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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Blab(s)=TheLaplace-Stieltjestrans<strong>for</strong>moftheprobabilitydensityfunctionofthelink Blpi(s)=TheLaplace-Stieltjestrans<strong>for</strong>moftheprobabilitydensityfunctionofblpi. blpi=Arandomvariablethatdenotesthelinkoccupancytimeoflinklpi.<br />
Slpi(s)=TheLaplace-Stieltjestrans<strong>for</strong>moftheprobabilitydensityfunctionofslpi. slab=Arandomvariablethatdenotestheservicetimeofawormatthebuer<strong>for</strong> slpi=Arandomvariablethatdenotestheservicetimeofawormviapathpatthe linklab. buer<strong>for</strong>theithlinkofpathp. occupancytimeatlinklab.<br />
3OrderingLinks Slab(s)=TheLaplace-Stieltjestrans<strong>for</strong>mofslab(). slab()=Theprobabilitydensityfunctionofslab.<br />
linkblockingfeature.Blockingoccursduetothesmallsizeoftheinputbu<strong>ers</strong>andresults aservedwormholdsthislink)isnotonlyafunctionofthewormsize,butalsoafunction inincreasedlinkoccupancytime.Thisoccupancytime(denedasthetimeintervalthat Awormholeroutingnetworkdi<strong>ers</strong>fromavirtualcut-throughnetworkbecauseofits Tp=Theaveragenetworkdelay<strong>for</strong>wormsviapathp.<br />
oftheblockingdelayinthesucceedinghops.Asaconsequence,itisimportanttond thedependencyamonglinks.Thelinkdependencyandthecycleoflinkdependencyare<br />
Notethatitispossiblethatlablefbutlabisnotasubsequentlinkoflefinanypath, Denition1Wesaythatlabdependsonlcd,if9p,suchthatlcdisasubsequentlinkof denedasfollows: lcdlef,thenwesaylablef,too(i.e.,itistransitive). labinpathp.Thisdependencyisrepresentedaslablcd.Moreover,iflablcd,and<br />
time.Inourearlierpaper[11],wedevelopedtherelationsbetweentheirdistributions andlcdlab. butreliedoniterativemethodstondthesolution.Actually,acomputationorder,which Denition2Wesaythatthereisacycleoflinkdependencyif9lab;lcdsuchthatlablcd accordingtothetransitiveproperty.<br />
indicatesthesequenceoflinks<strong>for</strong>blockingdelayanalysiscanbederivedifthereisnocycle oflinkdependency,asillustratedin[14].Themethodissimplythetopologicalsorting[15]. thecomputationorder,linkoccupancytimeandblockingtimecanbeevaluatedlinkby Forexamples,iflablcdlef,wehaveacomputationorder,lef!lcd!lab.Following Linkdependencyprovidestherelationshipbetweenlinkoccupancytimeandblocking<br />
link<strong>with</strong>outiterations.
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