationsweshowthat:(i)underlowload,SprayandWaitresultsinmuchfewertransmissionsandcompara-bleorsmallerdelaysthanooding-basedschemes,(ii)un-derhighload,ityieldssignicantlybetterdelaysandfewertransmissionsthanooding-basedschemes,(iii)itishighlyscalable,exhibitinggoodandpredictableperformanceforalargerangeofnetworksizes,nodedensitiesandconnectiv-itylevels;whatismore,asthesizeofthenetworkandthenumberofnodesincrease,thenumberoftransmissionspernodethatSprayandWaitrequiresinordertoachievethesameperformancedecreases,and(iv)itcanbeeasilytunedonlinetoachievegivenQoSrequirements,eveninunknownnetworks.WealsoshowthatSprayandWait,usingonlyahandfulofcopiespermessage,canachievecomparablede-laystoanoracle-basedoptimalschemethatminimizesdelaywhileusingthelowestpossiblenumberoftransmissions.Inthenextsectionwegooversomeexistingrelatedworkandsummarizeourcontribution.Section3presentsourpro-posedsolution,SprayandWait.Then,inSection4weshowextensivelyhowtooptimizeSprayandWaitinpracticalsit-uations,andalsoexamineitsscalability.SimulationresultsarepresentedinSection5,wheretheperformanceofallthestrategiesarecomparedwithrespecttomessagedeliverydelayandnumberoftransmissionspermessagedelivered.Finally,Section6concludesthepaper.2.RELATEDWORKANDCONTRIBUTIONSAlthoughasignicantamountofworkandconsensusex-istsonthegeneralDTNarchitecture[1],therehasntbeenasimilarfocusandagreementonDTNroutingalgorithms,especiallywhenitcomestonetworkswithopportunisticconnectivity.Thismightbeduetothelargevarietyofappli-cationsandnetworkcharacteristicsfallingundertheDTNumbrella.Alargenumberofroutingprotocolsforwirelessad-hocnetworkshavebeenproposedinthepast[6,20].However,traditionalad-hocroutingprotocolsarenotappropriateforthetypesofnetworkswereinterestedinhere.Theper-formanceofsuchprotocolswouldbepoorevenifthenet-workwasonlyslightlydisconnected.Toseethis,notethattheexpectedthroughputofreactiveprotocolsisconnectedwiththeaveragepathdurationandthetimetore-pairabrokenpathrepairwiththefollowingrelationship:throughput=min,raterepair [22].Whenthenet-workisnotdenseenough(asintheICMNcase),evenmod-eratenodemobilitywouldleadtofrequentdisconnections.Thisreducestheaveragepathdurationsignicantly.Addi-tionally,repairisatleast2timesthedelayoftheoptimalalgorithm.Consequently,inmostcasesrepairisexpectedtobelargerthanthepathduration,thiswayreducingtheexpectedthroughputtoalmostzero.Proactiveprotocols,ontheotherhand,wouldsimplydeclarelackofapathtothedestinationunderintermittentconnectivity,orresultintoadelugeoftopologyupdatesthatwoulddominatetheavail-ablebandwidthunderhighmobility.Anotherapproachtodealwithdisconnectionsordisrup-tions[2]istoreinforceconnectivityondemand,bybring-ingforexampleadditionalcommunicationresourcesintothenetworkwhennecessary(e.g.satellites,UAVs,etc.).Sim-ilarly,onecouldforceanumberofspecializednodes(e.g.robots)tofollowagiventrajectorybetweendisconnectedpartsofthenetworkinordertobridgethegap[28,18].Nevertheless,suchapproachesareorthogonaltoourwork;ouraimistostudywhatcanbedoneintheabsenceofsuchenforcedmobilityandconnectivity.AstudyofroutingforDTNnetworkswithpredictableconnectivitywasperformedin[15].There,anumberofalgorithmswithincreasingknowledgeaboutnetworkchar-acteristicslikeupcomingcontacts,queuesizes,etc.iscomparedwithanoptimalcentralizedsolutionoftheprob-lem,formulatedasalinearprogram.Althoughitisshownthatevenlimitedknowledgemightbeadequatetoecientlysolvethisproblem,thealgorithmsproposedapplytothetypesofDTNswereconnectivityisintermittent,butcanbepredicted(forexample,duetoplanetaryandsatellitemovementinIPN[7]).Inourcase,connectivityisratheropportunisticandsubjecttothestatisticsofthemobilitymodelfollowedbynodes.Anumberofroutingproposalsexistthatarespecicallytargetedtowardsthisnewcontextofintermittentlycon-nectedmobilenetworkswithopportunisticconnectivity.Manyofthemtrytodealwithapplication-specicproblems,es-peciallyintheeldofsensornetworks.In[23],anumberofmobilenodesperformingindependentrandomwalksservethatcollectdatafromstaticsensorsandde-liverthemtobasestations.EachDataMuleperformsDirectTransmission,thatis,willnotforwarddatatootherData-Mules,butonlydeliverittoitsdestination.Thestatisticsofrandomwalksareusedtoanalyzetheexpectedperfor-manceofthesystem.Theideaofcarryingdatathroughdisconnectedpartsusingavirtualmobilebackbonehasalsobeenusedin[5,13].Inanumberofotherworks,allnodesareassumedtobemobileandalgorithmstotransfermessagesfromanynodetoanyothernodearesoughtfor[3,8,11,14,17,18,19,Epidemicroutingextendstheconceptofoodinginintermittentlyconnectedmobilenetworks[27].Itisoneoftherstschemesproposedtoenablemessagedeliveryinsuchnetworks.Eachnodemaintainsalistofallmessagesitcarries,whosedeliveryispending.Wheneveritencountersanothernode,thetwonodesexchangeallmessagesthattheydonthaveincommon.Thisway,allmessagesareeventu-allyspreadtoallnodes,includingtheirdestination(inanepidemicmanner).Althoughepidemicroutingndsthesamepathastheoptimalschemeundernocontention[25],itisverywastefulofnetworkresources.Furthermore,itcreatesalotofcontentionforthelimitedbuerspaceandnetworkcapacityoftypicalwirelessnetworks,resultinginmanymessagedropsandretransmissions.Thiscanhaveadetrimentaleectonperformance,ashasbeennotedearlierin[19,26],andwillalsobeshowninoursimulationresults.Onesimpleapproachtoreducetheoverheadofoodingandimproveitsperformanceistoonlyforwardacopywithsomeprobability1[26].(Weshallrefertothispro-tocolasRandomizedFlooding.)Adierent,moresophis-ticatedapproachisthatofHistory-basedUtility-basedRouting[8,17,19].Here,eachnodemaintainsautilityvalueforeveryothernodeinthenetwork,basedonatimerindicatingthetimeelapsedsincethetwonodeslastencoun-teredeachother.Theseutilityvaluesessentiallycarryin-directinformationaboutrelativenodelocations,whichgetdiusedthroughnodesmobility.Therefore,aschemecanbedesigned,wherenodesforwardmessagecopiesonlyto nodeswithahigherutilitybyatleastsomepre-speciedthresholdvalueforthemessagesdestination.Suchaschemeresultsinsuperiorperformancethanooding[17,19],andmakesbetterforwardingdecisionsthanrandom-izedrouting[25].Thismethodhasalsobeenfoundtobequiteecientinthecontextofregular,connected,wirelessnetworks[11].Nevertheless,utility-basedschemesarestillooding-basedinnature.Whatisworse,theyarefacedwithanimportantdilemmawhenchoosingtheutilitythreshold.Toosmallathresholdandtheschemebehaveslikepureooding.Toohighathresholdandthedelayincreasessig-nicantly,asweshallsee.Single-copyschemeshavealsobeenextensivelyexploredin[23,25].Suchschemesgenerateandrouteonlyonecopypermessage(incontrasttooodingschemesthatessentiallysendacopytoeverynode),inordertosignicantlyreducethenumberoftransmissions.Althoughtheymightbeusefulinsomesituations,single-copyschemesdonotpresentdesir-ablesolutionsforapplicationsthatrequirehighprobabilitiesofdeliveryandlowdelays.Finally,anoptimaloracle-basedalgorithmhasbeende-scribedin[25].Thisalgorithmisawareofallfuturemove-ment,andcomputestheoptimalsetofforwardingdecisions(i.e.timeandnexthop),whichdeliversamessagetoitsdes-tinationintheminimumamountoftime.Thisalgorithmisofcoursenotimplementable,butisquiteusefultocompareagainstproposedpracticalschemes.Ourscheme,SprayandWait,managestosignicantlyre-ducethetransmissionoverheadofooding-basedschemeshavebetterperformancewithrespecttodeliverydelayinmostscenarios,whichisparticularlypronouncedwhencontentionforthewirelesschannelishigh.Further,itdoesnotrequiretheuseofanynetworkinformation,noteventhatofpastencounters.WealsoprovideanalyticalmethodstocomputethenumberofcopiespermessagethatSprayandWaitrequirestoachieveatargetaveragemessagedeliverydelay.Thesemethodsarecomplementedbyanalgorithmtoestimatenetworkparametersonline,likethetotalnumberofnodes,inordertobeabletooptimizeSprayandWaitwhentheseareunknown.Finally,wedemonstratethatSprayandWait,unlikeotherschemes,isremarkablyrobustandscal-able,retainingitsperformanceadvantageoveralargerangeofscenarios.3.SPRAYANDWAITROUTINGBasedonthepreviousexposition,wecanidentifyanum-berofdesirabledesigngoalsforaroutingprotocolininter-mittentlyconnectedmobilenetworks.Specically,ane-cientroutingprotocolinthiscontextshould:performsignicantlyfewertransmissionsthanepidemicandotherooding-basedroutingschemes,underallgeneratelowcontention,especiallyunderhightracachieveadeliverydelaythatisbetterthanexistingsingleandmulti-copyschemes,andclosetotheopti-behighlyscalable,thatis,maintaintheaboveper-formancebehaviordespitechangesinnetworksizeornodedensity.besimpleandrequireaslittleknowledgeaboutthenetworkaspossible,inordertofacilitateimplementa-Tothisend,weproposeanovelroutingscheme,calledSprayandWaitthatissimpleyetecient,andmeetstheabovegoals,aswewilldemonstrateinthenextsections.SprayandWaitroutingdecouplesthenumberofcopiesgen-eratedpermessage,andthereforethenumberoftransmis-sionsperformed,fromthenetworksize.ItconsistsoftwoDefinition3.1(SprayandWait).SprayandWaitroutingconsistsofthefollowingtwophases:sprayphase:foreverymessageoriginatingatasourcenode,messagecopiesareinitiallyspread forwardedbythesourceandpossiblyothernodesreceivingacopy todistinctrelays.(Detailsaboutdierentspray-ingmethodswillbegivenlater.)waitphase:ifthedestinationisnotfoundinthespray-ingphase,eachofthenodescarryingamessagecopyperformsdirecttransmission(i.e.willforwardthemes-sageonlytoitsdestination).SprayandWaitcombinesthespeedofepidemicroutingwiththesimplicityandthriftinessofdirecttransmission.Itinitiallyjump-startsspreadingmessagecopiesinaman-nersimilartoepidemicrouting.Whenenoughcopieshavebeenspreadtoguaranteethatatleastoneofthemwillndthedestinationquickly(withhighprobability),itstopsandletseachnodecarryingacopyperformdirecttransmission.Inotherwords,SprayandWaitcouldbeviewedasatrade-obetweensingleandmulti-copyschemes.Surprisingly,asweshallshortlysee,itsperformanceisbetterwithrespecttobothnumberoftransmissionsanddelaythanallotherpracticalsingleandmulti-copyschemes,inmostscenariosTheabovedenitionofSprayandWaitleavesopentheissueofhowthecopiesaretobespreadinitially.Anum-berofdierentsprayingheuristicscanbeenvisioned.Forexample,thesimplestwayistohavethesourcenodefor-wardallcopiestotherstdistinctnodesitencounters(SourceSprayandWait).Abetterwayisthefollowing.Definition3.2(BinarySprayandWait.).Thesourceofamessageinitiallystartswithcopies;anynodemessagecopies(sourceorrelay),andencountersanothernode(withnocopies),handsovertokeepsforitself;whenitisleftwithonlyonecopy,itswitchestodirecttransmission.ThefollowingtheoremstatesthatBinarySprayandWaitisoptimal,whennodemovementisIID.WhenallnodesmoveinanIIDmanner,BinarySprayandWaitroutingisoptimal,thatis,hastheminimumexpecteddelayamongallsprayandwaitroutingProof.Letuscallanodeactivewhenithasmorethanonecopiesofamessage.Letusfurtherdeneaspray-ingalgorithmintermsofafunctionasfollows:whenanactivenodewithcopiesencountersanothernode, ithandsovertoit)copies,andkeepstheremaining).Anysprayingalgorithm(i.e.any)canberep-resentedbythefollowingbinarytreewiththesourceasitsroot:assigntherootavalueof;ifthecurrentnodehasavalue1createarightchildwithavalueof1)andaleftonewithavalueof);continueuntilallleafnodeshaveavalueof1.Aparticularsprayingcorrespondsthentoasequenceofvisitingallnodesofthetree.Thissequenceisrandom.Nev-ontheaverage,alltreenodesatthesamelevelarevisitedinparallel.Further,sinceonlyactivenodesmayhandoveradditionalcopies,thehigherthenumberofac-tivenodeswhencopiesarespread,thesmallertheresidualexpecteddelay,let).Sincethetotalnumberoftreenodesisxed(21)foranysprayingfunction,itiseasytoseethatthetreestructurethathasthemaximumnumberofnodesateverylevel,alsohasthemaximumnum-berofactivenodes(ontheaverage)ateverystep.Thistreeisthebalancedtree,andcorrespondstotheBinarySprayandWaitroutingscheme. growslarger,thesophisticationofthesprayingheuris-tichasanincreasingimpactonthedeliverydelayofthesprayandwaitscheme.Figure1comparestheexpectedde-layofBinarySprayandWaitandSourceSprayandWaitasafunctionofthenumberofcopiesused,ina100networkwith100nodes.ThisgurealsoshowsthedelayoftheOptimalschemeintroducedin[25]. Delay of Spray and Wait50010001500200025003000350040005101520L (# of copies)time units source spray and wait binary spray and wait optimal Figure1:ComparisonofSourceSprayandWait,BinarySprayandWait,andOptimalschemes(networkwithnodes).4.OPTIMIZINGSPRAYANDWAITTOMEETPERFORMANCECONSTRAINTSBydenition,mostICMNnetworksareexpectedtooper-ateinstressedenvironmentsandbynaturebedelaytoler-.Nevertheless,inmanysituationsthenetworkdesignerortheapplicationitselfmightstillimposecertainperfor-mancerequirementsontheprotocols(e.g.maximumdelay,maximumenergyconsumption,minimumthroughput,etc.).Forexample,amessagesentoveranICMNofhandheldsinacampusenvironment,notifyinganumberofpeersaboutanupcomingmeeting,wouldobviouslybeofnouseifitarrivesafterthemeetingtime.ItisofspecialinterestthereforetoexaminehowSprayandWaitcanbetunedtoachievethedesiredperformanceinaspecicscenario.Beforewedosothough,wesummarizeinthefollow-inglemmaafewofourownresultsfrom[25]regardingtheexpecteddelayoftheDirectTransmissionandOptimalschemes:Letnodeswithtransmissionrangeper-formindependentrandomwalksona N× torus.Then:1.ThedelayofDirectTransmissionisexponentiallydis-tributedwithaverage34log 2.TheexpecteddelayoftheOptimalalgorithmis whereistheHarmonicNumber,i.e, =(logWehavealsocomputedatightupperboundfortheex-pecteddelayofSprayandWait.Duetolimitationsofspaceweomittheproofandonlystatetheresult.Theinterestedreadercanndtheproofin[24].TheexpecteddelayofSprayandWait,whenmessagecopiesareused,isupper-boundedby M1 Thisboundistightwhen4.1ChoosingLtoAchieveaRequiredExpectedDelayInthissectionweanalyzehowtochoose(i.e.thenum-berofcopiesused)inorderforSprayandWaittoachieveaspecicexpecteddelay.Notethattheissueofenergydissi-pationisalsodirectlytiedtothenumberofcopiesusedbySprayandWait,sinceSprayandWaitperformsexactlytransmissions.Letusassumethatthereisaspecicdeliverydelayconstrainttobemet.Thismightbe,forexample,amaximumexpecteddelaydictatedbytheapplication,asinthecaseofthemeetingnoticationmessage.Itisreasonabletoassumethatthisdelayconstraintisexpressedasafactortimestheoptimaldelay1),sincethisisthebestthatanyroutingprotocolcando.TheminimumnumberofcopiesneededforSprayandWaittoachieveanexpecteddelayatmostaEDisindependentofthesizeofthenetworktransmissionrange,andonlydependsonandthenum-berofnodesTheabovelemmastatesthattherequirednumberofcopiesonlydependsonthenumberofnodes,andisstraightforwardtoprovefromEq.(1).Furthermore,sincetheupperboundofEq.(1)istightforsmallL/Mvalues,ifthedelayconstraintisnottoostringent,wecanuseoneofthefollowingmethodstoquicklygetagoodestimatefor:(i)solvetheupperboundequationEq.(1)for,bylettingaEDandtaking,or(ii)approximatetheharmonicnumberinEq.(1)withitsTaylorSeriestermsuptosecondorder,andsolvetheresultingthirddegreepolynomial: 6)L2+ a+2M1 M(ML=M M1,rn=ni istheHarmonicnumberoforder Table1:minimumtoachieveexpecteddelay 3 4 5 6 7 8 9 13 6 5 4 3 3 3 2 bound N.A. 11 6 5 4 3 3 2 taylor N.A. 10 5 4 3 3 3 2 Onecouldalsoiterativelycalculatetheexactnumberofcopiesneeded,usingthesystemofrecursiveequationsfrom[24].Howeverthismethodisquitemorecumbersome.InTable1wecompareexactresultsfortotheonescalculatedwiththetwoapproximatemethodsfordierentvaluesof.Weassumethenumberofnodesequals100.N.AstandforNonAvailableandmeansthatsuchalowdelayvalueisneverachievablebythebound.Ascanbeseeninthistablethefoundthroughtheapproximationisquiteaccuratewhenthedelayconstraintisnottoostringent.4.2EstimatingwhenNetworkParametersareUnknownThroughoutthepreviousanalysisweveassumedthatnet-workparameters,likethetotalnumberofnodesandsizeofthenetwork,areknown.Thisassumptionmightbevalidinsomenetworksoperatedbyasingleauthority.Nev-ertheless,inmanyenvisionedICMNapplications,eitherorboth,mightbeunknown.Forexample,auserthatuseshisPDAtoexchangetextmessagesoveralowcostICMNnetworkformedbysimilarusers,maynotknowthenumberofothersuchusersoutthereortheirgeograph-icalspread,atthatspecictime.InordertomakeSprayandWaitecientinsuchscenariosaswell,wewouldliketoproduceandmaintaingoodestimatesofrelevantnetworkparameters,andadaptaccordingly.Astheanalysisoftheprevioussectionindicated,onlyanestimateofthenumberofnodesisrequiredtotune,inmostsituations.Thisproblemisdicultingeneral.AstraightforwardwaytoestimatewouldbetocountuniqueIDsofnodesencounteredalready.However,thismethodrequiresalargedatabaseofnodeIDstobemaintainedinlargenetworks,andalookupoperationtobeperformedeverytimeanynodeisencountered.Furthermore,althoughthismethodconvergeseventually,itsspeeddependsonnetworksizeandcouldtakeaverylongtimeinlargedisconnectednetworks.However,ifweassumethatnodesperformindependentran-domwalks,wecanproduceanestimateofbytakingad-vantageofinter-meetingtimestatistics.Specically,letusasthetimeuntilanode(startingfromthestation-arydistribution)encountersothernode.ItiseasytoseefromLemma4.2thatisexponentiallydistributedwithaverage1).Furthermore,ifwesimilarlyasthetimeuntiltwodierentnodesareencoun-tered,thentheexpectedvalueof M1+1 fromthesetwoequationswegetthefollow-ingestimatefor bytheprocedureabovepresentssomechal-lengesinpractice,becauseareensembleaverages.Sincehittingtimesareergodic[4],anodecancollectsampleintermeetingtimesandcalculatetimeaver-instead.However,thefollowingcomplica-tionarises:whenarandomwalkmeetsanotherrandomwalkbecomecoupled[12];inotherwords,thenextintermeetingtimeofisnotanymoreexponen-tiallydistributedwithaverage.Inordertoovercomethisproblem,eachnodekeepsarecordofallnodeswithwhichitiscoupled.Everytimeanewnodeisencountered,itisstampedascoupledforanamountoftimeequaltorelaxationtimeforthatgraph,whichistheexpectedtimeuntilawalkstartingfromagivenpositionreachesitsstationarydistribution[4].Then,whennodemeasuresthenextsampleintermeetingtime,itignoresallnodesthatitscoupledwithatthemoment,denotedasandscalesthecollectedsample .Asimilarprocedureisfollowedfor.Puttingitalltogether,aftersampleshavebeencollected: nnkMck M1T1T2=1 nnk Mck1 M1T11+ Mck inEq.(2)wegetacurrentestimate.AscanbeseenbyEq.(2),theestimatorforsensitivetosmalldeviationsoffromtheiractualvalues.Thereforeitisusefulforanodetoalsomaintainarunningaverageof.Specically,therunningestimateisupdatedwitheverynewestimateAlthoughtheexpositionofourestimationmethodhasbeenbasedontherandomwalkmobilitymodel,itisimpor-tanttonotethatthismethodholdsforanymobilitymodelwithapproximatelyexponentiallydistributedmeetingtimesThereasonforthisisthattheexpectedmeetingtime,whichdiersfrommobilitymodeltomobilitymodel,getscancelledfromtheequations.Whatisimportantisthattheexpectedtimeofmeetinganyofnodesisapproximately theex-pectedtimeofmeetingonenode.Figure2showshowtheonlineestimate,calculatedwithourproposedmethod,quicklyconvergestoitsactualvaluefora200200toruswith200nodes,forboththerandomwalkandrandomwaypointmodels.(Notethateveninthissmallscenario,ourmethodsconvergenceismorethantwotimesfasterthanID-counting.)Wehavetestedourestimatorindierentscenariosandhaveobservedsimilarconvergence,aswell.Inthefuture,weintendtoexaminehowourmethodperformswithothermobilitymodels,too.Asageneralrulethough,itisshownin[4]thatthehittingtimedistributionofgeneralrandomwalksongraphsalwayshasanexponentialtail,andinmanycasesisapproximatelyexponentialitself.Consequently,weexpectthatifagivenmobilitymodelcanbe(evenapproximately)representedasarandomwalkonanappropriategraph,thenourmethodshouldproducesucientlyaccurateestimates.Asanalnote,bothourmethodandID-countingcouldtakeadvantageofindirectinformationlearning,wherenodesexchangeknownuniqueIDsorindependentlycollectedsam-plestospeedupconvergence.4.3ScalabilityofSprayandWaitHavingshownhowtondtheminimumnumberofcopiestoachieveadelayatmosttimestheoptimal,itwouldbeinteresting,fromascalabilitypointofview,toseehow Estimation of M - Random Walk10020030040001000200030004000number of samplesM value Actual M = 200 Estimated M Estimation of M - Rand. Waypoint10020030040001000200030004000number of samplesM value Actual M = 200 Estimated M Figure2:Onlineestimatorofnumberofnodes(grid,transmissionrangemixingtime=4000thepercentageofnodesthatneedtoreceiveacopybehavesasafunctionof.Thereasonforthisisthefollowing:IfweassumethatalargeenoughTTLvalueisusedandnoretransmissionsoccur,ooding-basedschemeswilleventuallygiveacopytoeverynodeandthereforeper-formatleasttransmissions.Increasedcontentionandtheresultingretransmissionswillobviouslyincreasethisvaluesignicantly,asweshallsee.Evenutility-basedschemeswithreasonableutilitythresholdswillperform()trans-missions.Ontheotherhand,SprayandWaitperformstransmissions,andproducesverylittlecontentioncomparedtoooding-basedschemes.Consequently,thenumberoftransmissionsthatSprayandWaitperformspermessageisatmostafractionofthenumberoftransmissionspermessageooding-basedschemesperform.InFigure3wedepictthebehaviorofasafunc-tionoffordierentvaluesof.Itisimportanttonotetherethat,asthenumberofnodesinthenetworkincreases,thepercentageofnodesthatneedtobecomerelaysinSprayandWaittoachievethesameperformancerelativetotheoptimalisactuallydecreasing.Inotherwords,althoughtheperformanceoftheoptimalschemealsoimproveswiththeperformanceofSprayandWaitseemstoimprovefaster.Theintuitionbehindsthisinterestingresultisthefollowing:thedelayofSprayandWaitisdominatedbythedelayofthewaitphase;inthatcase,ifL/Miskeptcon-stant,thedelayofSprayandWaitdecreasesroughlyas1Ontheotherhand,thedelayoftheoptimalschemede-creasesmoreslowlyaslog(,ascanbeseenbyEq.(1).ThefollowingLemmagivesaformalproof. Percentage of Nodes Receiving a Copy 100100010000100000Number of Nodes (M)percentage (%) =10 Percentage of Nodes Receiving a Copy 100100010000100000Number of Nodes (M)percentage (%) =10 Figure3:Requiredpercentageofnodesceivingacopyforsprayandwaittoachieveanex-pecteddelayofaEDLetL/Mbeconstantandlet.LetdenotetheminimumnumberofcopiesneededbySprayandWaittoachieveanexpecteddelaythatisataED,forsome.Then isadecreasingfunctionofProof.wecanusetheupperboundofEq.(1)toexaminethebehaviorofSprayandWait: M1 =(log(=(log( Also,let,whereisaconstant(1).Replacinginthepreviousequationgivesus(log(1log(1 M1 1c c Now,forlarge 1.Therefore,keepingthesizeofthegridandtransmissionrangeconstantwegetthat=(1)+(1) =( Ontheotherhand,forconstantlog( )ascanbeeasilyseenfromEq.(1).Hence, 1 log((i.e.decreasingwith),ifL/Miskeptcon-stant.Thisimpliesthatifwerequire tobekeptcon-stantforincreasing,thenL/Mhastobedecreasing. Thisbehaviorof(combinedwiththeindepen-denceof)impliesthatSprayandWaitisextremelyscalable.Whilemostoftheothermulti-copyschemesperformarapidlyincreasingnumberoftransmis-sionsasthenodedensityincreases,SprayandWaitactuallydecreasesthetransmissionspernodeasthenumberofnodesincreases.Itsperformanceadvantageovertheseschemesbecomesevenmorepronouncedinlargenetworks.5.SIMULATIONRESULTSWehaveusedacustomdiscreteevent-drivensimulatortoevaluateandcomparetheperformanceofdierentrout-ingprotocolsunderavarietyofmobilitymodelsandundercontention.Aslotted,randomaccesswithcollisiondetec-tionMACprotocolhasbeenimplementedinordertoarbi-tratebetweennodescontentingforthesharedchannel.Theroutingprotocolswehaveimplementedandsimulatedarethefollowing:(1)Epidemicrouting(epidemic),(2)Ran-domizedoodingwith=(01)(random-ood),(3)Utility-basedroutingasdescribedin[19]withautility=(02)(utility-ood),(4)Optimal(binary)SprayandWaitwithcopies(spray&wait(L)),(5)SeekandFocussingle-copyrouting(seek&focus)[25],and(6)Oracle-basedOptimalrouting(optimal).(Wewillusetheshorternamesintheparenthesestorefertoeachroutingschemeinsimulationplots.)Inallscenariosconsidered,eachmessageisassignedaTTLvaluebetween40006000timeunits.Wehavetriedtotunetheparametersofeachprotocolineachscenariosep-arately,inordertoachieveagoodtradeofortheprotocolinhand.Finally,wedepicttwoplotsforSprayandWaitfortwodierentvalues,inordertogainbetterinsightintothetransmissions-delaytradeosinvolved.WechoosethesevaluesaccordingtothetheoryofSection4.(Specically,suchthatitsdelaywouldbeabout2thatoftheoptimalscheme,ifthenodeswereperformingrandomwalks.)Werstevaluatetheeectoftracloadontheperfor-manceofallroutingschemes(ScenarioA).Wethenexamine theirperformanceasthelevelofconnectivitychanges(Sce-narioB).5.1ScenarioA:EffectofTrafÞcLoad100nodesmoveaccordingtotherandomwaypointmodel[6]ina500500gridwithreectivebarriers.Thetransmis-sionrangeofeachnodeisequalto10.Finally,eachnodeisgeneratinganewmessageforarandomlyselecteddestinationwithaninter-arrivaltimedistributionuniformin[1]untiltime10000.Wevaryfrom10000to2000creatingaveragetracloads(totalnumberofmessagesgeneratedthroughoutthesimulation)from200(lowtrac)to1000(hightrac).Figure4depictstheperformanceofallroutingalgorithms,intermsoftotalnumberoftransmissionsandaveragede-liverydelay.Epidemicroutingperformedsignicantlymoretransmissionsthanotherschemes(from56000to144000),andatleastanorderofmagnitudemorethanSprayandWait.Therefore,wedonotincludeitinthetransmissionplots,inordertobettercomparetheremainingschemes.AsisevidentbytheseplotsSprayandWaitoutperformsallsingleandmulti-copyprotocolsdiscussedandachievesitsdesigngoalssetinSection3.Specically:(i)underlowtracitsdelayissimilartoEpidemicroutingandis2timesfasterthanallothermulti-copyprotocols;itperformsanorderofmagnitudelesstransmissionsthanEpidemicrouting,and56timeslesstransmissionsthanRandomizedandUtility-based,and(ii)underhightracitretainsthesameadvantageintermsoftotaltransmissions,andoutperformsallotherprotocols,intermsofdelay,byafactorof1Asanalnote,thedeliveryratioofalmostallschemesinthisscenariowasabove90%foralltracloads,exceptthatofSeekandFocuswhichwasabout70%,andthatofEpidemicroutingwhichplummetedtolessthan50%forveryhightrac,duetoseverecontention. 5000100001500020000250003000035000400004500050000Total Transmissions random-flood utility-flood seek&focus spray&wait(L=10) spray&wait(L=16) 50010001500200025003000350040004500 s eek&focu s Delivery Delay (time units) Increasing traffic Increasing traffic 5000100001500020000250003000035000400004500050000Total Transmissions random-flood utility-flood seek&focus spray&wait(L=10) spray&wait(L=16) 50010001500200025003000350040004500 s eek&focu s Delivery Delay (time units) Increasing traffic Increasing traffic Increasing traffic Increasing traffic Increasing traffic Figure4:ScenarioA-performancecomparisonofallroutingprotocolsundervaryingtracloads.5.2ScenarioB:EffectofConnectivityInthisscenario,thesizeofthenetworkis200200andweto4000(mediumtracload).Wevarythenumberofnodesandtransmissionrangetoevaluatetheperfor-manceofallprotocolsinnetworkswithalargerangeofcon-nectivitycharacteristics,rangingfromverysparse,highlydisconnectednetworks,toconnectednetworks.Beforeweproceed,itisnecessarytodeneameaningfulconnectivitymetric.Althoughanumberofdierentmet-ricshavebeenproposed(forexample[10]),nowidespreadagreementexists,especiallyifoneneedstocapturebothdisconnectedandconnectednetworks.Webelievethatameaningfulmetricforthenetworksofinterestistheex-pectedmaximumclustersizedenedasthepercentageoftotalnodesinthelargestconnectedcomponent(cluster).Thisindicateswhatpercentageofnodeshavealreadycon-glomeratedintotheconnectedpartofthenetwork,withoneimplyingaregularconnectednetwork.Figure5de-pictstheconnectivitymetricforthe200200torus,asafunctionoftransmissionrangefor2dierentvaluesof Percentage of Nodes in Max Cluster (200x200)0.20.40.60.81.25101520253035404550Tx Range (K)percentage M = 100 M = 200 Figure5:Expectedpercentageoftotalnodesinlargestconnectedcomponent,asafunctionofWehavepickedanumberofpointsfromFigure5thatspantheentireconnectivityrange,andhaveevaluatedtheperformanceofallprotocolsunderthesescenarios.Figure6andFigure7depictthenumberoftransmissionsandtheaveragedelay,respectively.Ascanbeseenthere,SprayandWaitclearlyoutperformsallprotocols,intermsofbothtransmissionsanddelay,foralllevelsofconnectivity.Mostimportantly,itisextremelyscalableandrobust,comparedtoothermulti-copyorevensingle-copyoptions.EpidemicroutingandtherestoftheschemesmanagetoachieveadelaythatiscomparabletoSprayandWaitforveryfewconnectivityvaluesonly,butperformquitepoorlyforthevastmajorityofscenarios.Furthermore,theirperformanceseemstovarysignicantlyfordierentconnectivitylevels(despiteoureorttotuneeachprotocolforthegivensce-nario,wheneverpossible).SprayandWait,ontheotherhand,exhibitsgreatstability.Itperformsaxednumberoftransmissionsacrossallscenarios,whileachievingadeliverydelaythatdecreasesasthelevelofconnectivityincreases,asonewouldexpect.6.CONCLUSIONInthisworkweinvestigatedtheproblemofecientrout-inginintermittentlyconnectedmobilenetworks.Wepro-posedasimplescheme,calledSprayandWait,thatman-agestoovercometheshortcomingsofepidemicroutingand Transmissions (thousands) K = 5 (2.5%) K = 10 (4.4%) K = 20 (14.9%) K = 30 (68%) K = 35 (92.5%) 100150200250300 Transmissions (thousands) K = 5 (1.9%) K = 10 (4.3%) K = 15 (13.3%) K = 20 (54.4%) K = 25 (95.5%) M = 100M = 200 Transmissions (thousands) K = 5 (2.5%) K = 10 (4.4%) K = 20 (14.9%) K = 30 (68%) K = 35 (92.5%) 100150200250300 Transmissions (thousands) K = 5 (1.9%) K = 10 (4.3%) K = 15 (13.3%) K = 20 (54.4%) K = 25 (95.5%) M = 100M = 200 Figure6:ScenarioB-Transmissionsasafunctionofnumberofnodesandtransmissionrange 5001000150020002500300035004000Delivery Delay (time units) K = 5 (2.5%) K = 10 (4.4%) K = 20 (14.9%) K = 30 (68%) K = 35 (92.5%) 5001000150020002500300035004000Delivery Delay (time units) K = 5 (1.9%) K = 10 (4.3%) K = 15 (13.3%) K = 20 (54.4%) K = 25 (95.5%) M = 100M = 200 5001000150020002500300035004000Delivery Delay (time units) K = 5 (2.5%) K = 10 (4.4%) K = 20 (14.9%) K = 30 (68%) K = 35 (92.5%) 5001000150020002500300035004000Delivery Delay (time units) K = 5 (1.9%) K = 10 (4.3%) K = 15 (13.3%) K = 20 (54.4%) K = 25 (95.5%) M = 100M = 200 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