An Analytical Model for Wormhole Routing with Finite Size Input Bu ers
An Analytical Model for Wormhole Routing with Finite Size Input Bu ers
An Analytical Model for Wormhole Routing with Finite Size Input Bu ers
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Position<br />
worm head worm tail<br />
sideofinequality(6)shouldbeadoptedtoapproximateblpiwhenthe<strong>for</strong>wardingdelay,xlpi, Figure2:<strong>An</strong>illustrationofthelinkoccupancytime. the<strong>for</strong>wardingdelayincreases,andmustbeatleastaslargeasthewormsize.Tosatisfy dominates.Also,theaveragelinkoccupancytimeshouldbemonotonicallyincreasingas<br />
filling buffer<br />
p<br />
Λ<br />
alloftheabove,thefollowingapproximationisproposed<strong>for</strong>thelinkoccupancytime<br />
i<br />
distribution: Sincebu<strong>ers</strong>tendtobefullyutilizedund<strong>ers</strong>everelyblockingconditions,thelefthand<br />
blocking<br />
p<br />
Blpi(s)=8>:Ylpi+Xlpi1hYlpiXlpiYlpi(s)+2XlpiL(s)iifL>Xlpi<br />
l i<br />
Ylpi+Xlpi1hYlpiXlpiYlpi(s)+2XlpiX<br />
(x<br />
p<br />
l ) worm size Time<br />
i<br />
lpi(s)iifLXlpi link occupancy time<br />
monotonicallyincreasing<strong>with</strong>Xlpi,asprovenin[14]. whereXlpiisth<strong>ers</strong>tmomentofX limXlpi!1Blpi(s)=X TheremainingW Itcanbeshownthatequation(7)hasthelimitvalues,limXlpi!0Blpi(s)=Ylpi(s)and lpi(s),sinceYlpi=Xlpi+L.Moreover,Blpiderivedbyequation(7)is lpj(s),Zlpj(s),Hlpj(s),andQlpj(s)quantitiesarediscussedinsection5. lpi(s),andsimilarly<strong>for</strong>YlpiandL.<br />
5<strong>Model</strong>ingthe<strong>Finite</strong><strong>Size</strong><strong>Bu</strong>er andvicev<strong>ers</strong>a.Toanalyzebothindependentlycouldresultinapoormodel.Forthe buerresemblesanitedamsystem.Awormowsinthebuerconstantlywhenitisnot sakeofaccuracyandsimplicity,weuseanalternativeapproachwhichtreatsbothlink full.However,theoutgoingowofthebuermaybeinterruptedduetowormblocking. Thequeueingmodel<strong>for</strong>anitedamsystemdevelopedin[16]cannotbeapplieddirectly inthiscase.Furthermore,thestatusofthebueristightlyrelatedtoitsupstreamnode, Sincebuercapacityisxedintermsofthenumberofits,thenatureoftheinput<br />
5.1M/G/1/KApproximation buerheadinthenexthop(i.e.,theone-hop<strong>for</strong>wardingdelay,!lpi)isexactlythewaiting contentionandtheinputbuerasonesinglequeue. capacityisK).Withinputports,theM/G/1/Kqueue(seegure3)hasthecapacity timeofanM/G/1queue<strong>with</strong>nitecapacity(denotedasM/G/1/K,<strong>for</strong>thecasethat Asshowningure3,thedelay<strong>for</strong>awormtoseizeitsoutputlinkandreachthe
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