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].