StochasticGeometryandWirelessNetworksVolumeIIApplications.pdf

StochasticGeometryandWirelessNetworksVolumeIIApplications.pdf

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StochasticGeometryandWirelessNetworksVolumeIIAPPLICATIONSFrancoisBaccelliandBartomiejBaszczyszynINRIA&EcoleNormaleSuperieure,45ruedUlm,Paris.Paris,December,2009.ThismonographisbasedonthelecturesandtutorialsoftheauthorsatUniversiteParis6since2005,Eu-random（Eindhoven,TheNetherlands）in2005,Performance05（JuanlesPins,France）,MIRNUGEN（LaPedrera,Uruguay）andEcolePolytechnique（Palaiseau,France）in2007.Thisworkingversionwascom-piledDecember4,2009.ToBeatriceandMiraiPrefaceAwirelesscommunicationnetworkcanbeviewedasacollectionofnodes,locatedinsomedomain,whichcaninturnbetransmittersorreceivers（dependingonthenetworkconsidered,nodesmaybemobileusers,basestationsinacellularnetwork,accesspointsofaWiFimeshetc.）.Atagiventime,severalnodestransmitsimultaneously,eachtowarditsownreceiver.Eachtransmitterreceiverpairrequiresitsownwirelesslink.Thesignalreceivedfromthelinktransmittermaybejammedbythesignalsreceivedfromtheothertransmitters.EveninthesimplestmodelwherethesignalpowerradiatedfromapointdecaysinanisotropicwaywithEuclideandistance,thegeometryofthelocationsofthenodesplaysakeyrolesinceitdeterminesthesignaltointerferenceandnoiseratio（SINR）ateachreceiverandhencethepossibilityofestablishingsimultaneouslythiscollectionoflinksatagivenbitrate.Theinterferenceseenbyareceiveristhesumofthesignalpowersreceivedfromalltransmitters,exceptitsowntransmitter.Stochasticgeometryprovidesanaturalwayofdefiningandcomputingmacroscopicpropertiesofsuchnetworks,byaveragingoverallpotentialgeometricalpatternsforthenodes,inthesamewayasqueuingtheoryprovidesresponsetimesorcongestion,averagedoverallpotentialarrivalpatternswithinagivenparametricclass.Modelingwirelesscommunicationnetworksintermsofstochasticgeometryseemsparticularlyrelevantforlargescalenetworks.Inthesimplestcase,itconsistsintreatingsuchanetworkasasnapshotofastationaryrandommodelinthewholeEuclideanplaneorspaceandanalyzingitinaprobabilisticway.Inparticularthelocationsofthenetworkelementsareseenastherealizationsofsomepointprocesses.Whentheunderlyingrandommodelisergodic,theprobabilisticanalysisalsoprovidesawayofestimatingspatialaverageswhichoftencapturethekeydependenciesofthenetworkperformancecharacteristics（connectivity,stability,capacity,etc.）asfunctionsofarelativelysmallnumberofparameters.Typically,thesearethedensitiesoftheunderlyingpointprocessesandtheparametersoftheprotocolsinvolved.Byspatialaverage,wemeananempiricalaveragemadeoveralargecollectionoflocationsinthedomainconsidered;dependingonthecases,theselocationswillsimplybecertainpointsofthedomain,ornodeslocatedinthedomain,orevennodesonacertainroutedefinedonthisdomain.Thesevariouskindsofiiispatialaveragesaredefinedinprecisetermsinthemonograph.Thisisaverynaturalapproache.g.foradhocnetworks,ormoregenerallytodescribeuserpositions,whenthesearebestdescribedbyrandomprocesses.Butitcanalsobeappliedtorepresentbothirregularandregularnetworkarchitecturesasobservedincellularwirelessnetworks.Inallthesecases,suchaspaceaverageisperformedonalargecollectionofnodesofthenetworkexecutingsomecommonprotocolandconsideredatsomecommontimewhenonetakesasnapshotofthenetwork.Simpleexamplesofsuchaveragesarethefractionofnodeswhichtransmit,thefractionofspacewhichiscoveredorconnected,thefractionofnodeswhichtransmittheirpacketsuccessfully,andtheaveragegeographicprogressobtainedbyanodeforwardingapackettowardssomedestination.Thisisrathernewtoclassicalperformanceevaluation,comparedtotimeaverages.Stochasticgeometry,whichweuseasatoolfortheevaluationofsuchspatialaverages,isarichbranchofappliedprobabilityparticularlyadaptedtothestudyofrandomphenomenaontheplaneorinhigherdimension.Itisintrinsicallyrelatedtothetheoryofpointprocesses.Initiallyitsdevelopmentwasstimulatedbyapplicationstobiology,astronomyandmaterialsciences.Nowadays,itisalsousedinimageanalysisandinthecontextofcommunicationnetworks.Inthislattercase,itsroleissimilartothatplayedbythetheoryofpointprocessesonthereallineinclassicalqueuingtheory.Theuseofstochasticgeometryformodelingcommunicationnetworksisrelativelynew.Thefirstpapersappearedintheengineeringliteratureshortlybefore2000.OnecanconsiderGilbertspaperof1961（Gilbert1961）bothasthefirstpaperoncontinuumandBooleanpercolationandasthefirstpaperontheanalysisoftheconnectivityoflargewirelessnetworksbymeansofstochasticgeometry.Similarobservationscanbemadeon（Gilbert1962）concerningPoissonVoronoitessellations.Thenumberofpapersusingsomeformofstochasticgeometryisincreasingfast.Oneofthemostimportantobservedtrendsistotakebetteraccountinthesemodelsofspecificmechanismsofwirelesscommunications.TimeaverageshavebeenclassicalobjectsofperformanceevaluationsincetheworkofErlang（1917）.Typicalexamplesincludetherandomdelaytotransmitapacketfromagivennode,thenumberoftimestepsrequiredforapackettobetransportedfromsourcetodestinationonsomemultihoproute,thefrequencywithwhichatransmissionisnotgrantedaccessduetosomecapacitylimitations,etc.Aclassicalreferenceonthematteris（Kleinrock1975）.Thesetimeaverageswillbestudiedhereeitherontheirownorinconjunctionwithspaceaverages.Thecombinationofthetwotypesofaveragesunveilsinterestingnewphenomenaandleadstochallengingmathematicalquestions.Asweshallsee,theorderinwhichthetimeandthespaceaveragesareperformedmattersandeachorderhasadifferentphysicalmeaning.Thismonographsurveysrecentresultsofthisapproachandisstructuredintwovolumes.VolumeIfocusesonthetheoryofspatialaveragesandcontainsthreeparts.PartIinVolumeIprovidesacompactsurveyonclassicalstochasticgeometrymodels.PartIIinVolumeIfocusesonSINRstochasticgeometry.PartIIIinVolumeIisanappendixwhichcontainsmathematicaltoolsusedthroughoutthemonograph.VolumeIIbearsonmorepracticalwirelessnetworkmodelingandperformanceanalysis.Itisinthisvolumethattheinterplaybetweenwirelesscommunicationsandstochasticgeometryisdeepestandthatthetimespaceframeworkalludedtoaboveisthemostimportant.Theaimistoshowhowstochasticgeometrycanbeusedinamoreorlesssystematicwaytoanalyzethephenomenathatariseinthiscontext.PartIVinVolumeIIisfocusedonmediumaccesscontrol（MAC）.WestudyMACprotocolsusedinadhocnetworksandincellularnetworks.PartVinVolumeIIdiscussestheuseofstochasticgeometryfortheivquantitativeanalysisofroutingalgorithmsinMANETs.PartVIinVolumeIIgivesaconcisesummaryofwirelesscommunicationprinciplesandofthenetworkarchitecturesconsideredinthemonograph.Thispartisself-containedandreadersnotfamiliarwithwirelessnetworkingmighteitherreaditbeforereadingthemonographitself,orrefertoitwhenneeded.Herearesomecommentsonwhatthereaderwillobtainfromstudyingthematerialcontainedinthismonographandonpossiblewaysofreadingit.Forreaderswithabackgroundinappliedprobability,thismonographprovidesdirectaccesstoanemerg-ingandfastgrowingbranchofspatialstochasticmodeling（seee.g.theproceedingsofconferencessuchasIEEEInfocom,ACMSigmetrics,ACMMobicom,etc.orthespecialissue（Haenggi,Andrews,Baccelli,Dousse,andFranceschetti2009）.Bymasteringthebasicprinciplesofwirelesslinksandoftheorgani-zationofcommunicationsinawirelessnetwork,assummarizedinVolumeIIandalreadyalludedtoinVolumeI,thesereaderswillbegrantedaccesstoarichfieldofnewquestionswithhighpracticalinterest.SINRstochasticgeometryopensnewandinterestingmathematicalquestions.ThetwocategoriesofobjectsstudiedinVolumeII,namelymediumaccessandroutingprotocols,havealargenumberofvariantsandofimplications.Eachofthesecouldgivebirthtoanewstochasticmodeltobeunderstoodandanalyzed.Evenforclassicalmodelsofstochasticgeometry,thenewquestionsstemmingfromwirelessnetworkingoftenprovideanoriginalviewpoint.AtypicalexampleisthatofrouteaveragesassociatedwithaPoissonpointprocessasdiscussedinPartVinVolumeII.ReaderalreadyknowledgeableinbasicstochasticgeometrymightskipPartIinVolumeIandfollowthepath:

PartIIinVolumeIPartIVinVolumeIIPartVinVolumeII,usingPartVIinVolumeIIforunderstandingthephysicalmeaningoftheexamplespertainingtowirelessnetworks.Forreaderswhosemaininterestinwirelessnetworkdesign,themonographaimstoofferanewandcomprehensivemethodologyfortheperformanceevaluationoflargescalewirelessnetworks.Thismethod-ologyconsistsinthecomputationofbothtimeandspaceaverageswithinaunifiedsetting.Thisinherentlyaddressesthescalabilityissueinthatitposestheproblemsinaninfinitedomain/populationcasefromtheverybeginning.Weshowthatthismethodologyhasthepotentialtoprovidebothqualitativeandquantitativeresultsasbelow:

Someofthemostimportantqualitativeresultspertainingtotheseinfinitepopulationmodelsareintermsofphasetransitions.Atypicalexamplebearsontheconditionsunderwhichthenetworkisspatiallyconnected.Anothertypeofphasetransitionbearsontheconditionsunderwhichthenetworkdeliverspacketsinafinitemeantimeforagivenmediumaccessandagivenroutingprotocol.Asweshallsee,thesephasetransitionsallowonetounderstandhowtotunetheprotocolparameterstoensurethatthenetworkisinthedesi