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1 BSFQ: Bin Sor Fair Quuing Shun Y Chung and Cornliu S nca Dparmn o Mahmaics and Compur Scinc, Emory Univrsiy, Alana, Gorgia Asrac Eising pack schdulrs ha provid air sharing o an oupu link can dividd ino wo classs: sord prioriy and ram-asd Sord prioriy mhods provid clln approimaion or ighd Fair Quuing (FQ) whil ramasd mhods ar mor compuaionally icin prsn a nw pack schduling algorihm calld Bin Sor Fair Quuing (BSFQ) ha comins h srnghs o oh yp o schdulrs As a rsul, BSFQ is highly scalal and can provid vry good approimaion or FQ prov ha BSFQ can provid nd-ond dlay and airnss guarans o conorman lows BSFQ also has a uil-in ur managmn uncion ha can proc packs o conorman lows rom non-conorman raic Th prormanc o BSFQ and is ailiy o dc non-conorman lows ar sudid using simulaions and compard o hos o h Dici Round Roin mhod Ind Trms Qualiy o Srvic, Dlay Guaran, Fairnss Guaran, FQ, BSFQ I INTRODUCTION THE Inrn has njoyd rmndous growh in h rcn yars and h raic on h Inrn is mor divrs han vr ranging rom radiional mail o ral im audio and vido applicaions Many novl applicaions ar wll-srvd i h nwork can provid a crain dgr o prormanc For insanc, h prormanc o audio applicaions ovr h Inrn would graly improvd i h nwork would al o provid dlay and/or ra guarans Qualiy o srvic provisioning o lows is ralizd y insulaing h lows rom on anohr Th idal pack schduling mhod o nsur airnss is h Fluid Fair Quuing (FFQ) mhod [2] In his hypohical pack srvic disciplin, daa o aciv lows ar ransmid on i a a im in a round roin ashion This srvic disciplin is no pracical as packs mus ransmid in hir niry Th ighd Fair Quuing (FQ) [2] and ors-cas Fair FQ (F Q) [3] ar pack schdulrs ha mimic FFQ as closly as possil Ths mhods ar compuaionally pnsiv and ohr mor icin schduling chniqus hav n dvlopd ha approima h havior o FQ Th mhods can cagorizd as sord-prioriy or ram-asd [5] In sord-prioriy schms, ach pack is assignd a prioriy valu and packs ar ransmid in incrasing ordr o hir prioriy To approima h ransmission ordr in h idalizd FFQ sysm, h prioriy valu assignd o a pack is som uncion o is dparur im in h FFQ sysm Eampls o sord-prioriy schms ar Virual Clock (VC) [6], Sl-Clock Fair Quuing (SCFQ) [7] and Lap-Forward Virual Clock (LFVC) [8] In h ram-asd approach, im is dividd ino rams and packs ar nrd ino a ram wihou cding a maimum An ampl o a ram-asd schm is h Dici Round Roin (DRR) mhod [9] Fram-asd mhods ar vry scalal as h pack procssing opraions hav consan im ( ) compliy, in conras o h sordprioriy schms ha rquir a soring opraion o insr a nw pack In his papr, w prsn h nw Bin Sor Fair Quuing (BSFQ) pack schduling mhod ha is a ram-asd mhod and uss prioriy assignmns in sord-prioriy schms o drmin h packs ha ar ransmid in ach round In BSFQ, h virual im is dividd ino slics o qual lngh calld ins As in sord-prioriy schms, ach arriving pack is sampd wih a prioriy valu which rprsns is virual dparur im Th pack is hn sord in h in ha corrsponds o h im slic conaining h virual dparur im o ha pack Th packs sord in h sam in ar quud in FIFO ordr or icincy rasons (u ohr mor sophisicad schduling mhod can opionally usd) As a rsul, BSFQ comins h srnghs o oh yps o schduling mhods: i is highly icin and provids r approimaion or FQ han ising ram-asd schms Th papr is organizd as ollows Scion II prsns an ovrviw o h rlad work prsn h BSFQ mhod in Scion III and show ha h BSFQ mhod can provid nd-ond dlay and airnss guarans Scion IV prsns a simulaion sudy o is prormanc Th papr is concludd in Scion V II RELATED ORK Th gnral procssor sharing (GS) [1] srvr is h idally air srvic disciplin ha can alloca a prdind shar o srvic capaciy o ach clin or nwork low I hr is a singl clin oaining srvic rom a GS srvr, h clin will srvd a h ra o h srvr Howvr, whn wo dirn clins ar prsn, oh will srvicd simulanously u ach will rciv hal h srvic ra Th srvic ras rcivd y ach clin can also wighd dirnly Th GS srvic mhod is also calld Fluid Fair Quuing (FFQ) in [2] /02/$1700 (c) 2002 IEEE

2 ( ( C E FFQ provids prc airnss o nwork lows Howvr, FFQ is no a pracical ransmission procdur caus packs mus ransmid as an aomic uni will rily rviw h asic concps o analyz h airnss o schduling algorihms A low is ackloggd i i has som packs in h oupu ur A pack schduling mhod is air i h dirnc in h normalizd srvic providd o any wo lows ha ar coninuously ackloggd ovr any inrval is oundd y som consan [7] Th normalizd srvic rcivd y low in is dind as, whr # is h amoun o daa o low ransmid in and is h rsrvd daa ra o # includs any par o packs rom ransmid in I a schduling disciplin is air hn hr is a consan such ha ) - ), or any wo lows and 0 ha ar ackloggd during h inrval Th consan ( is indpndn o h lngh o h inrval In h idally air FFQ mhod, w hav ha or any ims 1, - or any wo lows and 0 ha ar ackloggd during Bcaus packs mus ransmid as a uni, pack schdulrs will hav ) - ) 7 9 ack schdulrs ha approima h FFQ disciplin ar calld ack-y-pack Fair Quuing (FQ) [7] Th FQ [2] and F Q [3] pack schdulrs provid h mos accura approimaions o FFQ Ths mhods irs compu h virual inish im o a pack using h FFQ schdulr as rrnc and hn ransmis h packs in ascnding inish im valus Th FQ schdulr dos so or vry pack in h oupu ur whil h F Q schdulr only considrs hos packs ha would hav sard rciving srvic in h corrsponding FFQ sysm Mor compuaionally icin (u lss accura) pack schduling schms hav n dvlopd Th Virual Clock mhod [6] is on o h arlis mhods proposd o insula nwork lows Th VC mhod assigns h ; < pack > > o low wih h virual im samp A C D E H I K M > A C D E > O R T, or ; 7 9, whr M > is h arrival im o pack > and is h lngh o > Th virual im samp A C D E > Y is s o zro Xi and Lam showd in [11] ha i h sum o h ra o all lows sharing a link dos no cd h link capaciy, hn h dparur im > o pack > is oundd y > A C D E > R \ ] _ ` whr a d is h lngh o h largs pack o low and is h daa ra o h oupu link Alhough VC provids a dlay guaran o lows, i dos no provid any airnss guaran acks rom a low ha has n idl or a prolongd priod o im will assignd smallr virual im samp valus and can rciv or som priod o im a largr han rsrvd shar o srvic Th Lap Forward Virual Clock [8] mhod solvs h airnss prolm in VC y mporarily moving ovrsuscrid lows ino a low prioriy holding ara Only lows in h high prioriy ara will rciv srvic A low is ovrsuscrid i h dirnc wn h virual im samps o h currn pack o and h sysm im cds a crain hrshold In h cas whn all lows ar ovrsuscrid, h sysm clock is advancd orward o allow som lows o movd ack ino h high prioriy ara Th Sl-Clockd Fair Quuing mhod [7] uss an inrnal (virual) sysm clock o compu im samps or packs Th sysm clock is qual o h virual im samp o h pack ha is ing srvicd a im Th ; < pack > o low will rciv h virual im samp A C g E i k > H I K M > A C g E i k > O R T ; 7 9 Th valu A C g E i k > Y is s o zro and h packs ar ransmid in h ascnding ordr o hir virual im samp valus SCFQ can provid oh an nd-o-nd dlay guaran [12] and a airnss guaran [7] o conorman lows Th VC, LFVC and SCFQ schdulrs ar prioriy-asd schms whr packs ar ransmid in ascnding virual im samp valus rioriy-asd schms us som sor procdur o mainain a prioriy quu and hav non-consan pr pack run im compliis In conras, ram-asd mhods ransmi packs in rounds Thy do no us sor opraions and hav consan pr pack procssing im Th disadvanag o ram-asd mhods is h ac ha h dlay guaran hy provid hav a largr ound han sord-prioriy mhods Th Dici Round Roin (DDR) mhod [9] organizs packs o lows in spara quus and assigns a quanum siz o ach low Each low has a dici counr ha masurs h currn unusd porion o h allocad andwidh acks o ackloggd lows ar ransmid in rounds and in ach round, ach ackloggd low can ransmi up o an amoun o daa qual o h sum o is quanum and dici counr Th unusd porion o his amoun is carrid ovr o h n round as h dici counr valu Th Uniorm Round Roin (URR) mhod [10] is a cllasd mhod ha can rduc h jir o h DRR mhod y spacing cll ransmissions uniormly ovr a round For ampl, whn our lows M, p, q and r wih wih quanum valu s v clls and s s s y cll ar ackloggd, a possil ransmission ordr pr round in DRR is ( M, M, M, p, q, r ) In conras, h ordr o ransmission in URR is ( M, p, M, q, M, r ) Th ransmission ordring in URR mus rcompud whn a nw low is addd or an ising low is dld rom h schdul and his rcompuaion algorihm is whr C is h numr o slos in on round Ar h ordring is drmind, h pr pack procssing im is consan Th BSFQ mhod prsnd in his papr is a ram-asd mhod and uss virual im samps o drmin h schduling ordr Th virual im spac is dividd ino qual inrvals or ins acks ar assignd virual im samps and insrd ino hir corrsponding ins Th quuing ordr wihin a in /02/$1700 (c) 2002 IEEE

3 ˆ M ˆ is FIFO Th BSFQ mhod has consan run im compliy or conncion { salishmn and pack procssing A simulaion sudy will show ha BSFQ can provid a r approimaion or FQ han DRR using similar opraional paramrs FQ also has a uil-in ur managmn componn and is civnss o idniy packs rom non-conorman lows is illusrad using anohr simulaion primn Fig 1 τ( ) III BIN SORT FAIR QUEUEING = 1 [ 1, 1 ) [, ) [, ) [, ) Bin Sor Fair Quuing Currn in ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ƒƒ } } } ƒ } } } } } } } } } } } } } } } } } } } } } ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ } }} } ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒƒ }} (mpy in) 2 = 1 3= 2 = 3 Th BSFQ mhod is dind or oupu ur swichs Figur 1 shows h logical organizaion usd in BSFQ Th oupu ur is organizd ino ins and is a sysm dsign paramr mus s o a valu ha is qual o or grar han a crain hrshold o allow BSFQ o us all availal urs This hrshold valu will drivd lar in his scion Each in is implicily lald wih a virual im inrval and ach inrval has lngh Th paramr is anohr sysm dsign paramr o BSFQ and is valu has a signiican impac on BSFQ s prormanc Th inrvals o h ins ar disjoin and hir union spans a coninuous rang o h virual im spac Th ins ar ordrd according o hir virual im inrvals and ar srvicd in ha ordring Th currn in is iniially qual o h in wih virual im inrval [0, ) and only packs rom h currn in ar ransmid hn all packs in h currn in ar ransmid, h n in coms h currn on and h old currn in is lald wih an inrval ha ollows h las in Thus, hr ar always ins in h sysm BSFQ mainains a virual sysm clock which is qual o h l nd poin o h virual im inrval o h currn in a im Thus, h currn in has h lal and h ˆ < in ollowing h currn in has h lal ˆ Th virual im clock in BSFQ is a sp uncion similar o h virual clock in SCFQ and will incrmnd y whnvr all packs in h currn in ar ransmid Noic ha i a in is mpy whn i coms h currn in, hn is incrmnd y wihou ransmiing any packs will s lar ha whn is suicinly small, h prormanc o BSFQ will approima ha o FQ dno h daa ra o h oupu link y and ach low mus ngoia a guarand ra or saring is ransmission assum ha so ha h oupu link is no ovrsuscrid Th ; < pack > o low is assignd wih h virual im samp A C > whr: A C > H I K M > A C > O ; 7 9 (1) M whr > M is h sysm virual im a im > din A C > Y 9 Arriving packs ar sord in hir corrsponding ins in h FIFO ordr Th ind ˆ o h in usd o sor pack > is qual o: I ˆ A C > M > ; 7 9 (2) 9 hn > is sord in h currn in, and ohrwis, i ˆ 1, i is sord in h ˆ -h in ollowing h currn in I ˆ, h pack is discardd Furhrmor, i > has n discardd y BSFQ, hn h pack ind (; )isno incrmnd and n arriving pack o low will hav h sam ind as h discardd on Hnc, h BSFQ schdulr has a uil-in ur managmn componn A simulaion sudy will show ha h ur managmn uncion in BSFQ can civly proc packs o complian lows rom non-complian ons Th ollowing ampls show h pack schduling opraion in BSFQ and h c o h paramr on is prormanc Considr wo lows M and p wih rsrvd ras v ps and ps, rspcivly Assum ha a larg numr o packs rom oh lows arriv a h swich simulanously a im 0, and ach pack is 9000 is in lngh Th FQ schdulr will ransmi h packs as: >, >, >, > š, > œ, >, > ž, > š, c Th packs >, > š, > ž, > š, and so on, may ransmid in ihr ordr y FQ [0, 20) p1 p2 B (9) B (18) p1 p2 A p3 p p A p6 A (3) (6) A(9) 5 A (12) (15) A (18) [20, 0) p3 B p (27) B (36) p A 7 (21) p8 A p9 A p 10 (2) p11 p12 A (30) A A p13 (27) (33) (36) A (39) Fig 2 BSFQ using now considr h ransmission ordr in BSFQ Th virual im samps o M s and p s packs assignd y BSFQ ar: > v, >, >, c and >, >, > ª, and so on Figur 2 shows h quuing ordr in a BSFQ srvr or 9 hav assumd ha packs rom p arriv jus or hos o M and du o h FIFO ordring, p s packs ar quud or hos o M i hy ar nrd in h sam in can s rom Figur 2 ha h pack ransmission ordr in BSFQ is p«, p, >, >, >, > œ, >, > ž, p, p, >, > ±, > ², > Y, >, >, >, and so on BSFQ allocas or ach low is air shar o andwidh as h ransmission ra o low is hr ims ha o low p Howvr, h ransmission ordr /02/$1700 (c) 2002 IEEE

4 > > ¾ Ç r H 1 à ˆ Ý Ò Â Ó i [0, 5) [5, 10) Fig 3 BSFQ using [10, 15) [15, 20) p A 1 (3) p1 B (9) p2 A (6) p3 A(9) p A (12) p B 2 (18) p A 5 (15) p A 6 (18) [20, 25) p7 A (21) pa 8 (2) [25, 30) p3 B (27) p9 A (27) p 10 A (30) p11 [30, 35) A (33) p B (36) p12 A (36) p13 [35 0) A (39) dirs signiicanly rom h on in FQ aov For insanc, is ransmid or >, >, >, > œ and > Considr h sam arrival parn in h BSFQ sysm ha uss a smallr valu o Figur 3 shows h quuing ordr µ or hav again assumd ha all packs rom p ar quud or hos o M whn hy ar in h sam in can s rom Figur 3 ha h pack ransmission ordr is now >, p«, >, >, > œ, p, >, > ž, >, > ±, p, > ², > Y, >, p, >, >, and so on Th ordring is idnical o h on o a FQ srvr cp or > which is ransmid or > and ha is srvicd or > s rom hs wo ampls ha has a signiican impac on BSFQ s prormanc BSFQ can also accommoda lows ha do no mak rsrvaions din h rsidual ra o qual o all lows Noic ha will chang whn nw rsrvaions ar mad or an ising rsrvaion is rlasd acks rom any o h non-rsrvaion lows ar assignd a im samp valu using as rsrvd ra and hn schduld in h sam mannr as ohr rsrvaion lows In h rmaindr o his scion, w will prov ha BSFQ provids an nd-o-nd dlay and a airnss guaran A Dlay Guaran Th work in [12] prsns a class o schdulrs ha can provid an nd-o-nd dlay guarans or laky uck ra conrolld sourcs A pack schdulr is in class GR i h dparur im > o pack > is oundd y > or som consan º q q q > > º > H I K M > q or ; 7 9 is dind as: > O and q > Y 9 According o [12], h nd-o-nd dlay ½ o pack > ha ravrss a pah o ¾ nods, ach using a GR-schdulr, is oundd y ½ q > M > (3) H I K R  à º Å, whr q > is h valu o q > a nod 1, M > is h arrival im o > a h irs nod o h pah, º is h consan o nod Ç on h pah and Å is h propagaion dlay wn nods and Ç, or Ç Ê Ê Ê ¾ Furhrmor, i low is a laky uck complian low wih paramrs (Ë,Ì ), hn ½ Í O T Î Ï Ð Â Ñ Ã º Å will show ha BSFQ can provid an nd-o-nd dlay guaran y showing ha BSFQ longs o h class GR as dind in [12] irs show ha h numr o ys admid ino conscuiv ins is oundd and din h siz o a in r Lmma 1: Th numr o is o daa admid ino conscuiv ins is oundd y: r a d roo: L us considr h packs o a paricular low ha ar nrd ino h ins ˆ, ˆ,,ˆ, or an arirary ˆ 9 Assum ha h irs and las pack o nrd ino hs ins ar > i and >, rspcivly Bin ; conains only packs wih im samps wn ; and ;, or ; = ˆ,, ˆ Thror, w hav ha A C > i ˆ and A C > From (1), w hav: Thror, i A C Ê Ê Ê > A C A C Ê Ê Ê A C A C > O > O > i > O a d A C Ê Ê Ê > i a d Hnc, h numr o is o daa o low nrd ino ins ˆ hrough ˆ is a mos a d Th maimum numr o daa rom all lows ha ar nrd in hs ins is hus oundd y: r a d a d can considrd y Þ as h capaciy o conscuiv ins Sinc Ú Û Ü, w din h siz o a in à á as: à á () /02/$1700 (c) 2002 IEEE

5 ä Th numr o ins mus grar han a hrshold o allow BSFQ o us all availal urs I p is h siz o h oupu ur, hn à á p,or ` æ I 1 ` æ, hn a porion o h oupu ur will no usd y BSFQ can grar han ` æ sinc h ins ar logical Th valu o will ac h ailiy o BSFQ o dc packs o noncomplian lows: a largr will allow mor packs rom noncomplian lows o nr h swich Th ur managmn capailiis o BSFQ ar similar o hos o our piplin scion mhod in [1] and i can provid losslss guaran o laky uck complian lows Howvr, h oci o his papr ar on BSFQ s dlay guaran and airnss propris, and h analysis o h ur managmn capailiis o BSFQ is ousid h scop o h papr Bor w sa and prov Thorm 1, w will prsn h ollowing corollary ha is usd in h proo Corollary 1: Th numr o is o daa r admid ino conscuiv ins lald rom o, 7, is oundd y: r a d roo: Sinc h virual im inrvals o conscuiv ins incrass y, w can wri,, whr is h numr o conscuiv ins By Lmma 1, w hav ha: r a d a d can now prsn Thorm 1 which shows ha BSFQ longs o h GR class o schdulrs Thorm 1: Th dparur im is oundd y: g i k > q > g i k > o > in BSFQ a d roo: Th proo is similar o h on prsnd in [12] or M SCFQ din p Ç ) 9 1 Ç ; é > A C > O š L h largs ingr in p Sinc M > A C > Y, y h diniion o A C > Y, p conains a las on lmn and is always dind I ollows rom h diniion o p ha: M > A C > O M and > 1 A C > O or 1 ˆ ; Thror, rom (1), w hav ha: A C > M > Thus: and, A C > A C A C > > O M > ˆ Ê Ê Ê ; Now, assum ha > is nrd ino h in wih h lal ì ì A im M > whn pack > arrivs, h in ha M is ing srvicd y BSFQ has h lal > M > Thus, h maimum amoun o daa ha will ransmid wn > s arrival and > s dparur is all h daa in h M ins > M > o ì ì According o Corollary 1, h amoun o daa r sord in hs ins is a mos: r ì A C > From (5), w hav ha: r M > M > a d a d a d Th daa ransmission ra is and hror, h dparur im g i k > o pack > using a BSFQ schdulr is: g i k > M > r M > From h diniion o q Thus: q and, q > > q > M > q > From (6) and (7), i ollows ha: g i k > B Fairnss Guaran q > O in (3), w hav: M > > a d or 1 ˆ ; a d show ha h dirnc wn h normalizd srvic o any wo lows in BSFQ is oundd L > dno h dparur im o pack > o low A im >, (5) (6) (7) /02/$1700 (c) 2002 IEEE

6 Þ 7 1 BSFQ is srvicing a in wih lal > Sinc > is conaind y his in, w hav ha: > A C > 1 > > (8) or any pack > nd h ollowing auiliary lmma o show h airnss propry o BSFQ Lmma 2: I low is ackloggd a im A C > A C > O M > M >, hn roo: I is ackloggd a im, hn M > > O Bcaus is a monoonic (sp) uncion o,i M ollows ha > > O From (8), w hav ha M > A C > O and hror: A C > H I K A C > O M > A C > O Th ollowing horm shows ha BSFQ is air 0 Thorm 2: I lows and ar ackloggd during h inrval, hn ) - ) a d a d - - roo: L > Ê Ê Ê > š dno h s o packs rom dpar in Figur shows h iming rlaionships: Fig k p 1 k 11 p k p 2 k 21 p k 1 k L( ) p p 2 L( ) k p ha k 21 p L( ) L( ) acks srvicd during h usy priod ï ð ñ ò ð ó ô From his igur, w can conclud ha h arrival im M > o pack > mus lss han h dparur im > O o pack > O, or ˆ Ê Ê Ê, caus ohrwis low would no ackloggd hroughou h inrval Hnc, packs > Ê Ê Ê > arriv whil is ackloggd I also ollows rom Figur ha: ö irs ind an uppround or Sinc non-dcrasing uncion, w hav ha > > (9) is a From (8), w hav > A C > and hror: 7 A C 7 A C > > A C and Bcaus h packs >,, > arriv whil is ackloggd, w hav y Lmma 2 ha: and hror: A C > Using (9), w can ound A C > > rom aov y: 1 a d A C > O M > > > O > > O > A C > > O 7 A C > O 1 A C > A C > O A C > A C > (10) To oain a lowr ound or, w considr wo disinc cass Cas 1: Sinc hr is no pack dparur in ha Thror:,whav Using (8), w ind ha and hror: and Sinc packs,, arriv whn is ackloggd, w hav ha: I ollows rom h cas assumpion ha A C > A C > O R and hror: and, A C > A C > O > /02/$1700 (c) 2002 IEEE

7 þ s Þ s - Þ ] ý ý Using (9), w can ound Cas 2: A C > O 7 1 M > rom low y: a d (11) Sinc is ackloggd a im and hr ar no pack dparurs in >, i ollows ha M > Thror: A C A C > > > M > M > M > I ollows rom h cas assumpion ha A C > M > R and hror: and, a d From (9), w conclud ha h lowr ound or salishd in (11) is also valid or h scond cas I ollows rom (10) and (11) ha: ) - ) a d a d - - IV NUMERICAL EXAMLES hav sudid h prormanc o h BSFQ algorihm and compard i wih FQ and DRR Th FQ algorihm srvs as h rrnc mhod in h comparison sudy Th DRR mhod is slcd caus i has h sam run im compliy as BSFQ or oh pack procssing and conncion salishmn Th prormanc o h BSFQ and DRR mhods dpnd on h sing o som paramrs o h mhods In DRR, ach low is assignd a quanum siz s which is h amoun o crdis i rcivs pr round Th rsrvd andwidh or low is qual o ù, whr s all lows ù ú û ú ý Th rsrvd andwidh or h lows is unchangd i w us a quanum siz, 7 9, or ach low ; Using smallr quanum sizs will allow DRR o r approima h FQ schdulr, u i will also incras h pr pack procssing cos sinc h TABLE I FLOS USED IN THE SIMULATIONS Sourc ra Laky Buck Flow 1,2,3 16Mps 2Mps 50KB 2Mps,5,6 25Mps 5Mps 100KB 5Mps 7,8,9 0Mps 8Mps 200KB 8Mps DRR schdulr may nd o ira hrough a numr o rounds wihou ransmiing any packs Th BSFQ schdulr hiis a similar havior as DRR: using a smallr will allow BSFQ o r approima h FQ schdulr u i will also incras h liklihood ha a in is mpy Th pr pack procssing cos is incrasd in his cas sinc h BSFQ schdulr mus upda h sysm virual clock For a air comparison, w s à á s so ha ý h amoun o daa ransmid in ach round in oh schms ar approimaly qual Fig 5 low 1 low 2 low 3 low low 5 low 6 low 7 low 8 low 9 Oupu ur Nwork usd in h simulaion sudy 8 Mps irs compar h pack dlay wih a numr o simulaion primns Th nwork usd in h simulaion is shown in Figur 5 Th ransmission ra o h oupu link is 8 Mps Thr ar nin inpu lows whos propris ar shown in Tal I Column 1 in Tal I liss h low indics Columns 2 and 3 show h pak daa ra and h avrag daa ra - o ach low Flows 1, 2 and 3 hav low pak and urs ras, lows, 5 and 6 hav mdium pak and urs ras and lows 7, 8 and 9 hav high pak and urs ras acks hav id siz and ach pack is 53 ys Th pack arrivals o ach low ar gnrad y a Markovian on/o procss hn h Markovian procss is in h on sa, packs ar ransmid wih a consan ra ha is qual o h pak ra in Tal I No packs ar ransmid in h o sa Th duraion o h on and o priods or a low is such ha ` û Â, whr and ar h avrag lngh û Â o h on and o û priods, rspcivly Th avrag lngh o an on priod in h simulaions is s o 1 msc Th Markovian arrival gnraion procss is ollowd y a laky uck ra shapr wih h urs siz Ë and okn gnraion ra Ì givn in columns and 5 o Tal I Each yp o pack schdulr is prsnd wih h sam parn o pack arrivals and ach primn is run or 5000 (simulaion) sconds Tal II shows h rsrvaions or ach low For BSFQ, /02/$1700 (c) 2002 IEEE

8 8 Flow 1 Flow 1 Flow roailiy 002 FQ BSFQ - 5 ys/scion DRR - 5 crdis/round roailiy 002 FQ BSFQ - 5 ys/scion DRR - 5 crdis/round roailiy 002 FQ BSFQ - 5 ys/scion DRR - 5 crdis/round Dlay (in sconds) Fig 6 Dlay disriuion in Eprimn 1 ( and! $ & ( ) Dlay (in sconds) Dlay (in sconds) Flow 1 Flow Flow roailiy 002 FQ BSFQ ys/scion DRR crdis/round roailiy 002 FQ BSFQ ys/scion DRR crdis/round roailiy 002 FQ BSFQ ys/scion DRR crdis/round Dlay (in sconds) Fig 7 Dlay disriuion in Eprimn 2 ( and! $ & ) Dlay (in sconds) Dlay (in sconds) Flow 1 Flow Flow roailiy 002 FQ BSFQ ys/scion DRR crdis/round roailiy 002 FQ BSFQ ys/scion DRR crdis/round roailiy 002 FQ BSFQ ys/scion DRR crdis/round Dlay (in sconds) Fig 8 Dlay disriuion in Eprimn 3 ( and! $ & ( ) Dlay (in sconds) Dlay (in sconds) TABLE II FLO RESERVATION ARAMETERS IN EXERIMENTS BSFQ DRR Flows 1,2,3 2Mps ys,5,6 5Mps ys 7,8,9 8Mps ys h rsrvaion o low is qual o is avrag daa ra - in Tal I Th quanum o low in DRR is also proporional o - vary h paramr 8 in DRR and in BSFQ o sudy hir prormanc hav prormd hr µ ss o primns using h sings 8, and in DRR Th corrsponding sings or à á in 9 µ µ 9 µ BSFQ ar à á, à á 9 9 and à á Figur 6 shows h dlay disriuions in FQ, DRR and BSFQ or h lows 1, and 7 whn 8 in DRR and 9 µ à á in BSFQ Th graph is a hisogram To gnra h hisogram, w dividd h im ais wn 9 msc and = 50 msc ino inrvals o 01 msc in lngh and allid h numr o packs wih dlay ha all wihin h inrval acks wih dlays largr han 50 msc wr allid in anohr sum Each ally is hn dividd y h oal numr o packs and h rsuling valu is an sima o h proailiy ha h pack dlay alls wihin a givn inrval Th l mos plo in Figur 6 shows h dlay disriuions o low 1 using FQ, DRR and BSFQ Th < -ais in Figur 6 is h pack dlay masurd in sconds and h = -ais shows h racion o packs whos dlay alls wihin h givn inrval hav shown h dlay disriuions or dlays up o 002 scond caus h ail disriuions ar virually idnical (including h cas whr h dlay is grar han 005 sc) Th dlay disri /02/$1700 (c) 2002 IEEE

9 > J J J B ) < ) uions o lows 2 and 3 ar similar o ha o low 1 and hy ar no shown Th cnr and righ plos show h dlay disriuions or lows and 7, rspcivly Th disriuions o lows 5, 6, 8 and 9 ar omid or h sam rason can s in Figur 6 ha or 8 in DRR and à á ys in BSFQ, h 9 µ dlay princd y h packs using ihr mhod is virually idnical o ha in FQ I is vidn rom Figur 6 ha h DRR mhod using small quanum sizs and h BSFQ mhod wih small à á (or ) paramrs can civly approima FQ µ Figurs 7 and 8 show h dlay disriuions or (8 9 9, µ 9 µ à á 9 9 ys) and ( , à á ys), rspcivly can s ha h dlay disriuions in BSFQ and DRR dir signiicanly rom FQ or small dlays Also, i appars ha h dlay disriuions in h BSFQ mhod convrgs quickr o ha o FQ han DRR Th oal varianc disanc [13] is on usd o masur how ar away wo proailiy disriuions ar rom on anohr I > ar wo disriuions which pu proailiy mass on a ini s A, hn h oal varianc disanc wn hm is qual o: > d E F hav compud h oal varianc disanc o h dlay disriuion wn BSFQ and FQ, and wn DRR and FQ or ach o h aov cass Tals III and IV show h oal varianc disanc wn BSFQ and FQ, and DRR and FQ, rspcivly, or h disriuions givn in Figurs 6, 7 and 8 s ha h oal varianc disanc wn h 9 µ dlay disriuion o BSFQ (DRR) and FQ or à á (8 ) is vry small which conirms h ac ha h approimaion is vry good Tals III and IV show ha h oal varianc disancs wn BSFQ and FQ or à á 9 9 and µ 9 µ à á ar smallr han h disancs wn DRR and µ FQ or and , rspcivly Hnc, BSFQ provids a r approimaion or FQ in hs cass TABLE III TOTAL VARIANCE DISTANCES (BSFQ, FQ) aramr Sing Flow 1 Flow Flow 7 $ & K M $ & K O O $ & K M 6 O O O TABLE IV TOTAL VARIANCE DISTANCES (DRR, FQ) aramr Sing Flow 1 Flow Flow 7 K K 6 O O K O O O An addd advanag o BSFQ is h ac ha i has a uilin ur managmn uncion Rcall ha in BSFQ a pack is discardd whn is in ind is qual o or grar han h numr o ins acks o a low ha ransmis a a ra largr han is rsrvaion will assignd incrasingly largr virual ims and whn h virual im samp valu cds, hy will discardd Thus, BSFQ can provid procion agains non-conorman lows Howvr, proving ha BSFQ provids a losslss guaran similar o h work in [1] is ousid h scop o his papr will only dmonsra h ni o h uil-in ur managmn uncion o BSFQ using a simulaion primn In h n primn, w incras h pak and avrag daa ra o lows 7, 8 and 9 in Tal I o 50 Mps and 10 Mps, rspcivly Noic ha h oal daa ra o all lows is qual o 51 Mps which cds h link capaciy Th rsrvaion paramrs o h lows (s Tal II) rmains unchangd Th lows 7, 8 and 9 ar non-complian lows in h simulaion primn TABLE V NON-COMLIANT FLOS Fracion o packs droppd Flow 1 Flow Flow 7 BSFQ DRR µ Th paramrs usd in h primns ar à á in BSFQ and in DRR Th oupu ur siz is s o 1 Mys and a pack is admid i hr is availal spac Ar a pack has n admid, i is hn procssd y h pack schdulr In DRR, h pack is nrd ino is quu and will no discardd In conras, h pack is discardd y BSFQ i is in ind cds Tal V shows h prcnag o packs droppd y BSFQ and DRR Th al only shows h racion o h packs droppd rom lows 1, and 7 Th rsuls or lows 2 and 3, 5 and 6 and 8 and 9, ar similar o ha o low 1, and 7, rspcivly can s ha BSFQ only drops packs rom h non-complian lows In conras, all lows in DRR sur pack loss V CONCLUSION hav prsnd h BSFQ pack schduling algorihm ha comins h srnghs o ram-asd and prioriy-asd schdulrs BSFQ is a ram-asd mhod ha ransmis all packs in h currn in in ach round acks ar sord ino h ins asd on hir assignd virual im samps and wihin on in, packs ar ransmid in h FIFO mannr BSFQ is highly scalal, having consan run im compliy or pack procssing, as wll as or conncion salishmn Th virual im inrval paramr has a signiican impac on BSFQ s prormanc For vry larg valus o, h prormanc o BSFQ is idnical o FIFO, whil or small valus o, BSFQ is mor akin o SCFQ Thr ar a numr o acors ha hav o considrd in choosing h valu o /02/$1700 (c) 2002 IEEE

10 Th amoun o sa inormaion mainaind is invrsly proporional R o and a small will incras h numr o ins ndd Th icincy o BSFQ also dcrass whn is dcrasd caus i incrass h liklihood o ha som ins ar mpy This is similar o DRR whn small quanum valus ar usd and no packs ar sn in som rounds Drmining an opimal valu or is yond h scop o his papr hav shown ha BSFQ can provid nd-o-nd dlay guaran and airnss o lows ha shar a common link BSFQ also has a uil-in ur managmn uncion ha can insula conorman lows rom non-complian raic Rsuls o a simulaion sudy show ha BSFQ can provid r approimaion or FQ han DRR using similar opraional paramrs In summary, BSFQ has many dsiral srnghs, including scalailiy, ra and airnss guaran and uil-in ur managmn, o provid qualiy o srvic in high-spd nworks REFERENCES [1] L Klinrock, Quuing Sysms, Volum 1: Thory, John ily & Sons, Nw York, 1975, ISBN [2] A K J arkh and R G Gallagr A Gnralizd rocssor Sharing Approach o Flow Conrol in Ingrad Srvics Nworks: Th Singl Nod Cas, IEEE/ACM Transacion on Nworking, vol 1, no 3, pp 3 357, 1993 [3] H hang, F2Q: ors-cas Fair ighd Fair Quuing, IEEE Inocom 1996, pp , 1996 [] H hang, Srvic Disciplins For Guarand rormanc Srvic in ack-swiching Nworks, rocdings o h IEEE, 83(10), Oc, 1995 [5] D Siliadis and A Varma, Ra-proporional Srvrs: A Dsign Mhodology or Fair Quuing Algorihms, IEEE/ACM Transacions on Nworking, vol 6, No 2, pp 16 17, 1998 [6] L hang, Virual Clock: A Nw Traic Conrol Algorihm or ack Swiching Nworks, ACM SIGCOMM 1990, pp 19 29, 1990 [7] S J Golsani, A Sl-Clockd Fair Quuing Schm or Broadand Applicaions, IEEE Inocom 199, pp , 199 [8] S Suri, G Varghs and G Chandranmnon, Lap Forward Virual Clock: A Nw Fair Quuing Schm wih Guarand Dlays and Throughpu Fairnss, IEEE Inocom 1997, pp , 1997 [9] M Shrdhar and G Varghs, Eicin Fair Quuing using Dici Round Roin, IEEE/ACM Transacions on Nworking, vol, no 3, pp , 1998 [10] N Masuuru and R Aiara Eicin Fair Quuing or ATM Nworks using Uniorm Round Roin, IEEE Inocom 1999, Nw York, March 1999 [11] G G Xi and S Lam, Dlay Guaran o Virual Clock Srvr, IEEE/ACM Transacions on Nworking, vol 3, no 6, pp , 1995 [12] Goyal, S Lam and H Vin, Drmining End-o-End Dlay Bounds In Hrognous Nworks, roc 5h Inl orkshop on Nwork and Opraing Sysm Suppor or Digial Audio and Vido, Durham, Nw Hampshir, April 18-21, 1995, pp [13] S T Rachv, roailiy Mrics and h Sailiy o Sochasic Modls, ily Sris in roailiy and Mahmaical Saisics, Chichsr, Nw York, ily, 1991 [1] S Y Chung and C S nca, iplind Scions: A Nw Bur Managmn Disciplin or Scalal QoS rovision, IEEE Inocom /02/$1700 (c) 2002 IEEE