LatencyofWirelessSensorNetworkswithUncoordinatedPowerSavingMechanismsO - PDF document

LatencyofWirelessSensorNetworkswithUncoordinatedPowerSavingMechanismsOlivierDousseCH-1015LausanneSwitzerlandolivier.dousse@epß.chPetteriMannersaloVTTTechnicalResearchCentreofFinlandP.O.Box1202,02044VTTpetteri.mannersalo@vtt.ÞPatrickThiranCH-1015LausanneSwitzerlandpatrick.thiran@epß.chABSTRACTWeconsiderawirelesssensornetwork,wherenodesswitchbe-tweenanactive(on)andasleeping(off)mode,tosaveenergy. Correspondingauthor.PermissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforproÞtorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitationontheÞrstpage.Tocopyotherwise,torepublish,topostonserversortoredistributetolists,requirespriorspeciÞcpermissionand/orafee.May24Ð26,2004,Roppongi,Japan.Copyright2004ACM1-58113-849-0/04/0005...$5.00.1.INTRODUCTION andcommunicationenergyconsumptiondependsonthehardware,butalsoonthewaydataisaggregatedandmediumisaccessed.Thisconsumptionistheoneofferingprobablythelargestpotentialforreduction.Indeed,nodesspendaconsiderableamountofen-ergyinlisteningtotheirneighbours,andaslongasnoneoftheseneighbourstransmitsanydata,thisenergyissimplywasted.Datacollectionandmediumaccesscontrol(MAC)schemesneedthere-foretoincorporateenergysavingasaprimarygoal,andproposalsforsuchalgorithmshaverecentlyemergedintheliterature,whichshowindeedthatsigniÞcantenergysavingscanbeachieved.Inpar-ticular,mostproposalsforenergysavingMACschemesforsensornetworksintroduceasleepingmodefornodes,duringwhichprac-ticallynoenergyisspent[23,26].Introducingasleepingmodedoeshowevercomeatsomecost.AsolutionistouseaTimeDivisionMultipleAccess(TDMA)scheme,butthisrequiresnodestosynchronizewitheachotherquitetightly,whichcanbeaquitecomplextaskinlargenetworkswithrandomnodelocationsandimperfect(drifting)clocks.Lettingthenodessettheirwake-upandsleepingtimesinadecentralizedfashionreducesthiscomplexity,butthisincreasesthedelay(alsocalledlatency)totransferinformationbetweenthesinkandadis-tantnode.Pushingthedecentralizationtoanextremewherenodesgotosleepindependentlyfromeachother,whichisthesolutionweadoptinthepresentpaper,eliminatesthecomplexityofhavingsynchronizedclustersofnodes,butatthesametimeraisesconcernsaboutanincreaseofthelatencyofthenetwork.Moreimportantly,itwillnotonlyincreasetheaveragelatencyitself,butitwillalsoincreasethevarianceofthislatency.Forsomeapplications,suchasspatialdatacollectionforstatisticalpurposes,thisiscertainlyac-ceptable,butnotformanyothersthataremuchmoretime-critical.Atypicalandimportantexampleofsuchascenarioistheuseofsensornetworkformonitoringanareaandsendinganalarmwhenanabnormaleventissensedoccurs(suchasanintrusion,arapidlychangingvariable,etc.).EvenifsomeÞxedamountoflatencycanbetolerated,ahighlyvariablelatencyduetotherandompositionofthenodes,therandomradiorange,thenon-synchronizedorevenrandomsleepingandactiveperiodsismuchmoreproblematic.Isitpossibletoletnodesgointosleep,withoutanycoordinationbe-tweentheirschedules,andyethaverigorousboundsonthelatency?Thispaperprovidesapositiveanswertothisquestion,andpre-sentstheÞrstanalyticalboundsonthelatencyofasensorwithran-domi.i.d.activeandsleepingperiods.Weusedynamicpercolationtheorytoobtainthesebounds,whicharebasedonthefollowingNodesarerandomlyscatteredontheplaneasahomogeneousPoissonprocess;Nodescansenddatainonehoptoaneighbourwithinsomeprescribed,deterministicconnectivityrange;Nodesswitchbetweenanactive(ÒonÓstate)andasleeping(ÒoffÓstate)modeindependentlyfromeachother;Whenanodesensesanevent,orwhenitreceivesdata(mes-sage)fromoneofitsneighbours,itstaysactiveandbroad-caststhismessagetoallitsneighbourswithindirectreach,untilitissurethatthemessagehasreachedallitsneighbours,orthesink,withhighprobability;Aslongasnodesdonotsenseanyevent,ordonotreceiveanydatafromtheirneighbors,thedurationoftheironandoffperiodsaretwoindependentsequencesofi.i.d.randomvariables.Offperiodsareeitherconstantorexponentiallydistributed,whileonperiodsareidenticallydistributedbutnotnecessarilyexponentially;Thesensingrangeissmallerthanorequaltotheconnectivityrange.Thesensingrangeisrandomlydistributedbetweenaminimalnonzerovalue,andamaximumwhichisequaltotheconnectivityrange.ThetwoÞrstassumptionsmeanthatwemodelthesensornet-workasaPoissonBooleanmodel.Theassumptionofcirculardi-rectconnectivityareaisprobablythemostdebatableamongalltheassumptionsmadeabove[11].Inreallife,theconnectivityrangeisfarfrombeingacirclewithÞxedradiusinmanycircumstances:itvariesalot,dependingonmanyfactors,suchasinterferenceswithothernodes,backgroundnoise,time-varyingchannel,hard-waredefects,directionalantennas,etc.Amorerealisticmodelistosetathresholdonthesignaltonoiseandinterferenceratioatthereceiver,asin[11,1],whichdeÞnesadirectconnectivityareaaroundanodebycontourplots,whoseshapeisindeedhighlyvari-able.Weshouldnotehoweverthatpercolationdoesholdformodelstakinginterferencesintoaccount[7],undersomeconditions,whenitholdsforthePoissonBooleanmodel.Moreover,whennodeshavelongsleepingperiods,interferencesfromothernodesbecomelesscritical.Asecondremarkshouldbemaderelativetothehighlevelofdirectionalityinthereceptionsets,whichisnotcapturedbyanisotropicmodelliketheBooleanmodel.Quiteinterestingly,anisotropydoeshelptomakethenetworkpercolate,andhencetoincreaseconnectivity.Ithasbeenshown[10]thatgrainswithacir-cularshapesarethemostdifÞcult,amongallconvexgrainswiththesamesurface,tohavethenetworkpercolate.Consequently,al-thoughbeingcertainlyacrudemodeloftherealconnectivitygraphofasensornetworks,theBooleanmodelprovidesaconservativeestimateoftheactualconnectivityofasensornetwork.TheonlyaspectwhichisnotincludedinthestaticBooleanmodelisthetimevariabilityofthewirelesschannel.ItcanhoweverbeincludedinadynamicBooleanmodel,liketheblinkingPoissonBooleanmodelweproposeinthispaper.Theassumptionsonthedurationsoftheactiveandsleepingpe-riodsareratherweak,theyresumetoindependenceandidenticaldistributions,withoff-periodbeingconstant(correspondingtoaperiodicon/offschedule)orexponentiallydistributed(correspond-ingtoamemorylesssleepingschedule).WedonotrequirethattheonperiodsfollowanyspeciÞcdistribution.Theassumptionthatnodesbroadcastanydatatheysenseorreceivetoalltheirneigh-borsmakestheroutingsimpleandenablestocomputeanalyticallythebounds,whilestayingreasonableifincomingeventsrarelyoc-cur(inanintrusiondetectionscenarioforexample).Finally,itisreasonabletoassumethatthesensingradiusofasensorissmallerthanitstransmissionrange.Thismeansthattherewillbesupercrit-icalamountofdevicesformessagetransmissions.Inordertosavebatteries,itisnaturalthatonlyafractionofsensorarelisteningthetransmissionchannel.Sinceenergysavingiscentraltosensornetworks[9],ithasre-ceivedaconsiderableamountofattention,drivingroutingalgo-rithms(seee.g.[3,22]),scheduling(seee.g.[20,8]),datacollec-tionandaggregation(seee.g.[15])andMAC(seee.g.[22,27]).Schedulingstrategiestradingoffenergysavingandlatencyarede-visedin[28],whileaMarkovianmodelexploringtheperformanceofawirelessnetworkwithon/offperiodsisdescribedin[4],un-dertheassumptionthatthereisapathfromanysensortothesink.Herewedonotlookatthescheduling,routing,dataaggregationalgorithms,butatthelatencyofthenetwork.Contrarytoallthesepapers,wedonotassumethatthereisapathfromanysensortothesink.Onthecontrary,weshowthatevenifthereisnoconnectivity ofallsensorsatalltimesbecausemanyofthemaresleepingandonlyafewareactive,itispossibletotransferdatafromanysen-sortothesinkinaboundedtimewithprobabilityone,withoutanycoordinationbetweenthesensors.Thepaperisstructuredasfollows.Inthenextsection,weÞrstgiveanoverviewoftheschemeweproposetosaveenergycon-sumptionatthesametimeastokeepthelatencybounded.WeintroduceformallythenetworkmodelinSection3.Webeginbyre-mindingafewusefulfactsabouttheclassicstaticPoissonBooleanmodel,andwethenextendittoaccountfornodesswitchingbe-tweenonandoffperiods,inadynamicmodel,whichwenameblinkingPoissonBooleanmodel.Wegivesomeoftheirconnectiv-itypropertiesinthesamesection,andsomeoftheircoverageprop-ertiesinSection4.Section5isthemainresultofthepaper,namelytheproofthatthelatencyofthelineargrowslinearlywiththedis-tancebetweenthesinkandthesensorthatdetectedtheevent.Weprovebothalowerboundandanupperboundonthislatency,usingdifferentpercolationtechniques:theupperboundisestablishedus-ingLiggettÕsergodictheorem,whilethelowerboundisobtainedbyacouplingwiththecontinuumgrowthmodelproposedbyDeijfen[5].ThelinearityoftheseboundsisalsovalidatedbysimulationinSection5.3,andtheimpactoftheparametersofthenetwork(nodedensity,durationsofonandoffperiods)isdiscussed.2.DECENTRALIZEDENERGYSAVINGThegeneralstrategyweadopttosaveenergyandyetboundthelatencyofthenetworkisasfollows.Intheabsenceofanyincomingeventormessagefromaneigh-bour,nodesswitchbetweentheirsleepingandactivephasesinde-pendentlyfromeachother.Tosaveenergy,weassumethatthesleepingtimesaremuchlongerthantheactivetimes.Onceanodehassensedanevent,itstaysactive,andkeepssend-ingrepeatedlythisinformationtoitsimmediateneighbours,whichcanbereachedwithinasinglehop.Theneighbourswillonlybeabletohearthemessagewhentheyturntheirradioon.Whenalltheneighbourshaveheardthatmessage,thesensingnodecanturnoffitsradio,andresumeitsregularon/offschedule.Inordertoavoidsynchronization,thephaseoftheon/offscheduleisdrawnaccordingtothestationarydistribution.Theothernodeswillthenremainactiveuntilalltheirneighbourshavereceivedthemessage,andsoon,untilthesinkeventuallyreceivesthemessage.Rememberthatnodesarescatteredoverthedomainthatneedstobemonitored,accordingtoaPoissonprocess.Wesupposethatthereisalargenumberofnodes,sothatthesensingradiuscanbesettoasmallvaluewithoutimpactingthecoverageofthenet-work.However,sincenodesswitchbetweenthetwophasesin-dependentlyfromeachother,theremaybequitefewactivenodesatanygiventime,sothatthesetofsimultaneouslyactivenodesatanyparticulartimeisalwaysdisconnected.Thissetofsimul-taneouslyactivenodesrepresentshoweveronlyasnapshotofthenetworkatthisparticulartime.Bywaitinglongenough,asufÞ-cientlylargenumbernodeswillhavepassedbyanactivephase,sothatthesuperpositionofallthedifferentsnapshotsofthenetworkatalltimeswithinanintervalbecomessupercritical.Onthiscumu-lativegraph,thereisapathbetweenanysensorandthesinkwithhighprobability.Whileitisrelativelyeasytoshowthatindeedthenetworkwillpassbyasupercriticalphaseifwewaitlongenough(aswewillshowinProposition2),itismuchmoredifÞculttoÞndhowlongwehavetowaitbeforethispathisalmostsurelypresent,thatis,tocomputethelatency.Clearly,thelargerthedistancebe-tweenthesensingnodeandthesink,thelargerthelatency.Buthowdoesitdependonthisdistance?Theboundsonthelatency,whichweestablishinTheorem1,showthatitislinear,underthecondi-tionsstatedintheintroduction.Thesimulations,runundermoregeneralconditions(includingpropagationdelaysonthelinks,andarandomconnectivityradius)conÞrmthisÞnding.3.POISSONBOOLEANMODELS3.1StaticPoissonBooleanmodelInthestaticPoissonBooleanmodel,thelocationsoftheÒgrainsÓaredeterminedbypointsofastationaryPoissonpointprocessofintensity(seee.g.[24]).Inthispaper,weonlycon-anddiskshapedgrains.ThenthePoissonBooleanmodelisjustaunionofrandomlyscattereddisks(seeFigure1),i.e.,thecoverageprocessdeÞnestheoccupiedcomponentisthediskcenteredathavingradius,andwherethearei.i.d.,independenttopointprocess,anddis-tributedas -10 -5 5 10 -10 -5 5 10 -5 5 10 -10 -5 5 10 Figure1:PoissonBooleanmodelin.Sub-criticalintensityontheleftandsupercriticalintensityontheright.Thecoveragecanbemeasuredbythehittingprobability,whichistheprobabilitythatanarbitrarypointbelongstotheoccupiedregion.Inastationarysetting,thehittingprobabilitiesareequaltothemeanfractionofareaoccupiedby.ForthePoissonBooleanmodel,wehave))=))=expFormoreinformationseee.g.[24].Ontheotherhand,theoccupiedsetcanbedividedintodisjointclusterswhichareformedbytheoverlappingdisks.Onewaytomeasuretheglobalconnectivityisthesizeofthelargestcluster.Letusdenote,theunionofalltheoccupiedcomponentswhichintersect.Then,thecriticalintensitycanbedeÞnedbycheckingifthereisapositiveprobabilitythattheoriginbelongstoanunboundedcluster,i.e.,))=Underquiteweakassumptions,indimension2,thereex-ists0suchthatthelargestconnectedcomponentisun-boundeda.s.whenevertheintensity.Moreover,theinÞniteclusterisuniquea.s.Ontheotherhand,if,allthecon-nectedcomponentsareÞnitea.s.Thereisnoanalyticalformula,onlyboundsinsomespecialcases.However,itisrelatively 1E.g.,ER2d1 easytonumericallyestimate.Forexample,37foraÞxedradius,(seee.g.[25,21]).FormoredetailsrelatedtopercolationsinPoissonBooleanmodelsee[18].3.2BlinkingPoissonBooleanmodelInordertotakeintoaccountthealternationbetweenthesleepingandactivemodes,weintroduceadynamicPoissonBooleanmodel,wherethedynamicsisduetotheÒblinkingÓofthenodes.Asfarasweknow,onlyfewdynamicpercolationmodelshavebeenre-searchedmathematically.Inthelattice,dynamicbondpercolationshavebeenstudiedbyHŠggstršm,PeresandSteif[14,19].PoissonBooleanmodelswithmovingpointshavebeenstudiedbyvandenBergetal.[2].blinkingPoissonBooleanmodelisasimplemod-iÞcationofthestaticPoissonBooleanmodel.AsinthestaticPois-sonBooleanmodel,thepositionsofthenodesaredeterminedbyaPoissonpointprocesswithintensity.Ateachnode,weattachadiskwhoseradiusisdistributedas(deterministicori.i.d.randomradii).Thedynamicsfollowsfromtheassumptionthatthenodesalternatebetweenon-stateandoff-statewithperiodsdeterminedbythestationaryi.i.d.on/offprocesses.ThedistributionsoftheseprocessesareequaltothedistributionofprocessWeassumethatthelengthsoftheon/offperiodsareindependent,withtheoffperiodseitherconstantorexponentiallydistributedwithmeanoff,andtheonperiodsdistributedaccordingtoanar-bitrarydistributionwithmean.Theonlypurelytechnicalas-sumptionwerequirefortheonperiodsisthattheyhavealwaysnonzerolength,whichwecanwriteasass
s+)Zt=1Zs=1=1(1)Thisisalwaysthecaseinpractice.Ontheotherhand,assumingoff,theoffperiodssatisfysatisfys
s+)Zt=0Zs=0= etoffifexp-distributed, offifconstant.Thestationarydistributionofisgivenbyoffoff off offAssumethatthenodedensityissupercritical,thatis,Fromthepointofviewofsensornetworks,itisinterestingtoknowwhetherthenetworkremainsconnectedallthetime,despitethealternationbetweensleepingandactivestates.Ifthereisnoun-boundedconnectedcomponentofactivenodesatanysnapshot,thenanotherquestionishowlongonehastowaitinordertogetacompletelyconnectednetworkinthecumulativecoveragepro-cess.ThetwosituationsarevisualizedinFigure2andanswersaregivenbythefollowingpropositions.ROPOSITION.IftheZarestationaryon/offprocesseswithexponentialorconstantofftimesandonpe-riodssatisfying(1),thenaninniteclusterexistsforalltthereisnoinniteclusterforalltROOF.AslightlymodiÞedprooffrom[14].Letusassume.Let0suchthatandtake -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 Figure2:SnapshotsoftheBlinkingPoissonBooleanmodelatsomeparticulartime.Theblackdisksarethecoverageareaoftheactivenodes,thegrayonesofthesleepingnodes.Ontheleft,i.e.,thereisalwaysanunboundedactivecluster.Ontheright,,i.e.,alltheactiveclustersareÞnite0suchthatthat0
0+]Zt=1Z0=11.ThenThen0
]Z(i)t=1(1)on
c
foranarbitrary.Sinceeventsents0
]Z(i)t=1,i=1
2
,aremutuallyinde-pendent,wecanconsiderathinningofaPoissonprocesswherewetakeonlythenodeswhichareactivethewholeintervalal0
].ThesenodesaredistributedaccordingtoaPoissonprocesswithintensity.ThusaninÞniteclusterexistsforallall0
])=Theargumentcanberepeatedfortheintervalsalsk
(k+1)]withinteger.DenotetheeventthataninÞniteclusterexistsforallallk
(k+1)],anditscomplement.Then,thenweconsiderthenodeswhicharenotsleepingduringthewholeintervalal0
].ByEquation(2),forsmallenoughenough0
]Zt=1on+offoff.Thustheinten-sityofthosepointsislessthanthecriticalintensityandthereisa.s.nounboundedcluster.Again,thecountableadditivitycompletestheproof. Evenifwecanstillahavesomeformofconnectivityifwetakeintoaccountthecumulativecoverageprocess.Letusde-ÞnethecumulativeBlinkingBooleanmodelasthetheareawhichhasbeencoveredbysomeactivediskwithinwithin0
t].PROPOSITION.IftheZarei.i.d.station-aryon/offprocesseswithexponentialoff-times,thenthecumulativeconnectivitygraphonon0
t]hasalmostsurelyaninniteconnectedcomponentwhenevertoff Iftheoffperiodslastcon-stanttimethentheconditionistoff ROOF.Theprobabilitythatprocessvisitsatleastoncetheon-statewithinwithin0
t]isPmaxs[0
t]Zs=1=P(Z0=1)+Pmaxs[0
t]Zs=1Z0=0P(Z0=0)=1off ThepointsofthePoissonprocess,wherethecorrespondingon/offprocessesvisittheon-stateduringduring0
t],formathinnedPoissonpointprocesswithintensity1offf0
t]Zs=0Z0=0.ThecorrespondingPoissonBooleanmodelhasanunboundedclus-tera.s.ifofff0
t]Zs=0Z0=0
c.Inordertocompletetheproof,applyEquation(2). Thenextsectionshowshowtheseresultsareusefulfordeter-miningthecoverageandconnectivityofthenetwork.4.SENSINGCOVERAGEANDTRANSMIS-SIONCONNECTIVITY Figure3:Asensingcoverageontherightandthecorrespond-ingtransmissionnetworkontherightwhentransmissionra-diusis5timesbiggerthanthesensingradius.Weassumethattheradioequipmentsarealternatingbetweensleepandactivestateswithmeansojourntimesoffandsta-tionarydistributionsoff.Thesensingapparatusareeitheractiveallthetimeortheyalsointerchangeaccordingtosomeon/offprocesses,withparametersoffandstationarydistributionsoffForthesensingmodel,welettheradiiberandom,butforthetransmissionmodeltheradiiareassumedtobeconstantandde-notedby.Notethatiftwodisksofradiusoverlap,thenthecentersareatmost2apart.ThustheradioconnectivitygraphisdeterminedbyBooleanmodelwithdisksofradius4.1SensingareaForsomenetworks,thecoveredfractionoftheareaisthekeyproperty,forotherstheprobabilityofunwatchedroutesthroughthenetwork.Thus,dependingontheapplication,eitherthemeanareacoverageortheexistenceofapercolationclusterarethechar-acteristicswhichgivetheconditionsfortheminimaldensityoftheAssumethateachsensorcanmonitoradiskwhoseradiusisdis-tributedas.Asexplainedintheprevioussection,themeanfrac-tionofareacoveredisgivenby1exp.Toensureacoveredfractionatleastequalto,theaveragenumberofactivesensorsperunitareahastosatisfy Regardingunwatchedroutes,percolationtheorygivesthecriti-calintensitiesguaranteeingthatthereisanunbrokennetguardinganypassagethroughthenetwork.Toensureaconnectedcoveredareaatallinstants,itisenoughtohave,accordingtoProposition1.Ifalarmsaretriggeredby,forexample,slowlymovingevents,thenitisenoughtoobtainconnectivityoveralongertimeperiod.Inthiscase,Proposition2givesthesufÞcientconditionswithrespecttotheintensityandthemeanlengthsoftheon/offperiods.Moredetailsonthecoveragepropertiesofsensornetworkscanbefoundin[17].4.2RadioconnectivityConnectivitycanbeseenastheprobabilitythatanarbitrarynodeisconnectedtomostoftheothers.IfweassumethattheradiorangeofthedevicesissigniÞcantlylargerthantheirsensingrange,theconditionsonthenodedensitygivenintheabovesectionleadtoahighlysuper-criticalradiocon-nectivitygraph(seeFigure3).Infact,almostallthenodesarecon-nectedinthiscase,andthereexisthighlyredundantroutesbetweennodes.Thisredundancyisthemotivationforlettingthenodesturnofftheirradiodevicesporadically.Inourmechanism,ateachtimeinstant,thenumberofnodeswithactiveradiodeviceis.ThisdeÞnesanewstaticPois-sonBooleanmodel.Dependingonthevalueofthenewmodelcanbeeithersuper-critical,eithersub-critical,asshowninProposition1.Inordertospareasmuchbatteryaspos-sible,inthispaper,wechoosesolowthattheresultingprocessissub-critical.Eveninthiscase,messagescanbecarriedfromalmostanynodeofthenetworktoalmostanyothernode.Moreprecisely,messagescanbeexchangedbetweenanytwonodesthatbelongtotheinÞniteclusterof.ThetransmissionlatencywithrespecttothedistancebetweensourceanddestinationnodesisstudiedindetailinSection5.Thecasewhereishighenoughtokeepthenetworksuper-criticalatalltimeswillbeaddressedinfutureresearch.5.LATENCYAssumethatthenodesensingtheincomingeventisplacedattheorigin.Thisnodestartssendinganalarmmessageattime0.Alltheactivenodes,withintransmissionradius2orless,receivethemessageandtheyalsobegintobroadcastovertheirowntransmis-sionareas.Assumingnopropagationdelays,at0themessagehasspreadtotheclustercontainingtheorigin(seeFigure4a).Asleepingnodeinsidethesetthatwasalreadycoveredbythebroad-castmessageandthatchangesitsstateiscalledabridgeAfterthebridgehasstarteditsbroadcast,allthenewactivenodeswhore-ceivethemessage,eitherdirectlyfromthebridgeorviaamultihoppath,areaddedtotheoriginalcluster(seeFigure4b).Iftheori-ginbelongstotheinÞniteclusteroftheprocesscontinuesforever,otherwiseitstopsafterÞnitelymanysteps. Figure4:Messagespreadingisshowningrayarea.Inthispicturethediskradiiare.Whitenodesaresleeping,blacknodesactiveandabridgeisindicatedbythebox.5.1LinearspreadingofalarmmessagesInthissectionwestateourmainresult.AssumethatalarmmessagesaretransmittedoverablinkingPoissonBooleanmodel.Rememberthatmeansassumingaconstanttransmis-sionradius2foreachsensor.First,weshowthatifanalarmoc-cursinarandomplaceinsidetheinÞniteconnectedcomponentof,thelatencyisasymptoticallylinearwithrespecttothedis- tancetothesink.Secondly,themaximumdistancefromwherethealarmmessagecanbeheardisalsobehavingasymptoticallylin-early.Theseresultsallowustodimensionthetransmissionpartofasensornetwork.Firstofall,theygiveustoolstotuneupthealter-nationbetweensleepandactivephasesothatthespeedofalarmdetectionmeetsthepredeÞnedrequirements.Inaddition,ifonewantstoanalyzetheinterferenceduetoseparatealarmsappearingaboutthesametime,maximummessagedistancecanbeusedtoapproximatetheareawherethemessagescollide.Assumethattwonodes,locatedat,belongtotheinÞniteclusterof.Denotebythetimeittakestotransmitanalarmmessagefrom.TheÞrstpartofTheorem1showsisasymptoticallylinearindistance.Inthesecondscenario,weÞxonlythesource,anddenotebynodeswhichhavereceivedthemessageattimeiscalledthemessagecluster.Thenmaxthemaximaltransmissiondistance.Thisvalueisshowntobealsoasymptoticallylinear.Iftwonodes,locatedatXandY,belongtothein-niteclusterofthePoissonBooleanmodel,resultingfromablinkingPoissonBooleanmodelandZstationaryon/offprocesswithexponentialorconstantoff-times,thenthereisanitestrictlypositiveconstantsuchthat foranywheneverislargeenough.Moreover,therearenitestrictlypositiveconstants and suchthat maxGXt t whenevertislargeenough.Wewillprovethistheoreminthetwofollowingsubsections.ThelinearityofthetransmissiontimebetweenagivenpairofpointsisbasedonLiggettÕssubadditiveergodictheorem.Thelineargrowthofthemessageclusterisprovedbycouplingitwithacontinuumgrowthmodel.ItisimportanttonoticethattheconstantonlydependsontheparametersofthenetworkÑnamelyoffÑbutnotontherandomdispositionofthenodes.Thisvaluecanthusbeesti-matedbysimulation,givenasetofparameters,andusedtopredicttheperformanceofthenetworkbeforeitsdeployment.5.1.1ProofofEquation(3)WeconsiderÞrstpassagepercolationintherandomgraphdeter-minedby.Assumethattherandomvariablesofff0
t]Z0=0
i.e.,Tiisthetimeuntilanodeinsidethemessageclusterturnsac-tive.Ifneeded,thepropagationdelayscouldalsobeincludedintherandomvariables.Toeachorientededgeoftherandomgraph,weattachatimecoordinate(orÒdelayÓ).Itiseasytocou-plethemodelsinsuchawaythatthemessagetransmissiontimeinablinkingPoissonBooleanmodelandtheÞrstpassagetimeintheweightedgraphareequal,i.e.,isanarbitrarypathjoining.However,no-ticethattheotherpaths,exceptthefastestconnection,mayariselaterinthestaticdelaymodel.Intheblinkingmodel,whenanodeturnsactiveinsidethemessageclustermorethanonenewlinkcanjointheconnectivitygraph.Withoutlosinggenerality,weconsidertheÞrstpassagepercola-tioninthedirectionofthe-axis.Forany,wedenotetheindexofthenearestnodeintheinÞniteclusterargmin.Let(seeFigure5).NextdeÞnethecollectionofindexedvariablesbyforsomeconstantUsingLiggettÕssubadditiveergodictheorem(Theorem2),wecanprovethefollowingproposition.ROPOSITION n=limnT0
n where[16,Liggett’ssubadditiveergodictheorem]Letbeacollectionofrandomvariablesindexedbyintegerssat-n.Supposehasthefollowingproperties:(i)T(ii)Foreachn,cnforsomecon-stantc(iii)Thedistributionofdoesnotdependonm.(iv)Foreachkisastationarysequence.(b)Tnexistsa.s.Furthermore,ifkareergodic,then(d)TistheÞrstpassagetimefrom,andthepassagetimefrom,itisclearthatisatmost.ConditionisthusveriÞed. ~T(0,y)~T(x,y)~X(x)~X(y)~X(0)~ Figure5:Firstpassagepercolationpaths.Theroutescorre-spondtothefastestpathsbetweenpointsAsaÞrstpassagetimecannotbenegative,wehave.Tocomputeanupperboundof,weconsiderthe xd3d/2d/2y Figure6:Agoodrectangleassociatedwithahorizontaledge.Itcontainsaleft-rightcrossingandatop-bottomcrossingsinbothsub-squaresatitsendsshortestpath(indistance)from.Asub-optimalstrategyistofollowthispath,andwaitateachstepthatthenexthopbecomesactive.Intheworstcase,themessagehastowaitinaverageoffsecondsateachstep.Wehavethusoffdenotesthelengthinhopsoftheshortestpath.Proposition4willensurethatthelat-terexpectedvalueisalwaysÞnite,andthusthatConditionveriÞed.Toprovethisproposition,weneedthefollowingtwopre-liminarylemmas.(Seee.g.[12]p.295)Inanindependentbondper-colationmodelofopenedgedensity,wedenotebyAeventthatthereexistsanopenpathintherectangleRR0
4n][0
n]thatjoinsitsleftandrightborders(left-rightcrossing).Thereexistconstantssuchthat,wedenotebyCtheeventthattheori-ginissurroundedbyanoccupiedcircuit(i.e.acircuitentirelyin-cludedintheoccupiedregion)thatiscontainedintheframeFFn
n][n
n])([n2
n2][n2
n2]).Thereexistcon-suchthatROOF.WeusearenormalizationargumenttomaptheBooleanmodeltoadiscretemodel.Westartbyconstructingasquarelatticeovertheplane,withedgelength.Foreachedgeofthelat-tice,with,weconsidertherectanglerectanglex1d4
y1+d4][x2d4
y2+d4],asdepictedinFigure6.Wecallahorizontalrectangleifthereexistinanopenclusterthatcrossesitfromlefttoright,anopenclusterthatcrossescrossesx1d4
x1+d4][x2d4
x2+d4]frombottomtotop,andanopenclusterthatcrossescrossesy1d4
y1+d4][y2d4
y2+d4]frombottomtotop.WedeÞneverticalrectanglesinthesameway,exceptthatweexchangeleft-rightwithtop-bottom.Asissupercritical,foranyprobability1,onecanchooselargeenoughsothatarectangleisgoodwithprobabilityatleast(see[18,Corollary4.1]).Wethendeclareanedgeifitissurroundedbyagoodrect-angle,andclosedotherwise.Weobtainthusadependentbondper-colationmodel.However,iftwoedgeshavenocommonvertex,theirstatesareindependent.Ourmodelisthusa1-dependentper-colationmodel,whichisknowntopercolateifislargeenough.Moreprecisely,onecanÞndaproductmeasureonthismodel,whereeachedgeisopenwithprobability÷2,thatisstochas-ticallydominatedbyour1-dependentmeasuremeasureWeassumewithoutlossofgeneralitythattheverticesofthelat-ticehavetheform,for.Inthebondpercolationmodel 2k Y Figure7:a)Aframethatsurroundstheorigin.Thefourcross-ingsformacircuit.b)Theshortestpathfrom.Itismadeofachainofballs,thatcannotoverlapanyotherballthantheirpredecessorandsuccessor.withproductmeasure,welookforleft-rightcrossingsintherect-rect-k
k][k
k2]and[k
k][k2
k](theiractualsizeisthus22)forsome.TheprobabilitythatsuchacrossingexistsisgivenbyLemma1.Thesameistruefortop-bottomcrossingsinink
k2][k
k]and[k2
k][k
k].Theprobabilitythatthereisacrossingineachofthesefourrectanglessimultaneouslyisboundedby4crossingsAsthesecrossingsoverlapateachcorner,theoriginissurroundedbyanopencircuitthatiscontainedinink
k][k
k])([k2
k2][k2
k2])(seeFigure7a).Sincetheproductmeasureisdominatedbythe1-dependentmeasure,thecrossingsappearwithhigherorequalprobabilitythanintheindependentcase.Therefore,4crossings4crossingsMoreover,ourconstructionissuchthattheexistenceofanopencircuitinthediscretemodelimpliesthatthatkd
kd][kd
kd])([kd2
kd2][kd2
kd2])containsacomponentthatsurroundstheorigin,whichistheeventwewant.TheÞnalresultisobtainedbylettingin(5). ROPOSITIONIfXandYaretwopointsofaBooleanmodel,locatedatnitedistance,thatbelongtothesamecluster,andletbethenumberofhopsintheshortestpathbetweenthem.Thenisnite.ROOF.Weassumewithoutlossofgeneralitythat,andconsideraframe,asdeÞnedinLemma2,with .Ifthisframecontainsanoccupiedcircuit,thentheshortestpathfromisincludedinthesquaresquaren
n][n
n].Ontheotherhand,theshortestpathismadeofachainofballsofradius,anditisimpossiblethataballoverlapsmorethantwootherballsofthepath.Otherwise,onecouldremoveoneballandshortenthepath.Therefore,ahopspathmustcontainatleastdisjointballs(seeFigure7b).Thesurfaceoccupiedbythepathisthusatleast2.Asthepathiscontainedinasquareofsurface4,thelengthofthepathcannotexceed r2There-fore,PL8n2 .CombiningthiswithLemma2gives PL8n2 Thus,wecanÞndlargeenoughsothatfor k8e r WecanÞnallyupperboundtheexpectedvalueofthelengthoftheshortestpath: k8e r k8 areclearlyveriÞed,asisdeÞnedinastationaryway.Thefollowinglemmaistoprovethatthese-isergodic.Infact,weshowthatitismixing(i.e.,roughlyspeaking,asymptoticallyindependent),whichisastrongerproperty.Thesequenceismixing.ROOF.Wecomputebythefollowingconstruction:weconsiderthesquareofedgelengthcenteredattheorigin.Wedenotebythelargestoccupiedconnectedcomponentof,andbyargmintheclosestpointto.WethendeÞneasthetransmissiontime.WeobservethatwhengoestoinÞnity,issupercritical,thelargestoccupiedcomponentin.Moreover,asaconsequenceofProposition4,theshort-estpathbetweenisÞnite.Thuswehavethatalmostsurelyandtherefore,Weconsidernowthetranslationoverthevector.Clearlywehave.Similarly,wedeÞne.ThesamepropertyistrueforthetranslatedvariablesandforcombinationsoftwoeventsFinally,wecanshowthatthesequenceismixingbyset-Thesecondequalityfollowsfromthefactthatareindependent,astheydependontherealizationoftheblinkingBooleanmodelontwodisjointsquares. NowwehaveseenthatsatisÞesalltheconditionsofThe-orem2andthusprovedProposition3.Proposition5presentedinthenextsectionensuresthatFinally,weshouldshowthatdoesnotplayanyroleasymptoticallyandthatthediscretelimitcanbereplacedbyacontinuousone.Althoughbothclaimsarequiteevident,wegiveashortsketchoftheirproof.4.lim ROOF.(Sketch)Forindependentbondpercolationwith2,thereexists0suchthat(seee.g.[13]).UsingBorel-CantelliandthesamemappingfromPoissonBooleanmodeltodiscretepercolationasinLemma2,showsthat0a.s.Considerrationalanddenote Tt=÷T(0
.Then ,since isasubsequenceof whichconvergestobyProposition3.AlsobyProposition3, a.s.Thus a.s.Since Tnk ,theyhavethesamelimit,i.e.,.Thus tnx=(1)=a.s. 5.1.2ProofofEquation(4)ThelowerboundinInequality(4)followsdirectlyfrom(3).Thenextpropositiongivestheupperbound.Assumingasourceattheorigin,letdenotethenodeswhichhavereceivedthemessageattheareacoveredbythesenodesifadiskofradius2isattachedintoeachofthem.Naturally,ROPOSITIONConsidertheblinkingPoissonBooleanmodel.If,thenthereissuchthatalmostsurely forallsufcientlylarget.Theclaimisprovedbyshowingthatcanbeboundedabovebyacontinuumgrowthmodel(seeAppendixA)whichisdrivenbyaPoissonpointprocesswithintensityoffoffandanexponentiallyboundeddisksizedistribution.0.Attimeepochsoneoftheoff-nodesinside,denotedby,changesitsstateandgrows.determinesthenewareawhichreceivesthemessageattime(showninlightgrayinFigure8).WewillÞrstshowthatcanbeboundedbydiskswithi.i.d..ThisfollowsfromtheclustersizedistributioninasubcriticalregimeofaPoissonBooleanmodel.denotetheeventthatsetsintersectthesamecluster.[18,Theorem2.4]ConsiderPoissonBooleanmodelwhereRsatisesrforsomer.AssumingthenthereexistpositiveconstantsC,dependingonandthedimensiond,suchthatexpforanarbitraryboundedsetS.,thenCa.s.,whereDarei.i.d.withexpforsomepositiveconstantsCandC Figure8:Conditioned(lightgray),unconditionedincrements(blackcircles)andthecouplingdisk(thelargestcircle).TheradiiofthedisksareROOF.Letbeanarbitrarypointprocess.ForaBooleanmodelwithdiskscenteredaccordingtoandradii,letdenotetheunionoftheoccupiedclustersintersectingsetthesetwherethemes-sageisheard.growsat,allthepossiblenewpoints,exceptthe,areoutside.IntheactivenodesaredistributedaccordingtoastationaryPoissonpointprocesswithintensitywhichwedenoteby.ThusThissetcanbedrawnwithoutanyinformationaboutnodesinThustheincrementscanbedeterminedusingasequenceofi.i.d.Poissonprocesses.Moreover,sinceneglectingtheconditionswithrespectgivesnaturallyalargerset(seeFigure8).Finally,let))+sothatApplyingLemma5yieldsthatsatisÞes(6). NextweshowthattheprocessindicatingwhenandwhereabridgeappearscanbestochasticallyboundedbyaPoissonprocess.Inotherwords,thePoissonpointprocessincludesallthebridges(plusinÞnitelymanymore),itpreservestheopeningorder,andeachbridgeopensearlierthanitwasoriginallyscheduled.Ifthesleepingperiodsareeitherconstantorexpo-nentiallydistributed,thenthebridgeprocesscanbecoupledwithastationaryPoissonpointprocessinofintensityoffoffROOF.Letdenotethebridgesinwithopeningtimes.Attime0,thearedistributedaccordingtoPoissonpointprocesswithintensityoff.Ifoff,thenthen0
toffaredistributedaccordingtoaPois-sonpointprocesswithintensityoffoff.Thusiftheareuni-formlydistributedthenweknowthattheyallappearaccordingtoaPoissonprocessonon0
toff.Ontheotherhand,iftheexponentiallydistributed,theycanbecoupledwiththeoffoffandthuswiththePoissonpointprocesswithintensityoffoffThesamereasoningholdsforanynewsetaddedtotheclus-ter. ROOF.(ForProposition5)ByLemma6,weboundeachnewsetaddedtothemessageclusterbydiskswithi.i.d.radiiwithcumu-lativedistributionsatisfying(6).ByLemma7,eachofthebridgesoriginatinganincreaseofthemessagesetareincludedinaPois-sonpointprocessinwithintensityoffoff.ThusapplyingTheorem3(showninAppendixA)completestheproof. 5.2DurationofatransmissionphaseWhenanodereceivesorgeneratesamessage,itkeepstransmit-tinguntilitsneighborshavereceivedthemessage.Fromanen-ergyconsumptionpointofview,itisimportanttoknowhowlongthistransmittingphasewilllast.Assumingthatthenodeiscon-nectedtotheinÞnitecluster,thenÑinprincipleÑitwouldbeenoughthattheÒoptimallyÓlocatedneighboringnodesreceivethemessage.However,inordertomaximizethespeedatwhichmes-sagestravel,wedeÞnethebroadcastdurationasthetimeuntilalltheneighboringnodeshavereceivedthemessagewithprobability.Furthermore,weassumeherethattheemitterignoresitsneigh-borhoodandreceivesnofeedbackfromthereceivers.Althoughitwouldbeeasytoachievebetterconditionsbydesigninganappro-priateprotocol,weonlyconsiderthissimplemechanism.Giventhetransmissionradiusofthesensors,thereisaPoissondistributednumberofsensorsinsideitstransmissionrange.Inaworstcasescenario,weassumethatallthenodesareinsleep-ingmodeatthetimewhenthebroadcaststarts.Ifthesleepingpe-riodsareofconstantlength,thenthenaturaland100%safebroad-castingdurationisoff,whereisthepropagationdelay.Oth-erwise,weassumethattheoffperiodsoftheneighbours,,areexponentiallydistributedandInordertobesurethateverynodereceivesthemessage,thebroadcastingtimehastosatisfy offexpoffThus,ifthebroadcastingtimeoff thenitissurethateverynodeinitsneighborhoodgetsthemes-5.3SimulationstudiesWehaveperformedaseriesofsimulationstovalidatetheresultsconcerninglatencypresentedinSection5.1.Thesenumericalstud-iesclearlyagreewiththeanalyticalresultsonthelinearspreadingrate.AlthoughTheorem1isstatedonlyforconstanttransmissionradii,thesimulationssuggestthatitisalsovalidforrandomtrans-missionradii.Allthesimulationswererunwithintensity3.ThesleepingperiodsweredrawnfromanexponentialdistributionwithmeanoffThesizeoftheareawhichisreachedbythemessageattimemeasuredbymaxargmaxandthespreadingrateisestimatedby SimulationsofthegrowthofthemessageclusterwithvaryingperiodlengthsoffareshowninFigures9and10.Inprin-ciple,onlytheratiosoffoffoffmatter.Byasimpletimescalingargumentoff)canbetakenasthetime 5 10 15 20 25 30 35 40 50 75 100 125 150 175 200
Gt
0.3t 10 15 20 25 30 35 40 5 7.5 10 12.5 15 17.5 20  t1
0.3t1
0.2t1
0.1 Figure9:Maximumconnectiondistancefromtheoriginfordifferentactiveperiods.Tenindependentsimulationswithpa-off,and 0.1 0.2 0.3 0.4 0.5 10 15 20 25 30 35 40 t1 2 4 6 8 10 10 15 20 t1 Figure10:Estimationofgrowthrates.Parameters100independentsimulationsperestimate.unit.Thismapping,ofcourse,changestheactualvalueof.ForÞxedoff,decreasingthelengthsoftheactiveperiodsdoesnotre-allyworsentheperformancewhenissmallenough.Thisisnatural,sinceforverysmallvaluesof,almostallthesensorsaresleepingwhenthemessagearrivesintheirrangesothatthetimetowaitforawakeupdeterminesthetransmissionspeed.isÞxed,increasingoffnaturallydecreasesthemes-sagespreadingvelocity.However,offisthedeterminantfactorforenergysaving,especiallyifswitchingontheradioiscostly.There-fore,thelowerpartofFigure10presentstherealtrade-offinthismechanism.Weobservethatthecurvedecreasesveryfastatthebeginning,meaningthatallowingthenodestosleepforsometimecostsalotoflatency.Butafterward,increasingthesleepingperiodhaslessimpact,asthecurvebecomesßatter. 5 10 15 20 35 50 75 100 125 150 175 200
Gt
r
1r
U0.5,1.5 10 15 20 25 4 6 8 10  r
1r
U0.5,1.5 Figure11:Maximumconnectiondistancefromtheoriginforthemodelswithconstantordistributedradii.Tenindependentsimulationswithparametersoff 0.1 0.2 0.4 40 60 80 100 t1
U0.5,1.5r
1 Figure12:Modelswith.Es-timatedmaximumgrowthrates.ParametersoffTheeffectofallowingrandomradiicanbeseeninFigures11and12.Themaindifferencebetweenconstantandrandomradii modelsisthatinthelatterlongerjumpsarepossible.Theotherdif-ferenceisthatthemeancoveredareaislargerintherandomcase,.Thisnaturallyresultsinbetterconnectiv-ity.Howeverweobservethatthequalitativebehaviorissimilar.Actually,mostoftheproofspresentedinthispapercanbeeasilyextendedtotherandomradiimodel,andweconjecturethatallourresultsholdforthismodel. 5 10 15 20 25 30 35 40 50 75 100 125 150 175 200
Gt

0
0.05
0.5
5 20 30 40 4 6 8  
0
0.05
0.5
5 Figure13:Modelwithpropagationdelays.Maximumconnec-tiondistancefromtheorigin.Tenindependentsimulationswithoff 0.1 0.2 0.3 0.4 0.5 20 30 40 t1 0
0.05
0.5
5 Figure14:Modelwithpropagationdelays.Estimatedgrowthrates.Parametersoff,delays0.05,0.5and5.Estimatesbasedon100independentsimulations.Figure13showstheimpactofpropagationdelaysonthemes-sagespreading.Theasymptoticbehaviorisqualitativelyconserved,aspredicted.Thequantitativeimpactisalsotiny,ifweconsidertherealisticcasewheredelaysareshortcomparedtooff6.CONCLUDINGREMARKSTheblinkingPoissonBooleanmodelissuitedforsensornet-workswherenodesswitchbetweenasleepingandanactivephase.Eventhoughtheirswitchingon/offschedulesarenotcoordinatedatall,theirpositionsarerandom,andthedurationsoffsuchthatthenumberofactivenodesatanyparticulartimeissolowthatthenetworkisalwaysdisconnected,wehaveprovedthatanymessage(alarm)generatedbyasensorwillreachthesinkinatimeproportionaltothedistancebetweenthesensorandthesink.Thevalueoftherateofthislineargrowthdoesnotdependontheran-domlocationsofthenodes,butonlyontheparameters(connectivityrange),off(averageactiveandsleepingdurations).Inthispaper,wehaveonlyconsideredthecasewhere,atanyparticulartime,thenetworkisdisconnected(sub-criticalphase).Theothercase,wherethenetworkremainsinasupercriticalphaseatanytime,isalsointeresting.Indeed,thelatencywillbemuchsmallerthaninthesubcriticalcase,butitisnotclearhowtoboundit.Weleavethispartforfuturestudy.7.ACKNOWLEDGMENTSO.DousseÕsworkwassupported(inpart)bytheNationalCom-petenceCenterinResearchonMobileInformationandCommu-nicationSystems(NCCR-MICS),acentersupportedbytheSwissNationalScienceFoundationundergrantnumber5005-67322.ThisworkwasdonewhileP.MannersalowasvisitingEPFLasanERCIMfellow.HeisgratefulforthegenerousÞnancialsupportofSwissNationalScienceFoundation(PAER2-101377),EPFLandAcademyofFinland(#202218).8.REFERENCES[1]F.Baccelli,B.Blaszczyszyn,andF.Tournois.Spatialaveragesofcoveragecharacteristicsinlargecdmanetworks.WirelessNetworks,8:569Ð586,2002.[2]J.vandenBerg,R.Meester,andD.G.White.DynamicBooleanmodels.StochasticProcessesandtheir,69:247Ð257,1997.[3]J.-H.ChangandL.Tassiulas.Energyconservingroutinginwirelessad-hocnetworks.InProceedingsofIEEEInfocom,pages22Ð31,Jerusalem,2000.[4]C.-F.ChiasseriniandM.Garetto.Modelingtheperformanceofwirelesssensornetworks.InProceedingsofIEEEInfocom,2004.[5]M.Deijfen.Asymptoticshapeinacontinuumgrowthmodel.AdvancesinAppliedProbability,35(2):303Ð318,2003.[6]M.Deijfen,O.HŠggstršm,andJ.Bagley.AstochasticmodelforcompetinggrowthonMarkovProcessesandRelatedFields,2003.Toappear.[7]O.Dousse,F.Baccelli,andP.Thiran.Impactofinterferencesonconnectivityinadhocnetworks.IEEE/ACMTrans.,acceptedforpublication.[8]A.ElGamal,C.Nair,B.Prabhakar,E.Uysal-Biyikoglu,andS.Zahedi.Energy-efÞcientschedulingofpackettransmissionsoverwirelessnetworks.InProceedingsofIEEEInfocom2002,pages1773Ð1782,2001.[9]A.Ephremides.Energyconcernsinwirelessnetworks.WirelessCommunications,9(4):48Ð59,2002.[10]M.Franceschetti,L.Booth,J.Bruck,M.Cook,andR.Meester.Percolationinmulti-hopwirelessnetworks.IEEETransactiononInformationTheory,2003.Submitted.AshortversionappearedatISITÕ03.[11]D.Ganesan,B.Krishnamachari,A.Woo,D.Culler,D.Estrin,andS.Wicker.Complexbehavioratscale:Anexperimentalstudyoflow-powerwirelesssensornetworks.TechnicalReportUCLA/CSD-TR02-0013,UCLA,2002. [12]G.Grimmett.Percolation,volume321ofGrundlehrendermathematischenWissenschaften.Springer,Berlin,2ndedition,1999.[13]G.GrimmettandJ.Marstrand.Thesupercriticalphaseofpercolationiswellbehaved.Proc.RoyalSoc.LondonSer.A430:439Ð457,1990.[14]O.HŠggstršm,Y.Peres,andJ.Steif.Dynamicpercolation.Ann.IHPProbab.et.Statist.,33:497Ð528,1997.[15]B.Krishanamachari,D.Estrin,andS.Wicker.Theimpactofdataaggregationinwirelesssensornetworks.InProc.ofInternationalWorkshopofDistributedEventBasedSystems,Vienna,Austria,2002.[16]T.Liggett.Animprovedsubadditiveergodictheorem.AnnalsofProbability,13(4):1279Ð1285,1985.[17]B.LiuandD.Towsley.Onthecoverageanddetectabilityofwirelesssensornetworks.InProceedingsofWiOpt’03:ModelingandOptimizationinMobile,AdHocandWireless,SophiaAntipolis,France,2003.[18]R.MeesterandR.Roy.Continuumpercolation,volume119Cambridgetractsinmathematics.CambridgeUniversityPress,1996.[19]Y.PeresandJ.Steif.ThenumberofinÞniteclustersindynamicalpercolation.Probab.Th.Rel.Fields111:141Ð165,1998.[20]B.Prabhakar,E.Uysal-Biyikoglu,andA.ElGamal.Energy-efÞcienttransmissionoverawirelesslinkvialazypacketscheduling.InProceedingsofIEEEInfocom2001pages386Ð394,2001.[21]J.Quintanilla,S.Torquato,andR.M.Ziff.EfÞcientmeasurementofthepercolationthresholdforfullypenetrablediscs.J.Phys.A,33:399Ð407,2000.[22]S.Singh,MWoo,andC.S.Raghavendra.Power-awareroutinginmobileadhocnetworks.InProc.ofACMMobicom1998,pages181Ð190,1998.[23]A.SinhaandA.Chandrakasan.Dynamicpowermanagementinwirelesssensornetworks.IEEEDesignandTestofComputersMagazine,18(2):62Ð74,2001.[24]D.Stoyan,W.S.Kendall,andJ.Mecke.Stochasticgeometryanditsapplications.Wiley,Chichester,2ndedition,1995.[25]T.VicsekandJ.KertŽsz.MonteCarlorenormalisationgroupapproachtopercolationonacontinuum:Testofuniversality.J.Phys.A,14(L31),1981.[26]Y.Wei,J.Heidemann,andD.Estrin.Anenergy-efÞcientMACprotocolforwirelesssensornetworks.InProc.IEEE,NewYork,June2002.[27]W.YeandJ.Heidemann.Mediumaccesscontrolinwirelesssensornetworks.TechnicalReportISI-TR-580,USC,2003.[28]Y.Yu,B.Krishnamachari,andV.Prasanna.Energy-latencytradeoffsfordatagatheringinwirelesssensornetworks.InProceedingsofIEEEInfocom2004,2004.A.CONTINUUMGROWTHMODELContinuumgrowthmodelisamodelforaspreadinginfection.Assumingthataninfectionhasspreadtosetattime,thetimeuntilthenextoutburstoccurssomewhereinisexponentiallydis-tributedwithparameterandthelocationoftheoutburstisuniformlydistributedover.Outburstareassumedtobeballswithi.i.d.randomradii.ThismodelwasÞrststudiedbyDeijfen[5](seealsoDeijfen,HŠggstršmandBagley[6]).Formally,ContinuumgrowthmodelisdeÞnedasfollows.WestartwithastationaryPoissonpointprocess.ThePoissonprocesshasintensityandthepointslayinginsideasetaredenotedby.Foreachweattachaballwhichiscenteredathasarandomradius.Assumetheradiii.i.d.withacommoncumulativedistributionfunction

















































Figure15:Continuumgrowthmodelinisshownbytheshadedarea.Letusenumeratepointsofasfollows.Let,and.Given

Foreachthereis(a.s.unique)andarandomradiusContinuumgrowthprocessisaMarkovprocesswhichisconstructedfromthesequenceofthe.Figure15showsanexamplewhereForourpurposes,themainpropertyofContinuumgrowthmodelisthatthesizeoftheinfectedareagrowsasymptoticallylinearly.[6]Fixdandconsiderthed-dimensionalcontinuumgrowthmodelwithrate.AssumethatforsomeandletSbearbitrarybutboundedwithstrictlypositiveLebesguemeasure.Thenthereexistsarealnumbersuchthatforany,almostsurely forallsufcientlylarget.Moreover,thetimeconstantisgiven n=limn÷T(n) where