Multipath Network Flows: Bounded Buffers and Jitter

Transcription

1 Mulpah Nework Flow: Bounded Buffer and Jer Trcha Anjal, Aleander Forn Deparmen of Elecrcal and Compuer Engneerng Illno Inue of Technology Chcago, Illno Emal: {anjal, Grua Calnecu, Sanjv Kapoor, Nandakran Krubanandan Deparmen of Compuer Scence Illno Inue of Technology Chcago, Illno Emal: {calnecu, kapoor, Suep Tongngam School of Appled Sac Naonal Inue of Developmen Admnraon Bangkok, Thaland Emal: Abrac In h paper we addre he ue of degnng mulpah roung algorhm. Mul-pah roung ha he poenal of mprovng he hroughpu bu requre buffer a he denaon. Our model aume a nework wh capacaed edge and a delay funcon aocaed wh he nework lnk (edge). We conder he problem of eablhng a pecfed hroughpu from ource o denaon n he nework, gven bound on he buffer ze avalable a he denaon and a bound on he mamum delay pah are allowed o have. A relaed problem whch we conder o eablh bound on he delay varance (alo called jer) among he pah choen for he mul-pah roung cheme. We how ha he problem are NP-complee and preen peudo-polynomal algorhm baed on lnear programmng. We alo propoe praccal heurc and preen he epermenal reul on an eng nework opology. The reul are promng. I. INTRODUCTION Tradonal algorhm for eablhng nework connecvy beween a ource-nk par fnd a ngle pah of hore lengh (delay) beween he ource and denaon. However, h pah may be plagued by congeon problem nce h he pah ha wll alway be choen o forward he oal raffc beween he node par. On he oher hand, plng he raffc among mulple pah ulze he pah mulaneouly and reduce he congeon on any gven pah, whle ncreang he overall nework ranmon capacy. Mulpah roung proocol work on he prncple ha hgher performance can be acheved by ulzng more han one feable pah []. Mulpah roung can be effecvely ued for mamum ulzaon of nework reource by gvng he node a choce of ne hop for he ame denaon. Mulpah roung ha been propoed o ake advanage of nework redundancy, reduce congeon, and addre QoS ue [], []. Traffc engneerng, lower delay, ncreaed faul olerance, and hgher ecury are oher compellng reaon ha e for dcoverng and ulzng mulple pah. The man dadvanage aocaed wh he mulpah roung manfe a he packe dorderng a he recever, nce he raffc pl no hee mulple pah wh dfferen laence creang jer. Soluon o h problem for he TCP proocol have been propoed n he leraure uch a [4], [5]. The problem of dfferen laence onmulple pah can be reolved va eher boundng he jer or by he ue of buffer a he denaon. The choce of mulpah enal ha, o ynchronze acro varou pah, buffer need o be eablhed a he denaon o ha he daa packe can be ored and equenced appropraely. In h paper, we formulae opmzaon problem o model he problem n mulpah roung. We wan o eablh a pecfed hroughpu from ource o denaon n he nework, gven bound on he buffer ze avalable a he denaon or requremen on he jer. The pah o be eablhed are requred o be of bounded delay. Th formulaon general enough and can be appled o nework wh dfferen approache o mplemen mulpah roung. Our model aume a nework wh capacaed edge and a delay funcon aocaed wh he nework lnk (edge). We formulae he opmal Fed Buffer MulPah Flow (FBMPF) problem o deermne he e of mulpah beween he ourcedenaon par gven a fed buffer ze a he denaon. The relaed problem o eablh bound on he delay varance (alo called jer) among he pah choen for he mulpah roung cheme, ermed he Bounded Jer Mulpah Flow (BJMPF) problem. We how ha he problem are NPcomplee and degn a mnmum co nework flow problem whch model he bounded-buffer problem. Th lead o a peudo-polynomal algorhm, baed on lnear programmng. For he bounded-jer problem, we preen a faer peudopolynomal algorhm whch cloely appromae he requred hroughpu. The oluon of he lnear program brng ou an nereng feaure aocaed wh he FBMPF problem. We found ha may be benefcal o ue long pah n he nework n order o afy he buffer bound, nce pah whch have mlar delay would requre lower buffer ze. Long pah can be

2 obaned by accrung he delay n a cycle one or mulple me. Thu, he arfcal conruc of choong non-mple pah (pah repeang node, or n oher word, wh cycle) can lead o maller buffer ze a he recever. The eplc roung capably provded by echnologe uch a MulProocol Label Swchng (MPLS) can be ued o eablh hee nonmple pah n a nework. Before he acual daa ranfer, a Label Swched Pah (LSP) eablhed beween he endho. Durng he eablhmen of he LSP, label ranlaon able are creaed a each nermedae hop. The daa hen forwarded on he LSP by performng a label ranlaon. Relaed problem have been uded, bu we are no aware of any work dealng eplcly wh buffer ze conran. In he emnal work [6], he auhor addre mulpah roung ue a a non-lnear opmzaon problem. They are concerned wh delay and lo rae n an adapve eng, and her co per lnk a funcon of he raffc on ha lnk, an approach whch gnore he buffer a he recever. A fed number (k) of acual pah are conruced n [7] o mamze he oal flow (hroughpu). We alo mamze flow, bu we conder ogeher he buffer conran, and allow ofer (dcaed by oal co) conran on he pah. The paper organzed a follow. In Secon II, we nroduce he mahemacal model for he FBMPF problem, a well a he relaed Bounded Jer MulPah Flow (BJMPF) problem. Unforunaely, boh problem are NP-Complee a we prove n Secon III. In Secon IV, we provde an appromae oluon for BJMPF followed by he appromae FBMPF oluon n Secon V. The oluon we preen ue relaed dea. In Secon VI, we dcu he heurc mplemenaon followed by he epermenal reul n Secon VII and concluon n Secon VIII. II. MULTI-PATH PROBLEMS In h econ we defne relaed nework roung problem ha are of nere: We fr defne he bounded-laency nework flow problem: Conder a dgraph G = (V,E), wh n node and m edge (lnk), a capacy funcon c(u,v) : E R +, a lengh (delay) funcon l(u,v) : E R +, a ource, and a nk. Noe ha we do no aume ha delay or capace acro he edge o be equal. For a flow pah P from o, defne he oal lengh (delay) pah, L(P), a: L(P) = l(u, v) e=(u,v) P ha, he oal lengh (delay) of he pah equal o he um of he lengh of he edge on he pah. The mulpah flow problem (MPF), o generae a e of - pah P,P,...,P k wh correpondng flow value f,f,...,f k uch ha he followng condon are afed k = f = γ (demand conran),e=(u,v) P f c(u,v); e = (u,v) E conran) (capacy L(P ) L; P (pah lengh (delay) conran) where γ Z +. An nance of he mulpah flow problem denoed a MPF(N,γ,L), where N he npu nework characerzed by N = (G,c,l,,). Addonal conran for jer and buffer ze can be mpoed on he lengh (delay) of he pah eleced a oulned below. Bounded Jer MulPah Flow (BJMPF): The delay varance (jer) on he pah eleced, gnfcanly affec he buffer ze a he nk. Thu, advable o bound he mamum allowed delay varance (δ) for he pah choen, ha L(P ) L(P j ) δ; P,P j Fed Buffer MulPah Flow (FBMPF): Suppoe pah P,P...P k carry flow f,f...f k, where P he pah of mamum lengh (or delay), n order o acheve he dered hroughpu. The chedule o ranm he flow would requre ha a un of flow puhed on P a me would arrve a he denaon a me + L(P ). Daa packe ranmed a he rae f on pah P, k would accumulae a he denaon whn h me frame. The number of packe arrvng beween me + L(P ) and + L(P ) on pah P gven by f (L(P ) L(P )). Thee packe would requre o be buffered a he denaon ll he packe on he pah P arrve a he denaon, nce hee packe have arrved ou of equence. Aumng ha he buffer ze a he nk gven a a conan, we acheve he followng conran: f (L(P ) L(P )) B. () Here B he buffer ze a nk, and P he pah wh mamum lengh (delay). Due o he naure of he problem, we can ge oluon nvolvng cycle. In he followng eample, we how ha how cycle can gnfcanly mprove he performance. Conrucon z Fg.. u y Why cycle are good

3 In he fgure we have a nework wh all edge capace, and lengh (delay) a noed on each edge. The ource, he nk, and he flow demand. Conder he BJMPF problem wh jer 0. A flow of value acheved by ung he pah,,y,z,, and,u,, each of lengh (delay) 5 and carryng one un of flow. No oluon wh jer 0 e wh only mple pah (or, n oher word, whou cycle). Smlarly, n he ame eample, f he buffer ze 0, he only oluon for FBMPF wh flow demand he one gven above wh non-mple pah. Alo, a oppoed o clacal ma-flow problem, for neher BJMPF nor FBMPF rue ha a oluon carryng negral flow alway e. Theorem : BJMPF and FBMPF are NP-Complee. Proof: For boh problem we ue he ame reducon, from he well known Paron problem [8]: gven a mule X = {,,..., n } of neger, he decon problem ak here a paron of X no A and D uch ha A = D. We conruc our nework a follow: V = {v 0,v,...,v n }, = v 0, = v n, and for all =,,...,n, we pu wo edge e and e, boh wh al v, head v and capacy. We e l(e ) = + and l(e ) =. We e γ =, and for BJMPF we e δ = 0, whle for FBMPF B = 0. See Fgure for an eample Conrucon v y z y z v Fg.. Eample whou negral oluon Such a counereample (ee Fgure ) gven below: he nework N = (G, c, l,, ) wh G = (V,e) ha V = {,,v,,y,z,v,,y,z }, E = {,y,,y,y, y,yz,y z,v, v,v,z,v,z }, all edge e have c(e) =, and all egde e have l(e) = ecep l(v) = l( v ) = and l(y) = l(y ) = l(y) = l( y ) =. If γ =, and eher δ = 0 (for BJMPF) or B = 0 (for FBMPF), hen he only oluon ue each of he followng pah carryng0.5 un of flow:p =,,v,,p =,,v, P =,,y,z,, P 4 =,,y,z,, P 5 =,y,z,v,, and P 6 =,y,z,v,. One can check h by npecon, nong ha he pah,y,z, and,y z, of whch a lea one needed n any oluon wh demand γ =, have lengh 4, horer han he lengh of he pah,,v, and.,v, - of whch agan a lea one needed n any oluon wh demand γ =. In he counereample above all capace and lengh are non-negave neger bounded by. A we wll ee laer (Secon IV and V) we can olve olve uch nance n polynomal me. III. NP-COMPLETENESS In h econ, we how ha he problem of fndng mulpah wh bounded jer (or fed buffer) NP-Complee. Fg.. Eample of he NP reducon. If he nance of Paron {,,4,5,0}, hen we ge he graph above wh = v 0 and =v 5. Th nance can be paroned no A = {,4,5} and D = {,0}, and he mulpah oluon ue he pah wch ue he fr and la upper edge (wh he oher lower), and he pah wh fr and la lower edge (wh he oher hgher) - makng boh jer and buffer ze 0. If here a paron of X uch ha A = D, we ue wo pah P and P defned a follow: f A, we pu e n P and e n P, oherwe ( D), we pu e n P and e n P. The l(p ) = n + A, whle l(p ) = n+ D, afyng boh he jer and he buffer conran. Converely, any oluon ha afe he demand conran, and eher he jer or he buffer conran, mu con of a e of pah P, for {.,...,q} for ome q, of he ame lengh (delay) L. Every edge n he nework mu be ued a full capacy a appear n ome mn-cu. Then q l(p ) = =f n l(e) = n+. e E = Snce q = f = and all he pah have he ame lengh (delay), we oban l(p ) = n + (/) n =. Now f we le A = { n and P ue e }, mmedae ha A = (/) n =. Defnng D = S A gve u A = D ; hu he Paron nance feable. Noe: If he nework undreced, or ymmerc (each edge ha an oppoe edge wh he ame delay and capacy), he conrucon above (or he ymmerc varan) ll work a one canno hp wo un of flow from o whle ung pove flow from ome v j o v j. I urn ou ha NP-Compleene ndeed a conequence of havng large value of he delay funcon l(e) for e E. However, a we how below, he problem do adm peudopolynomal algorhm, precely polynomal me algorhm when he delay funcon can only aume value from a e of mall neger.

4 IV. SOLUTION TO BOUNDED JITTER MULTIPATH FLOW By roundng we aume ha he delay funcon gven al : E {,,...,k}. The roundng can caue mall error when compung he jer (or, n he ne econ, buffer ze), bu neceary o oban effcen oluon. We oban an algorhm polynomal n n,m,k, and U, where U = ma e E c(e). Fr hng we do, f a value L and n ha all pah P ued by he oluon have lengh (delay) beween L δ and L. In fac we do no know L and mu ry many value from a range - we delay for he momen he calculaon of h range. We denoe by P(L,δ) he e of uch pah. Wh he lengh (delay) rercon above, he problem become: Gven a nework N = (V,E,c,l,,) and neger L,δ,γ, fnd an neger q and for each {,,...,q}, a pah P P(L,δ) and pove flow value f uch ha: q f = γ = q f m(,e) c(e), e E = where m(,e) he number of me pah P ue edge e. I benefcal o rephrae h problem a he followng packng lnear program, wh eponenally many varable: mamze ubjec o P cp(l,δ) P P(L,δ) f P f P m(p,e) c(e) e E () f P 0 P P(L,δ) () where m(p,e) he number of me pah P ue edge e. If he objecve funcon a lea γ, he pah P wh f P > 0 gve a feable oluon o BJMPF. Th lnear program can be olved n me polynomal n n,m,l,logu by dong a layered conrucon a n [9] and n [7]. Whle h program reemble he clacal mamum flow problem, we do no have a combnaoral mehod for olvng eacly. Among oher dfference, a oppoed o mamum flow, no rue ha f all he npu capace are neger he opmum objecve an neger. If we are wllng o ele for an ε appromaon for he objecve funcon (o nead of γ un of flow, we only ge γ( ε), hen we can apply he Garg-Könemann algorhm [0] a eplaned ne. The lnear program above a packng LP. In general, a packng LP defned a ma{c T A b, 0} (4) where A, b, and c have pove enre; we denoe he dmenon of A a mn. In our cae he number of column of A prohbvely large (eponenal n number of edge). The algorhm of [0] aume ha he LP mplcly gven by a vecor b R m and here e an oracle whch fnd he column of A mnmzng a o-called lengh funcon. The lengh of column j wh repec o LP n Equaon (4) defned a lengh y (j) = ΣA(,j)y() c(j) for any pove vecor y. Th mean, for our parcular LP, ha we mu fnd, for a vecor y(e) : e E a pah P P mnmzng y(p) := e P y(e) where we make he convenon ha P a mule and each edge e couned every me P ue. Fndng uch a P can be accomplhed by a hore pah algorhm n he followng layered graph. See Fgure 4 for an eample. For each = 0,,,...,L, make V a copy of V. Conruc a new dreced graph ˆN wh ˆV = V( ˆN) = L =0 V We call u he copy of node u from V. For an edge e E wh al u and head v pu n ˆE = E( ˆN), for each = 0,,...,L l(e), an edge e wh al u and head v +l(e). Se y(e ) = y(e). Alo add edge f j, for j =,,δ, wh al L j and head L ; hee edge have y-value 0. Now fnd a hore pah, wh repec o y, from 0 o L n ˆN. y y Fg. 4. From he nework (V,E,,) on he rgh, he layered graph ˆN on he rgh when L = 7 and δ. A an eample, he pah,,y,, n he orgnal nework appear a 0,,y, 4, 7 n ˆN, and he pah,y, n he orgnal nework appear a 0,y, 5, 7 n ˆN. Snce ˆN acyclc, he runnng me of he hore pah algorhm O(mL). The runnng me of he Garg-Könemann algorhm [0] O((/ε) mt orc, nce m he number of conran (oher han non-negavy) and T orc he me requred o compue he mnmum lengh column - n our cae O(mL). Thu he oal runnng me, for one value of L, O((/ε) m L. I reman now o bound he range of poble value of L. Fr, we can aume γ > mn e E c(e), or ele we can ue one ngle pah. Second, we can aume δ nk, or ele we could mple ue ma-flow gnorng he delay conran, and hen decompoe he flow no mple pah of lengh (delay) beween and nk. Now, every pah P n a feable oluon ha lengh (delay) a lea L δ and ue f (L δ) lengh-capacy. The oal lengh-capacy of he graph e E l(e)c(e) mk e E c(e). A f = γ, we oban (L δ)γ mk e E c(e). We conclude L e E nk +mk c(e) mn e E c(e). Noe ha O((mkmU) ), we oban an algorhm producng ( ε)γ flow wh oal runnng me O((/ε) m (mkmu) ) = O((/ε) m 6 k U ), e E c(e) mn e E c(e) mu. Ung nk+mkmu L= L =

5 V. SOLUTION TO FIXED-BUFFER-MULTIPATH-FLOW Agan we aume ha he delay funcon gven a l : E {,,...,k},.e. only ake a value mall neger. Fr, we f a value L and gue ha he longe pah ued ha lengh (delay) eacly L. In fac we do no know L and mu ry many value from a range - we delay for he momen he calculaon of h range. We denoe by P(L) he e of uch pah. Wh he lengh (delay) rercon above, he problem become: Gven a nework N = (V,E,c,l,,) and neger L,γ,B, fnd an neger q and for each {,,...,q}, a pah P P(L) and pove flow value f uch ha: l(p ) = L (5) q f = γ (6) = q f (L L(P )) B (7) = q f m(,e) c(e), (8) = for all e E, where m(,e) he number of me pah P ue edge e. We olve he above problem by conrucng a lnear program whch reemble he mnmum co flow problem. Smlar conrucon appeared n [9], [7] for varaon of ma-flow. To decrbe and conruc he lnear program, we fr conruc from N a nework ˆN a follow. ˆV = V( ˆN) = L =0 V We call u he copy of node u from V. For an edge e E wh al u and head v pu n ˆE = E( ˆN), for each = 0,,...,L l(e), an edge e wh al u and head v +l(e). Alo add edge g j, for j = 0,,,L, wh al j and head L ; hee are he only edge wh co: co(g j ) = L j. The goal o hp a mnmum co γ un of flow from 0 o L n ˆN, ubjec o jon capacy conran a follow. For each edge e E, and each = 0,,,...,L l(e), we have f(e ) a he flow on edge e of ˆN. Alo, f(g j ) defned for he g j edge above. We ue he followng noaon: for a node u, δ + (u) he e of edge wh al u. and δ (u) he e of edge wh head u. The lnear program : ˆe δ (ˆv) L mnmze (L j)f(g j ) f(ˆe) = ˆe δ + (ˆv) j=0 L l(e) =0 ubjec o f(ˆe) ˆv ˆV { 0, L } (9) ˆe δ + ( 0) f(ˆe) = γ (0) f(e ) c(e) e E () f(ˆe) 0 ˆe ˆE () Conran 9, 0, and are flow conervaon, repecvely oal flow ou (replacng conran 6 ) and jon capacy conran (replacng conran 8). More precely: Clam : Aumng he longe pah ha lengh (delay) eacly L, he above lnear program ha objecve value a mo B f and only f he FBMPF nance ha a feable oluon. Proof: Aume fr ha he FBMPF ha a feable oluon: here an neger q and for each {,,...,q}, a pahp P(L) and pove flow valuef afyng Equaon 5, 6, 7 and 8. For a P P(L), le e(p,j) be he j h edge of P. We ue he convenon ha ummaon over an empy e 0. In he lnear program, for each e E and j = 0,,...,L l(e), defne Q(e,j) = {P P(L) r > 0 (e(p,r) = e and r j=l(e(p,r)) = j) } and f(e j ) = P f Q(e,j). In he feable oluon, Q(e,j) repren he e of pah n P(L) ha ue edge e afer a delay of j me un. In he lnear program, f(e j ) e o be he value of he flow carred by hee pah hrough edge e a he jh me ep. For each j = 0,,...,L, defne f(g j ) = P l(p f )=j. Then one can ealy check ha all he conran of he lnear program are afed, and ha he objecve value a mo B. Now aume ha he lnear program ha a feable oluon wh objecve funcon a mo B. Snce he flow conervaon conran are afed, h oluon can be decompoed n ˆN no a e of pahp. EachP drecly correpond o pah ofn afer, f neceary, we remove he la edge of P f ha edge of ype g j, for j {0,,...,L }. Smple calculaon how ha Equaon 5, 6, 7 and 8 are afed by hee value P. I reman now o bound he range of poble value of L. Fr, we can aume γ > mn e E c(e), or ele we can ue one ngle pah. Second, we can aume B nkγ, a any ma-flow ung mple pah ha lengh (delay) a mo nk. Thu: f (L l(p )) B nkγ. () Aume now we have a feable oluon wh pah P and flow f. We have ha f l(p ) canno eceed he oal lengh-capacy of he graph, e E l(e)c(e), whch upperbounded by mk e E c(e). Thu f l(p ) mk e Ec(e). (4) From Equaon and he prevou equaon we oban: Lf nkγ +mk e Ec(e). (5) Wh f = γ, we conclude ha for a feable oluon L nk +mk e Ec(e)/γ = O(m ku), (6) where a defned before, U = ma e E c(e). Thu FBMPF ha an algorhm polynomal n k,m,u.

6 oughpu Thro Buffer ze Fg. 5. Heurc reul for opology VI. HEURISTIC SOLUTION Heurc Heurc Opmal Becaue of he NP-complee naure of he BJMPF and FBMPF problem, we propoe a heurc algorhm o fnd feable pah ha afy he conran. The nereng apec of heurc ha hey allow u o ge faer, realme oluon wh reul cloe o he opmal, bede he fac ha we can work wh larger nework opologe and larger parameer han he opmal program. Our heurc am o mamze he roued flow ubjec o eher jer or buffer conran - n fac can handle boh conran f needed. The propoed heurc are greedy n naure. Gven he nework opology, we ue Sher algorhm [] o fnd he k hore pah n he nework. Repeaed node are allowed n each of he pah. Th algorhm or he pah n he ncreang order of her co. The lnk delay are ued a he co for he lnk. So, he pah obaned by h algorhm are n ncreang order of her delay. For he fr heurc, proceedng n a greedy manner, we ar elecng he pah from he fr one. We puh he mamum flow on h pah and hen conder he ne pah n he l. The pah wll be eleced uch ha he jer and buffer conran are repeced. Hence, he mo mporan conran of he heurc o keep he delay a low a poble by conderng he hore pah fr, and hen he capacy conran come no accoun and allow he flow o be en o he arge node. By proceedng n h way, f he um of he capace of he pah lower han he oal demand γ, hen we wll no be able o end he amoun of flow dered and he demand conran wll be appromaely repeced. For he econd heurc, we reorder hee pah n order of her capacy. In oher word, we elec pah wh hgher capace fr. Th done wh he objecve of elecng hgh capacy pah fr. The peudo-code for he fr heurc gven ne. I can be mnmally modfed for he econd heurc. The jer conran apple n lne 5-7, where Lengh Conran he lengh of he hore pah among our collecon of pah plu he δ, he mamum delay varance allowed. The buffer conran a gven n he decrpon of FBMPF enforced n lne 8-0. Noe ha P from he FBMPF decrpon, beng he longe pah, mu be updaed afer each added pah. If we wan o enforce only one of he jer or buffer conran, we make buffer ze or Lengh Conran very large. Algorhm : FBMPF : Fnd he k hore pah n he nework ung lengh (delay) a co : couner= : whle Pah are avalable and no all flow ha been roued do 4: Selec he pah n he l a locaon couner 5: f Lengh of h pah > Lengh Conran hen 6: break 7: end f 8: f Buffer conran volaed hen 9: break 0: end f : Add h pah o l of choen pah : Reduce he demand o be roued by he amoun end on h pah : Decreae every lnk capacy by h amoun a well 4: Increae couner 5: end whle VII. SIMULATION We have mplemened he opmal oluon and he heurc on a Java plaform. We have ued wo opologe o demonrae he reul. The fr one a mple 6 node opology for eay verfcaon of he reul and econd one a real opology, obaned from []. Th nework called GEANT a pan-european backbone whch connec Europe naonal reearch and educaon nework. The nework ha node and 94 lnk. The lnk delay are emaed baed on he lengh of he lnk and are agned beween and 0 mec. In he fgure 5 and 6, we how he oal flow ha can be en beween ource (node number ) and denaon (node number 6 for he fr opology and for he econd) a a funcon of he buffer ze a he recever, obaned wh he wo heurc, for he wo opologe repecvely. The fr daa pon where buffer 0 correpond o he lea co ngle pah flow, currenly beng ued n TCP/IP proocol. A can be een, ncreang he buffer ze ha ubanal mpac on he flow n he nework. For he GEANT opology, ung a buffer of appromaely 0.Gb, we are able o acheve hroughpu of 0Gb/ whch abou wce a compared o he ngle hore pah hroughpu of 0Gb/. The opmal program, on he oher hand, able o acheve a hgher hroughpu wh no buffer. Th becaue he opmal program allow cycle n he pah whch can adju he delay of he pah uch ha all pah have mlar delay. The heurc program ypcally run n le han ec wherea he opmal program ake abou hour o eecue. VIII. IMPLEMENTATION ISSUES We epec ha he problem menoned n h paper wll be appled o he core of he nework. Typcally, he acce node have only ngle connecon o he nework and mulpah are no poble. Alo, ypcally he acce lnk are he boleneck n he end-o-end pah. So, hee algorhm are only appled o he nework core where mulple pah

7 (Gb/) oughpu ( Thro Buffer ze (Gb) Heurc Heurc a greedy heurc whch ha been epermenally hown o ubanally mprove he hroughpu a compared o he curren TCP/IP proocol. Our heorecal oluon gve polynomal-me appromaon for he varaon of BJMPF and FBMPF when we have a fed number of ource-nk par. In he fuure, we wll develop opmal and heurc oluon for he cae when mulple uer are requeng ervce from he nework. Fg. 6. Heurc reul for GEANT opology are ncluded for redundancy and effcency, and here no boleneck lnk. Currenly, he algorhm are degned o be eecued by a cenral eny reponble for he nework. If here a drbued way of propagang nformaon abou he nework ae o all he node, hen he propoed algorhm can be deployed n a drbued manner. Alo, one aumpon ha he nework delay are no dynamc. In oher word, durng he calculaon of he pah, we aume ha he delay value do no change. We can ncorporae lgh varably dependng on he updae frequency. A new reque arrve n he nework, updaed lnk delay value can be ued o calculae he curren pah for he flow. IX. CONCLUSIONS In h paper, we have movaed he mulpah roung problem wh a buffer conran a he denaon. Th problem baed on he requremen on he denaon o buffer he packe receved along pah wh lower delay unl he flow on oher pah have arrved. The larger he dfference n delay of he pah, larger he buffer requred a he denaon. We have hown opmal heorecal reul o oban opmal buffer ze for a dered hroughpu. We have alo preened REFERENCES [] N. F. Maemchuk, Dpery Roung, n Proceedng of IEEE ICC, 975. [] H. Suzuk and F. A. Tobag, Fa Bandwdh Reervaon wh Mullne and Mul-pah Roung n ATM Nework, n Proceedng of IEEE Infocom, Florence, Ialy, 99, pp. 40. [] I. Cdon, R. Rom, and Y. Shav, Analy of Mul-Pah Roung, IEEE/ACM Tran. on Neworkng, vol. 7, pp , December 999. [4] S. Bohacek, J. Hepanha, J. Lee, C. Lm, and K. Obraczka., A New TCP for Peren Packe Reorderng TCP-PR, IEEE/ACM Tran. on Neworkng, o appear n Aprl 006. [5] H. Han, S. Shakkoa, V. Hallo C, R. Srkan, and D. Towley, Mul- Pah TCP: A Jon Congeon Conrol and Roung Scheme o Eplo Pah Dvery n he Inerne, vol. 4, pp. 60 7, December 006. [6] A. Elwald, C. Jn, S. Low, and I. Wdjaja, MATE: MPLS Adapve Traffc Engneerng, n Proceedng of IEEE INFOCOM 0, Anchorage, USA, Aprl 00, pp [7] R. Banner and A. Orda, Mulpah roung algorhm for congeon mnmzaon, n Proc. IFIP Neworkng 005, Waerloo, Onaro, Canada, May 005. [8] M. R. Garey and D. S. Johnon, Compuer and Inracably, W.H. Freeman and Co., NY, 979. [9] S.T. McCormck, The compley of ma flow wh bounded-lengh pah, Workng paper, 006. [0] N. Garg and J. Könemann, Faer and mpler algorhm for mulcommody?ow and oher fraconal packng problem, n Proceedng of 9h Annual IEEE Sympoum on Foundaon of Compuer Scence, 998. [] D. R. Sher, Ierave Mehod for Deermnng he k Shore Pah n a Nework, Nework, vol. 6, pp. 05 9, 976. [] GANT nework pan-european backbone, hp:// Oc 004.pdf.