Thoughslab()canberecoveredbyinvertingitsLaplace-Stieltjestrans<strong>for</strong>m,Slab(s),the tioncanbeexploited. inv<strong>ers</strong>ionisnotcompletelysystematic.Toeasethisdiculty,atwo-momentapproxima- Z1<strong>An</strong>otherchangeisabouttheintegration[18,Chapter5,equation(1.7)], 00labk k!e0labslab()d Figure4:Thetwo-stageapproximation<strong>for</strong>adistributionfunction. th<strong>ers</strong>ttwomomentsofslab,SlabandS2lab,theprobabilitydensityfunctionofslabcanbe approximatedas(gure4): slab()=8
5.2<strong>Bu</strong>eringDelayandMore asanM/G/1/Kqueue<strong>with</strong>K=,andthequeueservicetimeisexactlythelink hlpi,asshowningure3.Thecontentionblocking,Hlpi(s),canalsobeapproximated Thebueringdelay,Qlpj(s)isderivedsimplyasQlpi(s)=Wlpi(s) mustbeproperlyapproximatedrstinordertoderiveHlpi(s)andQlpi(s).Asimple known,whichrequiresknowledgeofHlpi(s)asshownintheabove.Consequently,Blab(s) occupancytime,Blab(s),iflpilab.However,Blab(s)isnotavailableuntilQlpi(s)is Hlpi(s)because!lpi=qlpi+ Blab(s)=ProbfbuerfullgSlab(s)+(1Probfbuerfullg)L(s) approximationisproposedasthefollowing: Thisapproximationisbasedonthefollowingobservations: 1.Whenthebuerisfull,itsimplyresemblesadatapipe|oneitofdataoutofthe thiscase. buercorrespondstooneitofdataenteringthebuer.Thus,Blab(s)=Slab(s),in (15) areintheM/G/1/KqueueusedtoapproximateW 2.Whenthereisspaceleftinthebuer,awormowsinthebuer<strong>with</strong>outinterruption.Thus,Blab(s)=L(s). lpi(s),namely,theprobabilitythatmorethan#worms Probfbuerfullg=1Xj=0#(j)0@(1PjB)j+1 thatisderivedwhenweanalyzeW thenitesizebuer.There<strong>for</strong>e,wehave, Thebuerfullprobabilitycanbecloselyestimatedfromthesteady-stateprobability wherejk,PjBdenotethek,PBoftheM/G/1/Kqueue<strong>with</strong>K=j+. Xk=jjk+PjB1A lpi(s).#isanequivalentqueuesizeof factthattheequivalentqueueusedtoapproximatethenitesizebuerdoesnotcount thebu<strong>ers</strong>pacethatcanonlyholdpartofaworm.Thisdelayisnotrecountedhere. AfterW Finally,Zlpi(s)isignored,sinceitissmallandimplicitlyincludedinHlpi(s)duetothe lpi(s),Hlpi(s)andQlpi(s)arederived,Blpi(s)isgivenbyequation(7). (16) inputbueris: hopinputbuerisderivedasslpi=hlpi+1+blpi+1,whichgivesus: Slpi(s)=Hlpi+1(s)Blpi+1(s) Now,theservicetimedistribution<strong>for</strong>awormthroughpathpattheheadofitsith Slab(s)=X Consideringwormsfromdierentpaths,theservicetimedistribution<strong>for</strong>anitesize p:lab2Lpp labSlpp(lab)(s) (17) wherep(lab)isafunctionwhichreturnsiiflinklabistheithlinkofpathp. (18)