On the Optimality and Interconnection of Valiant Load-Balancing Networks

Transcription

1 O the Optimality ad Itecoectio of Valiat Load-Balacig Netwoks Moshe Babaioff ad Joh Chuag School of Ifomatio Uivesity of Califoia at Bekeley Bekeley, Califoia Abstact The Valiat Load-Balacig (VLB) desig has bee poposed fo a backboe etwok achitectue that ca efficietly povide pedictable pefomace ude chagig taffic matices [1]. I this pape we show that the VLB etwok has optimal pefomace whe odes ca fail, i the sese that it ca suppot the maximal homogeeous flow fo ay umbe of ode failues. We geealize the VLB desig to eable itecoectio of multiple VLB etwoks, ad study itecoectio via bilateal peeig ageemets as well as tasit ageemets. We show that usig VLB as a tasit scheme yields the lowest possible etwok ad itecoectio capacities, while VLB peeig ca also achieve ea-optimal use of capacity. I. INTRODUCTION The Iteet coe cosists of multiple itecoected backboe etwoks, with each backboe etwok idepedetly povisioed, deployed, ad admiisteed by its owe. The itecoectio egime evolves ove time, as etwoks egotiate itecoectio ageemets with oe aothe. It has become so complex ad opaque that eseaches have to devise vaious pobig methods to ife the topology of the Iteet. O top of this, the taffic matices expeieced by the backboe etwoks ae becomig iceasigly vaiable at both lage ad small timescales. This is due to a umbe of factos, icludig the populaity of ew applicatio classes chaacteized by dyamic ovelay outig of lage data flows. This makes the tasks of taffic egieeig ad etwok povisio/upgade extemely challegig fo the etwok opeatos. Cosequetly, may backboe opeatos have esoted to ove-povisioig by a facto of up to te i ode to maitai low latecy i thei etwoks. This has led the etwokig commuity to evisit the desig of the backboe etwok achitectue. I paticula, eseaches at Stafod ad Bell Labs have sepaately put fowad two-phase load-balacig etwok desigs that ca povide pedictable pefomace fo highly vaiable taffic matices [1], [2]. Fo example, the Valiat Load-Balacig (VLB) desig fom Stafod [1] imposes a specific topological stuctue (logical full mesh) o the backboe etwok, ad outes data via exactly two hops ove the full mesh usig a simple load-balacig scheme (equal load-balacig o all odes). This allows the etwok to efficietly suppot chagig taffic matices with obustess agaist failues. The twophase outig scheme [2], [3] geealizes the above loadbalacig scheme to ay etwok topology, usig o-equal load balacig o itemediate odes. The VLB two-phase load-balacig scheme has may advatages fo the etwok opeato. It hadles the upedictability of taffic, by beig able to suppots ay taffic matix as log as it falls withi the capacity costaits of igess-egess capacity of each ode ( hose model ). Moeove, it avoids cogestio without ay dyamic o eal-time cofiguatio of the etwok. Routig is oblivious ad idepedet of the specific taffic matix. Fially, it is efficiet i the sese that it has miimal total capacity povisioed [1]. As etwok opeatos begi to cotemplate the VLB backboe desig, the atual ext questio aises: how should multiple VLB etwoks itecoect with oe aothe? This itecoectio poblem actually ecompasses may specific ope questios, icludig: how should the load-balaced outig algoithm be geealized acoss multiple VLB etwoks? How should the itecoectio poits be selected? Ca the VLB desig suppot diffeet itecoectio elatioships, e.g., tasit ad peeig? Ae the efficiecy ad obustess popeties of a sigle VLB etwok etaied fo multiple itecoected VLB etwoks? Diffeet methods of itecoectig VLB etwoks ae possible, ad they should be compaed agaist oe aothe alog dimesios such as efficiecy, obustess, evolvability, ad suppot fo competitio ad iovatio. I this pape, we will focus o establishig the optimality of the VLB desig fo a stadaloe etwok as well as fo vaious foms of itecoectios. Fist, we will exted the aalysis of [1], [4] ad establish the uivesal optimality of the VLB desig to ode failues (Sectio III). Next, we will popose a geealizatio of the VLB etwok that facilitates itecoectio, amely the m-hubs VLB etwok ad the m- hubs l-toleat VLB etwok (Sectio IV). This allows us to establish optimality esults fo the cases of tasit ad peeig betwee multiple VLB etwoks (Sectio V). A. Related Wok Ou pape follows a lie of eseach that aims to desig etwoks ude upedictable taffic, thus eplacig the assumptio of a fixed taffic matix by the hose-model [5] i which oly a boud o the igess ad egess ates of each ode i the etwok is kow. The goal is to build efficiet etwoks that suppot ay taffic matix that is cosistet with

2 the ate bouds. We ext discuss some elated liteatue o etwok desig ude the hose model. Ou pape is most closely elated to the papes by Zhag- She ad McKeow [1], [4] which coside the poblem of Iteet back-boe IP outig. They suggest to use a twophase outig, which they call the Valiat Load-Balacig (VLB) scheme (followig Valiat [6]), o a logical full mesh. Routig is doe i two phases, fist, each flow is equally split o all odes, ad the fowaded to its destiatio. This outig scheme is show to be optimal (with espect to total capacity). We show that it has the best pefomace with espect to ode failues, ad that its geealizatios ca be used fo efficiet itecoectio of etwoks. Kodialam, Lakshma ad Segupta [2] suggest two-phase outig schemes that ca be viewed as a geealizatio of the VLB scheme. They povide Liea Pogammig fomulatios fo vaious goals [3], [7] ad ague that it has may advatages ove diect outig. The pape [8] cosides the issue of esiliece to a sigle oute (ode) failue with thoughput as the optimizatio goal. Ou uivesal optimality of VLB to ode failues is, as fa as we ae awae of, the fist esult showig that the VLB scheme pefoms optimally with espect to ay umbe of failues, with total capacity as the optimizatio goal. Keslassy et al. [9] coside the use of a two-phase loadbalacig scheme iside a switch. While the applicatio is diffeet, the basic model ad goal is vey simila to the model of a sigle etwok that we coside. The pape also cosides outig ude the homogeeous hose model o a full mesh. The objective of Keslassy et al. is to miimize the sum of capacities of all edges i the etwok icludig self edges, while ou goal is to miimize the sum of capacities excludig self edges (as i [1], [4]), as this makes moe sese fo back-boe etwoks. [9] shows that optimizig fo the sum of all edges capacities esults with a uique optimal etwok that is biased (self edges have half the capacity of o-self edges), while fo ou optimizatio goal we obseve that thee ae multiple optimal etwoks, a fact that we show useful i etwoks itecoectio. The liteatue o badwidth povisioig fo Vitual Pivate Netwok (VPN) ude the hose model ca be viewed as a geealizatio of the sigle etwok model cosideed i this pape. This geealizatio allows fo heteogeeous ates at the odes, capacity bouds o the edges ad possibly diffeet cost of uit capacity fo diffeet edges. Gupta et al. [10] coside outig alog fixed outes ad show that the poblem of fidig a optimal tee outig o sigle-path outig is NP-had. Elebach ad Rüegg [11] allow fo multipath outig of splittable flows ad show that the optimal povisioig poblem ca be solved i polyomial time usig Liea Pogammig (LP). I cotast, ou pape addesses esiliece to failues ad itecoectio of etwoks, ad offes a simple outig scheme with load-balacig splits that ae idepedet of the souce ad destiatio. II. MODEL A. The Sigle Netwok Model We begi by pesetig the model of a sigle etwok with homogeeous access capacities. The etwok N cosist of odes.the etwok ca be epeseted as a diected gaph, with set of vetexes of size, ad a edge betwee each pai of odes. We assume that thee ae o costaits o the edges capacities. A taffic matix Λ is a matix such that a flow (ate) of size 0 eeds to be set fom ode i to ode j. We sometimes efe to as the steam fom i to j. Astafficis dyamic ad chages ove time, ou goal is to build a etwok that ca suppot a lage set of taffic matices. We adopt the hose-model ([5]) i which each ode i the etwok has a homogeeous boud o its igess ad egess ates, ad we wish to povisio the etwok to suppot ay taffic matix that is cosistet with the ate bouds. A outig scheme defies the way that taffic is outed i the etwok, ad give a outig scheme, we ca fid the miimal capacity o each edge that is equied to suppot all desied taffic matices. We assume that the flows ae splittable. Fomally, a taffic matix is legal with ate if fo ay i N, j N, ad fo ay j N, i N. Give a etwok with a capacity matix C, we say that the etwok ca suppot atafficmatixλ if thee exists a solutio to the multi-commodities flow poblem [12] defied by the demads Λ ad the capacity costaits C, o the diected gaph of the etwok. A etwok with capacity C ca suppot homogeeous ate of if fo ay legal taffic matix Λ with ate, the etwok ca suppot the taffic matix Λ. We defie the capacity of a etwok 1 with a capacity matix C, to be the sum of the edges capacity, without self edges: C = c ij i N j N A etwok is optimal if it has miimal capacity ove all etwoks that suppot all legal taffic matices with ate. All the etwoks that we peset will use local outig decisios (oblivious outig) ad will ot equie ay solvig of multi-commodities flow poblems (o ay othe Liea Pogammig), o the kowledge of the specific taffic matix. We also coside ode failues. A failue of a ode implies that it ca o loge geeate o eceive ay taffic, ad all edges that icidet o the ode ae ot used fo outig. Fomally, give that the odes F N failed, a legal taffic matix Λ with ate is legal afte F failed if fo ay i F, j N =0, ad fo ay j F, i N =0. A etwok with capacity matix C suppots homogeeous ate of afte odes F failed if ay Λ with ate that is legal afte F failed ca be outed without violatig the capacity costaits imposed by C, with flow of 0 o each edge that icidet o F. 1 We abuse otatio ad use C to deote the capacity of etwok with capacity matix C.

3 B. The Multiple Netwoks Itecoectio Model We ae iteested i the itecoectio of etwoks. Assume that thee ae q etwoks: X = {x 1,x 2,...x q }.Netwok x X has x odes, ad has homogeeous ate of x at each ode 2. Simila to the case of a sigle etwok, we do ot wat to assume kowledge of a specific taffic matix fo the itecoectio taffic. Rathe, we adopt a simila hosemodel fo the itecoectio taffic. We assume that thee exists a homogeeous boud R p o the igess ad egess ate of each etwok to the othe etwoks. The etwok opeatos decide o R p by egotiatio. Fomally, we call a itecoectio taffic matix legal if it espects the costait o local taffic: fo ay etwok x X ad fo ay ode i x, j x x, ad fo ay j x, i x x. espects the costait o itecoectio taffic: fo ay etwok x X it holds that i x j/ x R p, ad i/ x j x R p. We futhe assume that each etwok is able to geeate taffic of size R p, that is R p mi x X { x x }. While i the sigle etwok case we did ot estict the capacities of the edges iside the etwok, we impose atual estictios o the capacities o edges betwee diffeet etwoks. We assume that each ode has some locatio, ad the locatios of two odes i the same etwok ae diffeet. O the othe had, two odes fom diffeet etwoks ca shae a locatio. We oly allow such odes that shae a locatio to ceate a coectio betwee them. Fomally, let L beaset of locatios. Fo each ode i, letl(i) L be the locatio of ode i. Fox X ad i, j x, l(i) l(j). Fo two diffeet etwoks x, y X, x y ad odes i x, j y it holds that if l(i) l(j) (they ae ot i the same locatio) it implies that thee is o lik betwee the odes i ad j (c ij =0). Let S xy = {i x j ys.t.l(i) =l(j)} be the set of odes i etwok x that ca be peeed to odes i etwok y. That is, S xy is the set of odes fom which taffic ca be set fom etwok x to etwok y. We assume that these coectios ae bidiectioal, that is, evey ode i S xy is coected to a ode i S yx ad vise vesa. Fo the case of multiple etwoks we cae about the capacity of each etwok, that is fo etwok x X, C(x) = i x j x c ij. We also cae about the itecoectio capacity (total capacity betwee the etwoks), defied to be x X i x j/ x c ij. III. UNIVERSAL OPTIMALITY OF VLB TO NODE FAILURES I this sectio we coside a sigle etwok usig the VLB scheme. The VLB etwok suggested by Zhag-She ad McKeow [4] outes each steam i two stages. Fist 1/factio of each steam is set to each of the odes i the etwok (each steam is load balaced o all odes), ad the 2 All odes at the same etwok has the same ate, but diffeet etwoks might have diffeet ates. each steam is set to its destiatio. I [4] it is show that fo this outig scheme, capacity of 2/ o each edge is sufficiet, yieldig total capacity of 2( 1) fo the etie etwok. The pape also shows that total capacity of 2( 1) is ecessay to suppot all taffic matices. The followig is a coollay of Theoem 1 of [4]. Lemma 1: The capacity of a etwok with ñ odes that suppots homogeeous ate is at least 2 (ñ 1). The poof of the above is based o the obsevatio that i ode to suppot the matix i which each ode (1,...,ñ 1) is sedig to the ext ode, total fowad capacity j N,j>i c ij) of at least the size of the total fowaded ( i N flow of (ñ 1) is equied (as each fowaded flow must tavel fowad at least oce), ad the same fo the taspose matix ad backwad capacity. Note that the capacity eeded to suppot ay taffic matix is less tha twice the capacity eeded to suppot oe specific matix. What if odes ca fail? How well does the VLB etwok pefom whe odes might fail? I this sectio we show that VLB has the best possible pefomace with espect to ay umbe of failues, ad is the oly etwok with this popety. We show that fo ay give capacity of a etwok (total capacity of all o-self edges), the VLB etwok has the best esistace to failues ove all etwoks with the same capacity, i the followig sese. Assume that the capacity of the etwok is C =2 ( 1) (miimal capacity to suppot a ate of ). Fo ay l {1,...,}, aftel failues (wost case failues, doe by a advesay), the VLB etwok ca suppot the maximal possible homogeeous ate of, ad o othe etwok has this popety. Coside the VLB etwok i which each edge has a capacity of 2, this etwok ca suppot a homogeeous ate of at each ode. Now assume that thee ae l ode failues - as the etwok is symmetic it does ot matte which odes have failed. We ext show that afte the failues the etwok ca ow suppot a homogeeous ate of at each 1 ode. I the fist stage, each ode seds of each steam to each of the emaiig odes, ad at the secod stage all taffic is set to its destiatio. O each edge, at each stage, thee is a flow of at most ( ) ( 1 )=, thus thee is eough capacity fo this scheme. We ote that the VLB etwok afte ay l {1,...,} ode failues has a capacity of 2 ( l)( l 1). Next we pove that fo ay etwok with capacity C = 2 ( 1), aftel (wost case) failues, the etwok has at most the capacity of the VLB etwok afte l failues. Lemma 2: Give a etwok that has miimal capacity 2 ( 1) eeded to suppot homogeeous ate of, thee exists a set of odes of size l, such that afte all these odes fail, the total emaiig capacity is at most 2 ( l)( l 1). Poof: Let C deote the capacity matix of the etwok. Assume i cotadictio that fo evey set of size l, the total emaiig capacity is geate tha 2 ( l)( l 1). Let S deote the collectio of all sets of size l. By ou assumptio, fo ay set S S,

4 i S j S c ij > 2 ( l)( l 1) As the size of S is ( l), by summig ove all S S we get ( ) c ij > 2 ( l)( l 1) l S S i S j S We use the symmety betwee all odes to figue out how may times each c ij is couted i the above summatio. Fo ay give pai i<j, thee ae ( ) 2 l ways to chose the l odes to emove, out of all odes but i ad j (thee ae 2 such odes). Thus ( ) 2 c ij = c ij l S S i S j S We coclude that ( ) 2 c ij > l i N j N i N j N ( ) 2 ( l)( l 1) l As ( ) ( 2 l = l) ()( 1) ( 1) we deive c ij > 2 ( 1) i N j N which is a cotadictio. Coollay 3: A etwok that has capacity 2 ( 1) caot suppot homogeeous ate of moe tha fo evey l odes that fail. Poof: By Lemma 2 fo some set of l failig odes the 2 emaiig capacity is at most ( l)( l 1). By Lemma 1, if a etwok with capacity C has ñ = l odes, C it ca suppot homogeeous ate of at most 2(ñ 1). Thus if the etwok capacity is C 2 ()( 1), the etwok 2 ca suppot homogeeous ate of at most ()( 1) 2( 1) =. Give the coollay we ca ow defie optimal pefomace afte ay failues. Defiitio 4: A etwok that has miimal capacity 2 ( 1) eeded to suppot homogeeous ate of has optimal l-failues pefomace if fo ay set F of size l of odes that fail, it ca suppot the maximal homogeeous ate of afte F failed. We ext peset the mai esult of the sectio. Theoem 5: The VLB etwok has optimal l-failues pefomace fo ay l {1,...,}, ad is the oly etwok with this popety (ay etwok with this popety has the same capacities o all edges). Poof: The VLB etwok afte ay l {1,...,} ode failues has a capacity of 2 ( l)( l 1), ad as we have see above, ca suppot ate of afte ay failues of up to l odes. By Coollay 3 ay etwok that has capacity 2 ( 1) caot suppot homogeeous ate of moe tha fo evey l odes that fail, thus VLB has optimal l- failues pefomace. Next we show that o othe etwok has the popety (suppot the same maximal flow as the VLB, fo ay l). If a etwok does ot have exactly the same edge capacities as the VLB etwok, but has the same etwok capacity, this implies that thee is a edge (i, j) such that c ij < 2. Thus, fo l = 2, if all odes othe tha i, j fail, the etwok ca suppot less taffic tha VLB. VLB ca suppot a flow of ( 2) = 2, while the othe etwok caot. Thee is aothe way to view the above esults. Give a capacity budget C, the above esults gives the optimal use of such capacity, if oe wishes to build a etwok with maximal homogeeous ate fo ay umbe of ode failues. VLB etwok will eable a homogeeous ate of = C 2( 1) with = C 2( 1), o failues, ad a homogeeous ate of afte l failues (fo ay l). No othe etwok ca use the capacity budget i a bette way. Zhag-She ad McKeow [1] cosideed the poblem of desigig a etwok that ca suppot homogeeous ate of afte l ode failues (a l-toleat VLB etwok). The pape suggests to use capacity of 2 o each edge, ad load balace each steam o all suvivig odes, this gives total capacity 2 2 of ( 1). [1] poited out that the fuctio is vey flat fo small values of l, but gave o poof fo the optimality of this scheme. Ou esult shows that the etwok capacity of thei l-toleat VLB etwok is actually optimal. Coollay 6: Ay etwok that ca suppot ay legal taffic matix with homogeeous ate of afte ay l odes failues, 2 has capacity of at least ( 1). Poof: We have see that a etwok with odes ad ate, ca suppot homogeeous ate of at most afte l failues. If we like to suppot ate of afte the l failues, the = thus = ad by Lemma 1 the ecessay capacity to suppot this ate is 2 ( 1) = 2 ( 1) as equied. IV. GENERALIZATIONS OF THE VLB NETWORK We ow geealize the VLB scheme of [4] ad show that if each steam is load balaced o m odes (hubs) istead of all odes (this ca be viewed as a special case of the geealizatio of [2]), the total capacity of the etwok does ot chage. This m-hubs VLB etwok would be useful late i desigig optimal itecoectio etwok. Additioally, we discuss a geealizatio of this etwok that ca suppot up to l ode failues. Ou m-hubs l-toleat VLB etwok ca be viewed as a geealizatio of the scheme of Zhag-She ad McKeow [1] which hadles ode failues but load balaces each steam o all odes. Ou desig load balaces each steam oly o m odes, ad this implies some icease i capacity. Nevetheless, this scheme is also useful i desigig optimal itecoectio etwoks i the pesece of ode failues.

5 A. The m-hubs VLB Netwok The m-hubs VLB etwok load balaces each steam o m hubs. Defiitio 7: The m-hubs VLB etwok is a odes etwok whee m odes seve as hubs. Routig: Fo a give legal taffic matix Λ, it load balaces each souce-taget steam o each of the m hubs: At the fist stage each ode i seds 1 m of each steam to each of the m hubs, ad at the secod stage each steam is fowaded to its destiatio. Capacities: Let H be the set of hubs, H = m. The capacity of the edge (i, j) is 0 if i/ H, j/ H. m if i/ H, j H (fo the fist stage). m if i H, j/ H (fo the secod stage). 2 m if i H, j H ( m fo each of the two stages). The capacity of the etwok is 2( 1) as C = c ij =2( m)m m +m(m 1)2 m =2( 1) i N j N Note that a sta etwok is a special case whee m =1, ad the VLB scheme is the special case whee m =. Obseve that the capacity is optimal ad idepedet of m! Obsevatio 8: Fo ay m, them-hubs VLB etwok suppots homogeeous ate of. It has the same capacity of 2( 1), ad this capacity is ecessay to suppot ay legal taffic matix. Poof: At the fist stage, each ode i sed 1/m factio of each steam oigiatig fom i, to each ode k H. As j N, capacity of /m is sufficiet fo the taffic o the edge fom i to k H. At the secod stage, as each ode k H eceived 1/m factio of each steam with destiatio ode j, capacity of i N m m is sufficiet fom ode k to ode j fo the secod stage. The etwok has capacity of 2( 1), ad this capacity is show to be ecessay i [4]. We use the m-hubs VLB etwok to build optimal itecoectio of etwoks by peeig ageemets. I Sectio V-B.1 we show that if all etwoks has m>0 shaed locatio, o exta capacity i the etwoks is eeded to suppot peeig taffic. Each etwok us a m-hubs VLB etwok o the set of the m shaed locatios. Routig is doe by fist loadbalacig taffic o the hubs of the souce etwok, the peeig itecoectio taffic to the destiatio etwok, ad fially sedig all taffic to its destiatio ode. Not oly does this scheme has optimal capacity i each etwok, it also has optimal itecoectio (peeig) capacity. We ote that the m-hubs VLB etwok ca also be useful i cases whee oe wishes to educe the umbe of hubs to be maaged, as well as whe thee is ecoomics of scale with espect to edge capacities (as ow each o-zeo capacity edge has lage capacity). O the othe had, lowe umbe of hubs educes the toleace to failues. Next we coside ode failues, ad geealize the above defiitio. B. The m-hubs l-toleat VLB Netwok We begi by defiig toleace to at most l failues. Defiitio 9: A etwok is l-toleat if it ca suppot ay legal taffic matix afte F failed, fo ay F N of size at most l. The m-hubs l-toleat VLB etwok load balaces each steam o m hubs, ad is l-toleat. Defiitio 10: The m-hubs l-toleat VLB etwok is a odes etwok which has m hubs. Let H be the set of hubs, H = m. Assume that the set F of odes failed, ad that F l. Routig: Fo ay legal taffic matix Λ, it load balaces each souce-taget steam o each of the hubs that ae ot 1 i F. That is, factio H\F 1 of is set fom ode i to each ode k H \ F i the fist stage, ad fowaded to the destiatio i the secod stage. Capacities: The capacity of the edge (i, j) is 0 if i/ H, j/ H. if i/ H, j H (fo the fist stage). if i H, j/ H (fo the secod stage). 2 if i H, j H (fo the two stages). The capacity of the etwok is C =2( 1) m. Lemma 11 shows that the m-hubs l-toleat VLB etwok ca ideed suppot up to l ode failues. Lemma 11: The m-hubs l-toleat VLB etwok suppots homogeeous ate of afte F failed, fo ay set F N with F l. Poof: Afte the set F of odes failed ( F l) the flow set o the edges is as follows. At the fist stage, if i, k / F ad k H, i seds to k a flow of size m F H out of the 1 flow, fo ay j/ F (as it seds m F H factio of ay flow fom i/ F to j/ F though ay ode k H \ F ). This implies that o the edges fom i to k, the flow that is set is of size j/ F m F H = 1 m F H m F H m l j/ F Thus o the edge i to k, a capacity of is eough fo the fist stage. At the secod stage, if j, k / F ad k H, k seds to j all the flow it has eceived i the fist stage, that is destiated λ to j. k has eceived fom ode i/ F the flow ij m F H that is destiated to j. This implies that k has eceived at most i/ F m F H m l Thus o the edge k to j, a capacity of is eough fo the secod stage. We coclude that capacity of the m-hubs l-toleat VLB etwok is eough to suppot both stages. Defiitio 12: The l-toleat VLB etwok is defied to be the -hubs l-toleat VLB etwok. Note that the fuctio m is mootoically deceasig with m, thus load balacig o all odes (usig the l-toleat VLB

6 etwok fo which m = ) miimizes the etwok s capacity. The m-sta etwok is the m-hubs l-toleat VLB etwok. The above implies that the l-toleat VLB scheme (m = ) is bette tha the m-sta fo ay m<. Obsevatio 13: The l-toleat VLB etwok (with -hubs) has lowe capacity tha the m-hubs l-toleat VLB etwok (the m-sta etwok), fo m<. Moeove, Coollay 6 shows that the l-toleat VLB etwok is optimal, that is it has miimal capacity ove all etwoks that suppot homogeeous ate of afte l ode failues. V. INTERCONNECTION OF VLB NETWORKS We ae ow eady to peset itecoectio schemes fo multiple VLB etwoks based o tasit ad peeig ageemets. I additio to quatifyig the capacity equiemets fo diffeet schemes, we will also discuss the implicatios o the desig of itecoectio etwoks. Cosistet with established temiology, whe a VLB etwok plays the ole of a tasit etwok, it may cay taffic that eithe oigiates o temiates withi itself. With peeig, o-local taffic with oigi i etwok x X ad destiatio i etwok y X s.t. y x caot go though ay othe etwok z X s.t. z x, y. I this sectio (except i Sectio V-B.1) we coside that etwoks oly have pai-wise shaed locatios, i.e., o locatio is shaed by moe tha two etwoks. A. Tasit We fist coside a sigle tasit etwok to which multiple stub etwoks ae coected, ad show that usig the VLB scheme is optimal. This achitectue may be appopiate fo a atioal utility model o a egulated moopoly model, as the sigle tasit etwok execises moopolistic powe ove the stub etwoks. This scheme may also be appopiate fo a sigle etwok domai distibuted ove a lage geogaphic aea with low taffic volumes betwee egios, as it educes the latecy of taffic that is local to a egio. We also coside the case of two tasit etwoks, such that each etwok aloe ca suppot ay legal itecoectio taffic matix. This scheme esues that o tasit etwok execises moopolistic powe ove the stub etwoks. We ca also view this scheme as obust agaist failue of oe of the tasit etwoks. 1) A Sigle Tasit Netwok: Assume that thee ae q etwoks: X = {x 1,x 2,...x q } ad let z = x q be the tasit etwok. Recall that S xy deotes the odes of etwok x that shae commo locatios with odes of etwok y. Defiitio 14: The itecoectio etwok by VLB tasit is a iteetwok cosistig of a tasit etwok z ad q 1 stub etwoks X \{z} = {x 1,x 2,...x q 1 }. O each stub etwok x we build a S xz -hubs VLB etwok, usig the odes of S xz as the hubs. Let S z = x X S zx be the set of locatio i the tasit etwok that ae shaed with the stub etwoks. I the tasit etwok z we build a S z -hubs VLB etwok usig the odes of S z as the hubs. Routig: The VLB scheme i the tasit etwok us betwee the two stages of the VLB scheme of the stub etwoks. Steam goig fom ode i i etwok x to ode j i etwok y is outed as follows: 1) The steam is load balaced o the hubs of etwok x. If x = z go to step 4. 2) the steam is peeed to the tasit etwok z, foms xz to S zx. 3) It is load balaced fom S zx o all hubs S z of z (without load balacig fom S zx to S zx ) 3 4) If y = z go to step 7. 5) The steam is load balaced fom S z o the peeig odes S zy with etwok y (without load balacig fom S zy to S zy ). 6) It is peeed to the stub etwok y, foms zy to S yz,the hubs of y. 7) The load balaced taffic o the hubs of y is set to the destiatio j. Capacities: I etwok z capacity of 2 z ( z 1) (defied by the S z -hubs VLB etwok capacities) is eeded to suppot stages 1 ad 7. Additioal to the capacities equied fo local taffic, i ode to suppot stage 3 we add capacity of R p /( S zx S z ) fom each ode i S zx to each ode j S z \S zx. To suppot stage 5 we add capacity of R p /( S zy S z ) fom each ode i S z \ S zy to each ode j S zy.the capacity of the tasit etwok z is 2 z ( z 1)+2 R p (q 2). The capacity of each stub etwok x is 2 x ( x 1). The itecoectio capacity is 2 R p (q 1). Note that give a itecoectio etwok by VLB tasit with q 2 etwoks, addig a stub etwok causes a icease i capacity of 2 R p i the tasit etwok (this holds fo ay stub etwok othe tha the fist), ad additioal itecoectio capacity of 2 R p is eeded betwee the ew stub ad the tasit etwok. The tasit etwok opeato ca chage this exta cost to the stub etwok opeato. Theoem 15: The itecoectio etwok by VLB tasit ca suppot ay legal itecoectio taffic matix. Additioally, ay itecoectio etwok that ca suppot ay legal itecoectio taffic matix ad uses a tasit etwok has at least the same capacity i each etwok ad at least the same itecoectio capacity. Poof: Fist obseve that capacity of z ( z 1) is ideed sufficiet fo each of the steps 1 ad 7, to hadle taffic that its oigi o destiatio is etwok z. I ode to suppot stage 3, a capacity of R p /( S zx S z ) fom each ode i S zx to each ode j S z \S zx is sufficiet. This is tue because, afte the load balacig of step 1, each ode i S zx has 1/ S zx of at most R p of itecoectio taffic with oigi at x, ad it eeds to sed 1/ S z of it to j S z \ S zx. A simila agumet holds fo the capacity eeded to suppot stage 5. We ae left to show that the capacity allocated 3 This meas that each ode i S zx seds 1/ S z factio of each steam to each ode i S z \S zx. As each steam is aleady load balaced o S zx, thee is o eed fo load balacig taffic betwee odes i S zx: it uses capacity but at the ed of the stage each ode will have the same factio of each steam as i the absece of this load balacig.

7 i the tasit etwok z to suppot each of these two stages sums to R p (q 2). Ideed, fo stage 3 it holds that R p /( S zx S z )= x X\{z} i S zx j S z\s zx S zx S z \ S zx R p /( S zx S z )= x X\{z} R p R p q 1 x X\{z} x X\{z} (1 S zx / S z )= S zx / S z = R p (q 2). Simila calculatios give a capacity of R p (q 2) fo stage 5. As the taffic is load balaced i each stub etwok, a capacity of R p fom the stub to the tasit etwok is clealy sufficiet fo stage 2. The same capacity is sufficiet fo stage 6. As thee ae q 1 stub etwoks, the total itecoectio capacity is 2 R p (q 1). The ext lemma pesets lowe bouds o the capacity of each etwok, ad the itecoectio capacity. The lowe bouds match the capacities that we achieve usig the VLB scheme. Lemma 16: Give q etwoks such that o moe tha two etwoks shae a commo locatio. If we wish to suppot all legal itecoectio taffic matices usig a sigle tasit etwok, the it is ecessay to allocate capacity of at least 2 z ( z 1) + 2 R p (q 2) i the tasit etwok z. 2 x ( x 1) i each stub etwok x. 2 R p (q 1) fo itecoectio. Poof: By Lemma 1, i ode to suppot the local taffic i etwok x, we eed capacity of at least 2 x ( x 1). Additioally, each of the q 1 stub etwoks eeds a capacity of at least R p to ad fom the tasit etwok to suppot the itecoectio taffic to ad fom this etwok. This gives the lowe boud of 2 R p (q 1) o the itecoectio capacity. Fially, we coside the capacity of the tasit etwok. Coside some odeig ove the odes of z such that a ode coected to etwok x pecede all odes coected to etwok y, wheeve x < y. Fo ay legal itecoectio taffic matix, the total fowad capacity of the tasit etwok, defied to be i z j z,j>i c ij must be at least the size of the total fowad flow i z, plus the total fowad itecoectio flow betwee the stub etwoks, i.e., i z j z j>i c ij i z j z j>i + x z y z i x j y y>x This holds i paticula fo the matix i which each of the q 2 fist stub etwoks seds a combied itecoectio flow of R p to the ext stub etwok, ad each of the z 1 fist odes i z seds z to the ext ode i z. The total fowad flow of this matix is z ( z 1) + R p (q 2). The same boud holds fo the taspose matix ad backwad capacity, ad whe combied we coclude that the capacity of the tasit etwok must be at least 2 z ( z 1) + 2R p (q 2). 2) Two Tasit Netwoks: It is possible to geealize the costuctio peseted i the pevious subsectio to build a itecoectio etwok with multiple tasit etwoks 4. Assume that we would like to ceate a itecoectio etwok with two tasit etwoks ad q 1 stub etwoks (the umbe of stub etwoks is the same as i the pevious sectio), such that eve if oe of the tasit etwoks fails, the othe tasit etwok could suppot ay legal itecoectio taffic matix. Assume that o taffic fom oe tasit etwok is set to the othe tasit etwok (emovig this assumptio will cause mio chages i the followig obsevatios). I each of the tasit etwoks the same capacity as i the itecoectio etwok by VLB tasit etwok is ecessay ad sufficiet. Additioally, the itecoectio capacity will double (agai, it is ecessay ad sufficiet). At each stub etwok x, ifx has a shaed locatio with both tasit etwoks, we could build its hubs o the set of commo locatios to the thee etwoks, ad o exta capacity will be eeded at the stub etwok. If thee ae o locatios shaed by all thee etwoks, we ca coside eithe oe hop o two hops schemes fo outig of itecoectio taffic iside etwok x. I case of oe hop, capacity of 2 (2 x ( x 1)) is sufficiet (by allocatig capacities as if the hubs ae o the peeig odes with oe of the tasit ad with the othe). I case of two hops, we ca use the followig scheme. We build the hubs of x o a set of peeig odes with the two tasit etwoks, with equal umbe of peeig odes with each of the two. That is, let S xz1 ad S xz2 be the set of peeig odes with tasit etwok z 1 ad z 2, espectively. Assume w.l.o.g. that S xz1 S xz2, the we build the hubs o S xz1 ad a set of odes of size S xz1 fom the odes of S xz2.ifat some time tasit etwok z i is used (fo i {1, 2} ad j i), the outig is doe as follows. Fist taffic is load balaced o all hubs, the fom each hub that belogs to j, we sed the itecoectio taffic to a hub that belog to i (usig oeto-oe matchig betwee the two sets of hubs). This equies capacity of R p /(2 S xz1 ) o each edge, ad total capacity of R p /2. Whe itecoectio taffic is eceived fom z j,we do the same i evese ode. The total capacity i x that is sufficiet fo this scheme is 2 x ( x 1) + R p (if etwok j is used fo tasit, the usage of capacities betwee the hubs will be i evese ode, but the same capacities will be sufficiet). This is less tha the capacity eeded fo oe hop outig, sice istead of addig capacity of 2 x ( x 1) we ae addig R p x x (which is smalle fo ay x 2). B. Peeig We ow coside the itecoectio of multiple VLB etwoks usig oly peeig ageemets. We fist coside the case whee thee exists at least oe locatio that is shaed by all the etwoks. I this case, we show that o exta capacity is eeded i each etwok to suppot the peeig taffic. We 4 I the iteest of space we oly peset the basic ideas ad obsevatios, details ca be foud i the exteded vesio of the pape [13].

8 the coside the case whee each locatio is shaed by at most two etwoks. 1) Peeig with Uivesally Shaed Locatios: Defiitio 17: Assume that S, the set of locatio that ae shaed by all q etwoks, is ot empty. The peeig o uivesally shaed locatios VLB etwok is a itecoectio etwok i which each of the q etwoks u a S -hubs VLB etwoks with hubs o the odes of S. Theq etwoks pee at all odes of S. Routig: Routig i this etwok is doe i thee stages. I the fist stage each steam is load balaced o all hubs of the oigiatig etwok, the ay peeig taffic is set o the peeig edges to the destiatio etwok, ad i the fial stage all the taffic is fowaded to its destiatio. Capacities: Each etwok x has a capacity of 2 x ( x 1). Thee is itecoectio (peeig) capacity o R p betwee ay of the q(q 1) odeed pais of etwoks, thus the total itecoectio capacity is R p q(q 1). Theoem 18: The peeig o uivesally shaed locatios VLB etwok ca suppot ay legal itecoectio taffic matix by peeig. Additioally, ay etwok that ca suppot ay legal itecoectio taffic matix by peeig, has at least the same capacity i each etwok ad at least the same itecoectio capacity. Poof: The same agumets as the oes peseted i the poof of Obsevatio 8 also ca be used to pove that the peeig o uivesally shaed locatios VLB etwok ca suppot ay legal itecoectio taffic matix. (It makes o diffeece if the taffic with destiatio ode j is local o ot. The impotat obsevatio is that the ate of all the taffic with destiatio ode j, fom all oigis, is at most ). By Obsevatio 8, a capacity of 2 x ( x 1) is ecessay i etwok x to suppot ay local legal taffic matix, eve without ay peeig taffic. Thus this amout is clealy ecessay. Additioally, ecall that we assume that each etwok ca geeate ad eceive itecoectio taffic at the ate of R p. Thus fo ay of the q(q 1) pais of etwoks x y, a peeig capacity of R p is ecessay betwee x ad y. We coclude that if uivesally shaed locatios exist peeig usig the m-hubs VLB scheme o the shaed locatios is optimal. Moeove, a icease i R p oly esults i a icease i the itecoectio capacity, but ot i the capacities of the idividual etwoks. With some additioal assumptios, the above esult ca be geealized to the case of ode failues 5. If we like to suppot l < m ode failues, each etwok could use the m-hubs l-toleat VLB scheme with hubs o the peeig odes S ( S = m). Now taffic is load balaced o the hubs that suvived failue. The capacity of each etwok, as well as the m itecoectio capacity, gow by a facto of. Ude the stoge assumptio that R p is o lage tha the post-failue ate of each etwok, i.e., R p mi x X { x ( x l)}, by agumets simila to those peseted i the poof of Lemma 2 5 I the iteest of space we oly peset the basic ideas ad obsevatios, details ca be foud i the exteded vesio of the pape [13]. oe ca show that such peeig (itecoectio) capacity is ecessay to suppot failues of up to l odes (Lemma 22 i the Appedix). Moeove, if all etwoks have the same umbe of odes, ad each of the locatios is shaed by all etwoks, each etwok could u the l-toleat VLB etwok which has optimal etwok capacity (by Coollay 6). 2) Peeig with Pai-Wise Shaed Locatios: Above we obseved that if thee is at least oe itecoectio locatio uivesally shaed by all the etwoks, o exta capacity is eeded to suppot the peeig taffic. We ext coside the othe exteme, i which each locatio is shaed by at most two etwoks. We leave the itemediate cases fo futue eseach. Defiitio 19: Assume that ay locatio is shaed by at most two etwoks. The pai-wise peeig VLB etwok is a iteetwok i which each of the q etwoks us a S x -hubs VLB etwoks with hubs o the odes of S x = y x S xy,the set of locatios that x shae with othe etwoks. Netwoks x ay y pee at the set of commo locatios S xy ad S yx, espectively. Routig: Afte load balacig all taffic o the hubs, all peeig taffic is set to the odes that ae peeig with the destiatio etwok. The taffic is the haded off to the coespodig peeig odes, load balaced o the hubs of the destiatio etwok, ad set to the fial destiatio. Fomally, steam fom ode i i etwok x to ode j i etwok y is outed as follows: 1) The steam is load balaced o the hubs of etwok x. Ifx = y go to step 5. 2) The steam is fowaded fom the hubs of x to S xy,the peeig odes with y. 3) It is haded off to etwok y, foms xy to S yx. 4) It is load balaced fom S yx o all hubs of etwok y. 5) It is deliveed fom the hubs of y to the fial destiatio j. Capacities: I etwok x, a capacity of 2 x ( x 1) is sufficiet to suppot stages 1 ad 5. I ode to suppot stage 2 we add capacity of R p /( S xy S x ) fom each ode i S xy to each ode j S x \ S xy. To suppot stage 4 we add capacity of R p /( S yx S y ) fom each ode i S y \ S yx to each ode j S yx. Each etwok x has a capacity of 2 x ( x 1) + 2R p (q 2), ad the itecoectio capacity is of size R p q(q 1). While we caot pove that this peeig scheme yields optimal capacity i each etwok, we ca pove that it is withi a costat facto of the optimal. This costat is idepedet of q, the umbe of etwoks that ae peeig. Theoem 20: The pai-wise peeig VLB etwok ca suppot ay legal itecoectio taffic matix by peeig. Additioally, ay etwok that ca suppot ay legal itecoectio taffic matix by peeig, has at least 1/5 of the capacity i each etwok ad at least the same itecoectio capacity. Poof: Clealy the outig scheme ca oute ay legal itecoectio taffic matix, ad the capacities ae sufficiet fo this outig scheme. We oly eed to veify that the capacity of each etwok x is ideed 2 x ( x 1)+2R p (q 2).

9 The poof fo this is vey simila to the poof of the tasit etwok capacity of Theoem 15, ad is omitted i the iteest of space. Itecoectio capacity of R p q(q 1) is poved to be ecessay fo itecoectio by peeig i Theoem 18. We ext coside the capacity i each of the etwoks. By Lemma 21 below, if we like to suppot all legal itecoectio taffic matices usig peeig, the it is ecessay to allocate capacity of at least max{2 x ( x 1),R p (q 2)/2} i each etwok x. The capacity of etwok x i the pai-wise peeig VLB etwok is 2 x ( x 1) + 2R p (q 2). As 4 max{2 x ( x 1),R p (q 2)/2} 2 R p (q 2): 5 max{2 x ( x 1),R p (q 2)/2} 2 x ( x 1)+2R p (q 2) which implies that i ay itecoectio etwok by peeig, each etwok has at least 1/5 of the capacity of etwok x i the pai-wise peeig VLB etwok. Lemma 21: Give q 2 etwoks such that o moe tha 2 etwoks shae a locatio. If we like to suppot all legal itecoectio taffic matices usig peeig the it is ecessay to allocate capacity of at least max{2 x ( x 1),R p (q 2)/2} i each etwok x. Poof: By Obsevatio 8, a capacity of 2 x ( x 1) is ecessay i etwok x to suppot ay local legal taffic matix. If thee exists etwok y x such that the peeig odes of x with y ca geeate at least half of R p ( x S xy R p /2) the fo each of the q 2 etwoks w y, a capacity of at least x S xy R p /2 must ete S xw. Thus all the capacities that ete all the odes must sum to at least (q 2) R p /2. If fo ay etwok y x, x S xy <R p /2, the fo each y, the odes that ae ot i S xy ca geeate at least R p /2 (as we assume that all odes ca geeate R p ). Thus fo each of the q 1 etwoks y x, a capacity of at least R p /2 must ete S xy. Theefoe, all the capacities that ete all the odes must sum to at least (q 1) R p /2. We coclude that i ay case the capacity of etwok x is at least (q 2) R p /2. Note that if we assumed that fo each etwok y, the odes that ae ot i S xy could geeate a ate of R p, the we could impove the lowe boud to max{2 x ( x 1),R p (q 1)}. Usig this stoge assumptio, simila agumets show that the capacity of each etwok i the pai-wise peeig VLB etwok has at most 3 times the optimal capacity. By extedig the LP fomulatio of [11] we believe that the optimal povisioig ca be calculated i polyomial time, but we leave this fo futue eseach. Fially, we ote that oe might also coside peeig with oe hop i each etwok (istead of 2 hops as i the above scheme). Ude the mild assumptios that all itecoectio taffic caot be geeated by a sigle ode (R p x ) ad the itecoectio taffic is small eough (R p < x x 2 x ( x 1) q 2 ), this scheme causes a icease i capacity (which might be sigificat if R p is much smalle the the above expessio). If R p x this implies that fom each ode we eed capacity of x to each set of peeig odes with at least q 2 etwoks (excludig x ad possibly the etwok that this ode shae locatio with). Thus capacity of at least x x (q 2) is ecessay fo this scheme, ad if R p is small eough this capacity is much lage tha the capacity of the etwok whe usig 2 hops i each etwok. VI. CONCLUSIONS I this pape we have established the optimal esiliece of the Valiat Load-Balacig etwok to ode failues, ad its usefuless as a buildig block fo itecoected etwoks. I paticula, buildig a tasit-based itecoectio etwok usig the VLB scheme yields optimal capacity fo each etwok as well as optimal itecoectio capacity. This wok ca be exteded i the futue by cosideig heteogeeous ates, edge capacity costaits, as well as heteogeeous edge cost stuctues (possibly by extedig the LP fomulatio of [11]). It would also be impotat to coside the esiliece of the desig to edge failues i additio to ode failues. Additioally, etwoks may be itecoected usig a combiatio of tasit ad peeig ageemets. Theefoe we should exted, i futue wok, ou udestadig of the possible use of the VLB scheme i such a hybid eviomet. REFERENCES [1] R. Zhag-She ad N. McKeow, Desigig a pedictable iteet backboe etwok. i HotNets III, [2] M. Kodialam, T. V. Lakshma, ad S. Segupta, Efficiet ad obust outig of highly vaiable taffic, i HotNets III, [3], Guaateeig pedictable pefomace to upedictable taffic, i 43d Aual Alleto Cofeece o Commuicatio, Cotol, ad Computig, [4] R. Zhag-She ad N. McKeow, Desigig a pedictable iteet backboe with valiat load-balacig. i IWQoS, 2005, pp [5] N. G. Duffield, P. Goyal, A. Geebeg, P. Misha, K. K. Ramakisha, ad J. E. va de Meive, A flexible model fo esouce maagemet i vitual pivate etwoks, SIGCOMM Comput. Commu. Rev., vol. 29, o. 4, pp , [6] L. G. Valiat, A scheme fo fast paallel commuicatio, SIAM Joual o Computig., vol. 11, o. 2, pp , [7] M. Kodialam, T. V. Lakshma, ad S. Segupta, Maximum thoughput outig of taffic i the hose model, i IEEE INFOCOM, [8] M. Kodialam, T. V. Lakshma, J. B. Oli, ad S. Segupta, Pecofiguig ip-ove-optical etwoks to hadle oute failues ad upedictable taffic, i IEEE INFOCOM, [9] I. Keslassy, C.-S. Chag, N. McKeow, ad D.-S. Lee, Optimal loadbalacig, i INFOCOM, [10] A. Gupta, J. M. Kleibeg, A. Kuma, R. Rastogi, ad B. Yee, Povisioig a vitual pivate etwok: a etwok desig poblem fo multicommodity flow, i ACM Symposium o Theoy of Computig (STOC), 2001, pp [11] T. Elebach ad M. Rüegg, Optimal badwidth esevatio i hosemodel vps with multi-path outig. i INFOCOM, [12] R. Ahuja, T. Magati, ad J. Oli, Netwok Flows: Theoy, Algoithms, ad Applicatios. Petice Hall, [13] M. Babaioff ad J. Chuag, O the optimality ad itecoectio of valiat load-balacig etwoks, 2006, exteded vesio, o the authos web sites. APPENDIX Lemma 22: Assume that R p mi x X { x ( x l)}. If two etwoks ae peeig at m odes ad suppot ay legal itecoectio taffic matix afte F failed by peeig, whee F is ay set of odes with F l, the the peeig capacity betwee the two etwoks is at least 2R p m. Poof: I [13].