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Modeling and Performance Analysis of Multimedia Traffic over Communication Networks

Permanent Link: http://ufdc.ufl.edu/UFE0043986/00001

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Title: Modeling and Performance Analysis of Multimedia Traffic over Communication Networks
Physical Description: 1 online resource (95 p.)
Language: english
Creator: Kim, Kyung Woo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: delay -- multimodality -- throughput
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Applications of multimedia traffic over various communication channels need to share physically limited bandwidth efficiently and at the same time guarantee Quality of Service (QoS). As the size of multimedia data increases to quarantee a high quality, transmission delay also increases and this results in poor QoS over communication networks. For guaranteed QoS and low transmission delay in communication networks which provides an integrated multimedia service, it is desirable to study the statistical characteristics of multimedia traffic and is important to obtain an analytical and tractable model of compressed MPEG data. This dissertation presents a statistical approach to obtain an MPEG frame size model and estimates throughput and transmission delay over communication networks. For the study of multimedia traffic characteristics, MPEG-2 frames are extracted from typical DVD movies. To obtain candidate distributions, a probability histogram based on the Freedman-Diaconis method, which is used as a decision rule for a bin size, is considered and both single distributions and a mixed type distribution are also taken into account. In the single distribution case, distribution parameters are obtained from empirical data using the maximum likelihood estimation (MLE) method. The best fitted model for the multimedia traffic studied was found to be a Lognormal distribution. However, with this single distribution, we cannot explain the inherent multimodality clearly observed in the empirical multimedia frame data. Thus a Hyper-Gamma distribution is considered as an alternative model to explain its inherent multimodality. The Hyper-Gamma distribution parameters are obtained by means of an expectation maximization algorithm based on the K-means algorithm and a posteriori probability. Furthermore, the Bayesian Information Criterion (BIC) is used as a goodness of fit criterion. Single probability distributions are also considered to demonstrate the superiority of the proposed model in fitting MPEG-2 frame data. This dissertation shows that the Hyper-Gamma distribution is a good candidate for a stochastic model for MPEG-2 frame data. After obtaining a statistical model for multimedia traffic, the Hyper-Gamma service time distribution has been applied to network systems which are composed of nodes with a finite and infinite capacity and connected in tandem. The throughput for the infinite capacity system is equal to the average arrival rate because of the assumption for a Poisson arrival process and Burke's theorem, meanwhile the total delay is represented in terms of the average arrival rate, and the first and second moments of the Hyper-Gamma service time distribution. But, in case of the finite capacity system, the throughput at each node is represented in a product form of the arrival rate to that node and one minus its blocking probability. Furthermore, the delay in each node depends on the total time spent in the previous node. Therefore, this dissertation predicts throughput and transmission delay for multimedia traffic from the estimated Hyper-Gamma service time distribution and this result will be useful in the performance analysis of network systems based on multimedia traffic.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kyung Woo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Latchman, Haniph A.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043986:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043986/00001

Material Information

Title: Modeling and Performance Analysis of Multimedia Traffic over Communication Networks
Physical Description: 1 online resource (95 p.)
Language: english
Creator: Kim, Kyung Woo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: delay -- multimodality -- throughput
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Applications of multimedia traffic over various communication channels need to share physically limited bandwidth efficiently and at the same time guarantee Quality of Service (QoS). As the size of multimedia data increases to quarantee a high quality, transmission delay also increases and this results in poor QoS over communication networks. For guaranteed QoS and low transmission delay in communication networks which provides an integrated multimedia service, it is desirable to study the statistical characteristics of multimedia traffic and is important to obtain an analytical and tractable model of compressed MPEG data. This dissertation presents a statistical approach to obtain an MPEG frame size model and estimates throughput and transmission delay over communication networks. For the study of multimedia traffic characteristics, MPEG-2 frames are extracted from typical DVD movies. To obtain candidate distributions, a probability histogram based on the Freedman-Diaconis method, which is used as a decision rule for a bin size, is considered and both single distributions and a mixed type distribution are also taken into account. In the single distribution case, distribution parameters are obtained from empirical data using the maximum likelihood estimation (MLE) method. The best fitted model for the multimedia traffic studied was found to be a Lognormal distribution. However, with this single distribution, we cannot explain the inherent multimodality clearly observed in the empirical multimedia frame data. Thus a Hyper-Gamma distribution is considered as an alternative model to explain its inherent multimodality. The Hyper-Gamma distribution parameters are obtained by means of an expectation maximization algorithm based on the K-means algorithm and a posteriori probability. Furthermore, the Bayesian Information Criterion (BIC) is used as a goodness of fit criterion. Single probability distributions are also considered to demonstrate the superiority of the proposed model in fitting MPEG-2 frame data. This dissertation shows that the Hyper-Gamma distribution is a good candidate for a stochastic model for MPEG-2 frame data. After obtaining a statistical model for multimedia traffic, the Hyper-Gamma service time distribution has been applied to network systems which are composed of nodes with a finite and infinite capacity and connected in tandem. The throughput for the infinite capacity system is equal to the average arrival rate because of the assumption for a Poisson arrival process and Burke's theorem, meanwhile the total delay is represented in terms of the average arrival rate, and the first and second moments of the Hyper-Gamma service time distribution. But, in case of the finite capacity system, the throughput at each node is represented in a product form of the arrival rate to that node and one minus its blocking probability. Furthermore, the delay in each node depends on the total time spent in the previous node. Therefore, this dissertation predicts throughput and transmission delay for multimedia traffic from the estimated Hyper-Gamma service time distribution and this result will be useful in the performance analysis of network systems based on multimedia traffic.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kyung Woo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Latchman, Haniph A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043986:00001


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MODELINGANDPERFORMANCEANALYSISOFMULTIMEDIATRAFFICOVERCOMMUNICATIONNETWORKSByKYUNGWOOKIMADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012KyungwooKim 2

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Idedicatethisworktomyparentsandmywife. 3

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ACKNOWLEDGMENTS Firstandforemost,Iwouldliketoexpressmysinceregratitudetomyadvisor,Dr.HaniphLatchman.Ihavebenetedfromhispatience,guidance,andgreatkindness,Iwishtosayaheartfeltthankyoutohim.HisbeliefinmyabilityhasmademeachievethegoalsthatIpursued.Thisworkwouldnothavebeenpossiblewithouthisencouragementandfosteringmyeffort.Lookingback,manytimesIwaslostandfrustrated.Dr.Latchmanalwayslistenedtomywordsandencouragedandgiudedmeontherightpath.Again,Ideeplythankhimforhisinvaluableadvice.Ialsowouldliketothankallthemembersofmycommittee(Dr.JaniseMcNair,Dr.RichardNewman,Dr.AntonioArroyo)whogavemetheirinterestsandhelpfulsuggestionsonmyresearch.WhenIstartedmyacademiccareeratUniversityofFlorida,ItooktheDr.McNair'sclass.DeepknowledgeandkeeninsightintotheareaofstochasticanalysisandprobabilitytheorythatIhavebeengivenfromherclasscontributedgreatlytothesuccessoftheresearch.Dr.ArroyobroadenedmyhorizonsandgaveaveryhelpfulinsightintoproblemnaturethatIfrequentlyfacedwithduringmyacademictimesatUniversityofFlorida.IalsothankDr.Newmanforhiskindadviceandnumerousdiscussionswehad.Ialsothankallofmycolleagues,especiallymembersoftheLaboratoryforInformationSystemsandTelecommunications(LIST),Mr.JungeunSonandMr.YoungjoonLee,fortheirhelpfuldiscussionsonacademicquestionsthatariseinmanysituationsduringmytimesatLISTlaboratory.Asalways,Ithanktomyparentsandparents-in-lawwithallmyheartfortheirendlessloveandsupportduringthelongjourney.AndIamdeeplyindebtedtomywifeBomin.Shedeservesspecialpraise,asshealwayshasbeenwithmeandpatientlysupportedmyresearchandPh.D.study.TheloveIhavebeengivenfromherandmyfamilymakesmylifewonderful.ItgoesfarbeyondthewordsIcanexpress. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1AnOverviewofMultimediaTrafc ....................... 12 1.2NetworkSystemModeling ........................... 14 1.3DVDVideoStream:MPEG-2Standards ................... 16 2TRAFFICMODELINGOFMPEGFRAMESWITHSINGLEDISTRIBUTIONS 18 2.1AnOverviewofaStatisticalApproachtoMPEGFrames .......... 18 2.2StatisticalModelingofMPEGFrameswithSingleDistributions ...... 19 2.3ResultsandDiscussion ............................ 24 3STATISTICALANALYSISOFMULTIMEDIATRAFFICOVERCOMMUNICATIONNETWORKS ..................................... 33 3.1AnOverviewoftheStatisticalCharacteristicsofMultimediaTrafc .... 33 3.2Hyper-GammaDistribution .......................... 33 3.2.1ModelDescription ........................... 34 3.2.2TheFamilyoftheHyper-GammaDistributions ............ 36 3.3ExperimentalAnalysis ............................. 40 3.4ResultsandDiscusstion ............................ 47 4METHODSFORANANALYSISOFMULTIMEDIATRAFFIC .......... 58 4.1OtherApproachtoHandlingMultimediaTrafc:IEEE802.11n ....... 58 4.2AnOverviewofMeshNetworks ........................ 61 4.3AnalysisoftheDynamicsofMeshNetworks:Jackson'sModel ...... 62 4.4ResultsandDiscussion ............................ 66 5ANAPPROACHTOTHEPERFORMANCEANALYSISFORNETWORKSYSTEMSBASEDONMULTIMEDIATRAFFIC ........................ 67 5.1AnOverviewoftheHyper-GammaServiceTimeDistribution ....... 67 5.2ABackgroundforNetworkSystemswiththeHyper-GammaServiceTimeDistribution ................................... 67 5.3ThroughputandDelayforNetworkSystemsBasedonMultimediaTrafc 77 5

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5.4ResultsandDiscussion ............................ 85 6CONCLUSIONSANDFUTURERESEARCHDIRECTION ............ 87 6.1Conclusions ................................... 87 6.2FutureResearchDirection ........................... 89 REFERENCES ....................................... 91 BIOGRAPHICALSKETCH ................................ 95 6

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LISTOFTABLES Table page 2-1B-frameerrorsofthemovieMatrixReloaded. ................... 22 2-2P-frameerrorsofthemovieMatrixReloaded. ................... 22 2-3I-frameerrorsofthemovieMatrixReloaded. ................... 23 2-4Total-frameerrorsofthemovieMatrixReloaded. ................. 23 2-5EstimatedparametersforthemovieMatrixReloadedtotalframe. ........ 24 3-1Mean,varianceandCoVofempiricaldataandtheHyper-Gammadistribution. 41 3-2TheBICsofthreesingledistributions. ....................... 41 3-3TheBICsoftheHyper-Gammadistributionforthetwomovies. ......... 42 3-4ParametersofeachcomponentoftheHyper-GammadistributionforthemovieMatrix. ......................................... 44 3-5ParametersofeachcomponentoftheHyper-GammadistributionforthemovieLordoftheRingsII. ................................. 45 4-1WirelessLANthroughputbyIEEEStandard:Comparisonofdifferent802.11transferrates. ..................................... 58 7

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LISTOFFIGURES Figure page 2-1ProbabilitydensityfunctionsofthemovieMatrixReloadedB-frame. ...... 25 2-2CumulativedistributionfunctionsofthemovieMatrixReloadedB-frame. .... 26 2-3ProbabilitydensityfunctionsofthemovieMatrixReloadedP-frame. ...... 27 2-4CumulativedistributionfunctionsofthemovieMatrixReloadedP-frame. .... 28 2-5ProbabilitydensityfunctionsofthemovieMatrixReloadedI-frame. ....... 29 2-6CumulativedistributionfunctionsofthemovieMatrixReloadedI-frame. .... 30 2-7ProbabilitydensityfunctionsofthemovieMatrixReloadedtotalframe. ..... 31 2-8CumulativedistributionfunctionsofthemovieMatrixReloadedtotalframe. .. 32 3-1BICvaluesforeachcomponentdistributionsoftheB-framesofthemovieMa-trixandLordoftheRingsII. ............................. 49 3-2BICvaluesforeachcomponentdistributionsoftheP-framesofthemovieMa-trixandLordoftheRingsII. ............................. 50 3-3BICvaluesforeachcomponentdistributionsoftheI-framesofthemovieMa-trixandLordoftheRingsII. ............................. 51 3-4Statisticalcharacteristics(i.e.probabilitydensityfunction)ofB-frameofthemovieMatrix. ..................................... 52 3-5Statisticalcharacteristics(i.e.probabilitydensityfunction)ofP-frameofthemovieMatrix. ..................................... 53 3-6Statisticalcharacteristics(i.e.probabilitydensityfunction)ofI-frameofthemovieMatrix. ......................................... 54 3-7Statisticalcharacteristics(i.e.probabilitydensityfunction)ofB-frameofthemovieLordoftheRingsII. .............................. 55 3-8Statisticalcharacteristics(i.e.probabilitydensityfunction)ofP-frameofthemovieLordoftheRingsII. .............................. 56 3-9Statisticalcharacteristics(i.e.probabilitydensityfunction)ofI-frameofthemovieLordoftheRingsII. ................................. 57 4-1Meshnetworktopology. ............................... 62 4-2ThetopologicalstructureoftheJacksonnetwork. ................. 63 8

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5-1Innitecapacitynetworksystemsbasedonmultimediatrafcafterbuildingaroutingpath. ..................................... 78 5-2Finitecapacitynetworksystemsbasedonmultimediatrafcafterbuildingaroutingpath. ..................................... 84 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELINGANDPERFORMANCEANALYSISOFMULTIMEDIATRAFFICOVERCOMMUNICATIONNETWORKSByKyungwooKimMay2012Chair:HaniphA.LatchmanMajor:ElectricalandComputerEngineering ApplicationsofmultimediatrafcovervariouscommunicationchannelsneedtosharephysicallylimitedbandwidthefcientlyandatthesametimeguaranteeQualityofService(QoS).Asthesizeofmultimediadataincreasestoquaranteeahighquality,transmissiondelayalsoincreasesandthisresultsinpoorQoSovercommunicationnetworks.ForguaranteedQoSandlowtransmissiondelayincommunicationnetworkswhichprovidesanintegratedmultimediaservice,itisdesirabletostudythestatisticalcharacteristicsofmultimediatrafcandisimportanttoobtainananalyticalandtractablemodelofcompressedMPEGdata.ThisdissertationpresentsastatisticalapproachtoobtainanMPEGframesizemodelandestimatesthroughputandtransmissiondelayovercommunicationnetworks. Forthestudyofmultimediatrafccharacteristics,MPEG-2framesareextractedfromtypicalDVDmovies.Toobtaincandidatedistributions,aprobabilityhistogrambasedontheFreedman-Diaconismethod,whichisusedasadecisionruleforabinsize,isconsideredandbothsingledistributionsandamixedtypedistributionarealsotakenintoaccount.Inthesingledistributioncase,distributionparametersareobtainedfromempiricaldatausingthemaximumlikelihoodestimation(MLE)method.ThebestttedmodelforthemultimediatrafcstudiedwasfoundtobeaLognormaldistribution. However,withthissingledistribution,wecannotexplaintheinherentmultimodalityclearlyobservedintheempiricalmultimediaframedata.ThusaHyper-Gamma 10

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distributionisconsideredasanalternativemodeltoexplainitsinherentmultimodality.TheHyper-GammadistributionparametersareobtainedbymeansofanexpectationmaximizationalgorithmbasedontheK-meansalgorithmandaposterioriprobability.Furthermore,theBayesianInformationCriterion(BIC)isusedasagoodnessoftcriterion.SingleprobabilitydistributionsarealsoconsideredtodemonstratethesuperiorityoftheproposedmodelinttingMPEG-2framedata.ThisdissertationshowsthattheHyper-GammadistributionisagoodcandidateforastochasticmodelforMPEG-2framedata. Afterobtainingastatisticalmodelformultimediatrafc,theHyper-Gammaservicetimedistributionhasbeenappliedtonetworksystemswhicharecomposedofnodeswithaniteandinnitecapacityandconnectedintandem.ThethroughputfortheinnitecapacitysystemisequaltotheaveragearrivalratebecauseoftheassumptionforaPoissonarrivalprocessandBurke'stheorem,meanwhilethetotaldelayisrepresentedintermsoftheaveragearrivalrate,andtherstandsecondmomentsoftheHyper-Gammaservicetimedistribution.But,incaseofthenitecapacitysystem,thethroughputateachnodeisrepresentedinaproductformofthearrivalratetothatnodeandoneminusitsblockingprobability.Furthermore,thedelayineachnodedependsonthetotaltimespentinthepreviousnode.Therefore,thisdissertationpredictsthroughputandtransmissiondelayformultimediatrafcfromtheestimatedHyper-Gammaservicetimedistributionandthisresultwillbeusefulintheperformanceanalysisofnetworksystemsbasedonmultimediatrafc. 11

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CHAPTER1INTRODUCTION 1.1AnOverviewofMultimediaTrafc Overthepastfewyearscommunicationnetworkshaverapidlyevolvedtosatisfythecustomer'sneedsforintegratedservicessuchasvoiceoverIP,personalorindustrialvideo,andentertainmentmultimedia,etc.Thus,multimediatrafchasbecomeoneofthemajorsourcesofnetworktrafcloads.Moreover,multimediaservicessuchasreal-timestreamingvideo,onlinegames,IP-TV,andDigitalMultimediaBroadcasting(DMB)aresensitivetotransmissiondelayandademandforhighqualityofservice(QoS). Especially,incaseofahomenetworkwhichisdenedasasmallareanetworkwithinaresidentialunit,alotofhomenetworkstandardsandtechnologieshavebeendevelopedtoimprovethroughput,mitigatetransmissiondelayeffects,andenhanceQoSformultimediatrafcwhichallowtheuseofhomeHDTV,IPTV,interactiveonlinegames,etc.Thetypeofhomenetworkingstandardsareasfollows:HomePhonelineNetworkingAlliance(HomePNA)whichprovideshomenetworksystemoveratypicalphoneline,MultimediaoverCoaxAlliance(Moca)whichusestheequippedcoaxialcableoverhomenetworks,HomePlugPowerlineAlliance(HomePlugAV)whichusesinstalledpowerlines,IEEE802.11nstandardwhichisanamendedversionof802.11x,UltraWideband(UWB)whichemploystheIEEE802.15.3astandard,andG.hnwhichisdevelopedundertheInternationalTelecommunicationUnion(ITU-T)andsupportsnetworkingoverpowerlines,phonelinesandcoaxialcables. Moreover,inthedesignofsystemsthattransmitmultimediainformationthroughawiredorwirelesschannel,takingintoaccountthephysicallimitationsofchannelcharacteristics,itisdesirabletohaveananalyticalandtractablemodelofmultimediatrafccharacteristicsforguaranteedQoSandefcientmanagementofnetwork 12

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bandwidthincommunicationnetworks,sincetheframesizeofMPEGdataisrelativelylargeanductuatesconsiderably. Inresponsetotheevolutionofnetworktechnologiesandthefeatureofmultimediatrafc,researchershaveproposedaseriesofanalyticalMPEGsourcemodelsintheliterature.Nomuraandcolleague[ 1 ]introducedtherstorderAutoRegressive(AR)model,whichestimatedtheburstofvideosourceswithmeasuredautocorrelation,coefcientofvariation,andprobabilitydistribution.TheresultoftheirstudyisthatthestatisticalcharacteristicofVariableBitRate(VBR)videoinformationfollowsabell-shapeddistribution.ButNomura'smodelnownolongertsasinglegeneralvideosource.[ 2 3 ]Heyman,Frey,Lee,andO.Rosehavesuggestedagammadistribution-basedmodel.O.Rose[ 4 ]suggestedalayeredmodelingschemeforMPEGvideotrafc.AccordingtoRose'slayeredmodel,MPEGvideotrafciscomposedofthreelayers:Celllayer,Framelayer,andGOPlayer;andthestatisticalmodeloftheframeandGroupofPictures(GOP)sizecanbeestimatedbyGammaorLognormaldensityfunction.BasedonRose'slayermodel,itispossibletoobtainanoutlineofavarietyofstochasticmodulesandthedescriptionofhowtheyinteractinthecaseofvideotrafc.ButtherearedifcultiesinndingtrafcclassesforMPEGvideotrafc.ThestatisticalcharacteristicsoftheHeyman'sGammaBetaAutoRegressive(GBAR)model[ 5 6 ]haveageometricformofautocorrelationfunctionandaGamma(orNegative-Binomial)marginaldistribution.Heyman'sresearchisconductedbasedonalong(30min)sequenceofrealvideoteleconferencedataandtheGBARmodelistractablebecauseithasjustthreeparameterstobeestimated.HowevertheGBARmodelisnotadequatetotgeneralMPEGdatabecauseHeymanbuilttheGBARmodelbasedonlyontheVBRvideoconferencedata.[ 3 ]Lee[ 7 ]suggestedasumoftwogammadensityfunctionsmodel,inwhichheaddedindividualgammadensityfunctions,normalizedthem,andobtainedparametersusinganonlinearleastsquarealgorithm.Lee'smodelissimpleandeasilydealtwithbut,sometimes,hasweaknessinparameter 13

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estimation,sothatitoftenhasapoorperformanceinttingtovariousempiricaldata.Frey[ 3 ]proposedaGOPGBARmodel,whichisanupgradedversionofHeyman'sGBARmodel.FreyandhiscolleaguesuggestedthatthesizeofanMPEGB-frames,P-frames,andI-framescouldbemodeledasonegammarandomvariable,thesumoftwogammarandomvariables,andthesumofthreegammarandomvariables,respectively.TheGOPGBARmodelisalsosimpleandanalyticalbut,withtheaboveGammadistribution-basedmodels,cannotexplainthemultimodalpropertyobservedintheempiricalMPEGdatahistograms.ThemultimodalpropertyalsoarisesinanempiricalhistogramofMPEG-4andH.263framedataandonlinegametrafc.[ 8 9 ] 1.2NetworkSystemModeling Anetworksystemcanbeconsideredassystemsofow.Anumberofpacketsaretransferredthroughoneormorechannelswhicharelimitedincapacityfromonenodetoanother.Inthiskindofsituation,apacketservicerateinanodehastobealwaysbiggerthanapacketarrivalrateintoanodetoavoidapacketlossorguaranteethestablesystemow.However,ifthepacketarrivalrateisbiggerthanthepacketservicerate,thenthepacketbeginstooverowandcanbeblockedorresultinpacketlossatthenode.Moreover,thearrivalsorthesizeofpacketsoftenariseinanunpredictablefashion.Thus,conictsfortheuseofthechannelareinevitable,queuesofwaitingwillarise,andtheseenvironmentsbringaboutthenetworktrafcloads.Whenonewouldliketopredictorspecifythedynamicsofsystemsofow,andanalyzetheperformanceofnetworksystems,theinherentpropertyofrandomnessinthepacketarrivalprocessandservicetimedistributionmustbeconsidered. Anotherconsiderationofnetworksystemsisaperformanceofowintermsofthroughputanddelay.Beforedescribingthethroughputanddelay,itisnecessarytospecifyanaveragepacketarrivalrateandaverageservicerate.Theaveragearrivalrateandaverageserviceratearedenedastheexpectednumberofthepacketarrivalsperunittimeandtheexpectednumberofpacketsinserviceperunittime,respectively. 14

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Generally,throughputofanetworksystemmeasuredinbits/secondorframe/secondisdenedastheaverageratewithwhichpacketsaresuccessfullytransferredthroughthechannel,soactualthroughputisaproductofapacketdeparturerateandnon-blockingprobability.Delaycanbeconsideredasasumofaveragewaitingtimesinqueuesandservicetimesinnetworknodes. Thesimplestandconventionalwayofapproachingthiskindofdynamicsystemsisthequeueingtheory.Accordingtothebasicqueueingsystem(i.e.M/M/1queue),thearrivalprocessisaPoissonprocessandtheservicetimefollowsanexponentialdistributionwithasingleserver.Butwhenweventurebeyondtheclassicalnetworkmodel(i.e.M/M/1queue)intothemoregeneralandempiricalworld,thenrathercomplexphenomenaarise,whichimpliesthearrivalprocessandservicetimedistributionisnotalwaysaPoissonprocessandanexponentialdistribution,respectively.[ 10 12 ] Generally,amethodofanalyticallytractablemodelingofnetworksystemstoevaluateitsperformanceconsistsoftwoparts.Therstpartistondstatisticalcharacteristicsofnetworktrafc;thatis,whichprobabilitydensityfunctioncanspecifythestatisticalpropertyoftrafcow,andtheotherpartistogureoutarelationshipbetweenthestochasticmodelanditsresultingtrafccharacteristicssuchasthroughputoveranetworkchannelortransmissiondelay.Moreover,theMPEG2standardsfollowsthreetypesofframe(i.e.Intra-codedframe,Predictiveframe,andBidirectionalframe)whichhasrandomsizeofframes.Inaqueueingviewpoint,adifferentpacketsizeisdirectlyrelatedtotheaverageservicerateoritsservicetimedistribution.Therefore,thereisanessentialneedtoinvestigateastatisticalmodeloftheMPEGframesizetodetermineitsservicetimedistributionforMPEGtrafcow.Also,itisinevitabletotakeintoaccountthroughputandtransmissiondelayovercommunicationnetworksbasedonthequeueinganalysis.Thisdissertationproposessuchamodelwhichcanbeusedforaperformanceanalysisforcommunicationnetworksbasedonmultimediatrafc. 15

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1.3DVDVideoStream:MPEG-2Standards Nowadays,themostpervasiveopticaldiscstoragemediumwhichcanstoremultimediadataisDVD.DVDscanstoremorethansixtimesasmuchdatacomparedwithprevioustrendsstoragemediawhicharecalledCDs(CompactDisc).Moreover,DVD-Videobecomesthedominantformofhomevideodistributionworldwide.ThissectiongivesabriefoverviewofaDVDvideoformatinwhichMPEG-2standardsismostwidelyused.NotethatDVDvideodatausedinthisresearchwereextractedfromcommercialDVDs.Thisworkisdoneforacademicpurposesonlyandtherewerenoedits,distributions,orcollectionsafterthisresearch. MPEG-2standardsprovidethreetypesofmainframes.Intracodedframes(I-frames)aredirectlyencodedfromtheinformationinthepictureitself;thatis,theencodingprocessofI-framesisindependentofallotherframesandusestransformcodingwithonlymoderatecompression.I-framesproviderandomaccesspointstotheencodedvideosequencewheredecodingcanbegin.Predictivecodedframes(P-frames)areencodedbyusingmotioncompensation,whichiscalledforwardprediction,withrespecttothemostrecentI-framesorP-frames.ThecompressionrateofP-framesismoresubstantialthanI-frames.Bidirectional-predictivecodedframes(B-frames)areencodedbyusingabidirectionalpredictionrelativetoboththepreviousandsubsequentI-framesorP-framesasareferenceframe.B-framesprovidethehighestrateofcompressionofthethreeframetypes;however,theyhavethelargesttimetoencode.B-framescannotbeusedasreferencesforprediction.Theorganizationofthethreeframesinasequenceisveryexible.[ 3 13 16 ] TheMPEG-2videostreamstructuralhierarchyisasfollows:block,macroblock,slice,picture,GOP,videostreamsequence.Generally,adigitalimageisasetof2-dimensionalpictureelementswhicharecalledpixels,andpixelsbecomethesmallestunitofimageinformation.Sincerawimagedatahaveanenormousamountofinformation,imagecompressiontechniqueisinevitabletorepresentadigitalimage. 16

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Oneofthemostcommonlyusedmethodsforimagecompressionisthediscretecosinetransform(DCT).BasedontheDCTcodingscheme,compresseddataisstoredinablockwhichisasetof8by8arrayofpixelsor64coefcientsoftheDCT.TheblockiscalledthefundamentalcodingunitintheMPEGstandards.TheMPEG-2standardsdenesamacroblockasa16by16pixelsegmentinaframe,inotherwords,4blocksofluminanceand2blocksofchrominance.ThemacroblockplaysaroleofabasicunitformotioncompensationintheMPEG-2standards.Anarbitrarynumberofsequencesofmacroblockswhichstandinthesamerowarecalledaslice,inwhichmacroblocksarealignedfromlefttorightandtoptobottom.Apictureisdenedasencodedimagedata.Ingeneral,apictureisidenticaltoaframe,whichworksastheprimarycodingunitofavideosequenceinMPEG-2stream.TheencodedframesorpicturesinMPEG-2arearrangedingroupsofpictures(GOP).TheGOPalwaysstartswithanI-frame;theP-framesandB-framesareinsertedintothesequence.Therefore,ageneralstructureoftheGOPcanberepresentedbyaseriesofframes,IBBPBBPBBPBB,butthisisnotaregularformat.Avideostreamsequenceisthehighestsyntacticstructureofencodedvideostreams.Itstartswithasequenceheaderwhichisfollowedbyoneormorecontiguouscodedframes(oragroupofpictures),andceasesbyasequenceendcode.[ 7 14 17 ] ThetrafcmodelingoftheMPEGframeswithsingledistributionsisdescribedinChapter 2 .AnotherapproachtogureoutthestatisticalcharacteristicsofmultimediatrafcusingtheHyper-GammadistributionisintroducedinChapter 3 .Chapter 4 brieydemonstratessomemethodsforananalysisofmultimediatrafcandChapter 5 presentsanapproachtotheperformanceanalysisfornetworksystemsbasedonmultimediatrafc.Finally,Chapter 6 concludethisdissertationwithresultsfortheperformanceofnetworksystemssuchasthroughputanddelaywhichhavenodesconnectedinseries. 17

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CHAPTER2TRAFFICMODELINGOFMPEGFRAMESWITHSINGLEDISTRIBUTIONS 2.1AnOverviewofaStatisticalApproachtoMPEGFrames Forguaranteedqualityofservice(QoS)andsufcientbandwidthincommunicationnetworkswhichprovidesanintegratedmultimediaservice,itisimportanttoachieveananalyticalmodelforcompressedMPEGdata.Chapter 2 presentsastatisticalapproachtoanMPEGframesizemodeltoincreasenetworktrafcperformanceincommunicationnetworks.MPEGframedataareextractedfromcommercialDVDmoviesandempiricalhistogramsareconsideredtoanalyzethestatisticalcharacteristicsofMPEGframedata.SixcandidatesofprobabilitydistributionsareconsideredhereandtheirparametersareobtainedfromempiricaldatausingtheMaximumLikelihoodEstimation(MLE).Chapter 2 showsthattheLognormaldistributionisthebestttedmodelofMPEG-2totalframesasasingleprobabilitydistribution. Multimediainformationisstoredandtransmittedascompresseddatainadeviceandchannel,respectively.TwotypesofcommonlyusedcompressionmethodsareMPEG-2,whichisthesecondversionofstandardsdevelopedbytheMovingPicturesExpertGroup(MPEG)andH.264,whichisalsoknownasMPEG-4Part10orMPEG-4AVC(AdvancedVideoCoding).Thesetwoencodingmethodsarewidelyusedcommercially.Forexample,Blu-raydiscsanddigitalHDTVusebothMPEG-2andMPEG-4AVCasencodingmethod.AnMPEG-2formatisusedincurrentcommercialDVDmovies.ButifcompressedMPEG-2orMPEG-4AVC(H.264)dataistransmittedthroughawiredorwirelesschannel,takingintoaccountthephysicallimitationsofchannelcharacteristics,thetransmissionmustguaranteeQoS,whichmeansthatthechannelmustprovidesufcientthroughputandtolerabletransmissiondelay.ItisthereforeimportanttoachieveananalyticalandtractablemodelforadistributionofcompressedMPEGdatabecausetheframesizeofMPEGdataisrelativelymassiveanductuatesconsiderably. 18

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AsimplestochasticmodelfortheMPEG-2framesizeisproposedinChapter 2 toimprovebandwidthutilizationandtoreducetransmissiondelay.Inordertoobtainthemodelthatwelltsempiricaldata,sixdifferentprobabilitydistributionsinwhichtheirparametersaredeterminedbyanMaximumLikelihoodEstimation(MLE)methodisconsideredtocompareandanalyzethestatisticalcharacteristicsofMPEG-2frametrafc.Chapter 2 presentsthatthebestmodelforthedistributionofthetotalMPEG-2framesizeisaLognormaldistributionbymeansofaMeanSquaredError(MSE)method. 2.2StatisticalModelingofMPEGFrameswithSingleDistributions ThissectionpresentsanexperimentalanalysisofthestatisticalcharacteristicsofMPEG-encodedcommercialDVDmovieframedataandshowsthattheapplicationoftheresulttoempiricaldataisgoodenoughtospecifythecharacteristicsofMPEG-2totalframedata.Asmentionedpreviously,theprocedureoftheMPEGtrafcmodelingisgenerallycomposedoftwoparts.TherstpartistondthestatisticalcharacteristicsofMPEGtrafcandbuildaprecisemodelforastatisticalanalysis,andtheotheristogureoutarelationshipbetweenthestatisticalmodelanditstrafccharacteristics.Chapter 2 andChapter 3 areespeciallyfocusedontherstpart,astatisticalmodelingofMPEGtrafcwhichisbasedontheB-frames,P-frames,I-frames,andtotalframes.TheframesizemodelofMPEGtrafchasasimilarstatisticalcharacteristic(i.e.theshapeofpdf)ofaGammadistribution,aLognormaldistribution,aRayleighdistribution,aWeibulldistribution,aNakagamidistribution,andaRiciandistribution.Chapter 2 presentssixsimilarshapesofprobabilitydistributionsandcomparestheirpdfsandcdfswiththatoforiginaldata. Therststepofastatisticalanalysisforempiricaldataistobuildanempiricalhistogram.Conventionally,makingahistogramisanaturalandfundamentalwayofrepresentingasetofempiricaldatadrawnfromarealworldandonecanaffordtoestimateastatisticalcharacteristicofdatawiththistechnique.However,whenweare 19

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tryingtouseamethodofthehistogram,thereneedsacarefulconsideration;inordertomakeahistogramonemustdecidethenumberofbinstouse.Asanextremeexample,wecanconsiderthefollowingcase.Thereisasetofdata(e.g.10000samples)drawnfromtheGaussiandistribution.Ifweputallsamplesinonebinofthehistogram,thenwecanseethatthestatisticalcharacteristicsofempiricaldatalookslikeauniformdistribution,meanwhileifwechoosethesamenumberofbinsofthehistogramasthenumberofempiricaldatasamples(i.e.ananalysiswithrawdataitself),itcouldnotbeeasytocatchouttheinherentstatisticalcharacteristics.Therefore,thisdissertationusesaFreedman-Diaconismethodasadecisionruleforabinsizetogureoutaprobabilityhistogramfromframedata:[ 18 ] Bin Size=2IQR(x)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1=3(2) wheretheIQR(x)istheinterquartilerangeofempiricaldata,i.e.thedifferencebetweenthe75thand25thpercentileofempiricaldataandnisthenumberofobservationsinsamplex.TheFreedman-Diaconistechniquewasbasedonthegoalofminimizingthesumofsquarederrorsbetweenthehistogrambarheightandtheprobabilitydensityoftheunderlyingdistributionwhichgavethen)]TJ /F8 7.97 Tf 6.58 0 Td[(1=3partoftheequation.Theuseof2IQR(x)asameasureofspreadwasdeterminedfromtheirempiricalexperiments. TheparameterswhichspecifythestatisticalcharacteristicsofeachprobabilitydistributionareestimatedbytheMLEmethod.Inaddition,theMSEmethodisselectedtoevaluatethebestttedmodelforthedistributionoftheMPEGframesize.Thefollowingsaredensityfunctionformulaeofeachcandidatedistribution.ForaGammadistribution, fGAM(x)=xk)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(x= k\(k)x>0(2) 20

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wherekisashapeparameterandisascaleparameter.Bothparametersarepositiveandreal.ForaLognormaldistribution, fLOGN(x)=1 xp 2e)]TJ /F10 5.978 Tf 7.79 3.86 Td[((lnx)]TJ /F11 5.978 Tf 5.76 0 Td[()2 22x>0(2) whereisalocationparameter(mean)andisascaleparameter(standarddeviation).ForaNakagamidistribution, fNAKA(x)=2 !\()x2)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(x2=!x>0(2) whereisashapeparameterand!isaspreadparameter.Bothparametersarepositiveandreal(0:5).ForaWeibulldistribution, fWEIB(x)=k x k)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F8 7.97 Tf 6.59 0 Td[((x=)kx>0(2) whereisascaleparameterandkisashapeparameter.Bothparametersarepositiveandreal.ForaRayleighdistribution, fRAYL(x)=x 2e)]TJ /F11 5.978 Tf 9.61 3.26 Td[(x2 22x>0(2) whereisascaleparameteranditispositiveandreal.ForaRiciandistribution, fRICI(x)=x 2e)]TJ /F11 5.978 Tf 7.79 3.26 Td[(x2+2 22I0x 2x>0(2) whereI0)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(x=2isthemodiedBesselfunctionoftherstkindwithorderzero.isascaleparameter(0and0). HeymanandFreyalreadyproposedagamma-basedframesizemodel.Especially,FreyandhiscolleaguesuggestedthatthesizeofMPEGB-frames,P-frames,andI-framescouldbemodeledasonegammarandomvariable,thesumoftwogammarandomvariables,andthesumofthreegammarandomvariables,respectively.[ 3 6 ]However,thesemodelsarenotalwayssuccessfulinttinggeneralMPEGframes.Figures 2-1 and 2-2 showthemovieMatrixReloadedB-frameshistogram,pdfand 21

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Table2-1. B-frameerrorsofthemovieMatrixReloaded. ProbabilityDistributionspdferrors[bytes]cdferrors[bytes] Rician3:276210)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0017Nakagami3:630110)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0022Gamma4:188410)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0030Weibull5:735710)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0044Lognormal6:940910)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0070Rayleigh1:651110)]TJ /F8 7.97 Tf 6.58 0 Td[(100.0307 Table2-2. P-frameerrorsofthemovieMatrixReloaded. ProbabilityDistributionspdferrors[bytes]cdferrors[bytes] Rician2:943410)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0029Nakagami3:335510)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0037Gamma3:864210)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0051Weibull4:393210)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0054Lognormal6:006910)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0105Rayleigh1:140610)]TJ /F8 7.97 Tf 6.58 0 Td[(100.0324 itsempiricalcdf,respectively.ThesixprobabilitydistributionsareoverlappedonthehistogramandempiricalcdfsoftheB-framesinFigures 2-1 and 2-2 .Aswecansee,thehistogramofthemovieMatrixReloadedB-framesismorettedtoasingleRiciandistributionratherthanasingleGammadistribution.Wecanalsoconrmthisbythenumericalresults.Table 2-1 containstheMatrixReloadedB-frameerrorsofeachprobabilitydistributionmeasuredbytheMSEmethod.Asaresultofnumericalevaluations,thebestttedmodeltothestatisticalmodelofthemovieMatrixReloadedB-framesisnotasingleGammadistributionbutasingleRiciandistribution.Figuresfrom 2-3 to 2-6 showthestatisticalcharacteristicsofotherframedata(i.e.P-framesandI-frames).ThebestttedmodelsfortheP-framesandI-framesareasingleRiciandistributionandasingleNakagamidistribution,respectively.Tables 2-2 and 2-3 alsosupporttheseresultsnumerically.ThestatisticalmodelsforthemovieMatrixReloadedtotalframesinwhichB-frames,P-frames,andI-framesaremergedtogetherisshown 22

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Table2-3. I-frameerrorsofthemovieMatrixReloaded. ProbabilityDistributionspdferrors[bytes]cdferrors[bytes] Rician9:413510)]TJ /F8 7.97 Tf 6.58 0 Td[(120.0025Nakagami8:957710)]TJ /F8 7.97 Tf 6.58 0 Td[(120.0020Gamma9:194510)]TJ /F8 7.97 Tf 6.58 0 Td[(120.0020Weibull1:544410)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0045Lognormal1:210010)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0029Rayleigh3:952110)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0198 Table2-4. Total-frameerrorsofthemovieMatrixReloaded. ProbabilityDistributionspdferrors[bytes]cdferrors[bytes] Lognormal2:939110)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0017Gamma3:205510)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0020Nakagami4:075510)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0035Weibull4:837610)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0045Rayleigh5:756610)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0066Rician5:756610)]TJ /F8 7.97 Tf 6.58 0 Td[(110.0066 inFigures 2-7 and 2-8 .Apparently,asingleLognormaldistributioniswellttedtothehistogramoftotalframes.AndnumericalresultsinTable 2-4 supporttheresultsoftheseexperiments.InFigure 2-7 ,theRiciandistributioncoincideswiththeRayleighdistribution.ThereasonofthiscorrespondenceisthemodiedBesselfunctionwhichisappearsintheRiciandensityfunctionanditsparameter.Ingeneral,when=0,theRiciandistributionbecomestheRayleighdistribution.WecanmakesureofthisrelationshipinEquationsfrom( 2 )to( 2 ).AgeneralequationofthemodiedBesselfunctionoftherstkindis I(z)=z 21Xk=0)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(z2=4k k!\(+k+1)(2) TheorderzeroofthemodiedBesselfunctionoftherstkindis I0x 2=1Xk=0)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(z2=4k (k!)2z=(x)=2(2) 23

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Table2-5. EstimatedparametersforthemovieMatrixReloadedtotalframe. ProbabilityDistributionsParameters Gammak=6:3285=5:33103Lognormal=10:3457=0:4296Nakagami=1:6879!=1:34109Weibull=3:79104k=2:4164Rician=64:0363=2:59104Rayleigh=2:59104 IfweconsiderEquation( 2 )andEquation( 2 )alongwithestimatedparameters,wecanobtaintheresultoftheRayleighdistribution.Finally,Table 2-5 showstheestimatedparametersofthemovieMatrixReloadedtotalframes. 2.3ResultsandDiscussion Chapter 2 proposesasingleprobabilitydistributionasastatisticalmodeloftheMPEG-2totalframesizeformultimediatrafc.RigorousexperimentshasbeenconductedtogureoutthebestttedsingledistributionmodelfortheempiricalMPEG-2totalframeswhichareextractedfromthecommercialDVDmovieMatrixReloadedandtheevaluationforthetotalframeshasbeendonebymeansoftheMLEandMSEmethods.TheresultoftheseexperimentsissuchthatthebestttedmodelforthestatisticalmodeloftheMPEG-2totalframesturnsouttheLognormaldistribution.Moreover,extendedapplicationstootherDVDdataalsoshowthesameresults.ButsingledistributionscannotexplaintheinherentmultimodalpropertyinthehistogramofthemovieMatrixReloaded.Basedonthisresult,itturnsoutthatweneedamorerigorousanalysisonthestatisticalcharacteristicsofMPEGmultimediatrafc.Thisresearchwillhelpthedesignofmultimedianetworksystems. 24

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AGammadistribution,Lognormaldistribution,andNakagamidistribution BRayleighdistribution,Weibulldistribution,andRiciandistribution Figure2-1. ProbabilitydensityfunctionsofthemovieMatrixReloadedB-frame. 25

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AGammadistribution,Lognormaldistribution,andWeibulldistribution BRayleighdistribution,Nakagamidistribution,andRiciandistribution Figure2-2. CumulativedistributionfunctionsofthemovieMatrixReloadedB-frame. 26

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AGammadistribution,Lognormaldistribution,andNakagamidistribution BRayleighdistribution,Weibulldistribution,andRiciandistribution Figure2-3. ProbabilitydensityfunctionsofthemovieMatrixReloadedP-frame. 27

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AGammadistribution,Lognormaldistribution,andWeibulldistribution BRayleighdistribution,Nakagamidistribution,andRiciandistribution Figure2-4. CumulativedistributionfunctionsofthemovieMatrixReloadedP-frame. 28

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AGammadistribution,Lognormaldistribution,andNakagamidistribution BRayleighdistribution,Weibulldistribution,andRiciandistribution Figure2-5. ProbabilitydensityfunctionsofthemovieMatrixReloadedI-frame. 29

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AGammadistribution,Lognormaldistribution,andWeibulldistribution BRayleighdistribution,Nakagamidistribution,andRiciandistribution Figure2-6. CumulativedistributionfunctionsofthemovieMatrixReloadedI-frame. 30

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AGammadistribution,Lognormaldistribution,andNakagamidistribution BRayleighdistribution,Weibulldistribution,andRiciandistribution Figure2-7. ProbabilitydensityfunctionsofthemovieMatrixReloadedtotalframe. 31

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AGammadistribution,Lognormaldistribution,andWeibulldistribution BRayleighdistribution,Nakagamidistribution,andRiciandistribution Figure2-8. CumulativedistributionfunctionsofthemovieMatrixReloadedtotalframe. 32

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CHAPTER3STATISTICALANALYSISOFMULTIMEDIATRAFFICOVERCOMMUNICATIONNETWORKS 3.1AnOverviewoftheStatisticalCharacteristicsofMultimediaTrafc Chapter 2 wasdevotedtothestudyofthestatisticalMPEGframemodelwithasingleprobabilitydistribution;histogramsforeachofframedataandtotalframedatawerettedtosingleprobabilitydistributions.But,takingacarefullookattheresultsfromChapter 2 ,wecanrealizethattherearenoticeablephenomenainthehistograms.Thefactisthatthemodeofthehistogramisnotuniqueandwecaninterpretthesephenomenaasamixtureofmorethanoneunimodaldistributionwhichhasonlyonemode.Whenwestarttoanalyzethemixturetypeofdistributions,weneedtomakesurethatthemixturetypeofdistributionsisdifferentfromasumofindependentrandomvariables.Thesumofrandomvariableshasaprobabilitydensityfunctionwhichisgivenbytheconvolutionintegralofeachmarginaldensityfunction,whereasthedensityfunctionofthemixturetypeofdistributionshasaformofaweightedsumofeachdensityfunctionandweightcoefcientsshouldbeintherangebetweenzeroandunity.Anexampleofthesumofindependentrandomvariablesisthek-ErlangdistributionwhichisaddedupfromsingleexponentialrandomvariabletokexponentialrandomvariablesandanexampleofthemixturetypedensityfunctionistheHyper-ExponentialdistributionwhichisintroducedbyOrlilkandRappaport.[ 12 ] Chapter 3 introducesastatisticalmodeloftheMPEGframesasthemixturetypedistribution,namely,theHyper-Gammadistributionandpresentsanextensiveanalysisofitsadequatenessofthestatisticalmodelformultimediatrafc. 3.2Hyper-GammaDistribution TheHyper-GammadistributionisageneralizedformoftheHyper-ExponentialdistributionwhichwasintroducedbyRappaportandOrlik[ 12 ],theHyper-ErlangdistributionwhichwasintroducedbyFang[ 10 ]andtheHyper-Chi-Squaredistribution.ItisnotabnormalthataGammadistributioncanbeextendedtomanyotherprobability 33

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distributionsbyvaryingitsparametervalues.Forexample,ifashapeparameterofagammadensityfunctionisunity,thentheGammadistributionbecomesanExponentialdistribution.Iftheshapeparameterisapositiveintegerk,thenitbecomesasumofkindependent,identicallydistributedexponentialrandomvariable,inotherwords,k-Erlangdistribution.ThisversatilityofthegammarandomvariableholdsforthecaseoftheHyper-Gammarandomvariable.InthenextsectionwedenetheHyper-Gammadistributionanditsproperties. 3.2.1ModelDescription LetXbeaHyper-Gammarandomvariable.NotethattherandomvariableXisnotasumofgammarandomvariablesbutaweightedsumofgammadensityfunctions.Asmentionedpreviously,thedensityfunctionofasumofrandomvariablesisrepresentedbyaconvolutionintegralofthedensityfunctionofeachrandomvariable.But,inthiscase,thedensityfunctionoftheHyper-GammarandomvariablecanberepresentedasaweightedsumoftheNdifferentgammadensityfunction,whichisclosedunderconvexcombination;inotherwords,allcoefcientsforthedensityfunctionoftheHyper-Gammarandomvariablearenonnegativeandsumtounity.ThedensityfunctionoftheHyper-Gammarandomvariableisdenedas fhygam(x)=NXi=1ixki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F11 5.978 Tf 9.04 3.26 Td[(x i iki\(ki)x>0(3) wherekiistheithelementofasetofshapeparametersk=[k1;k2;k3;;kN]andiistheithelementofasetofscaleparameters=[1;2;3;;N].iistheithelementofasetofweightsofeachdensityfunction=[1;2;3;;N]andsumstounity.Eachparameterhasapositiverealvalue(i.e.ki>0,i>0and0i1).InEquation( 3 ),\(ki)isthegammafunctiondenedby \(k)=Z10tk)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(tdt(3) 34

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ThedistributionfunctionoftheHyper-Gammarandomvariablecanbeexpressedintermsofthelowerincompletegammafunction. Fhygam(x)=Zx0NXi=0itki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F11 5.978 Tf 9.7 3.26 Td[(t i kii\(ki)dt=NXi=1i(ki;x i) \(ki)(3) wherethelowerincompletegammafunctionisdenedas (k;x)=Zx0tk)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(tdt(3) WecanspecifythefundamentalstatisticalcharacteristicsoftheHyper-Gammarandomvariablebytheconceptofmoment.Ingeneral,adensityfunctionofarandomvariableXcanbecompletelydescribedprovidedtheexpectedvaluesofallthepowersofXaredened.ThenthmomentoftheHyper-Gammarandomvariableisgivenby Ehygam(Xn)=()]TJ /F5 11.955 Tf 9.3 0 Td[(1)ndn dsnFhygam(s)s=0(3) wheren=1;2;3;,andFhygam(s)isaLaplacetransformoftheHyper-Gammadistributionandcanbeinterpretedasanotherversionofacharacteristicfunction.ThecharacteristicfunctionanditsLaplacetransformoftheHyper-Gammadistributionisrepresentedby hygam(w)=NPi=1i(1)]TJ /F6 11.955 Tf 11.95 0 Td[(jwi))]TJ /F9 7.97 Tf 6.58 0 Td[(kiFhygam(s)=NPi=1i(1+is))]TJ /F9 7.97 Tf 6.59 0 Td[(ki(3) LetEhygam(X)andVARhygam(X)bethemeanandvarianceoftheHyper-Gammarandomvariable.TheexpectedvalueandvarianceoftheHyper-Gammarandomvariablecanbeexpressedastherstmomentandadifferenceofthesecondmomentandthesquareoftherstmoment.Wecanobtaintherstandsecondmomentsofthe 35

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Hyper-GammarandomvariablefromEquation( 3 ). Ehygam(X)=NXi=1ikii(3) Ehygam(X2)=NXi=1iki(ki+1)i2(3) VARhygam(X)=NXi=1iki(ki+1)i2)]TJ /F13 11.955 Tf 11.95 20.45 Td[("NXi=1ikii#2(3) 3.2.2TheFamilyoftheHyper-GammaDistributions Somemixed-typerandomvariablesarespecialcasesoftheHyper-Gammarandomvariable.AnappropriatechoiceoftheparameterskiandioftheHyper-Gammadensityfunctionmakesitpossibletoobtainanothermixed-typeofdistributions.Ifweselectandki=1andi=1=i,thentheHyper-GammadensityfunctionbecomestheHyper-ExponentialdensityfunctionwhichisproposedbyRappaportandOrlik.[ 12 ]TheHyper-ExponentialdensityfunctioncanbespeciedbyaweightedsumofNdifferentexponentialdensityfunctions.Eachofthemhasiwhichisarateorscaleparameter.iindicatesaweightoftheindividualexponentialdensityfunctionandsumstotheunity.Eachparameterhasapositiverealvalue(i.e.i>0and0i1).Letthedensityfunction,distributionfunction,meanandvarianceoftheHyper-Exponentialrandomvariablebefhyex(x),Fhyex(x),Ehyex(X),andVARhyex(X),respectively. fhyex(x)=NXi=1ixki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F11 5.978 Tf 9.04 3.26 Td[(x i kii\(ki)ki=1;i=1 i=NXi=1iie)]TJ /F9 7.97 Tf 6.59 0 Td[(ix(3) Fhyex(x)=NXi=1i(1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.58 0 Td[(ix)(3) Ehyex(X)=NXi=1i i(3) 36

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Ehyex(X2)=NXi=12i 2i(3) VARhyex(X)="NXi=12i 2i#)]TJ /F13 11.955 Tf 11.96 20.44 Td[( NXi=1i i!2(3) WecanobtainthesameresultwiththeHyper-ExponentialdistributionproposedbyRappaportandOrlikfromEquation( 3 )toEquation( 3 ).[ 12 ] Ifwesetki=miandi=1=i,thentheHyper-GammadensityfunctionbecomesthatoftheHyper-ErlangdistributionwhichisproposedbyFang.[ 10 ]ThedensityfunctionoftheHyper-ErlangrandomvariableisdenedasaweightedsumofNdifferenterlangdensityfunctionsandhasthefollowingform: fhyer(x)=NXi=1ixki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F11 5.978 Tf 9.04 3.26 Td[(x i kii\(ki)ki=mi;i=1 i=NXi=1imiixmi)]TJ /F8 7.97 Tf 6.59 0 Td[(1 (mi)]TJ /F5 11.955 Tf 11.96 0 Td[(1)!e)]TJ /F9 7.97 Tf 6.59 0 Td[(ix(3) where\(ki)=(mi)]TJ /F5 11.955 Tf 12.8 0 Td[(1)!providedki=miandmiisashapeparameterandtakesanonnegativeintegervaluefori=1;2;;N.iisascaleparameterandhasapositiverealnumber.ihasthesamemeaningasappearedintheHyper-Exponentialdistribution.Equation( 3 )isidenticaltotheHyper-Erlangdensityfunction,ifweselecti=miiwheremiisdenedbyanonnegativeintegerandiispositivenumberin[ 10 ].ThedistributionfunctionoftheHyper-Erlangrandomvariablecanbeacquiredfromtheintegrationofthedensityfunction. Fhyer(x)=Zx0NXi=1imiitmi)]TJ /F8 7.97 Tf 6.59 0 Td[(1 (mi)]TJ /F5 11.955 Tf 11.96 0 Td[(1)!e)]TJ /F9 7.97 Tf 6.59 0 Td[(itdt=NXi=1i(mi;ix) (mi)]TJ /F5 11.955 Tf 11.95 0 Td[(1)!(3) Therstmoment,thesecondmomentandvariancearegivenasfollows: Ehyer(X)=NXi=1imi i(3) 37

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Ehyer(X2)=NXi=1imi(mi+1)1 2i(3) VARhyer(X)=NXi=1imi(mi+1) 2i)]TJ /F13 11.955 Tf 11.96 20.44 Td[("NXi=1imi i#2(3) TheHyper-Chi-SquaredensityfunctioncanbeachievedfromtheHyper-Gammadensityfunctionbysettingtheparametersandi=2andki='i=2.ThedensityfunctionoftheHyper-Chi-SquarerandomvariableisalsospeciedbyaweightedsumofNdifferentChi-Squaredensityfunctionsandisgivenasfollows: fhycsq(x)=NXi=1ixki)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F11 5.978 Tf 9.04 3.25 Td[(x i kii\(ki)ki='i 2;i=2=NXi=1ix'i 2)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F11 5.978 Tf 7.78 3.26 Td[(x 2 2'i 2)]TJ /F13 11.955 Tf 8.76 9.69 Td[()]TJ /F9 7.97 Tf 6.68 -4.43 Td[('i 2(3) where'iisapositiveintegerandimpliesthenumberofdegreesoffreedom.irepresentsaweightofeachChi-Squaredensityfunctionandalsosumstounity.ThedistributionfunctionoftheHyper-Chi-Squarerandomvariablecanbeobtainedbyintegratingitsdensityfunction. Fhycsq(x)=NXi=1i('i 2;x 2) )]TJ /F13 11.955 Tf 8.77 9.69 Td[()]TJ /F9 7.97 Tf 6.67 -4.43 Td[('i 2(3) Therstmoment,thesecondmomentandvarianceoftheHyper-Chi-Squarerandomvariablearegivenasfollows: Ehycsq(X)=NXi=1i'i(3) Ehycsq(X2)=NXi=1i'i('i+2)(3) VARhycsq(X)=NXi=1i'i('i+2))]TJ /F13 11.955 Tf 11.95 20.44 Td[( NXi=1i'i!2(3) TheCoefcientofVariation(CoV)thatisadimensionlessvalueisonegoodwaytomeasurestatisticaldispersionforarandomvariable.Itisdenedastheratioofthe 38

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standarddeviationtothemean.TheCoVoftheexponentialdistributionisalwaysunitybecausethestandarddeviationoftheexponentialdistributionisequaltoitsmean.InthecaseoftheHyper-Exponentialdistribution,theCoVisgreaterorequaltoone,whichmeansthattheHyper-Exponentialdistributionhasgreaterdispersionthanthatoftheexponentialdistribution.[ 12 ]TheCoVoftheHyper-Erlangdistributioncanbeadjustabletothedesiredvaluebychangingtheparameters;thatis,itcanhavethevaluelessthan,equaltoorgreaterthanunity.[ 10 ]IncaseoftheHyper-Gammadistribution,theCoVisexpressedbyEquation( 3 ). CoV=s NPi=1iki(ki+1)i2)]TJ /F13 11.955 Tf 11.95 16.85 Td[(NPi=1ikii2 NPi=1ikii(3) TheCoVoftheHyper-Gammadistributionisalwayspositivesincethenumerator(i.e.thestandarddeviation)anddenominator(i.e.therstmoment)ofEquation( 3 )arepositive.Basically,therstandsecondmomentsoftheHyper-Gammadistributionisalwayspositive.IfwetakethesquareofCoV,then CoV2=NPi=1iki(ki+1)i2)]TJ /F13 11.955 Tf 11.96 16.86 Td[(NPi=1ikii2 NPi=1ikii2(3) IfwelookatthenumeratorofEquation( 3 )moreclosely,wecanrewriteitasfollows: 24NXi=1ik2ii2)]TJ /F13 11.955 Tf 11.96 20.44 Td[((NXi=1ikii)235+NXi=1ikii2(3) LetYbeadiscreterandomvariablewhichtakesonvaluesfromasetfk11;k22;k33;;kNNgwithprobability1inasimilarwayasappearedin[ 19 ].ThentherstandsecondtermsinEquation( 3 )canbeconsideredastheexpectedvaluesofarandomvariableY2andY,thustheterminbracketsinEquation( 3 )isconsideredasavarianceofYand 39

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isalwaysnonnegative.Moreover,thethirdtermisobviouslyapositiveandanincreasingfunction. 3.3ExperimentalAnalysis ThissectiondescribesanexperimentalanalysisoftheMPEG-encodedcommercialDVDmovieframesandshowsthatanapplicationoftheresultstoempiricaldataprovidesanacceptablemodelforthecharacteristicsofMPEGframetrafc.ThemethodsusedinthissectiontotraceastochasticmodelareempiricalhistogramsandBayesianInformationCriterion(BICs)basedonthemaximumlikelihoodvalueoftheMPEGframeswhichcamefromthefamouscommercialmovieMatrixandLordoftheRingsII.Toconstructaprobabilityhistogramfromrawframedata,theFreedman-Diaconismethodisalsoselectedasadecisionruleofabinsize[ 18 ]asappearedinEquation( 2 ).Moreover,forthepurposeofthestatisticalidentication,theBICvaluesareused[ 20 ]: BIC=ln(n))]TJ /F5 11.955 Tf 11.96 0 Td[(2ln(L)(3) wherenisthesameoneusedinEquation( 2 ),isthenumberofparametersinthestochasticmodelandLmeansthemaximumlikelihoodvalue.Gamma,Lognormal,andRayleighdistributionsarepresentedtocomparehowwelltheHyper-GammadistributionismatchedtothehistogramsoftheempiricalMPEGframes.Amongtheabovecompetingdistributions,theonewhichhasthesmallestBICvalueischosenasthebeststochasticmodelforMPEGframes.Table 3-1 presentsthemean,variance,andcoefcientofvariationforempiricaldataandtheestimatedHyper-Gammamodel,respectively,andonecanmakesurethattherearenobigdifferenceinCoVsbetweenempiricaldataandtheestimatedHyper-Gammamodel.Table 3-2 showstheBICsofthreesingledistributionsandwecanrecognizethattheGammadistributionhasthesmallestBICsforeachframeofthetwomoviesinsingledistributioncase.Figuresfrom 3-4 to 3-6 representthestatisticalmodelsofB-frames,P-frames,andI-frames 40

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Table3-1. Mean,varianceandCoVofempiricaldataandtheHyper-Gammadistribution. B-frameP-frameI-frame MatrixMean12:01041043:43841045:8147104EmpiricalVariance3:03371077:87081071:0255108DataCoV0.27400.25800.1742 HyperMean2:01041043:43841045:8147104GammaVariance3:06251078:03761071:0336108DistributionCoV0.27530.26070.1748 LoRIIMean2:03951043:96551047:2200104EmpiricalVariance1:00701082:78221087:4036108DataCoV0.49200.42060.3769 HyperMean2:03951043:96551047:2200104GammaVariance1:01001082:83101087:3988108DistributionCoV0.49280.42430.3767 1Unitsofmeanandvariancearemeasuredinbytes. Table3-2. TheBICsofthreesingledistributions. SingleMatrix DistributionsB-frameP-frameI-frame Gamma2614464.111027206.43351689.94Lognormal2620012.011032887.05354794.79Rayleigh2702306.101061707.03370863.32 SingleLordoftheRingsII DistributionsB-frameP-frameI-frame Gamma3636977.491443017.99498437.14Lognormal3660244.091461541.60499906.72Rayleigh3639226.321443057.18503278.33 ofthemovieMatrix,andFiguresfrom 3-7 to 3-9 areforthemovieLordoftheRingsII,respectively. Whenweexaminehistogramsoftheframesfromtwomovies,MatrixandLordoftheRingsII,wecanrecognizethatsimilarsituationarises.Ifwetakeaglanceatthem, 41

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Table3-3. TheBICsoftheHyper-Gammadistributionforthetwomovies. MatrixLoRII ]2B-frameP-frameI-frameB-frameP-frameI-frame 126144641027206351689363697714430184984372260258810231603469593629093143471349749132602209102277234693636193611433275496522426019561022696346952361787914331994962935260166210222503469273617844143310849632162601379102224634689736178461433086496335726011201022258346902361788514331134961388260107010222863469213617931143312849616092600770102232034694736179791433141496176102600756102236334697936180021433149496142112600665102240334700036180411433172496168122600653102242534702736180481433210496198132600662102245034705936180841433239496214142600679102247934708236181101433278496240152600718102250634707336181401433309496241162600726102253734709336181391433344496287172600730102256834712436181471433375496301182600758102260634714736181821433410496324192600777102264034717236182151433445496353202600822102267034719836182481433464496378212600826102270234721336182621433492496408222600862102273134724136182981433512496436232600893102276534727136183091433543496472242600911102266434729436183691433567496502252600950102282434732236183741433582496518262601003102285834734436184051433618496560272601020102288834737536184261433645496579282601044102278134739536184761433665496612292601079102281234742636184961433702496642302601105102284334744836185261433723496668 2The]symbolimpliesthenumberofclusters. wecaneasilyndoutthathistogramsofthemovieMatrixshowaunimodalproperty,whereastheLordoftheRingsIIhistogramshaveamultimodalproperty.Theunimodaldistributionhasonlyonemaximumvalueatitsmodeandnootherlocalmaxima;thus 42

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itshistogrammaybeapproximatedbyaunimodalfunction,asingleprobabilitydensityfunctioninthiscase.ButthedensityfunctionofeachframeofthemovieMatrixisnotcomposedofasingleprobabilitydistribution.WecanmakesureofthisfromBICvaluesinTable 3-3 showingthenumberofclusterswhichisreferredtoasthecomponentdistributionoftheHyper-GammadistributionandalsoconrmthisfromSubguresA'sinFiguresfrom 3-1 to 3-3 .WecanidentifythesmallestBICsofeachframeinTable 3-3 anditmeansthattheproperstochasticmodelforeachframeofthemovieMatrixisnotasingledistributionbutseveralormorenumbersofcomponentdistributionsaresynthesized.Meanwhile,themultimodaldistributionhasM(M=2;3;4;)distinctlocalmaximumsattheirmodesinitsdensityfunction.TheLordoftheRingsIIhistogramcannotbeexplainedwithaunimodaldistributionbutcanbeapproximatedbyamultimodaldistributionwithMdifferentmodesbecausetherearemorethanonelocalmaximuminitshistogram.WecanalsomakesureofthisfromBICvaluesinTable 3-3 andSubguresB'sinFiguresfrom 3-1 to 3-3 .Consequently,theycanbeinterpretedasamixedtypeofanumberofdifferentprobabilitydistributions.Thefollowingstepsareconductedtogureouthowmanydifferentdistributionsarecomposedoftheframedata. Step1 PerformaconventionalK-meansclusteringbasedonEuclideandistancewithrawframedata. Step2 Estimatetheparameters(i.e.weight,shapeandscaleparameters)ofeachclusterbymeansoftheMaximumLikelihoodEstimation. Step3 Recomposetheclustersofrawframedatabasedonaposterioriprobabilitywitheachparameterwhichisobtainedinthestep2. Step4 Estimatetheparameters(i.e.weight,shapeandscaleparameters)ofrawframedatabymeansoftheExpectationMaximizationalgorithmwiththeinitialvaluesobtainedfromthestep2. 43

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Table3-4. ParametersofeachcomponentoftheHyper-GammadistributionforthemovieMatrix. FramesWeightShapeScale 1=0:0449k1=2:84471021=9:71591012=0:0324k2=3:00691012=1:10901033=0:1181k3=1:24891023=1:25781024=0:1260k4=8:02401024=2:47851015=0:0785k5=5:59561025=4:1127101BFrame6=0:1198k6=4:00941026=4:29771017=0:1100k7=7:11881027=2:99901018=0:1220k8=6:90311028=2:68831019=0:0456k9=1:24611029=2:489210210=0:0069k10=1:245010010=1:618710411=0:0581k11=4:266610211=5:877210112=0:1377k12=1:933610112=7:2399102 1=0:2892k1=2:63611011=1:03981032=0:1545k2=1:17121022=3:1424102PFrame3=0:2868k3=1:15981023=2:71771024=0:0176k4=1:55831004=1:99381045=0:1464k5=6:96521015=6:17891026=0:1056k6=2:33611016=1:9811103 1=0:1728k1=2:87381011=1:76541032=0:0210k2=2:34211002=1:5161104IFrame3=0:1252k3=4:07221013=1:64691034=0:1969k4=2:85111024=2:05761025=0:2374k5=1:94591025=3:37911026=0:2468k6=1:50191026=3:5298102 Step5 Reshapetheclustersofrawframedatawithaposterioriprobabilitybasedontheparametersobtainedfromthestep4. Step6 ComparetheBICsofeachpartitioneddataobtainedfromthestep5withtheparametersobtainedfromthestep4. 44

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Table3-5. ParametersofeachcomponentoftheHyper-GammadistributionforthemovieLordoftheRingsII. FramesWeightShapeScale 1=0:1607k1=6:87261011=5:42501022=0:1366k2=4:86061012=5:5488102BFrame3=0:4387k3=1:23701013=1:08341034=0:1967k4=2:91051014=6:84201025=0:0673k5=1:21731005=1:1260104 1=0:0583k1=0:85431001=3:42001042=0:0876k2=1:73001012=3:4989103PFrame3=0:1592k3=3:79151013=9:71211024=0:1841k4=6:68171014=8:53011025=0:3519k5=1:34691015=1:94151036=0:1588k6=7:11641016=6:2715102 1=0:1044k1=1:03171021=9:18701022=0:3695k2=2:47671012=2:15571033=0:1737k3=1:27281023=6:3883102IFrame4=0:0407k4=2:70071004=1:19291045=0:2017k5=5:75821015=1:12151036=0:0914k6=1:04731026=1:17821037=0:0186k7=8:20961027=1:8589102 ThemethodofdeterminingthenumberofclustersdescribedintheaboveisoriginatedfromtheX-meansalgorithm.[ 21 ]PellegandMooreintroducedonewayofhowtochoosethenumberofclustersingivendata;theyappliedaconventionalK-meansmethodandBayesianInformationCriterion(BIC)togivendata.TheconventionalK-meansmethodisundoubtedlyoneofthewell-knownclusteringalgorithmbutitstillhassomemisclassieddatainitsresults.Toreducethenumberofmisclassieddata,twomorerepartitionstepsinwhichisdescribedinstep3andstep5areadded;adecisionruleofmembersofeachclusteristochoosetheclusterwhichhasthemaximumvalueofaposterioriprobability.Wecangeneralizethisbythefollowings;Wearealreadygivenasetofrawframedata,X=fx1;x2;;xj;;xng,nisthe 45

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totalnumberofsamplesinrawdataandthearbitrarynumberofclusters,C=fcig,i2f1;2;;N;gfromtheresultoftheconventionalK-meansclustering,whereNisthetotalnumberofclustersofwhichrawframedataarecomposed.MoreoverwewanttoknowtheclusterinwhicheachelementxjofXbelongs.Oneofthemostlikelycasesforustodoistochoosetheclusterwhichhasthebiggestposterioriprobabilityp[CjX].TheHyper-GammadistributionisalreadydenedinEquation( 3 )anditsposterioriprobabilityisgivenby f(xjW)=NXi=1ip(xji)=NXi=1ixki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F11 5.978 Tf 9.04 3.26 Td[(x i kii\(ki)(3) whereidenotesaprobability(i.e.weight)oftheithcomponentdensityfunctionp(xji),idenotesaparametervectorfullyspecifyingtheithcomponentdensityfunctionp(xji),andWdenotesasetofallparametersfullyspecifyingtheHyper-Gammadensityfunctionf(xjW),whereW=fwig,wi=fi;ig,i=fki;ig,andi2f1;2;;Ng.WhenweconsiderthesimplecaseoftwoclustersC1andC2,xjbelongstoC1ifp[C1jxj]>p[C2jxj],otherwisexjbelongstoC2.HereweapplytheBayes'theorem,wecanobtainthefollowings: xj2C1ifp(xjj1)1 p(xj)>p(xjj2)2 p(xj)xj2C2ifp(xjj1)1 p(xj)p(xjj2)2xj2C2ifp(xjj1)1
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bCirepresentstheclusterinwhichxjbelongs,whereiandjareintegersandjn.Aftergatheringclusterinformation,wecanobtainallparametersforeachcomponentoftheHyper-Gammadistributionthroughthestep2tothestep5.Table 3-4 andTable 3-5 showtheestimatedparametersofeachframeofthetwomovies.TheonewhichhasthesmallestBICischosenasthebestttedmodelforastochasticmodeloftwomovieframesanddeterminehowmanydifferentdistributionsconsistineachframeoftwomovies.ThestochasticmodelsforB-frames,P-frames,andI-framesofthemovieMatrixandLordoftheRingsIIcanbeapproximatedbytheHyper-Gammadistributioncomposedofasumof12,6,and6differentgammadensityfunctionsandasumof5,6,and7differentgammadensityfunctions,respectively.Thehistogramsofeachframeofthetwomovies,thedensityfunctionsofthreesingledistributions,thedensityfunctionsoftheHyper-GammadistributionsandtheirindividualcomponentsaregiveninFiguresfrom 3-4 to 3-9 .Finally,basedontheseresults,allsingleprobabilitydistributionsrepresentpoorperformancethanthatoftheHyper-Gammadistribution.Thusitturnsoutthatthebestttedmodelsforeachframeofthetwomoviesarenotsingleprobabilitydistributions,butrathertheHyper-Gammadistributions. 3.4ResultsandDiscusstion MPEGvideotrafcmodelingisanimportantissueforadesignofnetworksystemsbasedonmultimediatrafcsincethemassivenessofdatacancausehightransmissiondelayandlowQoS.Toresolvethisproblem,weneedtoestimateanaccuratemodelformultimediatrafc.Chapter 3 proposesastochasticmodelformultimediatrafc,theHyper-GammadistributionwhichisageneralizedformoftheHyper-Exponentialdistribution,Hyper-Erlangdistribution,andHyper-Chi-squaredistribution.ThemultimediatrafcmodelsproposedinthepreviousliteraturecannotexplaintheinherentmultimodalityinMPEG-2framedata,whereastheHyper-Gammadistributionmodelisabletopredictthestatisticalcharacteristicsofmultimediatrafcovercommunicationchannels.TheresultofthisstudiesestablishesthattheHyper-Gammadistribution 47

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modelenablestheconstructionofastochasticmodelformultimediatrafcbasedonMPEG-2framedata.Moreover,wecanpredictthroughputandtransmissiondelayofmultimediatrafcfromtheestimatedHyper-Gammadistributionandthisishelpfulwhenwetrytodesignanetworksystem. 48

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AThemovieMatrix BThemovieLordoftheRingsII Figure3-1. BICvaluesforeachcomponentdistributionsoftheB-framesofthemovieMatrixandLordoftheRingsII. 49

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AThemovieMatrix BThemovieLordoftheRingsII Figure3-2. BICvaluesforeachcomponentdistributionsoftheP-framesofthemovieMatrixandLordoftheRingsII. 50

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AThemovieMatrix BThemovieLordoftheRingsII Figure3-3. BICvaluesforeachcomponentdistributionsoftheI-framesofthemovieMatrixandLordoftheRingsII. 51

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AHistogram,singledensityfunctions,andHyper-Gammadensityfunction BEstimatedHyper-Gammadistributionanditscomponents Figure3-4. Statisticalcharacteristics(i.e.probabilitydensityfunction)ofB-frameofthemovieMatrix. 52

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AHistogram,singledensityfunctions,andHyper-Gammadensityfunction BEstimatedHyper-Gammadistributionanditscomponents Figure3-5. Statisticalcharacteristics(i.e.probabilitydensityfunction)ofP-frameofthemovieMatrix. 53

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AHistogram,singledensityfunctions,andHyper-Gammadensityfunction BEstimatedHyper-Gammadistributionanditscomponents Figure3-6. Statisticalcharacteristics(i.e.probabilitydensityfunction)ofI-frameofthemovieMatrix. 54

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AHistogram,singledensityfunctions,andHyper-Gammadensityfunction BEstimatedHyper-Gammadistributionanditscomponents Figure3-7. Statisticalcharacteristics(i.e.probabilitydensityfunction)ofB-frameofthemovieLordoftheRingsII. 55

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AHistogram,singledensityfunctions,andHyper-Gammadensityfunction BEstimatedHyper-Gammadistributionanditscomponents Figure3-8. Statisticalcharacteristics(i.e.probabilitydensityfunction)ofP-frameofthemovieLordoftheRingsII. 56

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AHistogram,singledensityfunctions,andHyper-Gammadensityfunction BEstimatedHyper-Gammadistributionanditscomponents Figure3-9. Statisticalcharacteristics(i.e.probabilitydensityfunction)ofI-frameofthemovieLordoftheRingsII. 57

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CHAPTER4METHODSFORANANALYSISOFMULTIMEDIATRAFFIC 4.1OtherApproachtoHandlingMultimediaTrafc:IEEE802.11n Inthissegmentoftheresearchweareinterestedinstudyingtheperformanceofmultimediatrafcoverwirelessmeshnetworks.Inthisregard,itisinstructivetoexaminehowmultimediatrafcishandledbynon-meshedwirelessnetworksusingthe802.11xsuiteofprotocol.Thisinformationwillproveusefulnessasweexaminethemorecomplexwirelessmeshstructures.Asmentionedinprevioussections,theperformanceenhancementfornetworksystemsisthenalgoalofallresearchersandengineersinvolvedinthestudyofnetworksystems.Thus,therehavebeenevolvedandintroducedanumberoftechnologiestosatisfythedemandforhigherperformancelocalareanetwork(LAN)systemsandtheonemainstreamofthedevelopmentofthesetechnologiesistheIEEE802.11standardseriesforawirelessLAN.Ifwehaveacarefullookatthetrafctypesovercommunicationnetworks,themultimediatrafcdominatesoverthewholecommunicationnetworktrafcduetoitsmassivenessofsize.Inaddition,whenthemultimediainformationistransferredthroughacommunicationchannelwhichhasaphysicallylimitedcapacity,thereneedstomakeaconsiderationforguaranteeingaQoSofthetransmission.InresponsetoincreasingthiskindofdemandsforachievingenhancedperformanceWLANs,theInstituteofElectricalandElectronicsEngineers-StandardsAssociation(IEEE-SA)hasbeenmakinganamendmentoftheIEEE802.11standards. Table4-1. WirelessLANthroughputbyIEEEStandard:Comparisonofdifferent802.11transferrates. IEEEWLANOver-the-AirMediaAccessControlLayerStandardEstimatesServiceAccessPoint(MACSAP)Estimates 802.11b11Mbps5Mbps802.11g54Mbps25Mbps(when.11bisnotpresent)802.11a54Mbps25Mbps802.11n200+Mbps100Mbps 58

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802.11(802.11legacy) TheIEEE802.11legacyistheoriginalversionofIEEE802.11standardwhichisreleasedin1997andclariedin1999.Itsupportupto2Mbpsdatarateandforwarderrorcorrectioncode.The802.11legacystandarddenesthreephysicallayerstechnologies;Direct-sequencespreadspectrum(DSSS)operatinginthe2.4GHzISMband,atdataratesof1Mbpsand2Mbps,Frequency-hoppingspreadspectrum(FHSS)operatinginthe2.4GHzISMband,atdataratesof1Mbpsand2Mbps,andInfraredoperatingatawavelengthbetween850and950nm,at1Mbpsand2Mbps.Today,802.11legacyisoutofdate,butisrenownedbyIEEE802.11b.[ 22 ] 802.11b TheIEEE802.11bstandardisanextendedversionoftheIEEE802.11legacyandusesthesamemediaaccesstechnologyspeciedintheIEEE802.11legacy.Itsupportsthedataratesfrom5.5Mbpsto11Mbps.Asthe802.11bstandarddirectlyextendsthemodulationtechniquespeciedinthe802.11legacystandard,whichiscalledcomplementarycodekeying,itcanobtainahigherdatarateinthesamebandwidthcomparedtothe802.11legacystandardandisthemostwidelyusedseriesofthe802.11standard.[ 22 23 ] 802.11a IncaseoftheIEEE802.11astandard,itemploysanorthogonalfrequencydivisionmultiplexing(OFDM)methodasamodulationschemeforatransmissionratherthanaspreadspectrumschemeastoincreasethedatarate.OFDMisknownasamulticarriermodulationwhichusesanumberoforthogonalsubcarriersignalsatcloselyspaceddifferentfrequenciesandtransmitssomeofbitsoneachchannel.ByusingtheOFDMtechnology,the802.11astandardcanobtainahightoleranceforseverechannelconditions,wherenarrowbandinterferenceorfrequencyselectivefadingduetothemulti-patheffect(i.e.inter-channelinterference)arises.TheIEEE802.11astandardoperatesinthe5GHzbandanditspossibledataratesperchannelare6,9,12,18,24,36,48,and54Mbps.[ 22 23 ] 59

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802.11g TheIEEE802.11gstandardwhichwasratiedinJune2003isanamendedversionoftheIEEE802.11bstandard,whichoperatesinthesamebaseband2.4GHzasthe802.11b,soitiscompatiblewiththe802.11bstandard.The802.11gstandardallowsthemaximumdataratetoupto54Mbps.Themaindifferencebetweenthestandards802.11gand802.11bisthemodulationtechnique.Thestandard802.11gemploysbothOFDMandDSSMforitsmodulationmethods,whereasthe802.11busesonlyDSSMmethod.[ 22 24 ] 802.11e In2005,theIEEE-SAintroducedanotheramendedversionof802.11standardwhichiscalledthe802.11estandard.ThepurposeofthisamendmentistoimproveandguaranteetheQoSforwirelessLANapplicationsthroughmodicationstotheMediaAccessControl(MAC)layer.Thus,modulesforadistributedcoordinationfunction(DCF)andpointcoordinationfunction(PCF)aresubstitutedforahybridcoordinationfunction(HCF)whichiscomposedofenhanceddistributed-channelaccess(EDCA)andHCF-controlledchannelaccess(HCCA).Bydoingthis,the802.11estandardhasbetterperformancethanthepreviousstandards,wheredelay-sensitiveapplicationssuchasvoiceoverWLANandstreamingmultimediaareused.[ 22 ] 802.11n TheIEEE802.11nstandardisthelatestamendedversionoftheIEEE802.11standard.Ithasdramaticallyenhancedthecapabilityofthedatatransmissionrate(i.e.throughput)from20Mbpstoaround200Mbpsinpracticeinaccordancewiththegrowingcustomer'sdemandforhigher-performancewirelesslocalareanetworks(WLANs).Theemergenceofthe802.11nstandardisverydesirableforthecurrentmarkettosatisfythecustomer'sneedsforintegratedservicessuchasVOIP,realtimevideostreaming,andentertainmentmultimedia,etcbecausethesekindsofservicesarehighlysensitivetotheQoS.Tofullltherequestsofthecurrentmarket,theIEEEstandardassociationemploystheOFDMmodulation 60

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withthemultiple-inputmultiple-output(MINO)antennatechnologyand40MHzbandwidthchannels.[ 25 26 ] Table 4-1 showstheaffordablethroughputbytheIEEE802.11standards.[ 27 ]WhenweconsidertheWLANenvironmentsalongwithnetworktrafc,the802.11nstandardis,recently,oneoftheremarkableenhancedtechnologiesinobtaininghigherthroughput.The802.11nstandardmakesitpossibletoextendarangeofcustomer'sdemands;HDTV,onlinegameservices,DMB,alineofinternetandmultimedia-enabledsmartphones,andetc,effectively.ItcouldalsoguaranteethehigherQoSthanthepreviousseriesoftheIEEE802.11standard.Ifweemploythe802.11nstandardintheirWLANenvironment,wecantakeadvantageofthebenetsthatitprovides;thefastconnectionspeed,thegreat,widereachingrange,thoughitisveryexpensiveascomparedwiththeotherstandards(i.e.802.11b/a/g)andalwaysrequiresaMIMOadaptertobeabletouseitsfullpotentials. 4.2AnOverviewofMeshNetworks Themeshnetworkisanetwork,whereeachnodecannotonlygeneratedatafortransmissionfromonenodetoanother,butalsorelaydatatoothernodesthroughoneormorechannelsandithaslatelyattractedconsiderableattentiononthenetworkandcommunicationresearchsociety.[ 28 32 ]Accordingtothetopologyofthemeshnetwork,itcanbecategorizedintotwotypesofconnection,fullmeshtopologyorpartialmeshtopology.Figure 4-1 illustratethetwotypesofthemeshnetworktopology.Incaseofthefullmeshtopology,allnodesareconnecteddirectlytoeachother,whereasthepartialmeshtopologyallowstheconnectiontypethatsomenodesareconnectedtoeachother,buttherearesomeofthenodeswhicharenotconnectedeachother.Thistopologycanmakethemeshnetworkmorereliableandprovidemuchredundancy.Whenonenodeisnotabletobefunctional,therestofthenodescanstillcommunicatewitheachother,directlyorthroughoneormoreintermediatenodes.Therefore,themeshnetworkcanmaintaincontinuousconnectionsandmakearecongurationaround 61

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brokenorblockedroutesbyhoppingfromnodetonodeuntilinformationisdeliveredtothedestination. Asdatatransmissioninthemeshnetworkariseinamulti-hopfashionfromthesourcetothedestination,weneedtoconsiderthehop-countsandthenumberofpacketsinthepopulationofeachnodebecausetheperformanceofmeshnetworkslikethroughputanddelaydependsonthehop-countsanditspopulationofdatawithineachnode.Ingeneral,whenthenumberofhop-countincreasesfromthedestinationnode,theper-nodethroughputmaydecrease,whereasendtoenddelayincreasesdramatically.Thiscancausethefairnessprobleminthemeshnetwork;thenodeswhichhavethesmallnumberofhop-countsfromthedestinationnodesufferfromalowerthroughputthanthenodesclosertothedestinationnode.[ 29 33 35 ] 4.3AnalysisoftheDynamicsofMeshNetworks:Jackson'sModel Thesection 4.2 providedanoverviewofmeshnetworks.Eventhoughmeshnetworkshaveagoodtopologicalstructureforthenextgenerationnetworkparadigm,itstillhasdrawbackstoovercome;thefairness(i.e.throughputanddelayperformance).Toimprovetheperformanceofmeshnetworks,thereisaneedtoanalyzethedynamiccharacteristicsofmeshnetworksrigorously.OnewayofapproachtothedynamicsofmeshnetworksisamethodwhichwasintroducedbyJackson.Accordingtothe AFullmeshtopology BPartialmeshtopology Figure4-1. Meshnetworktopology. 62

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Figure4-2. ThetopologicalstructureoftheJacksonnetwork. Jackson'stheorem,apacketarrivalprocessdependsonthetotalnumberofpacketsinthenetworkandservicerateatanynodemaybeafunctionofthenumberofpacketsinthatnode.[ 11 36 37 ]LetusconsiderthefullmeshnetworkdepictedinFigure 4-2 .ThereareNnumberofnodesinthesystemwheretheithnodehasmiexponentialserverseachwithservicerateparameterki,wheretherearekipacketsatthatnodeanddeneS(k)asthetotalnumberofpacketsinthesystem,wherethevectork=[k1;k2;;ki;;kN]representsthesystemstateandkiimpliesthenumberofpacketsintheithnodeincludingthepacket(s)inservice.Moreover,weneedtodeneapackettransitionprobabilitybetweennodes.rijdenotesatransitionprobabilityofapacketfromtheithnodetothejthnode,wherei;j=1;2;;N.Thenr0;imeansaprobabilitythatthenextexternallygeneratedarrivalwillenterthenetworkattheithnode,ri;N+1 63

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impliesaprobabilitythatapacketleavingnodeidepartsfromthenetwork,andr0;N+1representsaprobabilitythatthenextarrivalwillneednomoreservicefromthenetworkanddepartfromthenetworkinstantly.Thus,anarrivalratefromoutsidethesystemtothenodei,iisequaltor0;i(S(k))andfollowsaPoissonprocess,where(S(k))denotesthetotalexternalarrivalratetothenetworkwhenthenetworkstateisS(k)atthemoment.Ifweconsiderthetime-dependentnetworkstateprobabilitiesPk(t),thenadynamicequationofthemeshnetworkhastheformasfollows: d dtPk(t)=)]TJ /F13 11.955 Tf 11.29 20.45 Td[("(S(k))+NXi=1ki(1)]TJ /F6 11.955 Tf 11.96 0 Td[(rii)#Pk(t)+NXi=1(S(k))]TJ /F5 11.955 Tf 11.95 0 Td[(1)r0;iPk(i)]TJ /F8 7.97 Tf 6.25 -2.27 Td[()(t)+NXi=1ki+1ri;N+1Pk(i+)(t)+NXi=1NXj=1i6=jkj+1rjiPk(i;j)(t)(4) wheretermswhichhavethenegativevectorargumentsisequaltozeroandnotationsdenotethatk(i)]TJ /F5 11.955 Tf 7.09 -4.34 Td[()=[k1;k2;;ki)]TJ /F5 11.955 Tf 12.4 0 Td[(1;;kN],k(i+)=[k1;k2;;ki+1;;kN],andk(i;j)=[k1;k2;;ki)]TJ /F5 11.955 Tf 11.95 0 Td[(1;;kj+1;;kN](i6=j). TotraceEquation( 4 ),letusconsiderasimplebirth-deathprocesswhichisaMarkovchainX(t)withbirthratekanddeathratek.ThenPk(t)(=P[X(t)=k])canbeexplainedasaprobabilitythatthepopulationsizeisofkatarbitrarytimetandthefollowingprobabilitiesaretruebasedontheMarkovproperty: P[exactly1birthin(t;t+t)jkinpopulation]=kt+o(t) P[exactly1deathin(t;t+t)jkinpopulation]=kt+o(t) P[exactly0birthin(t;t+t)jkinpopulation]=1)]TJ /F6 11.955 Tf 11.95 0 Td[(kt+o(t) P[exactly0deathin(t;t+t)jkinpopulation]=1)]TJ /F6 11.955 Tf 11.95 0 Td[(kt+o(t) 64

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Inotherwords,exactlyonebirthoccursintheinterval(t;t+t)meansthatthereisnobirthin(0;t)andjustonebirthduringthesmalltimeperiodt.Thus, Phtt+tjt>ti=1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(t=1)]TJ /F5 11.955 Tf 11.96 0 Td[([1)]TJ /F6 11.955 Tf 11.95 0 Td[(t+(t)2 2!)-221(]=t+o(t)(4) andtheprobabilitythatnobirthoccursin(t;t+t)is Pht>t+tjt>ti=1)]TJ /F6 11.955 Tf 11.96 0 Td[(Phtt+tjt>ti=1)]TJ /F13 11.955 Tf 11.96 9.68 Td[()]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(t=e)]TJ /F9 7.97 Tf 6.59 0 Td[(t=1)]TJ /F6 11.955 Tf 11.96 0 Td[(t+(t)2 2!)-222(=1)]TJ /F6 11.955 Tf 11.96 0 Td[(t+o(t)(4) Finally,theprobabilitythatthetwoormorebirthoccurintheinterval(t;t+t)becomes P[twoormorebirthsin(t;t+t)]=1)]TJ /F6 11.955 Tf 11.95 0 Td[(P[zerobirthin(t;t+t)])]TJ /F6 11.955 Tf 11.95 0 Td[(P[onebirthin(t;t+t)]=1)]TJ /F5 11.955 Tf 11.95 -.17 Td[((1)]TJ /F6 11.955 Tf 11.95 0 Td[(t+o(t)))]TJ /F5 11.955 Tf 11.96 -.17 Td[((t+o(t))=o(t)(4) Thesamesituationariseinthedeathprocess. FromEquation( 4 )toEquation( 4 ),wecanobtainasetofdifferential-differenceequationswhichrepresentthedynamicsofthebirth-deathnetworksystem. (4a) d dtPk(t)=)]TJ /F5 11.955 Tf 11.29 -.17 Td[((k+k)Pk(t)+k)]TJ /F8 7.97 Tf 6.58 0 Td[(1Pk)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t)+k+1Pk+1(t)fork1 (4b) d dtP0(t)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0P0(t)+1P1(t)fork=0 65

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Now,goingbacktoEquation( 4 ),letusconsiderachangeofprobabilityduringasmallamountofintervalt.Thisquantityisgivenasfollows: Pk(t+t)="1)]TJ /F6 11.955 Tf 11.95 0 Td[((S(k))t)]TJ /F9 7.97 Tf 16.81 14.94 Td[(NXi=1ki(1)]TJ /F6 11.955 Tf 11.95 0 Td[(rii)t#Pk(t)+NXi=1(S(k))]TJ /F5 11.955 Tf 11.96 0 Td[(1)r0;itPk(i)]TJ /F8 7.97 Tf 6.26 -2.27 Td[()(t)+NXi=1ki+1ri;N+1tPk(i+)(t)+NXi=1NXj=1i6=jkj+1rjitPk(i;j)(t)(4) IfwesubtractPk(t),dividebytbothsidesandtakethelimitast!0,thenwe,nally,obtainEquation( 4 ). JacksonalreadystatedthatthesolutionofEquation( 4 ),ajointprobabilityofstatedescriptionisaproductofitsmarginaldistributionofndingkipacketsintheithnodegivenbypi(ki). p(k1;k2;;kN)=p(k1)p(k2)p(kN)(4) wherepi(ki)isaprobabilityofndingkipacketsintheclassicalM=M=mqueueingsystem.[ 11 ] 4.4ResultsandDiscussion Ingeneral,whenwearetryingtoobtaindynamicbehaviorfornetworksystemsandanalyzeit,ifwehavetheexponentialassumptionsforanarrivalprocessandservicetimedistribution,thenwecananalyzethenetworksystembyusingtheJackson'stheoremandhesaidthatajointprobabilityofstatedescriptionisaproductofitsmarginaldistributionofndingkipacketsintheithnode.Unfortunately,thelackoftheexponentialassumptioninaservicetimedistributionformultimediatrafcinarealworldcouldnotmakeitpossibletobuildananalyticalandtractablemodelformeshnetworks.Thusweneedtoconsideranothermethodtoobtainperformancefactorslikethroughputanddelayofnetworksystemsbasedonmultimediatrafc. 66

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CHAPTER5ANAPPROACHTOTHEPERFORMANCEANALYSISFORNETWORKSYSTEMSBASEDONMULTIMEDIATRAFFIC 5.1AnOverviewoftheHyper-GammaServiceTimeDistribution Uptothispoint,wehavestudiedthestatisticalmultimediatrafcmodelfromrealmoviedata,especiallyencodedinMPEG-2.Basedonanextensiveinvestigationofmovieframes,itturnsoutthatthestatisticalmodelforeachDVDmovieframecontributingtomultimediatrafcfollowstheHyper-Gammadistributionandit,morespecically,impliesthataprobabilitydensityfunctionoftheframesizeofcommercialDVDmoviesistheHyper-GammadistributionwhichisspeciedinEquation( 3 ).Itisalsodirectlyrelatedtoaservicetimedistributioninnetworksystems.Inotherwords,weagreethatwemayintuitivelyknowthefactthattheservicetimerequiredtoanodetohandlemovieframesinnetworksystemsisproportionaltothesizeofmovieframesandalsofollowstheHyper-Gammadistribution.Furthermore,thistypeofnetworksystemisrelevanttoanon-Markovianstochasticprocessforanetworkmodelbecauseofrelaxationforanexponentialassumptioninaservicetimedistribution.Thus,inordertoobtainperformancemeasures,likethroughputanddelayofnetworksystemsbasedonmultimediatrafc,wehavetoapproachthesenetworksystemswithoneofgeneraltypesofaservicetimedistribution,thatis,theHyper-Gammaservicetimedistributionmodel(i.e.M/HG/1queueingnetwork).AftercompletingtheanalysisofasinglenodeM/HG/1system,wewillapplytheresultsoftheM/HG/1systemtoaseries(ortandem)networktoobtainnetworkperformancefactors,namely,throughputanddelay. 5.2ABackgroundforNetworkSystemswiththeHyper-GammaServiceTimeDistribution TheresultsinChapter 3 describethefactthattheservicetimedistributionformultimediatrafcinnetworksystemsfollowstheHyper-GammadistributionwhichisexpressedinEquation( 3 )andhasnolongerMarkovianproperties.Thismakesitabitcumbersometoanalyzenetworksystemsbasedonmultimediatrafc.Moreover,since 67

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wedonothaveinformationaboutthearrivalprocessinrealnetworksystemsyet,weassumethatthearrivalprocesstoanetworknodefollowsaPoissonprocess.ThuswehaveanM/HG/1systemwhichisdenedasasingleserversystemwithPoissonarrivalswithanaveragerateofframespersecond,ameaninterarrivaltimeof1=secondandaHyper-Gammaservicetimedistribution.Furthermore,asystemutilizationfactorshouldbelessthanoneforastabilityofnetworksystems. =Ehygam[X](5) WecannowrewritethedensityfunctionoftheHyper-Gammadistributionwhichrepresentstheservicetimedistributionformultimediatrafcinnetworksystems. fhygam(x)=NXi=1ikii \(ki)xki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(ixx>0(5) TheonlydifferencebetweenEquation( 3 )andEquation( 5 )isasetofscaleparameters,namely,i=1=ifori=1;2;;N.WealsorewritetheLaplacetransformoftheHyper-Gammadistributionasfollows: FLhygam(s)=1Z0fhygam(x)e)]TJ /F9 7.97 Tf 6.59 0 Td[(sxdx=1Z0NXi=1ikii \(ki)xki)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(ixe)]TJ /F9 7.97 Tf 6.59 0 Td[(sxdx=NXi=1ikii \(ki)1Z0xki)]TJ /F8 7.97 Tf 6.58 0 Td[(1e)]TJ /F8 7.97 Tf 6.59 -.11 Td[((i+s)xdx=NXi=1ikii \(ki)1 (i+s)ki1Zoyki)]TJ /F8 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(ydy=NXi=1ikii(i+s))]TJ /F9 7.97 Tf 6.58 0 Td[(ki(5) Whenwearetryingtoanalyzethesekindsofsystems,wemustconsideravectorstatedescription[N(t);X0(t)]whichdescribesthenumberofframesinthesystemat 68

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timet,denotedbyN(t),andtheservicetimealreadyreceivedbytheframeinserviceattimet,denotedbyX0(t).Butifwelookatthesystemattheinstantofaframedeparturefromtheservice(orthesystem),thenthestatedescriptioncanbereducedtoN(t)sincetheexpendedservicetime,X0(t)atthisinstantisequaltozeroforaframeinservicebecauseitjustenteredtheservice.ThisisthefundamentalideabehindthemethodoftheimbeddedMarkovchaininwhichthedepartureinstantcorrespondstoanimbeddedpoint.Indeed,wealreadyknowforafactthatEquation( 5 )isalwaystrueforaPoissonprocessduetothePASTA(PoissonArrivalsSeeTimeAverages)property[ 38 ]. Pk(t)=P[N(t)=k]:thenumberofframesinsystemattimet Rk(t)=P[arrivalattimetndskframesinsystem] Pk(t)=Rk(t)(5) Moreover,onaverage,thelimitofthefollowingdistributionsarealsoidenticalintheM/HG/1system. pk=limt!1P[thenumberofframesinsystemattimet] rk=limt!1P[anarrivalndskframesinsystemattimet] dk=limt!1P[adepartureleaveskframesinsystemattimet] rk=dk(5) )rk=dk=pk(5) Consequently,thearrivalframestothesystem,departureframesfromthesystemandoutsideframeslookingintothesystemallseethesamedistributionofthenumberofframesintheM/HG/1system.[ 11 39 ] Letusnowconsiderthreerandomvariablesn,xn,andn.First,nisdenedasthenumberofframesleftbehindbyadepartureofthenthframefromservice.ThusP[n=k]impliesaprobabilityofknumbersofframesinthesystemjustafter 69

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adepartureofthenthframeanditslimitform(n!1)correspondstodk.AccordingtoEquation( 5 ),theprobabilitypk(=P[e=k])whichimpliestheaveragenumberofframesinsystemisequaltodk.Next,xnistheservicetimeforthenthframe,whichisdistributedaccordingtoFhygam(x)asappearedinEquation( 3 )andisindependentofn.Andthelastone,nisthenumberofframesarrivingduringtheservicetimexnofthenthframeanddependsonlyonthelengthofxn,notonn.Thelimitingdistributionsofthethreerandomvariables(n,xn,andn)areexpressedase,ex,ande.Furthermorewedenearandomvariableakasaprobabilitythatexactlykframesarriveduringtheservicetimeofaframe. ak=P[e=k]=1Z0(x)k k!e)]TJ /F9 7.97 Tf 6.59 0 Td[(xfhygam(x)dx(5) Additionally,wehavetodenetheone-steptransitionprobabilitieswhichisobservedonlyatthedepartureinstant. pij=P[n+1=jjn=i](5) Obviously,pijequalszeroforallj
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LetusnowtrytoobtaintheaveragequeuelengthatframedepartureinstantswhichcanberepresentedasE[e].Beforedoingthis,wehavetondtherelationshipbetweentworandomvariables,n+1andn.Inordertondanequationrelatingn+1andn,wemustconsidertwocasesattheinstantofthenthframedeparture,whichmeansthatthedepartureofthenthframeleavesbehindanon-emptysystemandanemptysystem.Inthenon-emptysystemcase(n>0),the(n+1)thframeisalreadyinthesystemwhenthenthframedeparts.Intheothercase(n=0),sincethedepartingframeleavesbehindanemptysystem,n+1isequaltothenumberofarrivalsinitsservicetime.Thus,wehaveEquation( 5 ). n+1=n)]TJ /F6 11.955 Tf 11.96 0 Td[(Dn+n+1(5)Dkisdenedasfollows: Dk=8><>:1fork>00fork0(5) WenowmakeanexpectationofbothsidesofEquation( 5 )andtakingthelimitasngoestoinnity,thenwehave E[De]=E[e](5) Sinceemeansthenumberofarrivalsduringaframe'sservicetimewhichisindependentofn,E[e]impliestheaveragenumberofarrivalsinaservicetime.Moreover,theleft-handsideofEquation( 5 )isgivenasfollowsandrepresentsaprobabilitythatthesystemisbusy. E[De]=1Xk=0DkP[e=k]=0P[e=0]+1P[e>0]=P[e>0](5) ThelasttermofEquation( 5 )alsoimpliestheprobabilityofabusysystemandcanbeinterpretedasasystemutilizationfactor.Thus,theaveragenumberofarrivalsper 71

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serviceintervalisidenticaltoandalsocanberepresentedintermsoftheaverageservicetime. E[De]=E[e]==E[ex](5) Letusnowproceedwithamethodforobtainingallthemomentsofarandomvariableeinequilibrium.Inordertodoso,wedenerstthemomentgeneratingfunction(z)whichisaz-transformfortherandomvariablee. (z),E[ze],1Xk=0P[e=k]zk(5) Thus, (z)=1Xk=0P[e=k]zk=1Xk=01Z0(x)k k!e)]TJ /F9 7.97 Tf 6.59 0 Td[(xfhygam(x)dxzk=1Z0e)]TJ /F9 7.97 Tf 6.59 0 Td[(x 1Xk=0(xz)k k!!fhygam(x)dx=1Z0e)]TJ /F9 7.97 Tf 6.59 0 Td[(xexzfhygam(x)dx=1Z0e)]TJ /F8 7.97 Tf 6.59 0 Td[(()]TJ /F9 7.97 Tf 6.58 0 Td[(z)xfhygam(x)dx(5)FLhygam(s)denotestheLaplacetransformoftheHyper-GammadistributionwhichisaservicetimedistributionasappearedinEquation( 5 ).Thuswehave (z)=FLhygam()]TJ /F6 11.955 Tf 11.96 0 Td[(z)(5) Equation( 5 )giveusaninsightthattherandomvariableeimpliesthenumberofarrivalsduringtheservicetimeintervalexwherethearrivalprocessisPoissonatanaveragearrivalrateofframespersecond.ThenthmomentoftheservicetimedistributionisgiveninEquation( 3 )andwealreadyknowtherelationshipbetweenthe 72

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z-transformandtheLaplacetransform,namely,s=0correspondstoz=1.Thuswehave FLhygam(k)(s)=()]TJ /F5 11.955 Tf 9.3 0 Td[(1)kdk dskFLhygam(s)s=0=()]TJ /F5 11.955 Tf 9.3 0 Td[(1)kE[exk](5) (1)(1)=d dz(z)z=1=E[e](5) (2)(1)=d2 dz2(z)z=1=E[e2])]TJ /F6 11.955 Tf 11.95 0 Td[(E[e](5) and FLhygam(0)=(1)=1(5) Ifweinvestigatethesecondmomentoftherandomvariablee,fromEquation( 5 )andEquation( 5 ),wecanobtainthefollowingrelationship. (2)(1)=d2 dz2(z)=d2 dz2FLhygam()]TJ /F6 11.955 Tf 11.95 0 Td[(z)=d dz)]TJ /F6 11.955 Tf 9.3 0 Td[(d dyFLhygam(y)=)]TJ /F6 11.955 Tf 9.3 0 Td[(d2 dy2FLhygam(y)dy dz=2d2 dy2FLhygam(y)y=0=2FLhygam(2)(0)=2E[ex2]=E[e2])]TJ /F6 11.955 Tf 11.95 0 Td[(E[e](5) whereweletybe)]TJ /F6 11.955 Tf 11.96 0 Td[(z. WenowattempttondtheaveragequeuesizeE[e]atframedepartureinstantsbysquaringEquation( 5 ). 2n+1=2n+Dn2+2n+1)]TJ /F5 11.955 Tf 11.96 0 Td[(2nDn+2nn+1)]TJ /F5 11.955 Tf 11.95 0 Td[(2Dnn+1(5) 73

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where(Dn)2equalsDnandnDnisidenticaltonfromEquation( 5 ).Alsothenumberofarrivalsduringthe(n+1)thserviceintervalisindependentofthenumberofframesleftbehindbythenthframe. Nowifwemakeanexpectationonbothsidesandtakealimit,thenwehave 0=E[De]+E[e2])]TJ /F5 11.955 Tf 11.96 0 Td[(2E[e]+2E[e]E[e])]TJ /F5 11.955 Tf 11.95 0 Td[(2E[De]E[e](5) FromEquations( 5 ),( 5 )and( 5 ),wehave E[e]=+E[e2])]TJ /F6 11.955 Tf 11.96 0 Td[(E[e] 2(1)]TJ /F6 11.955 Tf 11.95 0 Td[()=E[ex]+2E[ex2] 2(1)]TJ /F6 11.955 Tf 11.96 0 Td[(E[ex])(5) Consequently,fromEquations( 3 )and( 3 ),wehavetheaveragenumberofframesinthesystemformultimediatrafcwhichisspeciedbyPoissonarrivalprocessatanaveragerateofframespersecondandhavetheHyper-Gammaservicetimedistributionwithasingleserverisgivenasfollows: E[e]=NXi=1ikii+2NPi=1iki(ki+1)i2 2(1)]TJ /F9 7.97 Tf 15.13 11.36 Td[(NPi=1ikii)(5) LetusnowexamineLittle'slaw. N=T(5) where Nistheaveragenumberofframesinthesystem(includingaframeinservice)andandTistheaveragetimespentinthesystem(queueingtime+servicetime).Thus,fromEquation( 5 ),wehave N=E[e]=E[ex]+2E[ex2] 2(1)]TJ /F6 11.955 Tf 11.95 0 Td[(E[ex])=T(5) T=E[ex]+E[ex2] 2(1)]TJ /F6 11.955 Tf 11.96 0 Td[(E[ex])(5) 74

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Clearly,wecanmakesurethattheaveragetimespentinthesystemisidenticaltothesumoftheaveragetimespentintheserviceandtheaveragetimespentinthequeuefromEquation( 5 ).IfweapplyEquation( 5 )tooursystemwhichfollowsaPoissonarrivalprocessatarateofframespersecondandaHyper-Gammaservicetimedistribution,thenwehave T=NXi=1ikii+NPi=1iki(ki+1)i2 2(1)]TJ /F9 7.97 Tf 15.12 11.36 Td[(NPi=1ikii)(5) Ifwewanttoobtainthetheaveragenumberofframesinthequeue, Nq,(notincludingtheframeinservice),thenwehavetostartwiththedenitionof N. N=E[e]=1Xk=0kP[e=k](5) Thus, Nq=1Xk=1(k)]TJ /F5 11.955 Tf 11.95 0 Td[(1)P[e=k]=1Xk=1fkP[e=k])]TJ /F6 11.955 Tf 11.96 0 Td[(P[e=k]g=1Xk=0kP[e=k])]TJ /F7 7.97 Tf 16.35 14.94 Td[(1Xk=1P[e=k]= N)]TJ /F6 11.955 Tf 11.96 0 Td[((5) Finally,letusgoafterthelimitingdistributionfornwhichisrepresentedbypk(=P[e=k])inthesystem.Actually,weknowthattheprobabilitydkofndingkframesinthesystemjustafterthedepartureofaframe,isequaltopkfromEquation( 5 ).Therststepthatwehavetodoistodenethez-transformfortherandomvariablenanditslimitingrandomvariablee. Qn(z),E[zn],1Xk=0P[n=k]zk(5) 75

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Q(z),limn!1Qz(z)=1Xk=0P[e=k]zk=E[ze](5) Moreover,theM/HG/1queueingsystemisspeciedbyEquation( 5 ).Thus,weputbothsidesofEquation( 5 )toanexponentforzandtakeanexpectation. zn+1=zn)]TJ /F8 7.97 Tf 6.58 0 Td[(n+n+1(5) E[zn+1]=Qn+1(z)=E[zn)]TJ /F8 7.97 Tf 6.59 0 Td[(n+n+1](5) Sincetworandomvariablesnandn+1areindependenteachother,functionsofindependentrandomvariablesarealsoindependent,andwehavedened(z)inEquation( 5 )suchthat(z)=E[ze]=E[zn+1],wecanhavethefollowingform. Qn+1(z)=(z)E[zn)]TJ /F8 7.97 Tf 6.59 0 Td[(n](5) Now,wetakealookatthetermE[zn)]TJ /F8 7.97 Tf 6.58 0 Td[(n]. E[zn)]TJ /F8 7.97 Tf 6.59 0 Td[(n]=1Xk=0P[n=k]zk)]TJ /F8 7.97 Tf 6.58 0 Td[(k=P[n=0]+1Xk=1P[n=k]zk)]TJ /F8 7.97 Tf 6.58 0 Td[(1=P[n=0]+1Pk=0P[n=k]zk)]TJ /F6 11.955 Tf 11.95 0 Td[(P[n=0] z=P[n=0]+Qn(z))]TJ /F6 11.955 Tf 11.95 0 Td[(P[n=0] z(5) Thus,ifweputEquation( 5 )intoEquation( 5 ),takethelimitonbothsidesandsolveitforQ(z),thenwenallyhaveEquation( 5 )forQ(z),namely,thez-transformforthenumberofframesintheinnitecapacitysystem Q(z)=(z)P[e=0]+Q(z))]TJ /F6 11.955 Tf 11.95 0 Td[(P[e=0] z=(z)(1)]TJ /F6 11.955 Tf 11.96 0 Td[()(1)]TJ /F6 11.955 Tf 11.96 0 Td[(z) (z))]TJ /F6 11.955 Tf 11.95 0 Td[(z=FLhygam()]TJ /F6 11.955 Tf 11.96 0 Td[(z)(1)]TJ /F6 11.955 Tf 11.95 0 Td[()(1)]TJ /F6 11.955 Tf 11.95 0 Td[(z) FLhygam()]TJ /F6 11.955 Tf 11.96 0 Td[(z))]TJ /F6 11.955 Tf 11.96 0 Td[(z(5) 76

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wheretheprobabilityp0(=P[e=0])impliestheemptysystemandisequalto1)]TJ /F6 11.955 Tf 11.96 0 Td[(. 5.3ThroughputandDelayforNetworkSystemsBasedonMultimediaTrafc Sofar,whatwehaveobtainedistoanalyzetheperformanceofnetworksystemswhichisspeciedbymultimediatrafc.Letusnowconsiderthemeshnetworkinwhichweassumethataroutingpathhasalreadybeenestablished.Althoughanodeinmeshnetworksplaysbothrolesofaclientandarelaycenter,weassumethatallnodesinmeshnetworksworkonlyasanintermediatenode,namely,arelaycenterinwhichtheycannotgeneratetheirownframesformultimediatrafc,sincethereisusuallyjustonesourceforausertorequestoraccesstoamultimediasource.WealsoassumethatthearrivalprocessformultimediatrafcfollowsaPoissonprocessatarateofframespersecondsincewedonothaveinformationaboutthearrivalprocessinrealnetworksystemsyetandallnodesinmeshnetworksareidentical.Furthermore,wealsoassumethatthetypeofdataowisonewaytrafcwhichmeansthatthetrafcstreamisoneofup-streamingordown-streaming,thoughthereexistsabidirectionalstream(i.e.videoconference).WealsoconsiderthesituationwhichisdepictedinFigure 4-2 .Forexample,theupperexternalnodeisconsideredasakindofvideosourcenodeandthebottomexternalnodeistobeauserwhowantstorequestastreamingserviceforamovie.Additionally,theorderinwhichframesaretakenfromthequeue,allowedintoservice,anddeparturefromthesystemfollowstheFirstInFirstOut(FIFO)servicedisciplineonebyone.Consequently,basedontheaboveassumptions,wecancongureatandemmultimediatrafcnetworkasdepictedinFigure 5-1 ,wherethesourcenodeandthedestinationnodecorrespondtotheupperexternalnodeandauser,respectively.Letusnowattempttondthroughputanddelaywhicharecontributingtotheperformanceofnetworksystemsbasedonmultimediatrafc.Asdescribedinsection 1.2 ,throughputofanetworksystemisdenedastheaverageratewithwhichframesaresuccessfullytransferredthroughthechannelanddelaycanbeconsideredasasumofaveragewaitingtimesinqueuesandservicetimes 77

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Figure5-1. Innitecapacitynetworksystemsbasedonmultimediatrafcafterbuildingaroutingpath. innetworknodes.Moreover,incaseoftheM/HG/1system,weknowforafactthattheaveragearrivalratetoanodeisequaltotheaveragedepartureratefromthenodefromEquation( 5 )andeachnodehasaninnitecapacity,sothereisnopossibilitytodroparrivingframestoanynodes.Therefore,theaveragearrivalratetoanodeisidenticaltotheaveragedepartureratefromthatnodeandthisdepartureratebecomesanaveragearrivalratetothenextnodeandsoon.Consequently,throughputoftheM/HG/1systemisidenticaltotheaveragearrivalratetotherstnode,.Thenextonewehavetoconsiderisdelay.Theabovedenitionisconsideredonlyatasinglenode.But,inaviewpointofauseroradestinationnode,delayisconsideredasatotaltimespentfromthetimewhenthesourcenodestartstosendaframetothetimeuntilthedestinationnodereceivestheframe.SincethearrivalprocessfollowsaPoissonprocessatarateof,theinterarrivaltimeofeachframesisequalto1=.Therefore,inthecaseofthenetworkasinFigure 5-1 whichhasMintermediatenodes,thetotaltimespent(i.e.delay,Ttotal)fromthesourcenodetothedestinationnodeis Ttotal=(M+1) +M2664NXi=1ikii+NPi=1iki(ki+1)i2 2(1)]TJ /F9 7.97 Tf 15.12 11.35 Td[(NPi=1ikii)3775(5) InEquation( 5 ),theterminthesquarebracketscomesfromEquation( 5 )whichrepresentsatotaltimespent(queueandservice)atanodeintheM/HG/1system. Letusnowexaminenetworksystems(i.e.M/HG/1/K)basedonmultimediatrafc,whichhasnodeswithanitecapacityofsizeK(queue+service).Allother 78

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assumptionsremainsthesameasthecaseofnetworksystemswithaninnitecapacityofsize.Thiskindofsystemismorepracticalforrealnetworksystemsbecausetheinnitecapacityforanodeofnetworkscannotbeimplementable.Themainfeatureofthissystemisthatallnodesinthesystemcanhold,atmost,atotalofKframeswhichincludesaframeinserviceandanyfurtherarrivingframeswillbeblockedandmaybeconsideredasalostframe.Thus,anactualarrivalrateaisonlyafractionoftheaveragearrivalrateandisequaltotheactualdepartureratedwiththesamesituationinEquation( 5 ) a=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PB)=d(5) wherePBisablockingprobabilityduetothelimitationofanodecapacityK.Consequently,theactualthroughputfortheM/HG/1/Ksystemisaproductofaframearrival(=departure)rateandnon-blockingprobabilityforframes.Now,itisatimetoobtaintheblockingprobabilityPB.Fortunately,earlyresearchers,KeilsonandServi,havealreadyestablishedtheblockingprobabilityPBinthenitecapacitysystemwhichisrepresentedintermsoftheprobabilitythatanarrivingframendsgreaterthanequaltoKframesintheinnitecapacitysystem.[ 40 ]Letusnowdenethenotationsandtheirdenitionsoftheequilibriumprobabilitiesforthenitecapacitysystem: pkAprobabilityofthenumberofframesintheniteequilibriumsystem(k=0,1,2,,K)rkAprobabilitythatanarrivalndskframesintheniteequilibriumsystemwhetherornottheyenterthequeue(k=0,1,2,,K)dkAprobabilitythatadepartureleaveskframesintheniteequilibriumsystem(k=0,1,2,,K-1)rkAprobabilitythatanenteringframe(beingnotblocked)ndskframesintheniteequilibriumsystem(k=0,1,2,,K-1):Furthermore,weconsiderthefollowingrelationships.AccordingtothePASTAproperty[ 38 ],weobtaintheequalitypk=rk.BasedontheBurke'stheorem[ 41 ], 79

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wealsoacquiretheequalitydk=rkonaverage.Thus,fori
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Theonlydifferencebetweentwomatrices,PandPisafterthe(K)]TJ /F5 11.955 Tf 11.98 0 Td[(1)thcolumns,thusthestationaryprobabilityequationforanitecapacitysystemisidenticaltothatoftheinnitecapacitysystemuptothestate(K)]TJ /F5 11.955 Tf 12.06 0 Td[(1),whichimpliesthatthestateprobabilitiesinequilibriumforthenitesystemandtheinnitesystemareatworstproportionaluptothestate(K)]TJ /F5 11.955 Tf 11.95 0 Td[(1).Thuswehave dk=Cdk(5) and,theprobabilitiessumtoone,so C=1 K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Pj=0dj(5) Finally,wehave dk=dk K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Pj=0dj=dk d0+d1+d2++dK)]TJ /F8 7.97 Tf 6.59 0 Td[(1=pk p0+p1+p2++pK)]TJ /F8 7.97 Tf 6.59 0 Td[(1(5) wheredkandpkisdenedinEquation( 5 ).ThereforetheM/HG/1/KnitecapacitysystemisdirectlyrelatedtotheM/HG/1innitecapacitysystem.Ifwewanttondpk(=dk)whichrepresentsthesteady-stateprobabilityfortheinnitecapacitysysteminequilibrium,wecanobtainthefollowing. dk=d0ak+k+1Xj=1djak)]TJ /F9 7.97 Tf 6.59 0 Td[(j+1k=0;1;2;(5) Sinceak,whichcorrespondstoaprobabilitythattherandomvariableeisequaltok,isalreadydenedinEquation( 5 )andcanberepresentedasinEquation( 5 ),we 81

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obtain ak=1 k!dk dzk(z)z=0=1 k!dk dzkFLhygam()]TJ /F6 11.955 Tf 11.96 0 Td[(z)z=0=1 k!dk dzk"NXi=0ikii(i+)]TJ /F6 11.955 Tf 11.95 0 Td[(z))]TJ /F9 7.97 Tf 6.59 0 Td[(ki#z=0=1 k!NXi=0ikiik)]TJ /F8 7.97 Tf 6.59 0 Td[(1Yj=0(ki+j)k(i+))]TJ /F9 7.97 Tf 6.58 0 Td[(ki)]TJ /F9 7.97 Tf 6.58 0 Td[(kk>0(5) and a0=(z)jz=0=FLhygam()]TJ /F6 11.955 Tf 11.96 0 Td[(z)z=0=NXi=0ikii(i+)]TJ /F6 11.955 Tf 11.95 0 Td[(z))]TJ /F9 7.97 Tf 6.58 0 Td[(kiz=0=NXi=0ikii(i+))]TJ /F9 7.97 Tf 6.59 0 Td[(ki(5) Ifweattempttondthesteady-stateprobabilitydkinEquation( 5 )recursively,thenwecanobtainthefollowing: k=0;a0d1=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(a0)d0k=1;a0d2=d1)]TJ /F6 11.955 Tf 11.95 0 Td[(a1(d0+d1)k=2;a0d3=d2)]TJ /F6 11.955 Tf 11.95 0 Td[(a1d2)]TJ /F6 11.955 Tf 11.96 0 Td[(a2(d0+d1)k=3;a0d4=d3)]TJ /F6 11.955 Tf 11.95 0 Td[(a1d3)]TJ /F6 11.955 Tf 11.96 0 Td[(a2d2)]TJ /F6 11.955 Tf 11.96 0 Td[(a3(d0+d1)......(5) andingeneralform, a0dk+1="1)]TJ /F9 7.97 Tf 18.27 14.94 Td[(kXi=1ai# kXi=0di!+kXi=2dikXj=k)]TJ /F9 7.97 Tf 6.59 0 Td[(i+2aj(5) whered0isequalto1)]TJ /F6 11.955 Tf 12.13 0 Td[(,whichmeansthattheprobabilityofadepartureframeleavesbehindzeroframes,namely,anemptysystem. 82

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SofarwehavealltoolsrequiredinobtainingthroughputanddelayfortheM/HG/1/Ksystem.Theactualarrivalratea(=d)fortheM/HG/1/Ksystemisequalto(1)]TJ /F6 11.955 Tf 12.36 0 Td[(PB)asappearedinEquation( 5 )andtheprobability(1)]TJ /F6 11.955 Tf 12.72 0 Td[(PB)isgivenby1=(+d0)fromEquation( 5 ).InEquation( 5 )wehaved0=d0=K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Pi=0diandweacquireK)]TJ /F8 7.97 Tf 6.59 -.01 Td[(1Pi=0di=K)]TJ /F8 7.97 Tf 6.58 .01 Td[(1Pi=0 d0ai+K+1Pj=0djai)]TJ /F9 7.97 Tf 6.59 0 Td[(j+1!fromEquation( 5 ).Consequently,wehave a= +d0 K)]TJ /F10 5.978 Tf 5.76 0 Td[(1Pi=0 d0ai+K+1Pj=0djai)]TJ /F11 5.978 Tf 5.75 0 Td[(j+1!(5) whereakanddkcanbeobtainedfromEquation( 5 )toEquation( 5 ).Inequilibrium,theaveragenumberofframesNfortheM/HG/1/Ksystemisgiven N=KXk=0kpk=K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xk=0kpk+KpK(5) FromEquations( 5 ),( 5 )and( 5 ),wehave N=1 (+d0)K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xk=0kdk+KPB=1 (+d0)K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xk=0kdk+K1)]TJ /F5 11.955 Tf 33.09 8.09 Td[(1 (+d0)=K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Pk=0kdk+K[+d0)]TJ /F5 11.955 Tf 11.96 0 Td[(1] (+d0)(5) TheaveragetimespentWinthissystemcanbeacquiredbyusingLittle'slaw. W=N a=K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Pk=0kdk+K[+d0)]TJ /F5 11.955 Tf 11.95 0 Td[(1] (5)Letusnowdenethefollowingnotations. 83

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Figure5-2. Finitecapacitynetworksystemsbasedonmultimediatrafcafterbuildingaroutingpath. Arrivalratetonode1iArrivalratetonode(i+1)=DepartureratefromnodeiWiAveragetimespentinnodei(=queue+service)WTiAveragetimespentfromadepartureinstantatnode(i-1)toadepartureinstantatnodeiPBiBlockingprobabilityatnodei1(=E[ex])Utilizationfactoratnode1i(=i)]TJ /F8 7.97 Tf 6.59 0 Td[(1E[ex])Utilizationfactoratnodeidi;0Probabilitythatadeparturefromnodeileavesbehindanemptysystemdi;kProbabilitythatadeparturefromnodeileavesbehindkframesinthesystemTherefore,throughputateachnodeisgivenasfollows: 1=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PB1)= (1+d1;0)i=i)]TJ /F8 7.97 Tf 6.58 0 Td[(1(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PBi)=i)]TJ /F10 5.978 Tf 5.75 0 Td[(1 (i+di;0)M=M)]TJ /F8 7.97 Tf 6.59 0 Td[(1(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PBM)=M)]TJ /F10 5.978 Tf 5.76 0 Td[(1 (M+dM;0)(5) 84

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andtheaveragetimespentfromadepartureinstantatnode(i)]TJ /F5 11.955 Tf 12.86 0 Td[(1)toadepartureinstantatnodeiisgivenasfollows: WT1=1 +W1WTi=1 i)]TJ /F10 5.978 Tf 5.75 0 Td[(1+WiWTM=1 M)]TJ /F10 5.978 Tf 5.75 0 Td[(1+WM(5) where W1=1 K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Pk=0kd1;k+K)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1+d1;0)]TJ /F5 11.955 Tf 11.96 0 Td[(1Wi=1 i)]TJ /F10 5.978 Tf 5.75 0 Td[(1K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Pk=0kdi;k+K)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(i+di;0)]TJ /F5 11.955 Tf 11.95 0 Td[(1WM=1 M)]TJ /F10 5.978 Tf 5.76 0 Td[(1K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Pk=0kdM;k+K)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(M+dM;0)]TJ /F5 11.955 Tf 11.96 0 Td[(1(5) Thusthetotaldelayforthissystemis WT=1 M+MXi=1WTi(5) 5.4ResultsandDiscussion Sofar,wehavestudiedtwotypesofnetworksystems,M/HG/1andM/HG/1/K,whichhavetheHyper-Gammaservicetimedistributionwithaninnitecapacityandanitecapacity(ofsizeK)inthequeue.Moreover,weassumethatthearrivalprocesstoanetworknodefollowsaPoissonprocess,sincewedonothaveinformationaboutthearrivalprocessinrealnetworksystemsyet.BothsystemsareconsideredasaspecialcaseoftheM/G/1queueingsystemandthedynamicbehaviorofthissystemcanbespeciedatitsimbeddedpoint.(i.e.thedepartureinstantofaframefromtheserver.)Throughputoftheinnitecapacitysystemisequaltotheaveragearrivalratebecauseoftheassumptionforthearrivalprocessandenoughspacetoacceptarrivingframes.Totaldelayisrepresentedintermsoftheaveragearrivalrate,therst,andsecondmomentsoftheHyper-Gammaservicetimedistribution.But,inthenitecapacitysystem,throughputateachnodeisexpressedasaproductoftheaveragearrivalratetothatnodeanditsnon-blockingprobability.Furthermore,delayineachnodedependson 85

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thetotaltimespentinthepreviousnodeanditsexpressionhasaquitecomplicateformasgiveninEquationsfrom( 5 )to( 5 ). 86

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CHAPTER6CONCLUSIONSANDFUTURERESEARCHDIRECTION 6.1Conclusions Multimediatrafchasbecomeoneofthemajorsourcesofnetworktrafcloadsduetothemassivenessofsize.Soitisquitecumbersometoanalyzeandhandleitsproperties.Thusweneedananalyticalandtractablemodeltospecifyitsstochasticpropertieswhichdirectlyaffectstheperformanceofnetworksystems.Inthisdissertation,thestatisticalcharacteristicofmultimediatrafchasbeenstudiedforhowitcontributestodynamicbehaviorofnetworksystemsbasedonmultimediatrafcows. Astherststep,wehaveobtainedbasicelementswhichconsistofmultimediatrafc,namely,astreamofframesfromcommercialDVDmovies.Sincemakingahistogramisconventionallyanaturalandfundamentalwayofrepresentingasetofempiricaldatadrawnfromarealworldandonecanaffordtoestimateastatisticalcharacteristicofdatawiththistechnique,webuildanempiricalhistogramforastatisticalanalysisofempiricaldata.Theresultofempiricalhistogramsforthemultimediaframesizeshowsthattheinherentstatisticalcharacteristicisnotaunimodalproperty.Sinceamodeofhistogramsisnotunique,weinterpretthesephenomenaasamixtureofmorethanoneunimodaldistributionwhichhasonlyonemode.Amongthesekindsofmixturetypedistributions,weselecttheHyper-Gammadistributionasacandidatemodelformultimediatrafcbecauseofitsversatility.Byvaryingasetofshapeandscaleparameters,wecanobtaintheHyper-Exponential,theHyper-Erlang,andtheHyper-ChiSquaredistributionsand,moreover,theHyper-ExponentialandHyper-Erlangdistributionhavebeenintroducedinresearchareasofaperformanceanalysisforwirelesscommunicationnetworks.Consequently,afteranexaminationfortheframesizedistributionofalargequantityofDVDmovies,wearefacedwiththefactthatthestatisticalmodelfortheframesizedistributioncontributingtomultimediatrafcfollowstheHyper-Gammadistribution. 87

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Asthenextstep,wehaveappliedthisresulttonetworksystems.Whenwegenerallyattempttoanalyzethedynamicbehaviorofnetworksystems,weneedexponentialassumptionsinanarrivalprocessandaservicetimedistributiontoobtaintheoreticalresults.OnesuchnetworkwiththeseassumptionsistheJacksonnetwork.Unfortunately,oneoftheseassumptionsorbothofthemseldomhappeninarealworldandthelackoftheexponentialassumptionmakesitdifcultinanalyzingthenetworkperformance.Moreover,wemayintuitivelyknowthefactthattheservicetimerequiredforanodetohandlemovieframesinnetworksystemsisproportionaltothesizeofmovieframesandalsofollowstheHyper-Gammadistribution.Thus,wehavetondanothermethod,ratherthantheJackson'sapproach,toobtainaperformanceofsuchnetworksystembasedonmultimediatrafc.Consequently,weconsideratandemnetworkwithsomeassumptions;i)aroutingpathhasalreadybeenestablished.ii)allnodesinthenetworksystemareidenticalandplayaroleofarelaycenterinwhichtheycannotgeneratetheirowndataframeformultimediatrafc.iii)thearrivalprocessformultimediatrafcfollowsaPoissonprocessatarateofframespersecond.iv)atypeofdataowisonewaytrafcwhichmeansthatthetrafcstreamisoneofup-streamingordown-streaming.v)theservicedisciplinefollowsFIFOserviceonebyone.ThereforewehaveatandemnetworkwiththeM/HG/1andtheM/HG/1/Kqueueingnodes.Asaresult,throughputfortheinnitecapacitysystemisequaltoanaveragearrivalratetotherstnodebecauseoftheassumptionforthearrivalprocessandenoughroomforanentranceintoanodeforarrivingframes.Totaldelayisrepresentedintermsoftheaveragearrivalrate,therst,andsecondmomentsoftheHyper-Gammaservicetimedistribution.Inthecaseofthenitecapacitysystem,throughputateachnodeisrepresentedintheproductformoftheaveragearrivalratetothatnodeanditsnon-blockingprobability.Furthermore,delayineachnodedependsonthetotaltimespentinthepreviousnode. 88

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6.2FutureResearchDirection Ifwewanttomorespecicallydescribethedynamicbehaviorofnetworksystemswithmultimediatrafcinarealworld,weneedtogetridoftheassumptionswhichareconsidered.Therstassumptionisthataroutingpathhasalreadybeenestablished.Thisimpliesthatatransitionprobabilityforaframefromanodetoanodeisalreadyxedandisidenticaltoone.Ifthetransitionprobabilityistakenintoaccountforthenetworksystem,theactualarrivalratetoacertainnodewillbeslightlychangedsincethearrivalratetoacertainnodeinnetworksystemsisrepresentedasaproductofadeparturerateofthepreviousnodeandthetransitionprobabilityfromthatnodetoacurrentnode.Generally,thistransitionprobabilityisrelatedtoaMediumAccessControl(MAC)inarealworld.Thesecondonewehavetoconsideristhateachnodeinthenetworksystemworksonlyasarelaycenter.Butthenetworkfollowsadistributedsystemoralesharingarchitecture,forexample,PeertoPeer(P2P)system,weneedtoconsiderthequeuewhichisalreadystoredinacertainnode.Inthiscase,therearetwotypesofnodes;anodeinwhichhasnorelayedframesandanodeinwhichhasrelayedframes.Incaseofhavingnorelayedframesinanode,theactualarrivalratetothenextnodeisequaltotheinterarrivaltimeforframeswhicharealreadystoredinthepreviousnode.Incaseofanodeinwhichhasrelayedframes,theactualarrivalratetothenextnodecouldbeidenticaltothesumofaproductofeachprobabilitytotransferastoredframeandtotransferarelayedframeandaframedepartureratefromthatnode. Sofar,wehaveconductedananalysiswithaservicetimedistributionfromrealdataandthisisnotenoughtopredictadynamicbehaviorofthenetworksystembecauseaserviceprocessandanarrivalprocessareindependent.Therefore,thenextassumptionweneedtoconsideristhatthearrivalprocessformultimediatrafcfollowsaPoissonprocessatarateof.Unfortunately,wedonothaveinformationaboutthearrivalprocessinrealnetworksystemsyet,thusweneedtomeasuretheactualarrivalrateformultimediatrafctoacertainnodeinthenetworksystem.Forthispurpose,weneed 89

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toobtainanaverageinterarrivaltimebetweenframesbecausetheaveragearrivalrateisareciprocaloftheaverageinterarrivaltime.Thisworkcouldbeaccomplishedbyapacketsnifngmethodsuchasusinga'scapy'onlinuxoracommercialtrafcanalyzingprogram.Justforasimulation,onesimplewayofchoosingthearrivalprocessistomakeuseoftheeventnetworksimulatorsuchasNetworkSimulatorseries(i.e.NS2orNS3)tosynthesizetheaveragearrivalrate.Inthiscase,weneedtotakecareofthesystemutilizationfactor,sincethesystemutilizationfactormustbelessthanequaltounityforasystemstability.Thereforetheaveragearrivalratemustbelessthanequaltotheaverageservicetime.Finally,ifweconsiderallsituationsdescribedintheabove,wecanpredictthedynamicbehaviorofnetworksystemsspeciedwithmultimediatrafcmoreaccurately. 90

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[15] ISO,InformationtechnologyGenericcodingofmovingpicturesandassociatedaudioinformation:Video,ISO/IEC13818-2,2000. [16] A.Leon-GarciaandI.Widjaja,CommunicationNetworks:FundamentalConceptsandKeyArchitectures.McGraw-Hillseriesincomputerscience,McGraw-Hil,2006. [17] J.Mitchell,W.Pennebaker,C.Fogg,andD.LeGall,MPEGVideoCompressionStandard.ChapmanandHall,NY,1997. [18] D.FreedmanandP.Diaconis,Onthehistogramasadensityestimator:L2theory,ProbabilityTheoryandRelatedFields,vol.57,pp.453,1981. [19] P.OrlikandS.Rappaport,Amodelforteletrafcperformanceandchannelholdingtimecharacterizationinwirelesscellularcommunicationwithgeneralsessionanddwelltimedistributions,SelectedAreasinCommunications,IEEEJournalon,vol.16,pp.788,Jun.1998. [20] A.R.Liddle,Informationcriteriaforastrophysicalmodelselection,MonthlyNoticesoftheRoyalAstronomicalSociety:Letters,vol.377,no.1,pp.L74L78,2007. [21] D.PellegandA.Moore,X-means:ExtendingK-meanswithEfcientEstimationoftheNumberofClusters,inProceedingsoftheSeventeenthInternationalConferenceonMachineLearning,pp.727,Jul.2000. [22] W.Stallings,IEEE802.11:WirelessLANsfromaton,ITProfessional,vol.6,pp.32,Sep.-Oct.2004. [23] ISO/IECStandardforInformationTechnology-TelecommunicationsandInformationExchangeBetweenSystems-LocalandMetropolitanAreaNetworks-SpecicRequirementsPart11:WirelessLANMediumAccessControl(MAC)andPhysicalLayer(PHY)Specications(IncludesIEEEStd802.11,1999Edition;IEEEStd802.11A.-1999;IEEEStd802.11B.-1999;IEEEStd802.11B.-1999/Cor1-2001;andIEEEStd802.11D.-2001),ISO/IEC8802-11IEEEStd802.11Secondedition2005-08-01ISO/IEC880211:2005(E)IEEEStd802.11i-2003Edition,pp.1,2005. [24] IEEEStandardforInformationTechnology-TelecommunicationsandInformationExchangeBetweenSystems-LocalandMetropolitanAreaNetworks-SpecicRequirementsPartIi:WirelessLANMediumAccessControl(MAC)andPhysicalLayer(PHY)Specications,IEEEStd802.11g-2003(AmendmenttoIEEEStd802.11,1999Edn.(Reaff2003)asamendedbyIEEEStds802.11a-1999,802.11b-1999,802.11b-1999/Cor1-2001,and802.11d-2001),pp.i,2003. [25] T.PaulandT.Ogunfunmi,WirelessLANComesofAge:UnderstandingtheIEEE802.11nAmendment,CircuitsandSystemsMagazine,IEEE,vol.8,pp.28,quarter2008. 92

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BIOGRAPHICALSKETCH KyungwooKimreceivedtheB.S.degreeinelectricalandcomputerengineeringfromKyungpookNationalUniversity,Daegu,SouthKorea,in2000,theM.S.degreesinelectricalandcomputerengineeringfromKyungpookNationalUniversity,Daegu,SouthKorea,in2002,andUniversityofFlorida,Gainesville,Florida,USA,in2005,respectively.DuringhisPh.D.career,heparticipatedintheProgramCommitteeofthe15thWorldMulti-ConferenceonSystemics,CyberneticsandInformatics,RevieweroftheIEEEInternationalSymposiumonPowerLineCommunicationsandItsApplications,andRevieweroftheWorldMulti-ConferenceonSystemics,CyberneticsandInformatics.Hisresearchinterestsarenetworktrafcmodelingandanalysis,networkroutingalgorithm,andP2Pnetwork. 95