Revisiting an old friend_ on the observability of

TelecommunSyst(2010)43:147–165DOI10.1007/s11235-009-9205-6Revisitinganoldfriend:ontheobservabilityoftherelationbetweenlongrangedependenceandheavytailPatriceAbry·PierreBorgnat·FabioRicciato·AntoineScherrer·DarrylVeitchPublishedonline:21October2009©SpringerScience+BusinessMedia,LLC2009AbstractTaqqu’sTheoremplaysafundamentalroleinIn-ternettrafficmodeling,fortworeasons:First,itstheoreti-calformulationmatchescloselyandinameaningfulman-nersomeofthekeynetworkmechanismscontrollingtraf-ficcharacteristics;Second,itoffersaplausibleexplanationfortheoriginofthelongrangedependencepropertyinre-lationwiththeheavytailnatureofthetrafficcomponents.Numerousattemptshavesincebeenmadetoobserveitspre-dictionsempirically,eitherfromrealInternettrafficdataorfromnumericalsimulationsbasedonpopulartrafficmodels,yetrarelyhasthisresultedinsatisfactoryquantitativeagree-ments.Thisraisedintheliteratureanumberofcommentsandquestions,rangingfromtheadequacyofthetheoremtoP.Abry·P.Borgnat()·A.ScherrerCNRS,LaboratoiredePhysiqueENSLyon,UniversitédeLyon,46alléed’Italie,69007Lyon,Francee-mail:[e-mail]-mail:[e-mail]-mail:[e-mail]nto,Lecce,ItalyF.RicciatoForschungszentrumTelekommunikationWien(FTW),Vienna,Austriae-mail:[e-mail]CentreforUltra-BroadbandInformationNetworks(CUBIN),anaffiliatedprogramofNationalICTAustralia(NICTA),Dept.ofE&EEngineering,UniversityofMelbourne,Victoria,Australiae-mail:[e-mail]vanceofthestatisticaltoolsin-volvedinpracticalanalyses.Thepresentcontributionaimsatstudyingunderwhichconditionsthisfundamentaltheo-remcanbeactuallyseenatworkonrealorsimulateddata.Todoso,numericalsimulationsbasedonstandardtrafficmodelsareanalyzedinawaveletframework.Thekeytimescalesinvolvedarederived,enablingadiscussionoftheori-ginandnatureofthedifficultiesencounteredinattemptstoempiricallyobserveTaqqu’sTheorem.KeywordsHeavytail·Longrangedependence·Taqqu’sTheorem·Waveletanalysis·Scalesoftime·Internettrafficmodels1MotivationMorethanadecadeofresearchworkshaveshownthatmod-elingthestatisticalpropertiesofInternettrafficischalleng-ing.Indeed,traffictracesarecharacterizedbynontrivialstatisticalproperties,thetwomostprominentbeing:LongRangeDependence(LRD),orasymptoticself-similarity,ofthetime-seriescountingthe‘volume’ofcommunicationovertime(aggregatedtrafficcounts)[1–4];Heavy-tailness(HT)ofimportanttrafficstatisticssuchasthesizeoftheobjectssentthroughtheInternet(betheyimages,movies,)[5,6],orthesizeofcomputerac-tivitysessions,measuredeitherbydurationorinpacketorbytecount,orflowlengths[7–9].FromthefirstdisclosuresofLRDinInternettraffic[1],mechanismsexplainingitsori-ginhavebeensought.AtheoremduetoTaqquandcollaboratorsrelatedLRDandHTviaaninfinitesuperimpositionofOn/Offprocesses[1,2,10].Itwasimmediatelyrecognizedtoplayafunda-mentalroleforInternettrafficmodelingasitstheoretical148P.Abryetal.formulationcloselymatchedquantitieswhicharemeaning-fulinnetworkingterms,suchasflowsizes.Furthermoreus-ingobjectswithHTsizedistributionsastheinputofsomequeueingsystemyieldsoutputtraffictimesserieswithLRD.Moreover,thetheoremisappealingasitprovidesasimpleclosedformrelationH(α)=(3−α)/2betweentheLRDandHTparameters,respectivelyHandα.ApreciseanddetailedformulationispostponedtoSect.2.Thetheoreticalrelationabovepavedthewaytowardalargenumberofworkswhosegoalwasitsempiricalvalida-tion.Prominently,seminalworksonthisquestion[5,6]ob-tainedresultsinpartialandqualitativeagreementwiththetheoreticalprediction,firstforfic.However,numerousdifficultiesarose:obtainingandstoringsufficientlylongtraffictraceswasnon-trivial,andthevaluesofparametersfoundinrealtraffic,beingfixedbythedata,donotcoverthefullrangeallowedforbytheory.Practitionersthenresortedtonumer-icalsimulationstogeneratesynthetictrafficbasedeitheronpopulartimeseriesmodelsorontrafficsimulatorssuchasNS-2.Thisofferedthepossibilityofcontrollingandtuningthetrafficparameters,whilepreservinganumberofrealis-ticfeaturesofactualtraffic.AcomplementaryissuewasthefactthatstatisticaltoolsforHTandLRDmeasurementandanalysishadtheirownproblems.Forinstance,LRDmea-suredviaR/Sorvariogramprocedureshavebeenshownsincetosufferfrommanydrawbacksandtohavepoorper-formance(responsible,forinstance,ofthebiasofFig.3in[5]).Currently,theuseofthewaveletmethodology,pro-posedin[3],fortheanalysisofLRDiswell-assessedanddocumented,andintensivelyused.Yet,again,validationhasturnedouttobefarfromstraightforwardandquantitativesatisfactoryagreementshaverarelybeenreported.Often,asignificantmismatchbetweenthewaveletbasedestimatedLRDexponentandtheHTindexareobserved(forexam-pleseethedetailedandmotivatingcontributionsin[11,12],andtheanalysisof[13]).Thisleadspractitionerstoraiseanumberofquestions:IstheTheoremmisunderstood?misin-terpreted?misused?IsthewaveletbasedLRDanalysispro-cedureincorrect?ornotoperatinginthisframework?Isthesimulationset-upatfault?ThegoalofthepresentcontributionisobviouslynottocallTaqqu’sTheore