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A Framework Model for Packet Loss Metrics Based on Loss Runlengths

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A Framework Model for Packet Loss Metrics Based on Loss Runlengths
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  AFrameworkModelforPacketLossMetricsBasedonLoss Runlengths    H.SanneckandG.Carle GMDFokus,Kaiserin-Augusta-Allee31,D-10589Berlin,Germany  ABSTRACT  Forthesamelong-termlossratio,dierentlosspatternsleadtodierentapplication-levelQualityofService(QoS) perceivedbytheusers(short-termQoS).Whilebasicpacketlossmeasureslikethemeanlossratearewidelyused intheliterature,muchlessworkhasbeendevotedtocapturingamoredetailedcharacterizationofthelosspro- cess.Inthispaper,weprovidemeansforacomprehensivecharacterizationoflossprocessesbyemployingamodel thatcaptureslossburstinessanddistancesbetweenlossbursts.Modelparameterscanbeapproximatedbasedon run-lengthsofreceived/lostpackets.Weshowhowthemodelservesasaframeworkinwhichpacketlossmetrics existingintheliteraturecanbedescribedasmodelparametersandthusintegratedintothelossprocesscharacteri- zation.Variationsofthemodelwithdierentcomplexityareintroduced,includingthewell-knownGilbertmodelas aspecialcase.FinallyweshowhowourlosscharacterizationcanbeusedbyapplyingittoactualInternetlosstraces. Keywords:  PacketLoss,Short-TermQoSMetrics,LossBurstiness,InternetReal-TimeServices  1.INTRODUCTION  ExistingworkonInternetQualityofService(QoS)identieddelay,jitterandpacketlossasmetricsofinterestfor usersandoperators(  1  ).Amongthese,packetlossisanimportantmetricinthestatisticallysharedenvironmentof theInternet,wherethedemandforaresource(suchasbandwidthorbuer)mayexceedtheavailablecapacity.For multimediaapplicationssuchascodedaudioandvideo(liveorstored),lossmayalsoresultfrominordinatedelayin thenetworkwhichcausestheplay-outdeadlineofasampleoraframetobemissed.Whileexistingworkfocuseson capturingthemeanloss(long-termQoS),lessemphasisisputonmodellinglossdistribution(short-termQoS,cf.(  2,3  ) andthereferencestherein).Itisimportanttonotethatforthesamemeanloss,dierentlosspatternscanproduce dierentperceptionsofQoS,asdescribedin(  4{8  ).Also,asshownbyCidonetal.(  6  )andShacham/McKenney(  9  ),manyforwarderrorrecoveryapproachesbecomelessecientasthelossburstinessincreases.Thus,itisimportantto beabletocapturetheactuallossprocesswithsuitable(andsimple)metrics.Asanexample,Fig.1showsmeanloss rates   p  m  (  s  )foravoicestreamversusitssequencenumber  s  averagedwithaslidingwindowofveand100packets respectively(   p  5  (  s  ),  p  100  (  s  )).Itcanbeseenthatthedistributionoflossrates(andthustheperceptualimpact)over asmallwindowsize(   p  5  (  s  ))variesstrongly.Atthesametime,themeanlossrateevaluatedoverthelargerwindow size(   p  100  (  s  ))varieswithinamuchsmallerinterval.Themeanlossratecalculatedoveralargetimeintervalisthus notsuitabletorevealdierencesofperceivedQoS.Thus,ouraiminthispaperistodevelopamodelwiththefollowingproperties:   possibilityofexpressingwell-knownlong-andshort-termQoSmetrics(likee.g.thoseoftheGilbertmodel);   adjustablecomplexitydependentonspecicapplication/networkrequirements,supportingadditionalmetrics fordetailedshort-termQoSanalysis;   coverageofbothlossburstiness,aswellasdistancesbetweenlosses.Inthefollowingsection,wedescribeacharacterizationofthelossprocess,anddevelopananalyticalframework usingamodelbasedonlossandno-lossrun-lengths.Insection3,weshowtheapplicabilityofthedevelopedmeasures toactualInternetlosstraces.Finally,weconcludeinSection4withasummaryandpointoutdirectionsforfuture research.   Toappearin  ProceedingsoftheSPIE/ACMSIGMMMultimediaComputingandNetworkingConference2000(MMCN2000)  ,SanJose,CA,January2000  1   010020030040050060070080090000.10.20.30.40.50.60.70.80.91sequence number s   m  e  a  n   l  o  s  s  r  a   t  e  s  o  v  e  r  w   i  n   d  o  w  m p 100 (s)p 5 (s) Figure1.  MeanLossRatesforavoicestreamaveragedover5and100packetsrespectively  2.LOSSPROCESSCHARACTERIZATIONANDSHORT-TERMQOSMETRICS  Inordertocaptureandcontrolthelossprocessitself,short-termQoSmetricsareneeded.Suchmetricsareuseful inevaluatingthenetworkperformanceasperceivedbytheendusers.ExistingworkonQoSmetrics(  10{13  )does notprovideaframeworkthatintegratesthesemetrics,providesacommonnotationandallowsfortheirsimple computation.Inthefollowingwedeneamodelthatservesassuchaframework.Webeginwithneededdenitions:The  lossindicatorfunction  foracertainow  y  atacertainnodedependentonthepacketsequencenumber  s  is: l  (  s  )=    0:packet  s  isnotlost 1:packet  s  islost Wedenea  lossrunlength  k  forasequenceof  k  consecutivelylostpacketsdetectedat  s  j  (  s  j  >k>  0)with  l  (  s  j  ?  k  ?  1)=0  ;l  (  s  j  )=0and  l  (  s  j  ?  k  +  i  )=1  8  i  2  0  ;k  ?  1], j  beingthe  j  -th\burstlossevent".Theoccurenceof alossrunlength  k  isgivenby  o  k  .Thusforagivennumberofpackets  a  ofaowthatexperience  d  =  1  X    k  =1  ko  k  packet drops,wehavethe  relativefrequency   p  L;k  =  o  k  a  fortheoccurenceofalossburstoflength  k  andthemeanlossrate   p  L  =  1  X    k  =1  kp  L;k  . 2.1.Lossrun-lengthmodelwithunlimitedstatespace  Wedenetherandomvariable  X  asfollows: X  =0:\nopacketlost", X  =  k  :\  exactly  z  k  consecutivepacketslost",and  X    k  :\  atleast  x  k  consecutivepacketslost".Withthisdenition,weestablishalossrun-lengthmodel(Fig.2) withanunlimited(possiblyinnite)numberofstates,whichgiveslossprobabilitiesdependentontheburstlength  {  .Inthemodel,foreveryadditionallostpacketwhichaddstothelengthofalossburstastatetransitiontakesplace.Ifapacketissuccessfullyreceived,thestatereturnsto  X  =0.Thusthestateprobabilityofthesystemfor  k>  0is  P  (  X    k  ).Giventhecaseofanitenumberofarrivalsforaow  a  ,introducedinthepreviousparagraph,wecan approximatethestateprobabilitiesofthemodelfor  k>  0bythecumulativelossrate   p  L;cum  (  k  )=  1  X    n  =  k   p  L;n  (Table1). y  Inourdenition,aowisanapplication-layerdatastream.ForIPv4itcanbeidentiedbythetuple  (sourceaddress,destination address,protocolID,sourceport,destinationport)  . z  \Exactly"meansthatthetwopacketsimmediatelyprecedingandfollowingthe  k  lostpacketsarenotlostwithprobability1. x  \Atleast"meansthatthepacketimmediatelyprecedingthe  k  lostpacketsisnotlostwithprobability1. {  ThebasicmodelissimilartotheoneemployedbyVarma(  14  )andHsuetal.(  15  ). 2   ==X= mX= 2= 20m01223 pppp (m-1)m p 00X=001 p X= 1 p=1p 10 Figure2.  Lossrun-lengthmodelwithunlimitedstatespace(  m  !1  ) Lossrun-lengthmodel  a  arrivals  a  !1  (unlimitedstates) burstloss(  k>  0)   p  L;k  =  o  k  a P  (  X  =  k  ) meanloss   p  L  =  1  X     k  =1  kp  L;k  E    X  ]cumulativeloss(  k>  0)   p  L;cum  (  k  )=  1  X     n  =  k   p  L;n  P  (  X    k  ) (stateprob.) conditionalloss(  k>  1)   p  L;cond  (  k  )=   p  L;cum  (  k  )   p  L;cum  (  k  ?  1) =  P    1  n  =  k  o  n  P    1  n  =  k  ?  1  o  n  P  (  X    k  j X    k  ?  1) (statetransitionprob.  p  (  k  ?  1)(  k  )  ) burstlosslength(  k>  0)  g  k  =  o  k  P    1  n  =1  o  n  P  (  Y  =  k  ) meanburstlosslength  g  =  d  P    1  k  =1  o  k  =  P    1  k  =1  ko  k  P    1  k  =1  o  k  =  1  X     k  =1  kg  k  E    Y  ] Table1.  QoSmeasuresforthelossrun-lengthmodelwithunlimitedstatespace Thematrixofstatetransitionprobabilitiesforthismodelisgivenby  2  6 6 6 6 6 6 6 6 6 4    p  00   p  01  0   0   p  10  0   p  12  ...  p  20  00 ..................0   p  (  n  ?  2)0  0   0   p  (  n  ?  2)(  n  ?  1)   p  (  n  ?  1)0  00   0  3 7 7 7 7 7 7 7 7 7 5  Thetransitionprobabilitiesfor  k>  1whichcanalsobedescribedasconditionallossprobabilitiescanbecomputed easilyas:  p  (  k  ?  1)(  k  )  =  P  (  X    k  j X    k  ?  1)=  P  (  X    k  \  X    k  ?  1)  P  (  X    k  ?  1) =  P  (  X    k  )  P  (  X    k  ?  1) Again,iftheburstlossoccurences  o  k  constituteastatisticallyrelevantdataset,wecancomputeapproximations fortheconditionallossprobabilitiesasgiveninTable1.Additionally,wealsodenearandomvariable  Y  whichdescribesthedistributionofburstlosslengthswithrespect totheburstlossevents  j  (andnottopacketeventslikeinthedenitionof  X  ). E    Y  ]thenistheexpectedmeanburst losslength(lossgap).Table1showstheperformancemeasuresofthelossrun-lengthmodelforanitenumberof 3   =X= 2=20m01223 ppppp (m-1)mX=001 p = p 00 p mmX= 1X= m p 10 Figure3.  Lossrun-lengthmodelwithlimitedstatespace(  m  +1)states]arrivals  a  usingthelossrunlengthoccurences  o  k  ,aswellastherelationtothetransition/stateprobabilitiesofthe model(  a  !1  )withtherandomvariables  X  and  Y  .Thecumulativelossratefor  k  =0isdenedasthe\noloss" case(correspondingto  P  (  X  =0)):  p  L;cum  (  k  =0)=1  ?  1  X  k  =1   p  L;cum  (  k  )=1  ?  1  X  k  =1  1  X  n  =  k  o  n  a  =1  ?  1  X  k  =1  ko  k  a  =1  ?   p  L  2.2.Lossrun-lengthmodelwithlimitedstatespace  Toassesstheperformanceofanetworkwithrespecttoreal-timeaudioandvideoapplications,amodelwithalimited numberofstatesissucient.Thisisduetothefactthatreal-timeaudioandvideoapplicationshavestrictrequire- mentsandcannotuseanetworkservicewithasignicantnumberof\long"lossbursts.Fortheseapplicationsitis desirabletouseonlyfewmodelparameters,andtofocusonkeyaspectsofthelossprocess.Inaddition,memoryand computationalcapabilitiesofthesystemthatperformsmodellinghavetobetakenintoaccount(seealsoparagraph 2.6).Forthesereasonswederivefromthebasicmodelalossrun-lengthmodelwithanitenumberofstates(  m  +1).Fig.3showstheMarkovchainforthemodel.Table2givesperformancemeasuressimilartothoseinTable1  k  ,howeverthestateprobabilityforthenalstate  m  (  P  (  X  =  m  ))andtheprobabilityforatransitionfromstate  m  tostate  m  areadded.For0  <k<m  , X  =  k  representsasbefore\  exactly  k  consecutivepacketslost".Duetothe limitedmemoryofthesystem,thelaststate  X  =  m  isjustdenedas\  m  consecutivepacketslost".Thus  P  (  X  =  m  ) canbeseenasameasureforthe\lossoverawindowofsize  m  "(independentlyofactuallylargerlossrunlengths).Figure4showsthebasemeasuresusedtocomputethelossrun-lengthbasedmetrics.InFigure4(a),eachpoint indicateswhethertherewasaloss(1)ornot(0),representingthelossindicatorfunction.Figure4(b)showstheloss runlengths.Figure4alsoshowssomeofthestatetransitionswhenagivenlosstraceisappliedtoamodelofeither  m  =2or  m    4.With  m  =2foralossburstoflength  k  =4,thesystemisthreetimes(  k  ?  m  +1,seeFig.4(c)) instate2,andthustwo(  k  ?  m  )transitions  m  !  m  occur.Thisleadstothecomputationof   p  L;m  and   p  L;cond  (  m  ) (asapproximationsfor  P  (  X  =  m  )and  P  (  X  =  m  j X  =  m  )respectively)asgiveninTable2.Interestingly,Miyataetal.(  12  )proposeprecisely   p  L;m  asaperformancemeasureforFEC-basedaudioapplica- tions.This"slidingwindow"of  m  consecutivelylostpacketsallowstoreectspecicapplications'constraints,e.g. m  canbesettothelowestnumberofconsecutivelylostpacketsforwhichacompleteaudio\dropout"isperceived byauser.Then,largerlossburstsdonothaveahigherimpactandthusdonotneedtobetakenintoaccountwith theirexactsize.Weextendtheaboveapproachbylookingattheoccurenceofacertainnumberofpacketslost withinthewindowoflength  m  .Thisallowse.g.toassesshoweectiveFECprotectionappliedtogroupsofpackets wouldbewithoutkeepingtrackoftheactualApplicationDataUnit(ADU)associationoftheindividualpackets.In section1weintroducedthemeanlossrateoveraslidingwindowoflength  m  (   p  m  (  s  ))whichcanbedenedasthe convolutionoftheanalysiswindowwiththelossindicatorfunction(Fig.5): k  TheburstlosslengthmeasuresarecomputedasinTable1andarethereforenotshown. 4   Lossrun-lengthmodel  a  arrivals  a  !1  (  m  +1states) burstloss(0  <k<m  )   p  L;k  =  o  k  a P  (  X  =  k  ) burstloss(  k  =  m  )   p  L;m  =  1  X  n  =  m  (  n  ?  m  +1)  o  n  a  ?  m  +1  P  (  X  =  m  ) lossoverwindow  m  (stateprobability) meanloss   p  L  =  1  X  k  =1  ko  k  a E    X  ]cumulativeloss   p  L;cum  (  k  )=  1  X  n  =  k  o  k  a P  (  X    k  ) (0  <k<m  )(stateprobability) conditionalloss   p  L;cond  (  k  )=   p  L;cum  (  k  )   p  L;cum  (  k  ?  1) =  P  1  n  =  k  o  n  P  1  n  =  k  ?  1  o  n  P  (  X    k  j X    k  ?  1) (1  <k<m  )(statetransitionprob.  p  (  k  ?  1)(  k  )  ) conditionalloss   p  L;cond  (  m  )=  1  X  n  =  m  (  n  ?  m  )  o  n  d  ?  m P  (  X  =  m  j X  =  m  ) (  k  =  m  )(statetransitionprob.  p  mm  )  Table2.  QoSmeasuresforlossrun-lengthmodelwithlimitedstatespace(  m  +1)states] k   j 011000012340123414 j[m=2]  j (k -m+1) +  c)window m=2 1sl(s)  a)  j14  b) state transitions m>=4m=201100001  22 20 12 22 Figure4.  Basiclossmeasures 5 
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