Ingring ffciv-bndwidh-bsd QoS Rouing nd Bs ffor Rouing Sphn L. Spilr nd Dnil C. L Univrsiy of Souhrn Cliforni Dprmn of lcricl nginring Los Angls, CA, USA sphn.l.spilr@boing.com, dcl@usc.du Absrc A mhodology is prsnd for ingring ffciv-bndwidh-bsd rouing for QoS-snsiiv rffic nd dgrm rouing of h bs-ffor rffic. To prvn xcssiv dlys of bs-ffor rffic in nwork domin, w dvlop (1) consrin, sd in h form of rsidul link bndwidh, nd () cos funcion for pplicion o rouing of QoS conncions. Link-bsd nd ph-bsd problm formulions nd lgorihms r prsnd. or h cs h cos qunizion condiion holds, w dvlop n fficin implmnion of linkbsd rouing srgy h firs minimizs QoS cos, hn scondrily minimizs bs-ffor cos. Th prformnc of his pproch is furhr nhncd by xplicily ccouning for h diffrnc bwn h ffciv bndwidh nd h vrg bndwidh of rffic. Simulion rsuls illusr h pplicion of our -frindly mhod o n lgorihm for ph rouing wih rsorion. Kywords ffciv bndwidh; quliy of srvic; ph rsorion; dynmic rouing; consrin-bsd rouing; MPLS; bs-ffor rffic I. INTRODUCTION In rdiionl IP rouing ch pck ypiclly sks h shors ph o is dsinion hrough h conncionlss, hop-by-hop rouing; h id of conncion-bsd nd-o-nd rouing,.g., Muliproocol Lbl Swiching (MPLS) [1], hs bn subsqunly ddd. An dvng of h conncionbsd rouing would b is biliy o fcili h quliy of srvic (QoS) hrough rffic rgulion nd nginring for h rffic crrid hrough h conncion. In his ppr, w ddrss how o ingr h QoS rouing nd h bs-ffor () rouing in h sm IP dminisriv domin;.g., n inrn srvic providr (ISP). In ordr o mphsiz h pplicbiliy of h mhodology, w considr n inrn domin supporing MPLS nd us MPLS rminology. Howvr, h mhodology cn b gnrlly pplid o implmnions ohr hn MPLS, nd wh is dscribd by h rm, lbl swichd ph (LSP), in his ppr cn b rgrdd s ny ph nd bndwidh provision long h ph for micro-flow or s of micro-flows wih crin QoS rquirmn. W considr n inrn nwork domin wih hrognous rffic mix of vrious clsss using boh MPLS nd h dsinion-bsd, hop-by-hop IP rouing. Spcificlly, w ssum h LSPs r providd for rffic runks, i.., microflows or ggrgs of micro-flows h shr common clss of srvic (CoS). W ssum h h rffic runks roud rquir crin QoS gurns. Th nwork domin lso crris h (bs-ffor) clss, which dos no rquir ny QoS gurns. Alhough srvic providr my lso s up LSPs o crry rffic h rquirs no QoS gurns, in his ppr w us h rm, rffic, o rfr o rffic h is forwrdd vi rdiionl dsinion-bsd, hop-by-hop rouing bls. urhrmor, w ssum h, lhough rffic cn k dvng of unusd bndwidh h is llocd o LSPs for QoS runks, i uss i on sricly lowr prioriy bsis. Rfrring o ig. 1, suppos h QoS rffic runk,, nd rffic boh shr h minimum-hop ph from ingrss lbl swichd rour (LSR) s o grss LSR d, i.., ph s-u-d. urhr, suppos h ddiionl micro-flows r bing ddd o runk so h bcoms lrgr, i.., i uss mor bndwidh. Consqunly, rffic gs crowdd ou nd my bgin o suffr xcssiv dlys s bndwidh links (s, u) nd (u, d) is prmpd o suppor. Now suppos h h LSP rouing lgorihm ss up nohr ph from s o d o suppor nw rffic runk. If phs r slcd on h bsis of hop coun, hn s-v-w-d nd s-x-y-d r qully rciv choics. Howvr, if h rouing lgorihm rbirrily slcs s- x-y-d, hn h nw runk will impc rffic h is flowing from y o d whil slcion of s-v-w-d would hv lf h rffic undisurbd. s v x u igur 1. Shring of minimum-hop ph by rffic runk nd rffic. w y d 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
This ppr prsns mhodology o mk QoS rouing mor -frindly. W consrin QoS rouing so s o prvn i from cusing xcssiv rffic dlys bu miig h undsirbl consquncs for QoS rouing,.g., incrsd blocking probbiliy. W do his by mking us of wh w cll xcss ffciv bndwidh, h mrgin by which QoS runk s ffciv bndwidh xcds h vrg bndwidh h i cully consums. W k xcss ffciv bndwidh ino ccoun whn w pply our rffic dly consrin in drmining which phs r fsibl for QoS rouing. Also, w dfin cos qunizion condiion nd, for h cs h i pplis, xploi i o dvlop compuionlly fficin implmnion of wo-sg opimizion srgy for QoS rouing. This srgy firs slcs fsibl phs h minimiz QoS cos. As n xmpl, his QoS cos my b h ph s ovrll rsrvion of nwork bndwidh. Howvr, h opimizion mhod prsnd in his ppr will work wih gnrl clss of QoS cos funcions. Thn, in h cs h hr r mulipl phs h r id wih rspc o his QoS cos, h scond sg of h srgy is o slc from hs id phs on h minimizs msur of rffic dly. inlly, w dmonsr h prcicliy of our mhodology by ingring i ino wll-known lgorihm for rouing LSPs wih ph rsorion. Scion II inroducs our noions nd dfiniions. Scion III dvlops consrins nd coss o ccoun for rffic dly whn rouing LSPs. Our -frindly wo-sg opimizion srgy for QoS rouing, long wih n fficin implmnion of i, r inroducd in Scion IV. Scion V shows how our mhodology cn b pplid o LSP rouing wih ph rsorion nd includs simulion rsuls. inlly, Scions VI nd VII mnion rld work nd conclud h ppr, rspcivly. runk. I indics h moun of bndwidh h mus b rsrvd for runk in ordr o sisfy is QoS rquirmns. A runk s ffciv bndwidh my b dfind in rms of link opring poins s wll s h runk s rffic chrcrisics [8, 9]. Thrfor, w us α o rfr o h ffciv bndwidh of runk link. L rprsn rffic runk h is rqusing n LSP. Dnoing s dscripors using in plc of, w ssum h n LSP rqus provids h nwork dmission conrol nd rouing dcision mkr wih s rnsmission chrcrisics nd CoS which imply b. In gnrl, o spcify α, knowldg of link s opring poin is lso ncssry. or ch dircd link,, w dfin R ff C α, (1) { p } ff nd rfr o R s h rsidul bndwidh of xprssd in rms of ffciv bndwidh. On of h ncssry condiions for link o b fsibl for ssignmn o n LSP for runk is h mus sisfy h rsidul ffciv bndwidh consrin, α R ff. () A cndid ph, p, undr considrion for ssignmn o, is fsibl for if nd only if i is composd nirly of fsibl links. L Cos b msur of h cos o h nwork of using link in n LSP for runk, xclusiv of ny ffcs on rffic. W rfr o Cos s h bs cos of nd lso dfin II. PRLIMINARIS Th MPLS nwork domin is rprsnd by h dircd grph, G = (V, ), whr V is h s of vrics or nods (symbolizing LSRs) nd is h s of dircd links or dgs. ch dircd link,, hs bndwidh cpciy C. W lso rfr o links by hir nd-nods,.g., (i, j) o rprsn h link dircd from nod i o nod j. ch rffic runk,, is dscribd by is ingrss nod, s, grss nod, d, clss of srvic, c, supporing LSP, p, nd vrg rnsmission r, b. Th vrg rnsmission r, b, of is h sum of h vrg rnsmission rs of ll h micro-flows in runk. An LSP or ph, p, is dfind s n ordrd subs, { 1,,, h(p) }, of whr h(p) is h numbr of hops long p nd h ordr indics h squnc of links long p from ingrss nod s o grss nod d. W lso rfr informlly o phs by hir nod squncs ordrd from s o d,.g., s-u-d. W lso ssoci wih runk is ffciv bndwidh which w ssum o ccoun for h runk s QoS rquirmns (.g., consrins on pck dly, jir, or loss) [] nd o rprsn h moun of bndwidh h is rsrvd for h runk if i is roud. Alhough hr r diffrn dfiniions of ffciv bndwidh [3-9], in gnrl, i is vlu inrmdi bwn h mn nd pk flow rs of conncion or Cos p Σ p Cos. (3) s h corrsponding bs cos of slcing ph p o suppor. W cn lv h link bs coss s gnrl funcions o som xn. Howvr, w incorpor h rsidul ffciv bndwidh consrin ino h cos funcions by pnlizing h Cos o b infini for ny whr () is no sisfid. A simpl xmpl of link bs cos funcion is hop coun cos, implmnd by dfining Cos 1,, ff if α R ohrwis. Anohr link bs cos funcion, which is imd minimizing ph s ovrll rsrvion of nwork bndwidh, is h bndwidh rsrvion cos, implmnd by dfining Cos α,, ff if α R ohrwis. (4) (5) 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
W limi h scop hr o singl ph rouing for ch runk. Alhough muliph rouing my provid h ponil o improv nwork rsourc uilizion, i incrss h complxiy of h signling procss for LSP sup, incrss h liklihood of ou-of-ordr pck rrivls, nd cn limin h ddiiviy propry of ffciv bndwidhs [10]. on G/G/1 quuing dly bound. Ohr dly mrics cn b mployd in similr mnnr. or h M/M/1 dly mric, w follow [13, Ch. 5], dfining D o pproxim h vrg im h i ks for pck o rvrs h nwork from is sourc o dsinion. L III. TRAIC CONSIDRATIONS If rffic is no considrd whn rouing LSPs for QoS rffic runks, hn dmission nd rouing for n LSP rqus my consis of srching for ph h minimizs Cos p of (3). W nx dscrib LSP rouing consrins nd coss o ccoun for h ffcs of his rouing on rffic. whr M D γ 1 Σ M (9) C + d, (10) A. Trffic Avilbl Bndwidh W l C ΒΕ C b, (6) { p } nd rfr o C s h vrg bndwidh h is vilbl for srving rffic link. No h, by h noion of ffciv bndwidh, runk s vrg bndwidh consumpion dos no xcd is ffciv bndwidh, i.., α Thn, by (1), (6), nd (7), i follows h b, p,. (7) C ΒΕ R ff,. (8) Thrfor, C, h vrg bndwidh vilbl for rffic, my includ som bndwidh h is llocd o bu, on vrg, is no consumd by LSPs. Hr, w ssum h h rours provid pck schduling h nbls rffic o k dvng of his xcss ffciv bndwidh wihou inrfring wih h QoS runks. or xmpl, undr such schduling disciplins s wighd fir quuing (WQ) [11] nd dfici round robin (DRR) [1], low prioriy rffic cn b quud sprly from highr prioriy clsss nd srvicd whnvr h highr prioriy quus r mpy. Such disciplins lso provid h mns of disribuing o QoS runks hir llocd bndwidhs. B. Trffic Dly Mric Our pproch is o dfin mric h is indiciv of vrg rffic dly nd o us his mric o dfin rffic dly consrin h w pply in drmining which phs r fsibl for QoS rouing. Also, in h cs h hr r mulipl phs h minimiz h bs cos, Cos p, w hn slc from hm on h minimizs cos h is ssocid wih h rffic dly mric. In his ppr, w dmonsr how o uiliz wo xmpl rffic dly mrics, on bsd on n M/M/1 quuing modl pproximion nd h ohr, dscribd in Scion III., bsd is h vrg rffic lod link, d is h procssing nd propgion dly for, nd γ is h vrg ol r of rffic nring h nwork. Rfrring o (10), by Lil s horm, M rprsns h vrg moun of d h is ihr quud or bing rnsmid cross link. Thn Σ M in (9) rprsns h vrg ol moun of d in h nwork nd, gin by Lil s horm, D rprsns h vrg im pck spnds in h nwork. W ssum h h prmrs, γ, {d }, nd { }, r consn ovr n inrvl of inrs nd r pproximly known. or xmpl, migh rprsn h xpcd rffic lod on during givn busy hour of wkdy. C. Trffic Rsidul Bndwidh Consrin Implici in (10), h dfiniion of M, is h ssumpion h C >,. (11) To prvn xcssiv rffic dlys, w impos h furhr consrin h h dly cos b limid, i.., h D D (1) for som vlu, D. Morovr, from prcicl poin of viw, w would lik o gurn crin prformnc ch link. Thrfor, w rquir h M γ D,, (13) whr w hv slcd h limi on h righ-hnd sid of (13) such h (1) is sisfid if (13) is sisfid ch link. Thn (10) nd (13) imply h C +,. (14) γ D d Suppos h, o suppor nw rffic runk,, w r considring cndid ph h includs dircd link whos currn vilbl bndwidh for rffic is C. If 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
dos nd up supporing, hn, by (6), is vilbl bndwidh for rffic will bcom C b. Thn h rffic dly cos bound will sill b sisfid only if (14) holds wih C b subsiud for C, i.., No h h ls summion bov is indpndn of ph p. Consqunly, w dfin h rffic dly cos ssocid wih cndid ph, p, undr considrion for srving s C b γ D Inquliy (15) cn b xprssd s whr R v C +. (15) d b R v, (16) γ D d. (17) Lik R ff, R v rprsns rsidul bndwidh cpciy of ff dircd link. Whrs R limis h ffciv bndwidh, α, of nw runk h link cn suppor, R v limis h vrg rnsmission r, b, of nw runk h cn suppor. Thrfor, o b fsibl for supporing nw runk, dircd link mus sisfy boh rsidul bndwidh consrins, () nd (16). D. Trffic Cos of Cndid Ph Agin suppos h w slc ph, p, o suppor nw rffic runk,. Thn, for ch p, s vilbl bndwidh for rffic will chng from C o h nw vlu, C b. Consqunly, by (10), h nw vlu of M for ch p will b C + d, b nd, ling 1 {H} rprsn h indicor funcion, i.., 1 {Η} = 1, 0, hypohsis H is ru, ohrwis, by (9), h nw vlu of D wih p slcd will b D = 1 + d γ C 1{ p } b 1 = p γ C b C whr, Cos p Σ p Cos, (18) b Cos γ ( C b )( C ), if b R v ohrwis. (19) Cos, which w rfr o s h rffic dly cos ssocid wih using link o suppor runk, is h incrs in h vrg dly h rsuls if link is usd in n LSP o suppor runk. Agin, w dfin his cos o b infini if is corrsponding rsidul bndwidh consrin, (16) in his cs, is no sisfid.. An Alrniv Cos uncion A simpl lrniv o h pproximion of using n M/M/1 modl o rprsn rffic dly is o mk us of wll-known (.g., [13, Ch. 3]) uppr bound on h im spn wiing in G/G/1 quu. Ling W rprsn h vrg im h cusomr wis in G/G/1 quu, w hv h W W ub whr ( W ub σ σ ) λ + b, ( 1 λ µ ) σ is h vrinc of h cusomr inrrrivl im, nd σ b is h vrinc of h cusomr srvic im. Now w dfin h cos mric, whr W γ 1 Σ W (0) W ( σ + σ b ) ( 1 C ), (1) w hv w subsiud for λ nd C for µ, nd σ nd σ b cn b now usd o chrcriz h rriving nd dpring rffic of h link, rspcivly. Ling W b h imum llowbl vlu of W, w cn nsur h W W by rquiring h 1 + + d. γ C W γ W,, () which, long wih (1), implis h 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
C γ W γ W ( σ + σ ) b,. (3) In ordr for link o suppor nw rffic runk wih vrg rnsmission r b, (3) mus hold wih C b subsiud for C. This condiion is xprssd by (16) whr, in his cs, R v C γ W γ W ( σ + σ ) b. (4) W nx dfin rffic cos of cndid ph in rms of h G/G/1 bound. If ph p is slcd o suppor nw rffic runk wih vrg rnsmission r b, hn, by (0) nd (1) wih C - b subsiud for C, h nw vlu of W will b σ + σ b b W ( ) = p γ C b C σ + σ b C ( ) + γ C C. C W k h ph dpndn pr of W s h rffic dly cos of cndid ph, p, i.., using h G/G/1 bound, Cos p is givn by (18) wih Cos γ, 3 ( σ + σ b ) b ( C b )( C ) v, if b R (5) ohrwis. This opimizion srgy cn b implmnd dircly. Howvr, if crin qunizion condiion is sisfid, mor compuionlly fficin implmnion is possibl. Suppos h h bs cos of ny dircd link cn b xprssd s nonngiv ingr mulipl of posiiv rl, i.., for som qunum, q > 0, Cos {nq n Z + } { }. (6) Condiion (6) clrly holds for h simpl xmpl of (4) for q = 1 nd n = 1. Mor gnrlly, by slcing som q sufficinly smll, i my b possibl o pproxim bs cos by qunizd vrsion of Cos wihou hving ny pprcibl ffc on h rsuling rouing procss. Whnvr (6) holds, w cn us h following mhod o indircly implmn our wo-sg opimizion. Considr ph cos h is wighd sum of bs nd coss, i.., dfin Cos p WS Cos p + w Cos p, (7) whr w is posiiv wighing cofficin for h cos (wih uniy wighing of Cos p ). A ph h minimizs Cos WS p dos no ncssrily minimiz Cos p. Howvr, if w slc vlu for w h is sufficinly smll, hn w cn insur h ny ph h chivs fini minimum vlu of Cos WS p lso minimizs Cos p. Spcificlly, w hv h following Proposiion. Suppos h condiion (6) holds for q > 0 nd h w > 0 is such h w Cos p < q, for ny fsibl cndid ph, p. (8) If fsibl ph p 1 is such h IV. -RINDLY LSP ROUTING W nx dscrib our rouing opimizion srgy nd n fficin mhod for implmning i. Thn w prsn our frindly rouing wih problm formulions nd lgorihms. A. Opimizion Srgy An rriving LSP rqus for nw runk is dmid if hr xiss fsibl ph from s o d, i.., on composd nirly of links h sisfy rsidul bndwidh consrins () nd (16). or h gnrl cs in which hr r mulipl fsibl phs, w pply wo-sg opimizion srgy (which is concpully similr o on in [14]) for ph slcion. In h firs sg, w idnify ll fsibl phs h chiv fini minimum vlu of h bs cos, Cos p, of (3). Thn w pply h scond sg, which is o slc from ll such phs on h chivs fini minimum vlu of Cos p of (18). In h cs h such ph xiss, w ssign i o h LSP o suppor nw runk. hn i follows h WS Cos p 1 = min {p fsibl} Cos WS p, (9) Cos p 1 = min {p fsibl} Cos p. Proof: Suppos h (9) holds nd h hr xiss fsibl ph p such h Cos p < Condiion (6) hn implis h Cos p Cos p 1. Cos p 1 - q. (30) Thn 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
WS Cos p = Cos p + w Cos p (by (7)) Thn, by slcing Cos p 1 - q + w Cos p (by (30)) w δ q D (36) < Cos p 1 (by (8)) WS Cos p 1, (by (7)), for som δ (0, 1), w r ssurd h h cos conribuion o Cos p WS for ny fsibl ph, p, is lss hn q bcus conrdicing (9). w Σ x Cos w Cos urhrmor, sinc w is posiiv, if ph p is minimumbs-cos ph (i.., i minimizs Cos p ) nd p minimizs (7), hn p hs h minimum cos mong ll minimum bs cos phs. Now w commn on h vlus of w h sisfy condiion (8). I is h rsidul bndwidh consrin, (16), h llows us o slc w such h condiion (8) is sisfid. Th is, (16) forcs h rffic dly mric, nd hrfor Cos p, o b uppr boundd. Dfining for ny cndid ph, p, h vribls, x 1 { p},, (31) whr 1 {H} is gin h indicor funcion for hypohsis H, by (18), condiion (8) is quivln o w Σ x Cos < q. (3) or h cs h h rsidul bndwidh is givn by (17) (M/M/1 pproximion), i follows h, for ny fsibl link,, nd C - - b γ D d, (33) C - b. (34) Thrfor, h cos of using fsibl link is uppr boundd s Cos γ ( C b )( C ) D b (by (19)) d (by (33) nd (34)) γ D,. (35) w D (by (35)) < q (by (36)). (37) Alrnivly, for h cs h h rsidul bndwidh is givn by (4) (G/G/1 bound), similr rgumn shows h condiion (8) is sisfid if w is spcifid s w δ q for som δ (0, 1). γ W γ b ( σ + σ ) (38) B. LSP Rouing Problm ormulion Assuming h condiions (6) nd (8) hold, h wo-sg opimizion for rouing n LSP o suppor nw runk cn b dscribd s h slcion of {x } h minimizs subjc o Cos p WS = Σ x (Cos + w Cos ) (39) Σ j V x (i, j) Σ j V x (j, i) = 0, i s, d (40) Σ j V x (s, j) Σ j V x (j, s) = 1, (41) Σ j V x (d, j) Σ j V x (j, d) = 1, (4) x {0, 1},, d (43) whr x is dfind by (31), s nd d r h ingrss nd grss nods, rspcivly, Cos is givn by (19), nd h bs cos is dfind so h Cos = for ny link wih α > R ff. Th flow consrvion consrins, (40) for ll vrics in V ohr hn s nd d, (41) for s, nd (4) for d, forc {x } o dscrib singl ph from s o d. Thr my b ohr consrins plcd on his ph slcion problm,.g., dminisriv or srvic clss consrins. An xmpl of n dminisriv consrin is rsricion on wh links cn b usd in h ph. If only links in r dmissibl, hn w subsiu for in (39)- (43). An xmpl of srvic clss consrin is imum 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
hop coun, H(c ), ssocid wih h CoS, c, of h runk rqusing n LSP [14-16]. This consrin cn b ccommodd by dding h inquliy, Σ x H(c ), (44) o problm formulion (39)-(43). If hr is no fsibl {x } for h problm formulion, hn h LSP rqus is rjcd. Ohrwis, n opiml ph o suppor nw runk is consrucd s h ordrd s, p {(v n, v n+1 ) n = 1,,, Σ x * }, (45) whr {x * } is n opiml soluion o (39)-(43), v 1 = s, nd v n+1 * is h uniqu vrx such h x ( ) = 1. v n, v n+ 1 C. LSP Rouing Algorihm Alhough problm formulion (39)-(43) is 0-1 ingr progrmming problm, i hs spcil srucur h lnds islf o soluion by Dijksr s lgorihm. By dfining s our link mric, Cos WS Cos + w Cos, (46) w hv h ph from s o d wih shors possibl lngh undr mric Cos WS lso minimizs Cos WS p of (39). Th lgorihm, WS-OPT, shown in ig., compus WS mric Cos for ch link nd hn pplis Dijksr s lgorihm o srch for n opiml ph o ssign o h LSP for runk. In his figur, w dop h psudocod convnions of [17], omiing nd smns nd indicing block srucurs solly by indnion. WS-OPT(V,, s, d, α, b, {R ff }, {R v }, { }, D, {d }, w ) 1 for ch do compu Cos,.g., by (4) or (5) (Cos if α > R ff ) 3 compu Cos,.g., by (19) or (5) (Cos if b > R v ) 4 Cos WS Cos + w Cos 5 p Dijksr s lgorihm for (V, ), s, d, nd {Cos WS } 6 if no such p xiss 7 hn rjc LSP rqus 8 ls dmi LSP rqus, ssign ph p igur. WS-OPT o implmn wighd sum opimizion. A link s sisfying dminisriv consrins,, cn b subsiud for in WS-OPT. If hr is imum hop coun srvic clss consrin, H(c ), i cn b ccommodd wih sligh modificion of Dijksr s lgorihm. Considr is min whil loop, which coninus iring whil h prioriy quu of vrics o b procssd rmins nonmpy (.g., [17]). If w limi h numbr of loop irions o b qul o H(c ), hn h lgorihm will discovr only hos phs h do no xcd H(c ) hops. Rgrding im bound for WS-OPT, lins 1-4 k im O( ) nd Dijksr s lgorihm cn b run in im O( + V log V ) if is vrx prioriy quu is implmnd s iboncci hp [17]. Thrfor, WS-OPT ks im O( + V log V ). By comprison, dirc implmnion of h wo-sg opimizion would rquir n xhusiv srch for ll minimum-bs-cos phs, followd by slcion of minimum-bs-cos ph wih minimum cos. Thus, if condiions (6) nd (8) hold, hn indirc implmnion of h wo-sg opimizion by WS-OPT significnly rducs compuion complxiy sinc i only pplis Dijksr s lgorihm onc. D. Ph-bsd LSP Rouing Th problm formulion nd lgorihm prsnd in h prvious wo subscions r for link-bsd implmnion of -frindly LSP rouing. This pproch ssums knowldg of h nwork opology for rouing LSP rquss. An lrniv pproch is ph-bsd LSP rouing. In his cs, upon rrivl of n LSP rqus, ph is slcd from fixd pr-drmind s of cndid phs for h ingrssgrss pir nd srvic clss. Ph-bsd rouing my b prcicl in vry sprsly conncd nwork, or whn dminisriv or srvic clss consrins hv h ffc of winnowing down h cndid phs o mngbl numbr. Rprsning h cndid ph s by P nd ling z p 1 {ph p is slcd}, p P, (47) w pply h firs sg h wo-sg opimizion for phbsd rouing by slcing {z p } o minimiz subjc o Σ p P z p Cos p (48) Σ p P z p = 1, (49) z p {0, 1}, p P, (50) whr consrin (49) forcs slcion of singl ph. If (48) is infini for ll possibl slcions of {z p }, hn w rjc h LSP rqus. Ohrwis, w l {z p *} rprsn n opiml soluion o (48)-(50) nd implmn h scond opimizion sg by slcing nw s, {z p }, h minimizs subjc o (49), (50), nd Σ p P z p Cos p (51) Σ p P z p Cos p = Σ p P z p * Cos p. (5) 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
Th 0-1 ingr progrm of (48)-(5) is rdily solvd by srighforwrd lgorihm h (i) compus h bs cos, Cos p, of ch ph in P, (ii) forms s P* conining ll phs in P h hv fini minimum bs cos, (iii) compus h cos, Cos p, of ch ph in P*, nd (iv) slcs ph in P* h hs fini minimum cos (if i xiss). This lgorihm ks im O( P ). V. -RINDLY ROUTING WITH PATH RSTORATION A. Applicion o PI-R Rouing In his scion, w illusr h our -frindly QoSrouing cn b combind wih rsorbl rouing. Much rcn work hs bn don on rsorbl QoS rouing,.g., [18-0]. Rfrnc [0] dscribs n lgorihm clld pril informion wih xc rsrvions (PI-R) which dynmiclly rous LSPs wih 1:1 ph procion. Th rm, dynmic rouing, rfrs o rouing LSP rquss h rriv on-by-on; 1:1 ph procion rfrs o supporing ch civ ph wih link-disjoin bckup ph h is usd only if h civ ph fils. In procing gins singl link filurs, h PI-R lgorihm minimizs h bckup bndwidh rsrvions by shring bndwidh mong mulipl bckup phs. or xmpl, l us suppos h hr r only wo civ LSPs in h nwork. Th lgorihm my us priculr link, b, in h bckup ph for ch of wo runks. If hs runks hv muully link-disjoin civ phs, hn h lgorihm rsrvs n moun of bckup bndwidh b qul o h bndwidh rquirmn of h runk h rquirs h mos bndwidh. This rsrvion is nough for link b o suppor ihr runk long is bckup ph if singl link filur ffcs h runk s civ ph. On h ohr hnd, if h wo runks civ phs shr common link,, hn h lgorihm rsrvs nough bckup bndwidh link b o suppor h sum of h runks bndwidh rquirmns. This rsrvion procs h runks in h vn h link fils. Th pril informion usd by h PI-R lgorihm consiss of h following for ch link: h ol bndwidh rsrvion for civ phs, h ol bndwidh rsrvion for bckup phs, nd h rsidul bndwidh. Th link s rsidul bndwidh is h pr of is bndwidh (cpciy) h is no rsrvd for civ or bckup phs. xc rsrvions rfrs o how h bndwidh rsrvions for runk s bckup ph, onc is civ nd bckup phs hv bn roud, r xcly wh h runk rquirs. (Howvr, whn cully rouing runk, h PI-R lgorihm mks us of uppr bounds on h bckup bndwidh rsrvions h h runk will rquir.) Mor dils r rfrrd o [19,0]. W pply our link-bsd, wo-sg opimizion pproch o consrin PI-R rouing of civ phs o b -frindly. W do no impos -frindly rouing of h bckup phs s hy r usd only in h vn of link filur. Givn n LSP rqus for nw QoS rffic runk,, h rouing opimizion, frmd s 0-1 ingr progrmming problm s in [19, 0], is o minimiz Σ x + Σ c y, (53) subjc o (40)-(43), nlogous consrins for {y }, nd x + y 1,, (54) whr x (y ) is on if link lis in h civ (bckup) ph nd zro ohrwis, nd cofficins nd c, for link, boh dpnd on h runk s bndwidh rquirmn which w k o b is ffciv bndwidh. Boh nd c r clculd bsd on vribl, M, which is ird by h PI-R lgorihm hroughou h s of vlus of h n bndwidh rsrvion for civ phs ch link. Cofficin c is lso funcion of G, which rprsns h currn bndwidh rsrvion for supporing bckup phs. In our frindly vrsion of h PI-R lgorihm, w mk h simplifying ssumpion (s in [19, 0]) h h bndwidh rquirmn of ch runk dos no dpnd on link opring poins. Th is, w ssum h α = α, p,. W lso suppos h M, α, α, nd G r ll qunizd wih qunum, q. In his cs, for -frindly civ ph rouing wih pplicion of rsidul bndwidh consrins () nd (16), w dfin cofficin s α + w, Cos, if α M { p } nd () nd (16) hold ohrwis. (55) or xmpl, Cos is dfind by (19) nd w is dfind by (36) for h M/M/1 pproximion or (38) for h G/G/1 bound. As in [19, 0], 0, M + α G, c, if M + α G if M + α > G nd M + α G ohrwis. R Hr, nd in () s i is usd in (55), h rsidul ffciv bndwidh is R ff ff C α G,. (56) { p } Th dul-bsd soluion lgorihm of [19] is hn pplid. 1 1 In pplying his lgorihm, w runc o mulipls of q ch of h summnds h r summd o form h rducd cos, r ij, of [19]. This chnicl dil prvns cos from influncing h dul vribls, {σ ij}, of [19]. Thn cos dos no influnc h slcion of bckup phs. 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
B. Simulion of -rindly LSP Rouing To illusr oprion of -frindly PI-R lgorihm, w us 15-nod s nwork configurion [19, 0] shown in ig. 3 whr, in his cs, w l h bndwidh cpciy of ch dircd link b 100 Mb/s. (Th undircd links shown rprsn links h r dircd in boh dircions.) igurs 4 nd 5 dpic rsuls from mulipl simulions of scnrio wih Poisson rrivls of QoS LSP rquss nd xponnilly disribud LSP holding ims. In hs figurs, h blocking probbiliy (BP) fcd by n rriving LSP rqus is plod vrsus h vrg rffic lod,, which w k o b common o ll links. Th prmr, ρ, is h offrd lod in rlngs (), i.., h rio of h LSP rqus rrivl r o h dprur r of ch LSP. In hs simulions, w pproxim sdy s condiions by running ch simulion long nough for 30,000 LSP rquss o rriv. Th ingrss-grss pir of ch LSP rqus is rndomly chosn wih uniform probbiliy for ch pir. W ssum h h ffciv bndwidh for ch LSP rqus is uniformly disribud ovr {1,,, 15} Mb/s. In compuing cofficin v by (55), w us R s dfind by (17) (M/M/1 pproximion) in consrin (16). Ohr simulion dils r h w s q = 1 Mb/s, δ = ½, D = 1 s, d = 0, nd ssum h ffor pcks rvrs n vrg of 3 hops hrough h nwork so h γ = (1/3) Σ. 3 6 5 11 1 h, by xplicily ccouning for xcss ffciv bndwidh, w cn significnly rduc h BP pnly of implmning -frindly QoS rouing. 10 % 9% 8% 7% 6% 5% 4% 3% % 1% 0% BP ρ =80 BP 10 % 9% 8% 7% 6% 5% 4% 3% % 1% 0% 60 ρ =0 0 10 0 30 40 50 60 70 80 90 100 (Mb/s) 40 igur 4. BP vrsus for α = b p,. ρ =80 ρ=0 0 10 0 30 40 50 60 70 80 90 100 (Mb/s) 60 40 1 13 igur 5. BP vrsus for α = b p,. 4 8 7 igur 3. A 15-nod s nwork configurion [19, 0]. As shown in ig. 4, which is for h cs of α = b p,, hr is no much of pnly in rms of BP o implmning -frindly rouing (i.., spcifying cofficin by (55)) s incrss up o crin poin. or xmpl, for ρ = 40, h BP is wll undr 1% for ll rffic lods up o 50 Mb/s. Byond h poin, h BP driors rpidly. Whrs ig. 4 is for α = b p,. ig. 5 shows h corrsponding rsuls for α = b p,. In his cs, h BP of LSPs is ssnilly consn ovr n vn widr rng of rffic lods. or xmpl, for ρ = 40, h BP is wll undr 1% for ll rffic lods up o 75 Mb/s (s comprd o 50 Mb/s in ig. 4). Thrfor, h ig. 5 rsuls suggs 9 10 15 14 VI. RLATD WORK Thr r som works h hv ddrssd h consquncs of QoS rouing on rffic. Th virul rsidul bndwidh (VRB) concp [1-3] is o djus ch link s cul rsidul bndwidh (bndwidh unssignd o QoS conncions) o rflc h rffic lod on h link. A link s VRB is md lowr (highr) hn is rsidul bndwidh if h rffic lod on h link is hvy (ligh). QoS conncions cn hn b roud vi shors phs using link coss h r funcions of h VRB,.g., h rciprocl of h VRB [1, ]. A rld virul cos chniqu is lso usd in [3]. Anohr pproch, frmd in sing wih conncionlss, hop-by-hop QoS rouing, is h nhncd bndwidhinvrsion shors ph lgorihm [4]. I uss link cos h is h rciprocl of h rsidul bndwidh scld so s o pnliz longr phs. inlly, [14] prsns problm formulion for rouing bchs of QoS nd dmnds h imizs rvnus rnd by QoS dmnds nd, scondrily, imizs rvnus rnd by dmnds. 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003
Th mhodology h w prsn in his ppr is dircd owrd conncion-orind QoS rouing h uss ny link mric. W propos consrin on QoS rouing o prsrv minimum lvl of prformnc for low prioriy rffic. This consrin, which w s in rms of (scond) rsidul link bndwidh, is o kp h vrg dly from xcding som imum ccpbl limi. W miig h impc of his consrin on QoS rouing by xplicily ccouning for xcss ffciv bndwidh. Also, for h cs h qunizion condiion pplis, w propos n fficin implmnion of wo-sg opimizion srgy h minimizs h cos of QoS rouing nd, scondrily, minimizs dly. VII. CONCLUSIONS W hv proposd, for ingrion ino ffcivbndwidh-bsd QoS rouing lgorihms, mchnisms o quniivly ccoun for h impc h ny givn ph slcion hs on low prioriy rffic. Wih his mhodology, h dgr o which ph rouing is rsricd by -frindly consrins cn b rdd off, vi fw prmrs, wih rffic dlys. Our simulion rsuls for simpl s nwork configurion suggs h dgrdion of QoS rouing prformnc (.g., incrsd LSP blocking probbiliy) du o -frindly consrin nd no b prohibiiv, spcilly whn his consrin ccouns for xcss ffciv bndwidh bing uilizd by rffic. Our wo-sg opimizion srgy slcs from mong ohrwis qul-cos phs (h cknowldg rsidul bndwidh consrin du o rffic), on h hs h ls impc on dly. W hv shown h, ssuming qunizion of QoS link coss, h wo-sg opimizion srgy cn b implmnd vry fficinly, i.., wih h complxiy of Dijksr s lgorihm. RRNCS [1]. Rosn, A. Viswnhn, R. Cllon, Muliproocol Lbl Swiching Archicur, IT, RC 3031, Jnury 001. [] K. Kr, M. Kodilm, T. V. Lkshmn, Minimum inrfrnc rouing of bndwidh gurnd unnls wih MPLS rffic nginring pplicions, I Journl on Slcd Ars in Communicions, vol. 18, no. 1, Dcmbr 000. [3] G. d Vcin, G. Ksidis, J. Wlrnd, Rsourc mngmn in widr ATM nworks using ffciv bndwidhs, I JSAC, vol. 13, pp. 1081-90, 1995. [4] W. Whi, Til probbiliy wih sisicl muliplxing nd ffciv bndwidhs in muli-clss quus, Tlcommun. Sys., vol., pp. 71-107, 1993. [5] A. I. lwlid, D. Mir, ffciv bndwidh of gnrl Mrkovin rffic sourcs nd dmission conrol of high spd nworks, I/ACM Trnscions on Nworking, vol. 1, no. 3, pp. 39-343, Jun 1993. [6]. P. Klly, Nos on ffciv bndwidhs, in Sochsic Nworks,. P. Klly, S. Zchry, I. Zidins, ds., Oxford: Clrndon Prss, pp. 141-68, 1996. [7] C. S. Chng, J. A. Thoms, "ffciv bndwidhs in high-spd digil nworks," I JSAC, vol. 13, pp. 1091-1100, 1995. [8]. P. Klly, ffciv bndwidhs muli-clss quus, Quuing Sysms, vol. 9, no. 1, pp. 5-16, 1991. [9] C. Courcoubis, V.A. Siris, G. Smoulis, Applicion of h mny sourcs sympoic nd ffciv bndwidhs for rffic nginring, Tlcommunicions Sysms, vol. 1, pp. 167-191, 1999. [10] C. Courcoubis, V. A. Siris, Mnging nd pricing srvic lvl grmns for diffrnid srvics, Procdings of IWQoS '99, Jun 1999. [11] A. Dmrs, S. Kshv, S. Shnkr, Anlysis nd Simulion of irquuing Algorihm, Proc. ACM SIGCOMM '89 (Spmbr 1989): 1 1. [1] M. Shrdhr, G. Vrghs, fficin ir Quuing Using Dfici Round Robin, Proc. ACM SIGCOMM '95, vol. 5, no. 4 (Ocobr 1995):31 4. [13] D. Brsks, R. Gllgr, D Nworks, nglwood Cliffs, NJ: Prnic-Hll; 199. [14] D. Mir, K. G. Rmkrishnn, A cs sudy of mulisrvic, muliprioriy rffic nginring dsign for d nworks, Procdings of Globcom 99, Dcmbr 1999. [15] M. K. Girish, B. Zhou, J. Q. Hu, ormulion of h rffic nginring problms in MPLS bsd IP nworks, Procdings of ISCC 000, July 000. [16] Y. Sok, Y. L, Y. Choi, C. Kim, Dynmic consrind muliph rouing for MPLS nworks, I ICCCN 001, Ocobr 001. [17] T. H. Cormn, C.. Lisrson, R. L. Rivs, Inroducion o lgorihms, nd d., Cmbridg, Mss.: MIT Prss; Boson : McGrw-Hill; 001. [18] C. Hung, V. Shrm, Building rlibl MPLS nworks using ph procion mchnism, I Communicions Mgzin, vol. 40, no. 3, Mrch 00. [19] M. Kodilm, T.V. Lkshmn, Dynmic rouing of bndwidh gurnd unnls wih rsorion, Procdings of I INOCOM 000, vol.. [0] M. Kodilm, T.V. Lkshmn, Rsorbl dynmic quliy of srvic rouing, I Communicions Mgzin, vol. 40, no. 6, Jun 00. [1] Q. M, P. Snkis, Supporing dynmic inr-clss rsourc shring: muli-clss QoS rouing lgorihm, I INOCOM '99, vol., pp. 649-660, 1999. [] H. Kochkr, T. Ikng, Y. Oi, QoS rouing lgorihm bsd on muliclsss rffic lod, I GLOCOM '01, vol. 4, pp. 193 198, 001. [3] Y.-W. Chn, R.-H. Hwng, QoS rouing lgorihms for mulipl rffic clsss, I ICC 00, vol. 4, pp. 17-1, 00. [4] J. Wng, K. Nhrsd, Hop-by-hop rouing lgorihms for prmiumclss rffic in diffsrv nworks, I INOCOM 00, vol., pp. 705 714, 00. 0-7803-7753-/03/$17.00 (C) 003 I I INOCOM 003