Web caching and zipf-like distributions Evidence and implications

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WebCachingandZipf-likeDistributions:Evidenceand

Implications

LeeBreslauPeiCaoLiFanGrahamPhillipsScottShenker

Abstract

ThispaperaddressestwounresolvedissuesaboutWebcaching.ThefirstissueiswhetherWebre-questsfromafixedusercommunityaredistributedaccordingtoZipf’slaw[Zip29].Severalearlystudieshavesupportedthisclaim[Gla94,CBC95,ABCdO96]whileotherrecentstudieshavesuggestedother-wise[NHo98,ACC98].ThesecondissuerelatestoanumberofrecentstudiesonthecharacteristicsofWebproxytraces,whichhaveshownthatthehit-ratiosandtemporallocalityofthetracesexhibitcer-tainasymptoticpropertiesthatareuniformacrossthedifferentsetsofthetraces[CI97,RV98,DMF97,GB97a,KLM97].Inparticular,thesecondissueiswhetherthesepropertiesareinherenttoWebaccessesorwhethertheyaresimplyanartifactofthetraces.AnanswertotheseunresolvedissueswillfacilitatebothWebcacheresourceplanningandcachehierarchydesign.Weshowthattheanswerstothetwoquestionsarerelated.WefirstinvestigatethepagerequestdistributionseenbyWebproxycachesusingtracesfromavarietyofsources.Wefindthatthedistribu-tiondoesnotfollowZipf’slawprecisely,butinsteadfollowsaZipf-likedistributionwiththeexponentvaryingfromtracetotrace.Furthermore,wefindthatthereis(i)aweakcorrelationbetweentheaccessfrequencyofaWebpageanditssizeand(ii)aweakcorrelationbetweenaccessfrequencyanditsrateofchange.WethenconsiderasimplemodelwheretheWebaccessesareindependentandthereferenceprobabilityofthedocumentsfollowsaZipf-likedistribution.Wefindthatthemodelyieldsasymptoticbehaviorsthatareconsistentwiththeexperimentalobservations,suggestingthatthevariousobservedpropertiesofhitratiosandtemporallocalityareindeedinherenttoWebaccessesobservedbyproxies.Finally,werevisitWebcachereplacementalgorithmsandshowthatthealgorithmthatissuggestedbythissimplemodelperformsbestonrealtracedata.Theresultsindicatethatwhilepagerequestsdoindeedrevealshort-termcorrelationsandotherstructures,asimplemodelforanindependentrequeststreamfollowingaZipf-likedistributionissufficienttocapturecertainasymptoticpropertiesobservedatWebproxies.

1Introduction

DuetotheexplosivegrowthoftheWeb,Webproxycachinghasrecentlyreceivedconsiderableattention.

ItisconsideredoneofthemostimportanttechniquesatreducingWebtraffic,whichaccountsforalarge

percentageofInternettraffictoday.Severalresearchershaveobservedthattherelativefrequencywith

whichWebpagesarerequestedfollowsZipf’slaw[Zip29].Zipf’slawstatesthattherelativeprobabilityofarequestforthe’thmostpopularpageisinverselyproportionalto1/.Glassman[Gla94]wasperhapsthefirst

useofZipf’slawtomodelthedistributionofWebpagerequestsandseveralotherauthorshavealsoapplied

Zipf’slawtothedistributionofwebrequests[CBC95,ABCdO96].However,recentlyseveralstudieshave

questionedwhethertherequestsdoindeedfollowZipf’slawandconcludedotherwise[NHo98,ACC98]

.OneofourgoalsinthispaperistoinvestigatetheapplicabilityofZipf’slaw.Usingsixtracesfromprox-

iesatacademicinstitutions,corporationsandISPs,wefindthatthedistributionofpagerequestsgenerally

followaZipf-likedistributionwheretherelativeprobabilityofarequestforthe’thmostpopularpageis

inverselyproportionalto1/,withtypicallytakingonsomevaluelessthanunity.Theobservedvaluesof

theexponentvaryfromtracetotrace.Thatis,therequestdistributiondoesnotfollowthestrictZipf’slaw

2(forwhich),butinsteadfollowsamoregeneralZipf-likedistributionwithvarying.Moreover,we

findthatthereislittlecorrelationbetweentheaccessfrequencyofadocumentanditssize;thatis,popular

documentsarenotnecessarilysmallerthanunpopularones.Thecorrelationbetweentheaccessfrequency

ofadocumentanditsrateofmodificationvariesfromverylowtonone,dependingonthetraces.These

resultsraisethepossibilitythat,forsomepurposes,onemightbeabletosufficientlymodelwebaccessesbyasimplemodelthatassumesindependentreferencesfollowingaZipf-likedistributionandnocorrelation

betweenrequestfrequencyandresponsesizeorrateofchange.

InlookingatWebproxytraces,researchershavealsoinvestigatedhowthehit-ratiodepends,asymptoti-

cally,onthecachesizeandthenumberofrequests,andhaveexaminedthetemporallocalityoftheserequest

streams[CI97,RV98,DMF97,GB97a,KLM97].Althoughvariousstudieshaveuseddifferentsetoftraces,

thefollowingthreequalitativeasymptoticpropertieshavebeenidentified:

Foraninfinitesizedcache,thehit-ratioforaWebproxygrowsinalog-likefashion1asafunction

oftheclientpopulationoftheproxyandofthenumberofrequestsseenbytheproxy.Caoetal.

observedthispropertyinDigitalEquipmentCorporation’sproxytraces[CI97,KMM96],Gribbleetal.observedthispropertyinUniversityofCaliforniaatBerkeley’sproxytraces[GB97a,GB97b]

andDuskaetal.observedthispropertyinanumberoftracesfromuniversityproxiesandISPprox-

ies[DMF97].

Thehit-ratioofaWebcachegrowsinalog-likefashionasafunctionofthecachesize.ManyWeb

cachingstudiesreachthisconclusion[ABCdO96,Gla94,CI97,WAS96,GB97a,RV98,CBC95,DMF97].