Chapter 2 Valiant Load-Balancing: Building Networks That Can Support All Traffic Matrices

Transcription

1 Chapte 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices Rui Zhang-Shen Abstact This pape is a bief suvey on how Valiant load-balancing (VLB) can be used to build netwoks that can efficiently and eliably suppot all taffic matices. We discuss how to extend VLB to netwoks with heteogeneous capacities, how to potect against failues in a VLB netwok, and how to inteconnect two VLB netwoks. Fo the eades efeence, included also is a list of wok that uses VLB in vaious aspects of netwoking. 2.1 Intoduction In many netwoks the taffic matix is eithe had to measue and pedict, o highly vaiable ove time. In these cases, using Valiant load-balancing (VLB) to suppot all possible taffic matices is an attactive option. Fo example, even though the taffic in the Intenet backbone is extemely smooth due to high level of aggegation, it is still had to measue. Accuately measuing the taffic matix (e.g., using etflow) is too expensive to do all the time, and standad methods using link measuements give eos of 20% o moe. Even if the cuent taffic matix is satisfactoily obtained, extapolating it to the futue is faught with uncetainty, due to the unpedictable natue of Intenet taffic gowth. Finally, since Intenet taffic is dynamic, the taffic matix can deviate fom its nomal values at any time, possibly causing congestion. The taffic demand seen by a netwok can be epesented by the taffic matix, which indicates the ates at which each node initiates taffic to evey othe node. We say a netwok can suppot a taffic matix if fo evey link in the netwok, the load caused by the taffic matix is less than the capacity of the link. When a netwok cannot suppot the taffic matix pesented to it, at least one link in the netwok has a load highe than its capacity. Congestion occus and backlog in the buffe builds up on the congested link(s), causing packet dops, inceased delay, and high vaiations R. Zhang-Shen ( ) Google, Inc., ew Yok, Y uizhangshen@gmail.com G. Comode and M. Thottan (eds.), Algoithms fo ext Geneation etwoks, Compute Communications and etwoks, DOI / , c Spinge-Velag London Limited

2 20 R. Zhang-Shen in delay. Ideally, we would like to design a netwok that can suppot a wide ange of taffic matices, so that congestion occus only aely o not at all. In this pape, we discuss the use of VLB in building netwoks that can efficiently suppot all taffic matices which do not ove-subscibe any node. We fist biefly suvey the wide use of VLB in vaious aspects of netwoking, and descibe the basic scenaio of using VLB in a netwok. Section 2.2 extends VLB fom a homogeneous setting to netwoks with abitay capacities, Section 2.3 descibes how to potect against and ecove quickly fom failues in a VLB netwok, and Section 2.4 poposes to use VLB to oute taffic between two netwoks. Finally Section 2.5 discusses possible futue wok The Wide Use of VLB In the ealy 1980s, Valiant [19] fist poposed the scheme of outing though a andomly picked intemediate node en oute to a packet s destination. He showed that in an -node binay cube netwok, given any pemutation taffic matix, the distibuted two-phase andomized outing can oute evey packet to its destination within O(log ) time with ovewhelming pobability. This was the fist scheme fo outing abitay pemutation in a spase netwok in O(log ) time. Since then, such andomized outing has been used widely, and is often efeed to as (VLB), andomized load-balancing,otwo-phase outing. VLB has many good chaacteistics. It is decentalized, whee evey node makes local decisions. This also makes the scheme scalable. VLB is agnostic to the taffic matix because the andomness eases the taffic patten, and diffeent taffic matices can esult in the same load on the links. Soon afte its invention, was used in othe inteconnection netwoks fo paallel communication, to impove delivey time [1], and to elieve effects of advese taffic pattens [13]. In ecent yeas, it was adapted fo outing in tous netwoks [17, 18] in ode to povide wost-case pefomance guaantees without sacificing aveagecase pefomance. The key is to use VLB adaptively, based on the obsevation that unde low load, load-balancing only a small amount of taffic is sufficient to avoid congestion. VLB is also used in building netwok switches with geat scalability and pefomance guaantee, without the need of a centalized schedule. It was used in ATM switches [7], outes [4, 5], opticaloutes [3, 9], andsoftwaeoutes [2]. Inpaticula,theschemewas ediscoveedfodesigningouteswitch fabics [4] to mitigate outes scaling challenges, because it was difficult fo centalized schemes to keep up with the inceasing link speed. In this context, it was shown that splitting taffic in a ound-obin fashion has the same effect on link load as andom splitting [4], and that is the most efficient in tems of the total equied inteconnection capacity fo suppoting all taffic matices [8]. Almost simultaneously, seveal goups independently applied the idea of VLB to taffic engineeing and netwok design fo the Intenet, in ode to efficiently sup-

3 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices 21 pot all possible taffic matices. Kodialam et al. s two-phase outing [11, 12] isa taffic engineeing method, whee a full mesh of tunnels ae set up ove fixed capacity links and packets ae sent in two phases (i.e., two hops) in the netwok. Winze et al. s selective andomized load-balancing [14, 16, 21] used VLB and its vaiants to design cost-effective optical netwoks. Thei model assumes that a link s cost includes both the fibe and the teminating equipment, so thee is incentive fo having fewe links. In the optimal design, taffic is load-balanced only to a few intemediate nodes. Zhang-Shen and McKewon [23, 25] poposed using VLB ove a logical full mesh in a backbone netwok to suppot all taffic matices and to quickly ecove fom failues. In addition, VLB was used as an optical outing stategy in Ethenet LA [20], fo scheduling in meto aea WDM ings [10], fo cicuit-switched netwoks [22], and fo scaling and commoditizing data cente netwoks [6]. A study on the queueing popeties of a VLB netwok [15] found that VLB eliminates congestion in the netwok, and pseudo-andom (e.g., ound-obin) loadbalancing educes queueing delay. VLB was also shown to eliminate congestion on peeing links when used to oute taffic between netwoks [26] A Simple VLB etwok Conside a netwok of nodes, each with capacity, i.e., a node can initiate taffic at the maximum ate of, and can eceive taffic at the same maximum ate. We assume that the netwok taffic satisfies such node aggegate constaint, because othewise thee is no way to avoid congestion. A logical link of capacity 2 is established between evey pai of nodes ove the physical links, as shown in Figue 2.1. We use the convention that a flow in the netwok is defined by the souce node and the destination node, unless futhe specified. Evey flow enteing the netwok is equally split acoss two-hop paths between ingess and egess nodes, i.e., a packet is fowaded twice in the netwok: In the fist hop, an ingess node unifomly distibutes each of its incoming flows to all the nodes, egadless of 1 2/ 2 3 Fig. 2.1 VLB in a netwok of identical nodes each having capacity.afull mesh of logical links of capacity 2= connect the nodes... 4

4 22 R. Zhang-Shen the destinations. In the second hop, all packets ae sent to the final destinations by the intemediate nodes. Load-balancing can be done packet-by-packet, o flow-byflow at the application flow level. The splitting of taffic can be andom (e.g., to a andomly picked intemediate node) o deteministic (e.g., ound-obin). Assume we can achieve pefect load-balancing, i.e., can split taffic at the exact popotions we desie, then each node eceives exactly 1 of evey flow afte fist-hop outing. This means, all the nodes equally shae the buden of fowading taffic as the intemediate node. When the intemediate node happens to be the ingess o egess node, the flow actually taveses one hop (the diect link 2 between ingess and egess) in the netwok. Hence, of evey flow taveses the coesponding one-hop path. Such unifom load-balancing can guaantee to suppot all taffic matices in this netwok. Since the incoming taffic ate to each node is at most, andthetafficis evenly load-balanced to nodes, the actual taffic on each link due to the fist-hop outing is at most. The second-hop outing is the dual of the fist-hop outing. Since each node can eceive taffic at a maximum ate of and eceives 1 of the taffic fom evey node, the actual taffic on each link due to the second-hop outing is also at most 2. Theefoe, a full-mesh netwok whee each link has capacity is sufficient to suppot all taffic matices in a netwok of nodes of capacity. This is pehaps a supising esult a netwok whee any two nodes ae connected with a link of capacity 2 can suppot taffic matices whee a node can send taffic to anothe node at ate. It shows the powe of load-balancing. In VLB, each flow is caied by paths, and each link caies a faction of many flows; theefoe any lage flow is aveaged out by othe small flows. In a static full-mesh netwok, if all the taffic wee to be sent though diect paths, we would need a full-mesh netwok of link capacity to suppot all possible taffic matices; theefoe load-balancing is 2 times moe efficient than diect outing. 2.2 VLB in Heteogeneous etwoks Real-life netwoks ae often heteogeneous, i.e., the nodes in a netwok can have diffeent capacities. In this section we discuss how to extend the esult of unifom load-balancing to heteogeneous netwoks [24]. We fist intoduce notations. In a netwok of nodes, the taffic matix D f ij g is an matix, whee the enty ij indicates the dataate at which ode i initiates taffic destined to ode j. The taffic ate is typically aveaged ove a long peiod of time so we conside it as constant. Typically in a netwok thee ae buffes to absob shot-lived taffic fluctuations. Suppose ode i has capacity i, i.e., the node can initiate taffic at the maximum ate of i, and can eceive taffic at the same maximum ate. So in ode fo the taffic matix to not ove-subscibe any node, it must satisfy X X ij i ; 8i and ji i ; 8i: (2.1) j j

5 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices 23 Without loss of geneality, assume that the nodes have been soted accoding to deceasing capacities, i.e., 1 2, so ode 1 is the lagest node and ode the smallest. Let R be the total node capacity, i.e., R D P id1 i.we assume 1 P id2 i because even if 1 > P id2 i, ode 1 cannot send o eceive taffic at a ate highe than P id2 i, because that would ove-subscibe some nodes. Suppose that a full mesh of logical links ae set up to connect these nodes. Let c ij epesent the equied link capacity fom ode i to ode j and C the link capacity matix fc ij g.havingc ij D 0 means that link.i; j / is not needed. The simple homogeneous netwok pesented in Section has that c ij D 2 ; 8i j, and i D ; 8i. In a netwok with identical nodes, it is natual to load-balance unifomly. But unifom load-balancing seems too estictive in a heteogeneous netwok because it does not take into account the diffeence in node capacities. A natual solution is to load-balance popotionally to the capacity of the intemediate node, i.e., ode i eceives faction i R of evey flow. This is a diect genealization fom unifom multicommodity flow in the homogeneous case to poduct multicommodity flow [16]. The equied link capacity is c ij D 2 i j =R [24]. We can futhe genealize VLB to allow any flow splitting atio and let some extenal objective to detemine the optimal atios. We intoduce a set of loadbalancing paametes p i such that p i 0, foalli, and P id1 p i D 1. An ingess node splits each flow accoding to fp i g and sends p i of evey flow to ode i. This gives us the feedom of, fo example, letting the lage nodes fowad moe taffic than the smalle nodes, o not using some of the nodes as intemediates (by setting the coesponding p i to zeo). If thee ae some objectives to be optimized, thee ae paametes (p i, i D 1;2;:::;) that can be tuned, but if moe feedom is needed, we can, fo example, let each flow have its own set of load-balancing paametes. We now find the equied link capacity. The fist-hop taffic on link.i; j / is the taffic initiated by ode i that is load-balanced to ode j,andtheateisatmost i p j. The second-hop taffic on the link is the taffic destined to ode j that is loadbalanced to ode i, and the ate is at most j p i. Theefoe the maximum amount of taffic on link.i; j / is i p j C j p i, which is also the equied capacity on link.i; j / c ij D i p j C j p i : (2.2) The equied (outgoing) inteconnection capacity of ode i is l i D X c ij D i C Rp i 2 i p i ; j Wj i and the total equied inteconnection capacity of the netwok is L D X l i D id1 X i;jwi j c ij D 2 R X i! i p i : (2.3)

6 24 R. Zhang-Shen Thus if we want to minimize the total equied inteconnection capacity of the netwok, we need to maximize P i i p i, subject to the constaints on p i. The optimal solution has the fom that p i >0only if i D max j j, i.e., taffic is only load-balanced to the lagest node(s). Thus we have min L D 2.R max i /: (2.4) i We can show that Equation (2.4) is also the minimum total inteconnection capacity equied by any netwok to suppot all taffic matices [24], and hence is the necessay and sufficient condition. One netwok that minimizes the total equied inteconnection capacity is a sta with ode 1 at the cente, i.e., p 1 D 1, p i D 0 fo i 2. In the case whee all nodes have the same capacity, to achieve minimum L, p i can take any non-negative values as long as they sum up to 1. Thus the unifom load-balancing pesented in Section is optimal in tems of total equied inteconnection capacity. Splitting flows popotional to node capacities, i.e., p i D i =R, is not optimal when nodes have diffeent capacities. In ode to use the minimum amount of inteconnection capacity, only nodes with the lagest capacity can act as intemediate nodes. This can be limiting, especially if only one node has the lagest capacity, because a sta netwok is not good fo fault toleance. A sta netwok is efficient but not balanced, because the cente node acts as the intemediate node fo all taffic. We need a scheme that is not only efficient but also balanced, and we found one by minimizing the netwok fanout. The fanout of node i is f i D l i i, the atio of node i s inteconnection capacity to its node capacity. Since the inteconnection capacity is used both fo sending taffic oiginated fom the node and fo fowading taffic fo othe nodes, the fanout measues the amount of esponsibility the node has to fowad othe nodes taffic elative to its size. If the fanouts of the two nodes ae the same, then the lage node fowads moe taffic, which is a desied popety. Thus, to have a balanced netwok, we minimize the maximum fanout ove all nodes, which esults in all nodes having equal fanout. The esulting load-balancing paametes and fanout ae p i D i R 2 P i k ; k R 2 k f i D 1 C i D 1;2;:::;; 1 P j ; 8i: j D1 R 2 j The optimal load-balancing paametes ae almost popotional to the node capacities: p i / i R 2 i. The paamete p i is a stictly inceasing function of i. Theefoe a lage node has geate esponsibility fo fowading taffic. We can futhe show that the total inteconnection capacity used in this scheme is no moe than 1 2.p 2 C 1/ D 1:207 times the minimum total capacity [24]. Thus, the scheme of minimizing maximum fanout is not only balanced, but also efficient. One can of couse choose to optimize othe citeia that ae moe suitable fo the paticula situation. Kodialam et al. [11] assume that the undelying physical links have fixed capacities, constaining the logical link capacities. They give efficient

7 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices 25 algoithms fo calculating o appoximating the optimal load-balancing atios which allow the netwok to suppot the most taffic. Shephed and Winze [16] use a ealistic cost model that takes into account both the fibe cost and the equipment cost. They opt to load-balance to only a subset of the nodes so that a full mesh is not needed. 2.3 Fault-Toleance in a VLB etwok Most netwoks need to accommodate planned and unplanned inteuptions. As we show in this section, VLB netwoks, due to thei ich connectivity and path divesity, have many advantages in tems of toleating failues, such as All of the woking paths between a pai of nodes ae used all the time, and flows ae load-balanced acoss all woking paths. Most othe schemes equie potection paths that ae idle duing nomal opeation. All paths ae used all the time, so thee is no need to set up new paths upon failue. In ode to potect against k failues, 1 the faction of exta capacity equied is appoximately k. This is extemely efficient compaed to othe fault toleance schemes. VLB natually potects against multiple failues. One can decide duing design what failue scenaios the netwok should toleate, such as k abitay link o node failues o a paticula set of failue pattens. When thee ae no failues, each flow is load-balanced ove all the paths accoding to some atio. The same mechanism can be used when thee ae failues and some flows have fewe than woking paths. We assume that a node keeps tack of the available paths fo each flow oiginating fom it, and load-balances each flow ove the available paths, accoding to the same atios. The netwok is self-healing because the paths ae known pio to failues and taffic can be eouted as soon as failue is detected. We deive how much link capacity is needed so as to suppot all taffic matices unde k n abitay node failues 2 and k l abitay link failues. Fo simplicity, we assume the simple netwokmodelwheeall nodeshave the same capacity.dueto the symmety of the topology, all links in the fault-toleant netwok have the same capacity equiement, epesented by C.;k n ;k l /. We fist conside node failues and link failues sepaately and then combine them. By definition, C.; 0; 0/ D 2. ode failues ae elatively easy to analyze. When a node fails, it takes down all the links connecting to it and stops sending o eceiving taffic. The netwok becomes an. 1/-node full-mesh netwok, and a link capacity of is sufficient We focus on failues in the logical topology, and since seveal logical links can shae a physical link, a physical failue can coespond to multiple logical failues. 2 If a node fails, we discad the taffic oiginating fom o teminating at this node.

8 26 R. Zhang-Shen fo suppoting all taffic matices. In geneal, when thee ae k node failues, the netwok becomes an. k/-node full mesh, so the link capacity equied to toleate k node failues is C.;k;0/ D C. k; 0; 0/ D 2 k : (2.5) Link failues ae a little moe complicated, as they can destoy the symmety of the topology, so that it is no longe a full mesh. We only conside the wost-case failue scenaios (advesaial link failues) hee. We omit the details and only give the final esult. The amount of capacity equied on each link to toleate k abitay link failues, fo 1 k 2,is 8 < 2 C k D 1 C.;0;k/ D : k 1 C k D 2 o 2;o 6 1 othewise: 2 k (2.6) Fo k 1, the netwok becomes disconnected in the wost-case failue scenaio, theefoe we cannot guaantee a congestion-fee netwok. The significance of Equation (2.6) is that all enties on the ight-hand side ae vey close to 2 k. Figue 2.2 plots Equation (2.6)fo D 50, and as we can see, the cuve is vey flat fo small values of k. ow assume that thee ae k n node failues and k l link failues. This is equivalent to having k l link failues in a. k n /-node netwok. So C.;k n ;k l / 2 k n k l D 2 k ; (2.7) whee k D k n C k l. This means that the cuve fo Equation (2.7) is oughly the same as that fo Equation (2.6), shown in Figue 2.2. So we conclude that in a VLB netwok, a small amount of ove-povisioning goes a long way to make the netwok Fig. 2.2 The equied link capacity vs. the numbe of link failues in a 50-node netwok Requied capacity of each link (multiples of ) umbe of link failues, k

9 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices 27 fault toleant. Fo example, if the links in a 50-node netwok ae ove-povisioned by just about 11%, the netwok can toleate any five (node o link) failues. 2.4 VLB fo Peeing Taffic Today, most congestion in Intenet backbone takes place on the peeing links connecting them. When a link between two netwoks is congested, vey likely some othe peeing links between these two netwoks ae lightly loaded. That is, the peeing links between two netwoks ae usually not evenly utilized. We popose to use a technique simila to VLB to oute peeing taffic, so as to eliminate congestion on the peeing links. We show that if taffic is load-balanced ove all the peeing links between two netwoks, thee will be no congestion as long as the total peeing capacity is geate than the total peeing taffic. Even though the two netwoks using VLB to oute thei peeing taffic do not need to use VLB intenally, we assume they do and analyze how the peeing scheme affects how the netwoks un VLB. Suppose two VLB netwoks ae connected by a subset of thei nodes (the peeing nodes), as shown in Figue 2.3. Fo the ease of desciption, we use the same numbeing fo the peeing nodes in both netwoks. (ote that this convention is diffeent fom Section 2.2.) The taffic exchanged between the two netwoks is called peeing taffic and assume the total amount is no moe than R p in each diection. In the netwok of nodes, we intoduce the peeing load-balancing paametes q i ;i D 1;2;:::;, such that a potion q i of the peeing taffic between the two netwoks is exchanged at node i. atually, q i D 0 if i is not a peeing node. The peeing load-balancing paametes, q i, togethe with the maximum peeing taffic between the two netwoks, R p, detemine the sizes of the peeing links: the equied capacity of the peeing link at node i is R p q i. Suppose the peeing links have the equied capacities, then if the peeing taffic is load-balanced acoss the peeing links accoding to the popotions q i, and the total amount of peeing taffic between the two netwoks does not exceed R p, thee will be no congestion on the peeing links. etwok 1 R p etwok 2 q q q 4 4 Fig. 2.3 Two VLB netwoks connect at a set of peeing nodes. The total amount of taffic exchanged between the two netwoks is no moe than R p, and a potion q i of the peeing taffic is exchanged at node i.

10 28 R. Zhang-Shen The exta equiement of outing peeing taffic may esult in highe capacity equiements inside the netwoks. If we teat the peeing taffic that oiginates fom the netwok as taffic destined to the peeing nodes, and the peeing taffic that entes the netwok as taffic oiginated fom the peeing nodes, then the peeing taffic may have to tavese two hops in each netwok. Altenatively, we can load-balance peeing taffic ove the peeing points only, instead of all the nodes. Thus, we equie that peeing taffic taveses at most one hop in each netwok, and at most two hops altogethe. Suppose the peeing load-balancing paametes ae fixed, fo example, though negotiation between the two netwoks. Then we can vay p i, i.e., how non-peeing taffic is outed intenally, to minimize the equied link capacities. We obseve that R p is likely to be bigge than the node capacities i. R p is the total amount of taffic the two netwok exchanges and can be a lage potion of the netwok s total taffic R, while the node capacities ae likely to make up only a small faction of R, onthe ode of R. If we assume that R p i fo all i,thenwehave c ij D i max.p j ;q j / C j max.p i ;q i /; and the minimum c ij is achieved fo all links when p i D q i fo all i. So the optimal capacity allocation in a netwok with peeing taffic is c ij D i q j C j q i : (2.8) Since q i is zeo if node i is a non-peeing node, c ij D 0 if both ode i and ode j ae non-peeing nodes. The netwok is now a two-tieed one: in the cente is a full mesh connecting the peeing nodes, and on the edge ae the non-peeing nodes, each connected to all the cente nodes. Setting local load-balancing paametes to be the same as peeing load-balancing paametes means that only the peeing nodes will seve as intemediate nodes to fowad taffic. Peeing nodes ae often the lagest nodes in the netwok, so they should have lage esponsibilities in fowading taffic. The analysis shows that the optimal way is to only let the peeing nodes fowad taffic. This has the additional benefits of equiing fewe links and educing netwok complexity. 2.5 Discussions Using VLB to oute taffic in a netwok has many advantages, such as efficiency and fast failue ecovey. Thee ae some open questions as well. Some examples ae Sending packets though two hops may incease the popagation delay, while sending packets though the diect link may cause inceased capacity equiement. Thee should be a tadeoff between packet delay and equied capacity.

11 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices 29 Diffeent paths that a flow taveses may have diffeent delays, and hence packets in the flow may not aive at the destination in the oiginal ode. This is usually not a poblem in the Intenet, since a flow may consist of many application-level flows, and can be split accodingly fo load-balancing. But it can be a poblem if a flow cannot be easily split. It is unclea whethe VLB can be incementally deployed in a netwok. Despite the challenges, VLB is apidly gaining populaity in netwoking eseach. This suvey pape can be a stating point fo those inteested in applying VLB to othe situations. Refeences 1. R. Aleliunas. Randomized paallel communication (peliminay vesion). In PODC 82: Poceedings of the fist ACM SIGACT-SIGOPS symposium on Pinciples of distibuted computing, pages 60 72, K. Agyaki, S. Baset, B.-G. Chun, K. Fall, G. Iannaccone, A. Knies, E. Kohle, M. Manesh, S. edevschi, and S. Ratnasamy. Can softwae outes scale? In PRESTO 08: Poceedings of the ACM wokshop on Pogammable outes fo extensible sevices of tomoow, pages 21 26, P. Benasconi, J. Gipp, D. eilson, J. Simsaian, D. Stiliadis, A. Vama, and M. Zingibl. Achitectue of an integated oute inteconnected spectally (IRIS). High Pefomance Switching and Routing, 2006 Wokshop on, pages 8 pp., June C.-S. Chang, D.-S. Lee, and Y.-S. Jou. Load balanced Bikhoff-von eumann switches, Pat I: One-stage buffeing. Compute Communications, 25(6): , C.-S. Chang, D.-S. Lee, and C.-M. Lien. Load balanced Bikhoff-von eumann switches, Pat II: Multi-stage buffeing. Compute Communications, 25(6): , A. Geenbeg, P. Lahii, D. A. Maltz, P. Patel, and S. Sengupta. Towads a next geneation data cente achitectue: scalability and commoditization. In PRESTO 08: Poceedings of the ACM wokshop on Pogammable outes fo extensible sevices of tomoow, pages 57 62, M. Henion, K. Schodi, D. Boettle, M. De Some, and M. Dieudonne. Switching netwok achitectue fo ATM based boadband communications. Switching Symposium, XIII Intenational, 5:1 8, I. Keslassy, C.-S. Chang,. McKeown, and D.-S. Lee. Optimal load-balancing. In Poc. IEEE IFOCOM, I. Keslassy, S.-T. Chuang, K. Yu, D. Mille, M. Hoowitz, O. Solgaad, and. McKeown. Scaling Intenet outes using optics. Poceedings of ACM SIGCOMM 03, Compute Communication Review, 33(4): , Octobe I. Keslassy, M. Kodialam, T. Lakshman, and D. Stiliadis. Scheduling schemes fo delay gaphs with applications to optical packet netwoks. High Pefomance Switching and Routing (HPSR), pages , M. Kodialam, T. V. Lakshman, J. B. Olin, and S. Sengupta. A Vesatile Scheme fo Routing Highly Vaiable Taffic in Sevice Ovelays and IP Backbones. In Poc. IEEE IFOCOM, Apil M. Kodialam, T. V. Lakshman, and S. Sengupta. Efficient and obust outing of highly vaiable taffic. In Hotets III, ovembe D. Mita and R. A. Cieslak. Randomized paallel communications on an extension of the omega netwok. J. ACM, 34(4): , H. agesh, V. Poosala, V. Kuma, P. Winze, and M. Zingibl. Load-balanced achitectue fo dynamic taffic. In Optical Fibe Communication Confeence, Mach 2005.

12 30 R. Zhang-Shen 15. R. Pasad, P. Winze, S. Bost, and M. Thottan. Queuing delays in andomized load balanced netwoks. In Poc. IEEE IFOCOM, May F. B. Shephed and P. J. Winze. Selective andomized load balancing and mesh netwoks with changing demands. Jounal of Optical etwoking, 5: , A. Singh. Load-Balanced Routing in Inteconnection etwoks. PhD thesis, Depatment of Electical Engineeing, Stanfod Univesity, A. Singh, W. J. Dally, B. Towles, and A. K. Gupta. Locality-peseving andomized oblivious outing on tous netwoks. In SPAA 02: Poceedings of the fouteenth annual ACM symposium on paallel algoithms and achitectues, pages 9 13, L. G. Valiant. A scheme fo fast paallel communication. SIAM Jounal on Computing, 11(2): , R. van Haalen, R. Malhota, and A. de Hee. Optimized outing fo poviding Ethenet LA sevices. Communications Magazine, IEEE, 43(11): , ov P. J. Winze, F. B. Shephed, P. Oswald, and M. Zingibl. Robust netwok design and selective andomized load balancing. 31st Euopean Confeence on Optical Communication (ECOC), 1:23 24, Septembe R. Zhang-Shen, M. Kodialam, and T. V. Lakshman. Achieving bounded blocking in cicuitswitched netwoks. IEEE IFOCOM 2006, pages 1 9, Apil R. Zhang-Shen and. McKeown. Designing a Pedictable Intenet Backbone etwok. In Hotets III, ovembe R. Zhang-Shen and. McKeown. Designing a pedictable Intenet backbone with Valiant Load-Balancing. Thiteenth Intenational Wokshop on Quality of Sevice (IWQoS), R. Zhang-Shen and. McKeown. Designing a Fault-Toleant etwok Using Valiant Load- Balancing. Poc. IEEE IFOCOM, pages , Apil R. Zhang-Shen and. McKeown. Guaanteeing Quality of Sevice to Peeing Taffic. Poc. IEEE IFOCOM, pages , Apil 2008.

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