On the Optimality and Interconnection of Valiant Load-Balancing Networks

Transcription

1 O the Optiality ad Itecoectio of Valiat Load-Balacig Netwoks Moshe Babaioff ad Joh Chuag School of Ifoatio 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 pefoace ude chagig taffic atices [1]. I this pape we show that the VLB etwok has optial pefoace whe odes ca fail, i the sese that it ca suppot the axial hoogeeous flow fo ay ube of ode failues. We geealize the VLB desig to eable itecoectio of ultiple VLB etwoks, ad study itecoectio via bilateal peeig ageeets as well as tasit ageeets. We show that usig VLB as a tasit schee yields the lowest possible etwok ad itecoectio capacities, while VLB peeig ca also achieve ea-optial use of capacity. I. INTRODUCTION The Iteet coe cosists of ultiple itecoected backboe etwoks, with each backboe etwok idepedetly povisioed, deployed, ad adiisteed by its owe. The itecoectio egie evolves ove tie, as etwoks egotiate itecoectio ageeets with oe aothe. It has becoe so coplex ad opaque that eseaches have to devise vaious pobig ethods to ife the topology of the Iteet. O top of this, the taffic atices expeieced by the backboe etwoks ae becoig iceasigly vaiable at both lage ad sall tiescales. This is due to a ube of factos, icludig the populaity of ew applicatio classes chaacteized by dyaic ovelay outig of lage data flows. This akes the tasks of taffic egieeig ad etwok povisio/upgade exteely challegig fo the etwok opeatos. Cosequetly, ay backboe opeatos have esoted to ove-povisioig by a facto of up to te i ode to aitai low latecy i thei etwoks. This has led the etwokig couity 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 pefoace fo highly vaiable taffic atices [1], [2]. Fo exaple, the Valiat Load-Balacig (VLB) desig fo Stafod [1] iposes a specific topological stuctue (logical full esh) o the backboe etwok, ad outes data via exactly two hops ove the full esh usig a siple load-balacig schee (equal load-balacig o all odes). This allows the etwok to efficietly suppot chagig taffic atices with obustess agaist failues. The twophase outig schee [2], [3] geealizes the above loadbalacig schee to ay etwok topology, usig o-equal load balacig o iteediate odes. The VLB two-phase load-balacig schee has ay advatages fo the etwok opeato. It hadles the upedictability of taffic, by beig able to suppots ay taffic atix as log as it falls withi the capacity costaits of igess-egess capacity of each ode ( hose odel ). Moeove, it avoids cogestio without ay dyaic o eal-tie cofiguatio of the etwok. Routig is oblivious ad idepedet of the specific taffic atix. Fially, it is efficiet i the sese that it has iial total capacity povisioed [1]. As etwok opeatos begi to coteplate the VLB backboe desig, the atual ext questio aises: how should ultiple VLB etwoks itecoect with oe aothe? This itecoectio poble actually ecopasses ay specific ope questios, icludig: how should the load-balaced outig algoith be geealized acoss ultiple 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 ultiple itecoected VLB etwoks? Diffeet ethods of itecoectig VLB etwoks ae possible, ad they should be copaed agaist oe aothe alog diesios such as efficiecy, obustess, evolvability, ad suppot fo copetitio ad iovatio. I this pape, we will focus o establishig the optiality of the VLB desig fo a stadaloe etwok as well as fo vaious fos of itecoectios. Fist, we will exted the aalysis of [1], [4] ad establish the uivesal optiality of the VLB desig to ode failues (Sectio III). Next, we will popose a geealizatio of the VLB etwok that facilitates itecoectio, aely the -hubs VLB etwok ad the - hubs l-toleat VLB etwok (Sectio IV). This allows us to establish optiality esults fo the cases of tasit ad peeig betwee ultiple VLB etwoks (Sectio V). A. Related Wok Ou pape follows a lie of eseach that ais to desig etwoks ude upedictable taffic, thus eplacig the assuptio of a fixed taffic atix by the hose-odel [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 atix that is cosistet with

2 the ate bouds. We ext discuss soe elated liteatue o etwok desig ude the hose odel. Ou pape is ost closely elated to the papes by Zhag- She ad McKeow [1], [4] which coside the poble of Iteet back-boe IP outig. They suggest to use a twophase outig, which they call the Valiat Load-Balacig (VLB) schee (followig Valiat [6]), o a logical full esh. Routig is doe i two phases, fist, each flow is equally split o all odes, ad the fowaded to its destiatio. This outig schee is show to be optial (with espect to total capacity). We show that it has the best pefoace with espect to ode failues, ad that its geealizatios ca be used fo efficiet itecoectio of etwoks. Kodiala, Laksha ad Segupta [2] suggest two-phase outig schees that ca be viewed as a geealizatio of the VLB schee. They povide Liea Pogaig foulatios fo vaious goals [3], [7] ad ague that it has ay advatages ove diect outig. The pape [8] cosides the issue of esiliece to a sigle oute (ode) failue with thoughput as the optiizatio goal. Ou uivesal optiality of VLB to ode failues is, as fa as we ae awae of, the fist esult showig that the VLB schee pefos optially with espect to ay ube of failues, with total capacity as the optiizatio goal. Keslassy et al. [9] coside the use of a two-phase loadbalacig schee iside a switch. While the applicatio is diffeet, the basic odel ad goal is vey siila to the odel of a sigle etwok that we coside. The pape also cosides outig ude the hoogeeous hose odel o a full esh. The objective of Keslassy et al. is to iiize the su of capacities of all edges i the etwok icludig self edges, while ou goal is to iiize the su of capacities excludig self edges (as i [1], [4]), as this akes oe sese fo back-boe etwoks. [9] shows that optiizig fo the su of all edges capacities esults with a uique optial etwok that is biased (self edges have half the capacity of o-self edges), while fo ou optiizatio goal we obseve that thee ae ultiple optial etwoks, a fact that we show useful i etwoks itecoectio. The liteatue o badwidth povisioig fo Vitual Pivate Netwok (VPN) ude the hose odel ca be viewed as a geealizatio of the sigle etwok odel 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 poble of fidig a optial tee outig o sigle-path outig is NP-had. Elebach ad Rüegg [11] allow fo ultipath outig of splittable flows ad show that the optial povisioig poble ca be solved i polyoial tie usig Liea Pogaig (LP). I cotast, ou pape addesses esiliece to failues ad itecoectio of etwoks, ad offes a siple outig schee with load-balacig splits that ae idepedet of the souce ad destiatio. A. The Sigle Netwok Model II. MODEL We begi by pesetig the odel of a sigle etwok with hoogeeous 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 assue that thee ae o costaits o the edges capacities. A taffic atix Λ is a atix such that a flow (ate) of size 0 eeds to be set fo ode i to ode j. We soeties efe to as the stea fo i to j. As taffic is dyaic ad chages ove tie, ou goal is to build a etwok that ca suppot a lage set of taffic atices. We adopt the hose-odel ([5]) i which each ode i the etwok has a hoogeeous boud o its igess ad egess ates, ad we wish to povisio the etwok to suppot ay taffic atix that is cosistet with the ate bouds. A outig schee defies the way that taffic is outed i the etwok, ad give a outig schee, we ca fid the iial capacity o each edge that is equied to suppot all desied taffic atices. We assue that the flows ae splittable. Foally, a taffic atix is legal with ate if fo ay i N,, ad fo ay j N, i N. Give a etwok with a capacity atix C, we say that the etwok ca suppot a taffic atix Λ if thee exists a solutio to the ulti-coodities flow poble [12] defied by the deads Λ ad the capacity costaits C, o the diected gaph of the etwok. A etwok with capacity C ca suppot hoogeeous ate of if fo ay legal taffic atix Λ with ate, the etwok ca suppot the taffic atix Λ. We defie the capacity of a etwok 1 with a capacity atix C, to be the su of the edges capacity, without self edges: C = i N A etwok is optial if it has iial capacity ove all etwoks that suppot all legal taffic atices with ate. All the etwoks that we peset will use local outig decisios (oblivious outig) ad will ot equie ay solvig of ulti-coodities flow pobles (o ay othe Liea Pogaig), o the kowledge of the specific taffic atix. We also coside ode failues. A failue of a ode iplies that it ca o loge geeate o eceive ay taffic, ad all edges that icidet o the ode ae ot used fo outig. Foally, give that the odes F N failed, a legal taffic atix Λ with ate is legal afte F failed if fo ay i F, = 0, ad fo ay j F, i N = 0. A etwok with capacity atix C suppots hoogeeous 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 iposed 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 atix C. c ij

3 B. The Multiple Netwoks Itecoectio Model We ae iteested i the itecoectio of etwoks. Assue that thee ae q etwoks: X = {x 1, x 2,... x q }. Netwok x X has x odes, ad has hoogeeous ate of x at each ode 2. Siila to the case of a sigle etwok, we do ot wat to assue kowledge of a specific taffic atix fo the itecoectio taffic. Rathe, we adopt a siila hoseodel fo the itecoectio taffic. We assue that thee exists a hoogeeous 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. Foally, we call a itecoectio taffic atix 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 i/ x j x R p. j / x R p, ad We futhe assue that each etwok is able to geeate taffic of size R p, that is R p i x X { x x }. While i the sigle etwok case we did ot estict the capacities of the edges iside the etwok, we ipose atual estictios o the capacities o edges betwee diffeet etwoks. We assue that each ode has soe locatio, ad the locatios of two odes i the sae etwok ae diffeet. O the othe had, two odes fo diffeet etwoks ca shae a locatio. We oly allow such odes that shae a locatio to ceate a coectio betwee the. Foally, let L be a set of locatios. Fo each ode i, let l(i) L be the locatio of ode i. Fo x 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 sae locatio) it iplies that thee is o lik betwee the odes i ad j (c ij = 0). Let S xy = {i x j y s.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 fo which taffic ca be set fo etwok x to etwok y. We assue 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 ultiple etwoks we cae about the capacity of each etwok, that is fo etwok x X, C(x) = i x j x i x c ij. We also cae about the itecoectio capacity (total capacity betwee the etwoks), defied to be x X j / x c ij. III. UNIVERSAL OPTIMALITY OF VLB TO NODE FAILURES I this sectio we coside a sigle etwok usig the VLB schee. The VLB etwok suggested by Zhag-She ad McKeow [4] outes each stea i two stages. Fist 1/factio of each stea is set to each of the odes i the etwok (each stea is load balaced o all odes), ad the 2 All odes at the sae etwok has the sae ate, but diffeet etwoks ight have diffeet ates. each stea is set to its destiatio. I [4] it is show that fo this outig schee, 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 atices. The followig is a coollay of Theoe 1 of [4]. Lea 1: The capacity of a etwok with ñ odes that suppots hoogeeous ate is at least 2 (ñ 1). The poof of the above is based o the obsevatio that i ode to suppot the atix i which each ode (1,...,ñ 1) is sedig to the ext ode, total fowad capacity,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 ust tavel fowad at least oce), ad the sae fo the taspose atix ad backwad capacity. Note that the capacity eeded to suppot ay taffic atix is less tha twice the capacity eeded to suppot oe specific atix. What if odes ca fail? How well does the VLB etwok pefo whe odes ight fail? I this sectio we show that VLB has the best possible pefoace with espect to ay ube 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 sae capacity, i the followig sese. Assue that the capacity of the etwok is C = 2 ( 1) (iial capacity to suppot a ate of ). Fo ay l {1,...,}, afte l failues (wost case failues, doe by a advesay), the VLB etwok ca suppot the axial possible hoogeeous 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 hoogeeous ate of at each ode. Now assue that thee ae l ode failues - as the etwok is syetic it does ot atte which odes have failed. We ext show that afte the failues the etwok ca ow suppot a hoogeeous ate of at each 1 ode. I the fist stage, each ode seds of each stea to each of the eaiig 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 ost ( ) ( 1 ) =, thus thee is eough capacity fo this schee. 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), afte l (wost case) failues, the etwok has at ost the capacity of the VLB etwok afte l failues. Lea 2: Give a etwok that has iial capacity 2 ( 1) eeded to suppot hoogeeous ate of, thee exists a set of odes of size l, such that afte all these odes fail, the total eaiig capacity is at ost 2 ( l)( l 1). Poof: Let C deote the capacity atix of the etwok. Assue i cotadictio that fo evey set of size l, the total eaiig capacity is geate tha 2 ( l)( l 1). Let S deote the collectio of all sets of size l. By ou assuptio, fo ay set S S,

4 i S S S i S j S j S c ij > 2 ( l)( l 1) As the size of S is ( l), by suig ove all S S we get ( ) c ij > 2 ( l)( l 1) l We use the syety betwee all odes to figue out how ay ties each c ij is couted i the above suatio. Fo ay give pai i < j, thee ae ( ) 2 l ways to chose the l odes to eove, 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 i N ( ) 2 ( l)( l 1) l As ( ) ( 2 l = l) ()( 1) ( 1) we deive c ij > 2 ( 1) i N which is a cotadictio. Coollay 3: A etwok that has capacity 2 ( 1) caot suppot hoogeeous ate of oe tha fo evey l odes that fail. Poof: By Lea 2 fo soe set of l failig odes the 2 eaiig capacity is at ost ( l)( l 1). By Lea 1, if a etwok with capacity C has ñ = l odes, C it ca suppot hoogeeous ate of at ost 2(ñ 1). Thus if the etwok capacity is C 2 ()( 1), the etwok 2 ca suppot hoogeeous ate of at ost ()( 1) 2( 1) =. Give the coollay we ca ow defie optial pefoace afte ay failues. Defiitio 4: A etwok that has iial capacity 2 ( 1) eeded to suppot hoogeeous ate of has optial l-failues pefoace if fo ay set F of size l of odes that fail, it ca suppot the axial hoogeeous ate of afte F failed. We ext peset the ai esult of the sectio. Theoe 5: The VLB etwok has optial l-failues pefoace fo ay l {1,...,}, ad is the oly etwok with this popety (ay etwok with this popety has the sae 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 hoogeeous ate of oe tha fo evey l odes that fail, thus VLB has optial l- failues pefoace. Next we show that o othe etwok has the popety (suppot the sae axial flow as the VLB, fo ay l). If a etwok does ot have exactly the sae edge capacities as the VLB etwok, but has the sae etwok capacity, this iplies 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 optial use of such capacity, if oe wishes to build a etwok with axial hoogeeous ate fo ay ube of ode failues. VLB etwok will eable a hoogeeous ate of = C 2( 1) with o failues, ad a hoogeeous ate of = C 2( 1), 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 poble of desigig a etwok that ca suppot hoogeeous 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 stea o all suvivig odes, this gives total capacity 2 2 of ( 1). [1] poited out that the fuctio is vey flat fo sall values of l, but gave o poof fo the optiality of this schee. Ou esult shows that the etwok capacity of thei l-toleat VLB etwok is actually optial. Coollay 6: Ay etwok that ca suppot ay legal taffic atix with hoogeeous 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 hoogeeous ate of at ost afte l failues. If we like to suppot ate of afte the l failues, the = thus = ad by Lea 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 schee of [4] ad show that if each stea is load balaced o 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 -hubs VLB etwok would be useful late i desigig optial itecoectio etwok. Additioally, we discuss a geealizatio of this etwok that ca suppot up to l ode failues. Ou -hubs l-toleat VLB etwok ca be viewed as a geealizatio of the schee of Zhag-She ad McKeow [1] which hadles ode failues but load balaces each stea o all odes. Ou desig load balaces each stea oly o odes, ad this iplies soe icease i capacity. Nevetheless, this schee is also useful i desigig optial itecoectio etwoks i the pesece of ode failues.

5 A. The -hubs VLB Netwok The -hubs VLB etwok load balaces each stea o hubs. Defiitio 7: The -hubs VLB etwok is a odes etwok whee odes seve as hubs. Routig: Fo a give legal taffic atix Λ, it load balaces each souce-taget stea o each of the hubs: At the fist stage each ode i seds 1 of each stea to each of the hubs, ad at the secod stage each stea is fowaded to its destiatio. Capacities: Let H be the set of hubs, H =. 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 each of the two stages). The capacity of the etwok is 2( 1) as C = i N c ij = 2( ) +( 1)2 = 2( 1) Note that a sta etwok is a special case whee = 1, ad the VLB schee is the special case whee =. Obseve that the capacity is optial ad idepedet of! Obsevatio 8: Fo ay, the -hubs VLB etwok suppots hoogeeous ate of. It has the sae capacity of 2( 1), ad this capacity is ecessay to suppot ay legal taffic atix. Poof: At the fist stage, each ode i sed 1/ factio of each stea oigiatig fo i, to each ode k H. As, capacity of / is sufficiet fo the taffic o the edge fo i to k H. At the secod stage, as each ode k H eceived 1/ factio of each stea with destiatio ode j, capacity of i N is sufficiet fo 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 -hubs VLB etwok to build optial itecoectio of etwoks by peeig ageeets. I Sectio V-B.1 we show that if all etwoks has > 0 shaed locatio, o exta capacity i the etwoks is eeded to suppot peeig taffic. Each etwok us a -hubs VLB etwok o the set of the 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 schee has optial capacity i each etwok, it also has optial itecoectio (peeig) capacity. We ote that the -hubs VLB etwok ca also be useful i cases whee oe wishes to educe the ube of hubs to be aaged, as well as whe thee is ecooics of scale with espect to edge capacities (as ow each o-zeo capacity edge has lage capacity). O the othe had, lowe ube of hubs educes the toleace to failues. Next we coside ode failues, ad geealize the above defiitio. B. The -hubs l-toleat VLB Netwok We begi by defiig toleace to at ost l failues. Defiitio 9: A etwok is l-toleat if it ca suppot ay legal taffic atix afte F failed, fo ay F N of size at ost l. The -hubs l-toleat VLB etwok load balaces each stea o hubs, ad is l-toleat. Defiitio 10: The -hubs l-toleat VLB etwok is a odes etwok which has hubs. Let H be the set of hubs, H =. Assue that the set F of odes failed, ad that F l. Routig: Fo ay legal taffic atix Λ, it load balaces each souce-taget stea o each of the hubs that ae ot 1 i F. That is, factio H\F 1 of is set fo 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). Lea 11 shows that the -hubs l-toleat VLB etwok ca ideed suppot up to l ode failues. Lea 11: The -hubs l-toleat VLB etwok suppots hoogeeous 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 F H out of the 1 flow, fo ay j / F (as it seds F H factio of ay flow fo i / F to j / F though ay ode k H \F). This iplies that o the edges fo i to k, the flow that is set is of size j / F F H = 1 F H F H 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 fo ode i / F the flow ij F H that is destiated to j. This iplies that k has eceived at ost i/ F F H l Thus o the edge k to j, a capacity of is eough fo the secod stage. We coclude that capacity of the -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 is ootoically deceasig with, thus load balacig o all odes (usig the l-toleat VLB

6 etwok fo which = ) iiizes the etwok s capacity. The -sta etwok is the -hubs l-toleat VLB etwok. The above iplies that the l-toleat VLB schee ( = ) is bette tha the -sta fo ay <. Obsevatio 13: The l-toleat VLB etwok (with -hubs) has lowe capacity tha the -hubs l-toleat VLB etwok (the -sta etwok), fo <. Moeove, Coollay 6 shows that the l-toleat VLB etwok is optial, that is it has iial capacity ove all etwoks that suppot hoogeeous ate of afte l ode failues. V. INTERCONNECTION OF VLB NETWORKS We ae ow eady to peset itecoectio schees fo ultiple VLB etwoks based o tasit ad peeig ageeets. I additio to quatifyig the capacity equieets fo diffeet schees, we will also discuss the iplicatios o the desig of itecoectio etwoks. Cosistet with established teiology, whe a VLB etwok plays the ole of a tasit etwok, it ay cay taffic that eithe oigiates o teiates 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 oe tha two etwoks. A. Tasit We fist coside a sigle tasit etwok to which ultiple stub etwoks ae coected, ad show that usig the VLB schee is optial. This achitectue ay be appopiate fo a atioal utility odel o a egulated oopoly odel, as the sigle tasit etwok execises oopolistic powe ove the stub etwoks. This schee ay also be appopiate fo a sigle etwok doai distibuted ove a lage geogaphic aea with low taffic volues 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 atix. This schee esues that o tasit etwok execises oopolistic powe ove the stub etwoks. We ca also view this schee as obust agaist failue of oe of the tasit etwoks. 1) A Sigle Tasit Netwok: Assue 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 coo 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 schee i the tasit etwok us betwee the two stages of the VLB schee of the stub etwoks. Stea goig fo ode i i etwok x to ode j i etwok y is outed as follows: 1) The stea is load balaced o the hubs of etwok x. If x = z go to step 4. 2) the stea is peeed to the tasit etwok z, fo S xz to S zx. 3) It is load balaced fo S zx o all hubs S z of z (without load balacig fo S zx to S zx ) 3 4) If y = z go to step 7. 5) The stea is load balaced fo S z o the peeig odes S zy with etwok y (without load balacig fo S zy to S zy ). 6) It is peeed to the stub etwok y, fo S 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 ) fo 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 ) fo 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. Theoe 15: The itecoectio etwok by VLB tasit ca suppot ay legal itecoectio taffic atix. Additioally, ay itecoectio etwok that ca suppot ay legal itecoectio taffic atix ad uses a tasit etwok has at least the sae capacity i each etwok ad at least the sae 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 ) fo 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 ost 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 siila aguet holds fo the capacity eeded to suppot stage 5. We ae left to show that the capacity allocated 3 This eas that each ode i S zx seds 1/ S z factio of each stea to each ode i S z \S zx. As each stea 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 sae factio of each stea as i the absece of this load balacig.

7 i the tasit etwok z to suppot each of these two stages sus 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). Siila 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 fo the stub to the tasit etwok is clealy sufficiet fo stage 2. The sae 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 lea pesets lowe bouds o the capacity of each etwok, ad the itecoectio capacity. The lowe bouds atch the capacities that we achieve usig the VLB schee. Lea 16: Give q etwoks such that o oe tha two etwoks shae a coo locatio. If we wish to suppot all legal itecoectio taffic atices 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 Lea 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 fo the tasit etwok to suppot the itecoectio taffic to ad fo 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 soe 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 atix, the total fowad capacity of the tasit etwok, defied to be i z j z,j>i c ij ust 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 atix i which each of the q 2 fist stub etwoks seds a cobied 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 atix is z ( z 1) + R p (q 2). The sae boud holds fo the taspose atix ad backwad capacity, ad whe cobied we coclude that the capacity of the tasit etwok ust 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 ultiple tasit etwoks 4. Assue that we would like to ceate a itecoectio etwok with two tasit etwoks ad q 1 stub etwoks (the ube of stub etwoks is the sae 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 atix. Assue that o taffic fo oe tasit etwok is set to the othe tasit etwok (eovig this assuptio will cause io chages i the followig obsevatios). I each of the tasit etwoks the sae 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, if x has a shaed locatio with both tasit etwoks, we could build its hubs o the set of coo 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 schees 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 schee. We build the hubs of x o a set of peeig odes with the two tasit etwoks, with equal ube 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. Assue 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 fo the odes of S xz2. If at soe tie 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 fo each hub that belogs to j, we sed the itecoectio taffic to a hub that belog to i (usig oeto-oe atchig 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 fo z j, we do the sae i evese ode. The total capacity i x that is sufficiet fo this schee 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 sae 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 salle fo ay x 2). B. Peeig We ow coside the itecoectio of ultiple VLB etwoks usig oly peeig ageeets. 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 ost two etwoks. 1) Peeig with Uivesally Shaed Locatios: Defiitio 17: Assue that S, the set of locatio that ae shaed by all q etwoks, is ot epty. 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. The q etwoks pee at all odes of S. Routig: Routig i this etwok is doe i thee stages. I the fist stage each stea 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). Theoe 18: The peeig o uivesally shaed locatios VLB etwok ca suppot ay legal itecoectio taffic atix by peeig. Additioally, ay etwok that ca suppot ay legal itecoectio taffic atix by peeig, has at least the sae capacity i each etwok ad at least the sae itecoectio capacity. Poof: The sae aguets 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 atix. (It akes o diffeece if the taffic with destiatio ode j is local o ot. The ipotat obsevatio is that the ate of all the taffic with destiatio ode j, fo all oigis, is at ost ). By Obsevatio 8, a capacity of 2 x ( x 1) is ecessay i etwok x to suppot ay local legal taffic atix, eve without ay peeig taffic. Thus this aout is clealy ecessay. Additioally, ecall that we assue 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 -hubs VLB schee o the shaed locatios is optial. 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 soe additioal assuptios, the above esult ca be geealized to the case of ode failues 5. If we like to suppot l < ode failues, each etwok could use the -hubs l-toleat VLB schee with hubs o the peeig odes S ( S = ). Now taffic is load balaced o the hubs that suvived failue. The capacity of each etwok, as well as the itecoectio capacity, gow by a facto of. Ude the stoge assuptio that R p is o lage tha the post-failue ate of each etwok, i.e., R p i x X { x ( x l)}, by aguets siila to those peseted i the poof of Lea 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 (Lea 22 i the Appedix). Moeove, if all etwoks have the sae ube of odes, ad each of the locatios is shaed by all etwoks, each etwok could u the l-toleat VLB etwok which has optial 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 extee, i which each locatio is shaed by at ost two etwoks. We leave the iteediate cases fo futue eseach. Defiitio 19: Assue that ay locatio is shaed by at ost 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 coo 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. Foally, stea fo ode i i etwok x to ode j i etwok y is outed as follows: 1) The stea is load balaced o the hubs of etwok x. If x = y go to step 5. 2) The stea is fowaded fo the hubs of x to S xy, the peeig odes with y. 3) It is haded off to etwok y, fo S xy to S yx. 4) It is load balaced fo S yx o all hubs of etwok y. 5) It is deliveed fo 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 ) fo 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 ) fo 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 schee yields optial capacity i each etwok, we ca pove that it is withi a costat facto of the optial. This costat is idepedet of q, the ube of etwoks that ae peeig. Theoe 20: The pai-wise peeig VLB etwok ca suppot ay legal itecoectio taffic atix by peeig. Additioally, ay etwok that ca suppot ay legal itecoectio taffic atix by peeig, has at least 1/5 of the capacity i each etwok ad at least the sae itecoectio capacity. Poof: Clealy the outig schee ca oute ay legal itecoectio taffic atix, ad the capacities ae sufficiet fo this outig schee. 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 siila to the poof of the tasit etwok capacity of Theoe 15, ad is oitted i the iteest of space. Itecoectio capacity of R p q(q 1) is poved to be ecessay fo itecoectio by peeig i Theoe 18. We ext coside the capacity i each of the etwoks. By Lea 21 below, if we like to suppot all legal itecoectio taffic atices usig peeig, the it is ecessay to allocate capacity of at least ax{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 ax{2 x ( x 1), R p (q 2)/2} 2 R p (q 2): 5 ax{2 x ( x 1), R p (q 2)/2} 2 x ( x 1)+2R p (q 2) which iplies 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. Lea 21: Give q 2 etwoks such that o oe tha 2 etwoks shae a locatio. If we like to suppot all legal itecoectio taffic atices usig peeig the it is ecessay to allocate capacity of at least ax{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 atix. 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 ust ete S xw. Thus all the capacities that ete all the odes ust su 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 assue 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 ust ete S xy. Theefoe, all the capacities that ete all the odes ust su 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 assued that fo each etwok y, the odes that ae ot i S xy could geeate a ate of R p, the we could ipove the lowe boud to ax{2 x ( x 1), R p (q 1)}. Usig this stoge assuptio, siila aguets show that the capacity of each etwok i the pai-wise peeig VLB etwok has at ost 3 ties the optial capacity. By extedig the LP foulatio of [11] we believe that the optial povisioig ca be calculated i polyoial tie, but we leave this fo futue eseach. Fially, we ote that oe ight also coside peeig with oe hop i each etwok (istead of 2 hops as i the above schee). Ude the ild assuptios that all itecoectio taffic caot be geeated by a sigle ode (R p x ) ad the itecoectio taffic is sall eough (R p < x x x ( x 1) q 2 2 ), this schee causes a icease i capacity (which ight be sigificat if R p is uch salle the the above expessio). If R p x this iplies that fo 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 schee, ad if R p is sall eough this capacity is uch lage tha the capacity of the etwok whe usig 2 hops i each etwok. VI. CONCLUSIONS I this pape we have established the optial 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 schee yields optial capacity fo each etwok as well as optial 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 foulatio of [11]). It would also be ipotat to coside the esiliece of the desig to edge failues i additio to ode failues. Additioally, etwoks ay be itecoected usig a cobiatio of tasit ad peeig ageeets. Theefoe we should exted, i futue wok, ou udestadig of the possible use of the VLB schee i such a hybid evioet. REFERENCES [1] R. Zhag-She ad N. McKeow, Desigig a pedictable iteet backboe etwok. i HotNets III, [2] M. Kodiala, T. V. Laksha, ad S. Segupta, Efficiet ad obust outig of highly vaiable taffic, i HotNets III, [3], Guaateeig pedictable pefoace to upedictable taffic, i 43d Aual Alleto Cofeece o Couicatio, Cotol, ad Coputig, [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. Raakisha, ad J. E. va de Meive, A flexible odel fo esouce aageet i vitual pivate etwoks, SIGCOMM Coput. Cou. Rev., vol. 29, o. 4, pp , [6] L. G. Valiat, A schee fo fast paallel couicatio, SIAM Joual o Coputig., vol. 11, o. 2, pp , [7] M. Kodiala, T. V. Laksha, ad S. Segupta, Maxiu thoughput outig of taffic i the hose odel, i IEEE INFOCOM, [8] M. Kodiala, T. V. Laksha, 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, Optial loadbalacig, i IEEE INFOCOM, 2005, pp [10] A. Gupta, J. M. Kleibeg, A. Kua, R. Rastogi, ad B. Yee, Povisioig a vitual pivate etwok: a etwok desig poble fo ulticoodity flow, i ACM Syposiu o Theoy of Coputig (STOC), 2001, pp [11] T. Elebach ad M. Rüegg, Optial badwidth esevatio i hoseodel vps with ulti-path outig. i IEEE INFOCOM, [12] R. Ahuja, T. Magati, ad J. Oli, Netwok Flows: Theoy, Algoiths, ad Applicatios. Petice Hall, [13] M. Babaioff ad J. Chuag, O the optiality ad itecoectio of valiat load-balacig etwoks, 2006, exteded vesio, o the authos web sites. APPENDIX Lea 22: Assue that R p i x X { x ( x l)}. If two etwoks ae peeig at odes ad suppot ay legal itecoectio taffic atix 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 Poof: I [13]..