Rethinking Virtual Network Embedding: Substrate Support for Path Splitting and Migration

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1 Rethinking Virtul Network Emedding: Sustrte Support for Pth Splitting nd Migrtion Minln Yu, Yung Yi, Jennifer Rexford, Mung Ching Princeton University Princeton, NJ ABSTRACT Network virtuliztion is powerful wy to run multiple rchitectures or experiments simultneously on shred infrstructure. However, mking efficient use of the underlying resources requires effective techniques for virtul network emedding mpping ech virtul network to specific nodes nd links in the sustrte network. Since the generl emedding prolem is computtionlly intrctle, pst reserch restricted the prolem spce to llow efficient solutions, or focused on designing heuristic lgorithms. In this pper, we dvocte different pproch: rethinking the design of the sustrte network to enle simpler emedding lgorithms nd more efficient use of resources, without restricting the prolem spce. In prticulr, we simplify virtul link emedding y: i) llowing the sustrte network to split virtul link over multiple sustrte pths nd ii) employing pth migrtion to periodiclly re-optimize the utiliztion of the sustrte network. We lso explore node-mpping lgorithms tht re customized to common clsses of virtulnetwork topologies. Our simultion experiments show tht pth splitting, pth migrtion, nd customized emedding lgorithms enle sustrte network to stisfy much lrger mix of virtul networks. Ctegories nd Suject Descriptors C.2.5 [Computer-Communiction Networks]: Locl nd Wide-Are Networks; G.1.6 [Numericl Anlysis]: Optimiztion Generl Terms Algorithms; Design Keywords Virtul Network Emedding; Pth Splitting; Pth Migrtion; Network Virtuliztion; Optimiztion 1. INTRODUCTION Network virtuliztion hs emerged s powerful wy to llow multiple network rchitectures, ech customized to prticulr ppliction or user community, to run on common sustrte. For exmple, virtuliztion cn enle multiple reserchers to evlute new network protocols simultneously on shred experimentl fcility [3, 7, 2, ]. In ddition, network virtuliztion could serve s the foundtion of future Internet tht llows multiple service providers to offer customized end-to-end services over common physicl infrstructure [14, 27]. For exmple, Voice over IP (VoIP) could run on virtul network tht provides predictle performnce (y llocting dedicted resources nd employing routing protocols tht ensure fst recovery from equipment filures), wheres online nking could run on virtul network tht provides security gurntees (through self-certifying ddresses nd secure routing protocols). Mking efficient use of the sustrte resources requires effective techniques for virtul network (VN) emedding mpping new virtul network, with constrints on the virtul nodes nd links, on to specific physicl nodes nd links in the sustrte network. However, the VN emedding prolem is extremely chllenging, for four min prcticl resons: Node nd link constrints. Ech VN request hs resource constrints, such s processing resources on the nodes nd ndwidth resources on the links, tht the emedding must stisfy. For exmple, to run controlled experiment, resercher my need 1 GHz of CPU for ech virtul node nd Mps for ech virtul link. In ddition, the VN my impose dditionl constrints on node loction or link propgtion dely. For exmple, commercil gming service my need virtul nodes in severl mjor cities, s well s virtul links with propgtion delys less thn 50 msec. The comintion of node nd link constrints mke the emedding prolem computtionlly difficult to solve. Admission control. Since the sustrte resources re limited, some VN requests must e rejected or postponed to void violting the resource gurntees for existing virtul networks. Tht is, the sustrte must reserve node nd link resources, nd perform dmission control on new requests to ensure tht sufficient resources re ville. For exmple, virtul network tht requires 1 GHz of CPU for ech virtul node my e rejected if no physicl nodes hve enough unllocted processing cpcity. Once ccepted, the virtul networks receive their gurnteed resources through scheduling techniques for shring the node nd link resources. Online requests. The VN requests re not known in dvnce, nd my rrive dynmiclly nd sty in the network for n ritrry period of time efore deprting. For exmple, resercher my strt new experiment t ny time, to run for some durtion sed on the needs of the experiment. Similrly, service provider my deploy new service t ny time, nd continue supporting the service indefinitely, possily discontinuing the service when it is no longer profitle. To e prcticl, the emedding lgorithm must hndle VN requests s they rrive, rther thn hndling lrge collection of requests t once. Online prolems re typiclly much ACM SIGCOMM Computer Communiction Review 19 Volume 38, Numer 2, April 08

2 more difficult to solve, ecuse the emedding lgorithm hs little (if ny) visiility into the future request rrivls. Diverse topologies. The virtul networks my hve diverse topologies. For exmple, reserchers my run experiments under vriety of topologies to explore how their protocol performs in different settings. Also, service provider my tilor the virtul-network topology to the ppliction, such s hu-nd-spoke to provide customers with ccess to centrlized server, or tree to distriute streming video to group of receivers. Although virtul networks my hve wide vriety of structures, certin kinds of topologies such s hu-nd-spoke or tree my e especilly common in prctice. Hndling ritrry topologies, while efficiently supporting the most common topologies, introduces n dditionl chllenge for the emedding lgorithm. These four properties mke the VN emedding prolem very difficult. In fct, the prolem is computtionlly intrctle, even if some of these four properties re ignored. Due to the comintion of node nd link constrints, the VN emedding prolem is NP-hrd, even in the offline cse. (For exmple, ssigning virtul nodes to the sustrte network without violting ndwidth constrints cn e reduced to the multiwy seprtor prolem, which is NP-hrd [6].) Even if the loctions of the virtul nodes re pre-determined, emedding the virtul links with ndwidth constrints is still NP-hrd, s discussed in more detil in Section 3. The online prolem is even more difficult to solve. Trditionl techniques for solving online prolems (e.g., dynmic progrmming) re imprcticl here ecuse the properties of incoming VN requests re generlly unpredictle nd the serch spce is prohiitively huge when the sustrte network is lrge. Previous reserch hs ddressed these computtionl chllenges y restricting the prolem spce in one or more dimensions to enle efficient heuristics [31, 13, 23, 26], t the expense of limiting the prcticl pplicility of the solutions. For exmple, the ppers either solve n offline vrint of the prolem [31, 23], consider only ndwidth constrints [13, 23], or do not perform dmission control [31, 23, 13]. In this pper, we tke different pproch we reconsider the cpilities of the underlying sustrte network, to mke the sustrte network more supportive of the VN emedding prolem. This llows us to crete simpler emedding lgorithms tht mke more efficient use of the sustrte resources, without compromising on the four chllenges listed ove. Since network virtuliztion is still in its infncy, we elieve it is importnt to explore how to design the sustrte to est stisfy its gols. In prticulr, we investigte how to simplify the prolem of virtul-link emedding y llowing the sustrte network to mp virtul link to multiple sustrte pths with flexile pth-splitting rtio. In ddition, for efficient hndling of online requests, we llow the sustrte to periodiclly re-optimize the mpping of existing virtul links, either y selecting new underlying pths or dpting the splitting rtios for the existing pths. Flexile pth splitting is relizle in prctice without disrupting the sic properties of link such s predictle ndwidth, propgtion dely, nd in-order pcket delivery, s discussed in more detil in Section 3.3. Flexile pth splitting llows us to mp virtul links to the sustrte in polynomil time, while mking much more efficient use of sustrte ndwidth nd incresing roustness to sustrte filures. This feture llows us Tle 1: Nottions of VN Emedding Prolem G s Sustrte network N s L s A s N A s L P s G v N v L v C v N C v L R N R L Nodes of sustrte network Links of sustrte network Node ttriute of sustrte network Link ttriute of sustrte network Pths on sustrte network Virtul network Nodes of virtul network Links of virtul network Node constrint of sustrte network Link constrint of sustrte network Resources llocted for virtul network nodes Resources llocted for virtul network links VN request 1 c VN request d e Atlnt New Jersey 70 A Sustrte network B C D E c F 60 G H I 60 Figure 1: An exmple of VN emedding to stisfy the first three of the four chllenges listed ove. To ddress the fourth chllenge, we introduce customized node-emedding lgorithms for common topologies like hu-nd-spoke. The reminder of this pper is orgnized s follows. In Section 2, we define the VN emedding prolem nd present simple emedding lgorithm tht does not ssume ny specil cpilities from the sustrte. This lgorithm is similr to the techniques proposed in previous reserch, extended to support dmission control nd online requests. The lgorithm serves s sis of comprison for our new lgorithm for sustrtes tht support pth splitting nd migrtion, s discussed in Section 3. Section 4 presents simultions tht evlute our lgorithm nd quntify the enefits of more flexile sustrte. In Section 5, we present our customized node-emedding lgorithm for hu-nd-spoke topologies, nd promising initil simultion results. Section 6 compres our lgorithms with relted work, nd Section 7 concludes the pper. 2. VIRTUAL NETWORK EMBEDDING In this section, we first descrie the generl VN emedding prolem. Then, we present n originl solution to this prolem without ssuming tht the sustrte cn split virtul link over multiple underlying pths. 2.1 Virtul Network Emedding Prolem Sustrte network. We denote the sustrte network y n undirected grph G s =(N s,l s,a s N,A s L), where N s nd L s refer to the set of nodes nd links, respectively. We use superscript to refer to sustrte or virtul network, nd use 40 d e ACM SIGCOMM Computer Communiction Review Volume 38, Numer 2, April 08

3 suscript to refer to nodes or links, unless otherwise specified. Sustrte nodes nd links re ssocited with their ttriutes, denoted y A s N nd A s L, respectively. In this pper, we consider CPU cpcity nd loction for node ttriutes, nd ndwidth cpcity for link ttriutes. We lso denote y P s the set of ll loop-free pths in the sustrte network. The right side of Figure 1 shows sustrte network. The numers ner the links represent ville ndwidths nd the numers in rectngles re the ville CPU resources t the nodes. Virtul network request. We denote y n undirected grph G v =(N v,l v,cn v,cl) v virtul network request. A VN request typiclly hs link nd node constrints tht re specified in terms of ttriutes of the sustrte network. We denote y CL v nd CN v the set of link nd node constrints, respectively. Figure 1 depicts two VN requests: the VN request 1 requires the ndwidth over the links (, ) nd (, c), nd the CPU resource t ll nodes,,, ndc; the VN request 2 is: connect two nodes d, e N v with constrints tht node d should e in Atlnt (where sustrte nodes D nd G re locted), nd node e should e in New Jersey (where sustrte nodes E nd I re locted), with ten units of ndwidth on the virtul link etween them. VN emedding. A virtul network emedding for VN request is defined s mpping M from G v tosusetof G s, such tht the constrints in G v re stisfied, i.e., M : G v (N, P,R N,R L), where N N s nd P P s, nd R N nd R L re the node nd link resources llocted for the VN requests. The VN network emedding cn e nturlly decomposed into node nd link mpping s follows: Node Mpping: M N :(N v,cn v ) (N,R N ), Link Mpping: M L :(L v,cl) v (P,R L). The right side of Figure 1 shows the VN emedding solutions for the two VN requests. For exmple, the nodes,, nd c in VN request 1 re mpped to the sustrte nodes A, E, nd F, nd the virtul links (, ) nd(, c) rempped to the sustrte pths (A,D,E) nd (A,D,F) with the CPU nd ndwidth constrints ll stisfied. A similr mpping occurs for VN request 2. Ojectives. Our min interest is to propose n efficient emedding lgorithm for the online prolem, where VN requests rrive nd deprt over time. From the sustrte network provider s point of view, nturl ojective of n online emedding lgorithm would e to mximize the revenue. We introduce the notion of revenue tht corresponds to the economic enefit of ccepting VN requests. We denote y R(G v (t)) the revenue of serving the VN requests t time t. Then, our ojective is to mximize the long-term verge revenue, given y the following: P T t=0 lim R(Gv (t)). (1) T T The revenue cn e defined in vrious wys ccording to economic models. In this pper, we focus on ndwidth nd CPU s the min sustrte network resources. Then, nturl choice of the revenue for VN request would e the weighted sum of revenues for ndwidth nd CPU, ech of which is proportionl to the mount of the requested resources. Similr to the work in [31], we introduce tunle weight α tht llows the sustrte provider to strike Req. Queue Reqs postponed Reqs postponed new Reqs. Req comes Node Mpping Req leves fixed nodes Link Mpping time window Emedding output for this time window Figure 2: The seline lgorithm overview lnce etween the reltive costs of the two clsses of resources. Thus, for VN request G v, we define its revenue R(G v (t)) t ny prticulr time t tht the virtul network G v is running s: R(G v (t)) = X X w(l v ) + α CPU(n v ), (2) l v L v n v N v where w(l v )ndcpu(n v ) re the ndwidth nd CPU requirements for the virtul link l v nd the virtul node n v, respectively. We note tht the ndwidth revenue (i.e., the first term in Eqution (2)) is not ffected y the sustrte pths tht the virtul links re mpped to, in prticulr, physicl distnce or the numer of hops of the mpped pths. This seems to e resonle, since VN requests only cre out the stisfiility of their constrints in the sustrte network nd will not py for longer distnce. To chieve the gol in Eqution (1), it is crucil to emed incoming VN requests efficiently, such tht the sustrte resource is minimlly occupied. This is ecuse n inefficient emedding of virtul network t time t my restrict the sustrte s ility to ccept future requests. Section 4.2 shows the reltionship of revenue nd efficiency. 2.2 Bseline VN Emedding Algorithm In this susection, we propose simple emedding lgorithm tht does not exploit ny specil cpilities from the sustrte network. The lgorithm is motivted y the techniques proposed in relted work (e.g., [31, 26]) with extensions to perform dmission control nd hndle online requests. Figure 2 depicts our lgorithm. Our lgorithm collects group of incoming requests during time window nd then tries to llocte sustrte resources to stisfy the constrints required y the requests. Some requests my e deferred due to lck of ndwidth or CPU resources in the sustrte network, nd returned to the request queue. The requests in the queue re dropped if they cnnot e served within some dely, which, specified y request, corresponds to the time tht request is willing to wit. The requests in the request queue re processed gin in the susequent time windows. We process ll VN requests rriving within the time window s well s in the request queue, in decresing order of their revenues. We mp virtul nodes onto the sustrte for ll the considered VN requests, nd then mp the virtul links for the requests tht successfully finish the node mpping stge. An lterntive wy, which we do not choose, ACM SIGCOMM Computer Communiction Review 21 Volume 38, Numer 2, April 08

4 Algorithm 1 Greedy Node Mpping Algorithm Step 1 Sort the requests ccording to their revenues. Step 2 If no requests left, stop. Step 3 Tke one request with the lrgest revenue. Step 4 Find the suset S of sustrte nodes tht stisfy restrictions nd ville CPU cpcity (lrger thn tht specified y the request.) If S ==, store this request in the queue, nd GOTO Step 2. Step 5 For ech virtul node, find the sustrte node in S with the mximum ville resources H (defined in Eqution (3)), nd GOTO Step 2. is to mp the nodes nd links of one request first, efore mpping the other requests. In the seline VN emedding lgorithm, oth methods produce similr mpping results, ut our method is more efficient ecuse of tch processing in the node/link mpping stge. The optiml emedding lgorithm is computtionlly intrctle s discussed in Section 1. Our seline VN emedding lgorithm heuristiclly tries to chieve the gol in Eqution (1) over ech time window. Indeed, the lgorithm contriutes to instntneous revenue mximiztion y giving higher priority to the requests with more revenue nd ccepting s mny requests s possile in the node mpping. Additionlly, the lgorithm tends to mke efficient utiliztion of the sustrte ndwidth resources y mpping virtul links to shortest pths in the sustrte network, leving more resources for future requests Node Mpping Algorithm We employ greedy node mpping lgorithm, since it is computtionlly too expensive to employ other strtegies, such s itertive methods [23] nd simulted nneling [13, ]. The motivtion of the greedy lgorithm is to mp the virtul nodes to the sustrte nodes with the mximum sustrte resources so s to minimize the use of the resources t the ottleneck nodes/links [31]. This is eneficil to future requests which require specific sustrte nodes with scrce resources. In our lgorithm, we collect ll outstnding requests, nd then mp ll the virtul nodes in these requests to the sustrte nodes. VN requests sometimes impose some restrictions on their nodes. The exmples of node restrictions include geogrphic loction nd specil functionlity t the sustrte node. These node restrictions re quite common in prctice, e.g., servers ner their customers in content-delivery service, progrmmle routers, nd node with Internet-2 network connectivity. Requests with restrictions reduce the serch spce for plcing the virtul nodes (Step 4). For exmple, loction-specific requests usully limit their virtul nodes to prticulr geogrphic regions. Then, we keep trck of the ville node/link resources of the sustrte network. Note tht for sustrte node n s N s,wedonotusecpu(n s ) lone s the metric of ville resource, ecuse we not only wnt to mke sure tht there is enough CPU cpcity ville, ut lso consider ndwidth cpcity to prepre for the susequent link mpping stge. Therefore, we define the mount of ville resources for sustrte node n s y: X H(n s )=CPU(n s ) w(l s ), (3) l s L(n s ) where L(n s ) is the set of ll djcent sustrte links of n s, Algorithm 2 Link Mpping Algorithm Step 1 Sort the requests tht successfully completed the node-mpping stge y their revenues. Step 2 If no requests left, stop. Step 3 Tke one request with the lrgest revenue. Step 4 For ech virtul link of the request, we serch the k-shortest pths for incresing k, nd stop the serch if we cn find one with enough ndwidth cpcity. Step 5 If fil in Step 3 for some virtul link, then defer this request, nd store it in the request queue. Step 6 GOTO Step 2. CPU(n s ) is the remining CPU resource of n s,ndw(l s ) is the unoccupied ndwidth resource for the sustrte link l s. The definition in (3) is similr to tht in [31] with slight difference tht the numer of virtul links nd nodes re used to mesure the resources, not the ctul mount of CPU nd ndwidth resources. With this definition, for virtul node, we find the sustrte node with the mximum ville resources (Step 5) Link Mpping Algorithm When the sustrte nodes re selected for mpping, we mp the virtul links to specific sustrte links. Finding n optiml mpping from virtul link to single sustrte pth with fixed node mpping reduces to the Unsplittle Flow Prolem (UFP), which is NP-hrd [21, 22]. Therefore, we use the k-shortest pth lgorithm s n pproximtion pproch in order to minimize ndwidth consumption y the virtul network. We serch the k-shortest pths for incresing vlues of k, until we find pth which hs enough ndwidth to mp the corresponding virtul link. Our k-shortest-pth linkmpping lgorithm cn e solved in O(M +N log N +k) time in sustrte network with N nodes nd M links [12]. Both for computtionl efficiency nd efficient use of sustrte resources, k should e kept smll. 3. PATH SPLITTING AND MIGRATION Restricting ech virtul link to single sustrte pth mkes the link-emedding prolem computtionlly intrctle, nd the resulting emeddings inefficient. In this section, we first rgue tht the sustrte network should support flexile splitting of virtul links over multiple sustrte pths, nd present new link-emedding lgorithm tht cpitlizes on the flexiility. Next, we descrie how to periodiclly re-optimize the mpping of existing virtul links to llow the sustrte network to ccept more new requests. Finlly, we explin how sustrte support for pth splitting nd migrtion cn e implemented in prctice. 3.1 Pth Splitting Motivtion for Flexile Pth Splitting To motivte sustrte support for pth splitting, consider the exmple in Figure 3. Initilly the sustrte network runs single virtul network with three virtul nodes nd two virtul links tht ech require units of ndwidth. The virtul nodes re mpped to physicl nodes A, E, nd F, nd the two virtul links re mpped to the pths (A,D,E) nd (A,D,F), s shown in the lower left prt of the figure. Now, suppose new VN request rrives with single virtul ACM SIGCOMM Computer Communiction Review 22 Volume 38, Numer 2, April 08

5 Existing VN Req 1 A B D E c F G H I Sustrte network t time t-1 c C New rriving VN Req 2 d 30 Accept A e Without pth splitting Pth splitting B C Reject D E c F G H I Sustrte network t time t Figure 3: Illustrtion of the enefit of pth splitting link tht requires 30 units of ndwidth. Unfortuntely, no one pth in the sustrte network cn ccommodte the new request. However, the new VN could e mpped to nodes D nd E, if the sustrte could llocte units of ndwidth onthepth(d,e)ndonthepth(d,g,h,i,e).tht is, directing two-thirds of the trffic over the (D, E) pth nd one-third over the (D, G, H, I, E) pth would llow the sustrte to ccept the second request. Pth splitting enles etter resource utiliztion y hrnessing the smll pieces of ville ndwidth, llowing the sustrte to ccept more VN requests. In ddition, flexile pth splitting mkes the link-emedding prolem computtionlly trctle. A virtul link l with some cpcity constrint, sy C l, is mpped into multiple pths in the sustrte network, such tht the sum of reserved end-toend ndwidth long the multiple pths is equl to C l.the division of trffic over the sustrte pths is specified s splitting rtio, such s rtio of 2:1 in the exmple in Figure 3. Under flexile splitting over multiple pths, the link-emedding prolem cn e reduced to the Multicommodity Flow Prolem (MFP) [5], which cn e solved in polynomil time. The enefits of hving multiple pths hve een estlished in other contexts, such s lod lncing nd reliility. In fct, even hving just two pths cn significntly reduce the mximum lod on network, compred to solutions tht limit the trffic flow to single pth [24, 19]. Hving multiple pths lso enles fster recovery from network filures. For exmple, if link or node fils, the network cn quickly switch the ffected trffic to other pths simply y chnging the splitting rtios. In contrst, in singlepth setting, filure requires estlishing new end-toend pth, leding to more severe service disruption. Due to the computtionl, performnce, nd reliility enefits, we elieve flexile pth splitting should e key feture in future virtulized network infrstructures, nd the rest of this pper will provide the lgorithmic nd simultion-sed evidence to support this view Link Mpping Algorithm with Pth Splitting We descrie the link mpping lgorithm supporting pth splitting to enle efficient solutions in Algorithm 3. In Step 1, we first construct liner constrints for the virtul links. For simplicity, consider request with only one link l v with the cpcity constrint C, where two end nodes of l v re denoted y n v 1 nd n v 2. We denote y M N (n v 1)=n s 1 nd M N (n v 2)=n s 2 the sustrte nodes chosen for n v 1 nd n v 2, Req. Queue new Reqs. Reqs postponed Reqs postponed Reqs postponed Req comes Link Mpping Link Mpping Node Mpping Req leves fixed nodes Pth Migrtion time window Unsplittle Node Rempping Splittle Emedding output for this time window Figure 4: Algorithm for pth splitting nd migrtion respectively, y the node-mpping lgorithm in Section 2.2. The pir of sustrte nodes (n s 1, n s 2) is commodity, nd finding multiple sustrte pths for l v is equivlent to finding flows from source n s 1 to destintion n s 2 in the sustrte network with ville ndwidth on the sustrte links. Thus, group of, sy r, virtul links genertes group of r commodities. The lgorithm tries to find ll the pths for r commodities sed on the following liner constrints: l s L s, rx f(c i,l s ) w(l s ) (4) i=1 where f(c i,l s ) is the ndwidth on the sustrte link l s tht we llocte to commodity c i, or its corresponding virtul link. After generting the r commodities nd the liner constrints, we solve the resulting multicommodity flow prolem (Step 2). Even with flexile pth splitting, the MFP prolem my not hve fesile solution ecuse one or more sustrte links do not hve enough ville cpcity. The lgorithm revisits the node-mpping decisions for these virtul links (Steps 4, 5, nd 6). The filure in the MFP computtion implies tht one or more sustrte links violte the liner constrints in Eqution (4). Fortuntely, the MFP lgorithm cn esily output the sustrte links tht violte the constrint, s well s the extent of the violtion. The noderempping stge focuses its ttention on the sustrte link with the lrgest violtion (ottleneck link), i.e., the l s L s with the highest vlue of P r i=1 f(ci,ls ) w(l s ). We rndomly choose one virtul link tht is originlly mpped to the pth including the ottleneck link, nd mp one end of this virtul link to nother sustrte node with mximum remining resource H (defined in Eqution (3)), in order to void occupying this ottleneck link. The node rempping revisits the node mpping decision for the new requests. However, the rempping process does not chnge the resource lloction for virtul networks lredy running in the sustrte. We try this node rempping for predefined numer of times T try nd mke sure ech time we choose different ottleneck link. If the MFP is still infesile fter T try trils, we defer the request tht requires the most ndwidth on the ottleneck sustrte link nd return ACM SIGCOMM Computer Communiction Review 23 Volume 38, Numer 2, April 08

6 Algorithm 3 Link Mpping Algorithm for Requests with Pth Splitting MFP Computtion: Step 1 For ll requests with splittility, construct liner constrint on the commodities for ech sustrte link. Step 2 Solve MFP (Multicommodity Flow Prolem). Step 3 If fesile, stop. Node Rempping: Step 4 If infesile, find the ottleneck sustrte link. Step 5 Rndomly choose one virtul link tht is originlly mpped t the ottleneck link, pick one end of the virtul link nd mp it to nother sustrte node with mximum remining resource H (defined in Eqution (3)). Then GOTO Step 2 with new liner constrints. Step 6 If rempping of virtul nodes for T try times does not produce fesile solution, eliminte one of the VN requests hving the lrgest impct on infesiility. Then, construct the liner constrins only with the remining requests, nd GOTO Step 2. it to the request queue, nd then try to solve the MFP with the remining requests gin. Lrger vlues of T try increse the computtionl overhed ut improve the likelihood of finding successful emedding. In prctice, some virtul networks my hve strict requirements tht preclude pth splitting As such, we envision our lgorithm would hndle mix of oth kinds of VN requests. As illustrted in Figure 4, we first pply Algorithm 2 for requests tht do not llow pth splitting, efore pplying Algorithm 3 for requests tht llow pth splitting. Unfortuntely, node-rempping is difficult to perform for the unsplittle virtul links, since the emedding lgorithm processes one virtul link t time. For the collection of virtul links tht fil to find suitle pth, we cnnot esily identify the most congested sustrte link in computtionlly efficient mnner. To mintin computtionl simplicity, we do not consider node-rempping for these requests. In the evlution, we quntify the enefits of pth splitting, with nd without the node-rempping step. 3.2 Pth Migrtion To del with the online nture of the VN emedding prolem, we introduce the ide of pth migrtion, i.e., chnging the route or splitting rtio of virtul link. This turns out to e nother dvntge of llowing multipth in the sustrte network Motivtion for Pth Migrtion Since VN requests rrive nd deprt over time, the sustrte network cn esily drift into n inefficient configurtion, where resources re incresingly frgmented, forcing the sustrte to reject future requests or route new virtul links over more expensive (longer) pths. Theoreticlly, one could try to ddress these chllenges with predictive models of future requests, coupled with mthemticl techniques like dynmic progrmming. However, the rrivl nd deprture of requests is unpredictle nd the underlying serch spce is too lrge for dynmic progrmming to e prcticl. Insted, we rgue the sustrte network should e le to relnce the mpping of virtul networks to mke more efficient use of the sustrte resources nd to mximize the A Existing VN Req 1 B C 5 5 d D E A D 30 c F Sustrte network t time t-1 c New rriving Without VN Req 2 d 30 migrtion e Reject Accept With migrtion c B F Sustrte network t time t Figure 5: Illustrtion of the enefit of migrtion chnce of ccepting future requests. In prticulr, the ility to migrte virtul links to different sustrte pths while keeping the node mpping intct cn further improve the sustrte s ility to ccept future requests. To motivte sustrte support for pth migrtion, consider the exmple in Figure 5. Initilly the sustrte networks runs single virtul network with three nodes (mpped to physicl nodes A, E, nd F) nd two virtul links tht ech require units of ndwidth (mpped to the pths (A,D,E) nd (A,D,F)). Now, suppose new VN request rrives with single virtul link tht requires 30 units of ndwidth. Unfortuntely, no pir of nodes in the sustrte network cn ccommodte the new request, even if pth splitting is permitted. For exmple, in the left side of Figure 5, nodes D nd E hve pth (D,E) with units of ndwidth, pth (D,B,C,E) with 5 units of ndwidth, nd pth (D, A, B, C, E) with 0 units of ndwidth not enough to support virtul link requiring 30 units of ndwidth. However, migrting some of the trffic for the first virtul network to different pth would enle the sustrte to ccept the new request. In prticulr, the sustrte could crry hlf of the trffic for virtul link (,) on new pth (A, B, C, E) to free up dditionl cpcity on the sustrte link (D, E). Then, the second virtul network cn hve link (d,e) mpped to sustrte pth (D,E) Migrtion Algorithm In the migrtion lgorithm descried in Algorithm 4, we fix the node mpping of the virtul networks lredy running on the sustrte. We perform pth migrtion y rerunning the link-mpping lgorithm with requests tht llow pth splitting (Algorithm 3). Pth migrtion is performed y either chnging the splitting rtios for the existing pths or selecting new underlying pths. If only djusting the splitting rtios is llowle rther thn setting up ny new pths, we hve to mke sure tht the flows coming from commodity only trverse the sustrte pths originlly tken in the link mpping stge. Thus, we dd the following liner constrints to the constrints in Eqution (4) (Step 2): f(c i,l s )=0, l s L s, c i,l s / P s (c i), (5) where P s (c i) is the set of originl sustrte pths the virtul link (or corresponding commodity c i) ws mpped to. Then we solve the MFP prolem gin with the new constrints of oth Equtions (4) nd (5). If we re llowed to select C E e ACM SIGCOMM Computer Communiction Review 24 Volume 38, Numer 2, April 08

7 Algorithm 4 Pth Migrtion Algorithm For ll the served requests, Step 1 Select the request set S whose durtions re lrger thn threshold T dur. Step 2 If only chnging splitting rtio is llowed, dd liner constrints (Eqution (5)), so tht ech virtul link is forced to e mpped to the pths it originlly tke in the link mpping step. If setting up new pth is lso llowed, skip this step. Step 3 Rerun the link mpping lgorithm with pth splitting, nd migrte the relted pths. new underlying pths, we rerun the link-mpping lgorithm with only the constrints in Eqution (4). Pth migrtion llows us to (periodiclly) tret the online emedding prolem s n offline prolem, to cpitlize on the efficiency gins tht re possile when hndling lrge collection of requests together. As such, we expect the enefits of pth migrtion to e highest when the time window (for grouping requests) is smll, nd less significnt s the window grows lrger. In prctice, migrting pths introduces overhed to estlish new pths, switch the trffic onto the new pths, nd ter down the old pths. As such, the enefits of pth migrtion should e weighed ginst the overheds. To illustrte this, we expect tht VN requests would e quite diverse in their durtions, which corresponds to their running time in the sustrte network, rnging from few months to severl hours. As n exmple, content distriution network like Akmi [1] my run infinitely, wheres n impromptu conference or video gme my lst for few hours. The lgorithm should not migrte short-lived virtul networks tht re likely to exit the system soon fter the migrtion completes. Thus, our lgorithm only considers the requests whose durtions re lrger thn some threshold T dur (Step 1). Fortuntely, migrting long-running virtul networks should offer mple enefits in prctice, since mny shortlived virtul networks will come nd go while they run. Virtul-network requests would indicte their likely durtion, or we cn infer tht virtul network tht hs run for long time is likely to continue running for long time, nlogous to previous reserch on migrtion in the context of jo scheduling [17]. 3.3 Implementtion Issues Pth splitting cn e implemented in the sustrte network without significnt overhed. When the virtul node directs pcket over the virtul link, the sustrte sends the pcket over one of the pths sed on the trget splitting rtio. Pth splitting my cuse out-of-order pcket delivery. Some virtul networks do not cre out out-of-order pckets; or they cn reorder the out-of-order pckets y themselves, e.g. those pplictions with only UDP flows. We cn lso mke the virtul networks olivious to the trffic splitting y preventing the disruptions of out-of-order pckets in the sustrte, e.g., using hsh-sed splitting. Out-of-order delivery is primry concern for pckets in the sme flow group of pckets etween the sme end hosts or prt of the sme trnsport-level connection. Hshsed splitting prevents out-of-order delivery y directing ll pckets from the sme flow to the sme pth. The sustrte router first divides the hsh spce into weighted prtitions tht ech correspond to one sustrte pth. Then, we pply hshing to the pckets sed on their heder its nd forwrd the pckets to the corresponding sustrte pth. This hsh-sed scheme is efficient nd, in fct, is widely used in IP networks to split trffic evenly over equl-cost multipth [, 8]. For those non-ip pckets, the virtul network would need to tell the sustrte which its in the heder indicte pckets in the sme flow, so tht the hshing cn e sed on those fields. There re lso techniques for more generlized, enhnced multipth routing to relize pth splitting, see e.g., [18]. Pth migrtion is closely relted to pth splitting, nd is esily implemented either y selecting new underlying pths or dpting the splitting rtios for the existing pths. In ddition, pth migrtion will not cuse significnt service disruptions for two resons: (i) we only need slight chnge of flow splitting rtio for the lredy-existing pths; (ii) we cn crete the new pth in dvnce efore moving the trffic to void service disruption. The sustrte router cn use consistent hshing to minimize the frction of flows tht must chnge pths when the splitting rtio chnges or new pths re creted [9]. Therefore, pth migrtion should not unduly influence the performnce experienced y the virtul network. In our current work, we hve focused on pth migrtion while keeping the node-mpping intct, to minimize the disruption experienced y the virtul networks. However, in ongoing work we pln to explore node migrtion to provide even greter flexiility in hndling new VN requests. We elieve node migrtion should e fesile for severl resons. First, long-running services usully hve their own mintennce windows, where they drin trffic off server to upgrde the softwre. These mintennce windows cn e used for node migrtion. Second, with mple wrning nd prior plnning, we cn minimize the negtive effects of node migrtion on n ongoing service. Node migrtion cn e done quite quickly in prctice, e.g., within few seconds [29], nd the virtul node cn continue running in the old loction until the migrtion completes. 4. PERFORMANCE EVALUATION In this section, we first descrie the performnce evlution environment, nd then present our min evlution results. Our evlution focuses primrily on quntifying the enefits of sustrte support for flexile pth splitting nd migrtion in the VN emedding prolem. 4.1 Evlution Environment We implemented VN emedding simultor (pulicly ville t [4]) to evlute our emedding lgorithm nd the dvntges of pth splitting nd migrtion. The ctul chrcteristics of sustrte nd virtul networks re not well understood since network virtuliztion is still n open field. Therefore, we use synthetic networks to study the trends nd quntify the enefits of pth splitting nd migrtion. Sustrte network. We use the GT-ITM tool [30] to generte sustrte network topology. The GT-ITM tool hs een populrly used in reserch tht requires prcticl network topology genertion. The sustrte network is configured to hve 0 nodes nd round 500 links, scle tht corresponds to medium-sized ISP The CPU resources t nodes nd the link ndwidths t links follow uniform distriution from 0 to 0 units. ACM SIGCOMM Computer Communiction Review 25 Volume 38, Numer 2, April 08

8 Averge Revenue Over Time E[w] = 50, Bseline Algorithm E[w] = 50, Pth Splitting E[w] = 50, Pth Splitting & Migrtion E[w] = 25 Averge Revenue Over Time Bseline Algorithm Pth Splitting Pth Splitting & Migrtion Averge Revenue Over Time Ttry = 0 Ttry = 1 Ttry = Splitting Rtio (%) Figure 6: Revenue chnges with RPS(%) (E[CPU]:0, DELAY:3, α:0, T try:1) Splitting Rtio (%) Figure 7: Revenue chnges with RPS(%) (E[BW]:50, E[CPU]:0, DELAY:6, α:0, T try:1) Splitting Rtio (%) Figure 8: Influence of Node Rempping (E[BW]:50, E[CPU]:0, DE- LAY:3, α:0) Virtul network request. In one VN request, the numer of VN nodes is rndomly determined y uniform distriution etween 2 nd in Figures 6 12, following similr setup to previous work [31]. We lso test lrger requests whose numer of re chosen uniformly etween 2 nd in Figure 11. Ech pir of virtul nodes re rndomly connected with proility 0.5. This mens tht for n-node virtul network, we hve n(n 1)/4 links on verge. The rrivls of VN requests re modeled y Poisson process with men five requests per time window. The durtion of the requests follows n exponentil distriution with time windows on verge. We run ll of our simultions for 500 time windows, which corresponds to out 2500 requests on verge in one instnce of simultion. The prmeters nd their symols tht we vry in ll our simultions re summrized in the following tle: E[CPU] verge CPU requirement on virtul node E[BW] verge ndwidth requirement on virtul link RPS(%) percentge of the requests llowing pth splitting DELAY time request is willing to wit (see Section 2.2) α weight constnt in revenue function (Eqution (2)) T try numer of rounds in node rempping Comprison method. Compring our lgorithm with previous work is difficult ecuse these erlier emedding lgorithms do not strt with the sme prolem formultion. They do not hndle one or more of the first three chllenges in Section 1 (i.e., comined node nd link constrints, dmission control, or online requests). Insted, we use the lgorithm in Section 2.2, which emodies mny of the key ides from prior work, s seline for comprison. 4.2 Evlution Results Our evlution results quntify the enefits of pth splitting nd migrtion in vrious environments. We present our simultion results y summrizing the key oservtions. (1) More requests llowing pth splitting leds to lrger revenues, which is further improved y pth migrtion. Figures 6 shows the long-term verge revenue with incresing percentges of the requests permitting pth splitting for different verge link ndwidth requirements. In these experiments, we remove the influence of CPU in the constrint nd the revenue (i.e., E[CPU]=0 nd α=0). Ech request which cnnot e served immeditely will wit for t most 3 time windows in the queue (DELAY=3) nd node rempping in Algorithm 3 is tried just once, i.e., T try =1. In Figure 6, the performnce of the seline lgorithm in Section 2 does not depend on RPS(%), since the seline lgorithm mps ech virtul link into singlepthin the sustrte network. However, with more requests llowing pth splitting, the sustrte network resources re efficiently utilized t current time window, which enles the system to ccept more requests, leding to n increse in the verge revenue. When ll the requests llow pth splitting, our lgorithm chieves out 1% revenue increse over the seline lgorithm. Even with hlf of the requests permitting pth splitting, we still gin out 65% revenue increse. Figure 6 lso shows tht pth migrtion further increses the revenue. For exmple, when RPS(%)=0, our lgorithm with pth migrtion chieves dditionl % revenue increse over the lgorithm only with pth splitting. This implies tht pth splitting is dominnt fctor in the revenue increse, nd pth migrtion further uilds on pth splitting to dpt to the online VN emedding prolem more flexily. More enefits re expected to e otined y node migrtion, t the expense of more service disruption. When the ndwidth requirement is low (E[BW]=25) nd sustrte resources re mple, we cn ccept ll the requests for oth lgorithms. Nturlly, the revenue remins the sme, whether the requests llow pth splitting or not. We will show lter in Figure 12 tht in this cse, our lgorithm reduces cost more thn the seline lgorithm. Note tht in Figure 6, the revenues differ when E = 25 nd E =50 due to its dependence on the mount of required (verge) ndwidth in the requests. (2) Pth splitting still increses revenue when CPU requirements re considered. Figure 9 shows the longterm revenues with oth CPU nd ndwidth requirements, where the verge CPU requirement is set to e 25 nd other prmeters re the sme s those in Figure 6. We oserve similr increse in revenue from pth splitting. We chieve more thn 0% revenue increse over the seline lgorithm, when RPS(%)=0; nd out 50% when RPS(%)=50. However, the enefits from migrtion re less drmtic. This is due to the fct tht we only employ pth migrtion, which does not offer ny enefits when the node CPU resource is the ottleneck. Note tht revenue increse with the CPU requirement is less thn tht without the CPU requirement. This is nticipted, ecuse when ACM SIGCOMM Computer Communiction Review 26 Volume 38, Numer 2, April 08

9 Averge Revenue Over Time Bseline Algorithm Pth Splitting Pth Splitting & Migrtion Splitting Rtio (%) Averge Revenue Over Time Bseline Algorithm Pth Splitting Pth Splitting & Migrtion Weight Constnt in Revenue () Averge Revenue Over Time Bseline Algorithm Pth Splitting Pth Splitting & Migrtion Splitting Rtio (%) Figure 9: Effect of CPU requirement (E[BW]:50, E[CPU]:25, DELAY:3, α:0, T try:1) Figure : Effect of α (RPS(%):0, E[BW]:50, E[CPU]:25, DELAY:3, T try:1) Figure 11: Effect of lrger virtul networks (nodes:2-, E[BW]:25, E[CPU]:0, α:0, T try:1) CPU requirements tends to reduce the numer of ccepted requests. To further evlute effects of CPU resource constrints, in Figure, we hve tested different weight constnts α in the revenue definition (Eqution (2)) while keeping ll the other prmeters the sme. The enefits of pth splitting over the seline lgorithm decrese s α increses, since pth splitting nd migrtion only improve the ndwidth resource utiliztion in the link mpping stge. For exmple, when CPU nd ndwidth re lmost eqully evluted in the revenue function, we chieve more thn 0% of revenue increse over the seline lgorithm; when CPU resource ecomes the min fctor (α=1), our lgorithm with pth splitting still chieves round 60% more revenue thn the seline lgorithm. (3) Node rempping contriutes modestly to revenue increse. Revenue is not significntly influenced y the dely we choose. All previous experiments were mde with T try =1, i.e., we rn one round of node rempping in Algorithm 3. In Figure 8, we show the result of the pth splitting lgorithm without node rempping (i.e., T try = 0), where the revenue only decreses y 4%. This implies tht the revenue increse shown in erlier simultions minly comes from pth splitting itself. With incresing vlues of T try, we could chieve more sustntil increses in revenue t the expense of computtion time, ecuse we must rerun the link mpping stge for T try times more thn the pth splitting solution without node rempping. Figure 7 shows tht our enefits of pth splitting nd migrtion re not influenced y dely we choose (DELAY=3), since the result for DELAY=6 is similr to Figure 6 where DE- LAY=3. This is ecuse the sustrte resources re lmost fully used with requests coming nd deprting over time, so tht the deferred requests cnnot e ccepted even if it wits for more time. (4) Pth splitting nd migrtion cn still help improve revenue for lrger requests. Figure 11 exmines the enefits of pth splitting nd migrtion for lrger requests, i.e., the numer of nodes in ech VN request uniformly rnges etween 2 nd. We first keep the totl mount of resource requirement the sme s tht in the erlier experiments. Figure 11 shows tht with ll the requests llowing pth splitting, the revenue increses y 50% with pth splitting. This enefit is less thn the 1% increse of revenue in Figure 6 with E[BW] = 50. This is due to the fct tht without chnge of totl resource requirement, lrger requests led to more links nd thus less ndwidth per link. In other experiments when we increse the scle of requests without chnging the verge ndwidth per link, then we re le to chieve more enefits. We omit this result due to spce limittion. Averge Bndwidth Cost Over Time Bseline Algorithm Pth Splitting Pth Splitting & Migrtion Splitting Rtio (%) Figure 12: Effect on cost (E[BW]:25, E[CPU]:0, DELAY:3, α:0, T try:1) (5) Without dmission control, pth splitting nd migrtion reduces cost. In cse when the sustrte network resources re sufficient, ut the numer of incoming VN requests per time-window is smll, we cn proly service ll the requests, irrespective of using of pth splitting or not, i.e., the long-term revenue chieved will e the sme for the seline lgorithm nd the lgorithm with pth splitting. However, the lgorithm llowing pth splittility sves sustrte network resources. First, we discuss notions of cost to quntify efficiency in resource utiliztion. The ndwidth cost for VN request should e defined to reflect the entire mount of ndwidth used to mp the request to the physicl sustrte network. For single virtul link in the request, it would e nturl to use its required ndwidth multiplied y the length of the sustrte pth tht the virtul link is mpped to. With pth splitting, we count the ndwidth llocted on ech pth of the virtul link nd sum them up. Thus, we define ACM SIGCOMM Computer Communiction Review 27 Volume 38, Numer 2, April 08

10 Algorithm 5 Customized Node Mpping Algorithm for Requests with Hu-nd-spoke Steps 1, 2, nd 3: Sme s in Greedy Node Mpping (Algorithm 1). Step 4 If the request hs hu-nd-spoke topology, 4.1 For ech hu node, find the sustrte node with the mximum ville resource in S. 4.2 For ech spoke node, find the shortest pth etweensustrtenodeins nd the sustrte node mpped to the corresponding hu node. else, pply Step 4 in Greedy Node Mpping. Step 5 Sme s in Greedy Node Mpping. the cost of virtul network G v y: C w (G v )= X X hops(p)w(p, l v ), (6) l v L v p P s (l v ) where P s (l v ) is the pth(s) the virtul link l v is mpped on, hops(p) is the numer of hops of pth p, ndw(p, l v ) is the mount of ndwidth llocted to tht virtul link. Similrly, we define the CPU cost of virtul network G v y: C CPU(G v )= X CPU(n v ), (7) n v N v where CPU(n v ) is the mount of CPU virtul node n v requires. In Figure 12, we hve simulted the cse where the resource requirement of requests is low (E[BW]=25, E[CPU]=0), where other prmeters re the sme s efore, i.e., (DE- LAY=3, α=1, T try=1). We oserve tht with the increse in percentge of requests permitting pth splitting, we reduce the ndwidth cost C w over the seline lgorithm y mking more efficient use of the network. When RPS(%)=0, we reduce % cost thn the seline lgorithm. Pth migrtion further reduces the ndwidth cost y 7%. The CPU cost C CPU remins the sme (C CPU =3.1) with the increse of RPS(%), since ll the requests re ccepted, whether they llow pth splitting or not. 5. CUSTOMIZED NODE MAPPING Although virtul networks my hve ritrry topologies, we expect some clsses of topologies to e reltively common, since they meet the needs of the key pplictions in network virtuliztion. For exmple, hu-nd-spoke topology is commonly used to connect multiple sites to centrlized server, e.g., gming nd CDN (Content Distriution Network), nd tree topology is commonly used to distriute content efficiently to lrge collection of receivers, e.g. multicst distriution of IPTV. The populrity of smll set of topologicl structures cn e leverged for etter solutions to the VN emedding prolem. In our ongoing work, we present node-mpping techniques tht re customized to specific topologies, strting with the simple hu-nd-spoke topology. As n exmple, we propose customized node mpping (which is extended from the Greedy Node Mpping in Algorithm 1) with hu-nd-spoke topologies, s summrized in Algorithm 5. The customized node mpping lgorithm differs from the greedy node mpping in tht we choose the sustrte nodes differently for hu nd spokes nodes. The mximum ville resource is llocted only for the hu nodes (Step 4.1), nd the spoke nodes re mpped into the sustrte nodes tht hve the shortest pth to the sustrte node hosting the virtul hu node (Step 4.2). This is motivted y the fct tht the hu node hndles much more trffic thn the individul spokes. In Step 4.2, we lso chieve significnt cost reduction, since cost is generlly proportionl to the distnce (i.e., numer of hops), wheres the greedy lgorithm lloctes lrge sustrte resource to unimportnt nodes (i.e., the spokes). The wsted resources keep the greedy node mpping lgorithm from leving enough resources ville to stisfy the future requests. We did the evlution on our customized node mpping lgorithm with the requests of hu-nd-spoke topologies nd compre it with the greedy node mpping. Our preliminry experiments show tht tht our customized lgorithm performs etter thn the greedy lgorithm when the percentge of hu-nd-spoke topologies mong ll the requests increses. By tking dvntge of the topology informtion, our customized lgorithm lloctes the hu-nd-spoke request more efficiently thn the greedy lgorithm. Thus, our lgorithm cn llow more requests nd chieve higher verge revenue over time. This result is omitted due to pge limit. We re currently exploring nd evluting other lgorithms tht re customized for different common topologies, like trees. 6. RELATED WORK Previous reserch hs explored how to emed Virtul Privte Networks (VPNs) in shred provider topology [11, 16]. VPNs usully hs stndrd topology, such s full mesh nd hu-nd-spoke [25]. The resource constrints in VPN re typiclly just ndwidth requirements, specified y trffic mtrix (i.e., the trffic volume for ech pir of nodes), rther thn node constrints (e.g., processing resources). VN emedding prolem is different with VPN design prolem in tht VN emedding prolem must del with oth node nd link constrints for ritrry topology. Relted work on VN emedding ddresses the hrdness of the prolem y relxing one or more of the key properties of the prolem. These key properties include (i) whether requests re processed online or not, (ii) whether the requests hve link constrints, node constrints, or oth, (iii) whether dmission control is performed to reject requests when resources re insufficient, nd (iv) wht virtul topologies re supported. Severl of the previous studies focus on the offline prolem, where ll VN requests re known in dvnce. Zhu nd Ammr [31] ssume tht the sustrte network resources re unlimited, nd im t chieving lod lncing in the sustrte network without the need for dmission control. The VN-emedding prolem for the requests with generl topology is solved y sudividing the requests into multiple str topologies to llocte more sustrte resource to the center of ech decomposed str topology. Our greedy node emedding lgorithm (Algorithm 1) is sed on the key ide of this pper. Lu nd Turner [23] lso consider n offline prolem for only single virtul network with ckone-str topology, where their gol is to minimize the cost. They ssume tht only ndwidth constrints re imposed, nd the sustrte network resources re unlimited with no dmission control needed. In regrd to the online prolem, Fn nd Ammr [13] consider dynmic topology reconfigurtion policies for virtul ACM SIGCOMM Computer Communiction Review 28 Volume 38, Numer 2, April 08

11 networks with dynmic communiction requirements, ut no considertion of the node constrints such s CPU. They lso ssume tht sustrte network resources re unlimited to ccept ll requests (i.e., no dmission control) nd try to find strtegy to minimize cost. Zhu nd Ammr [31] lso solve the online prolem y reclculting the whole emedding solution periodiclly. The Assign lgorithm [26] used in Emul tested considers the online emedding prolem with the ndwidth constrint. The node constrint in Emul is provided s the exclusive use of nodes, i.e., different virtul networks cnnot shre sustrte node. Admission control is not explicitly ddressed in [26], ut it cn e inferred tht emul rejects request if the sustrte ndwidth/node resources re insufficient. Emul s PlnetL portl [28] provides resource lloction service for users to ccess to PlnetL tested. It serches for the PlnetL nodes with low CPU nd memory lods for the users. Our work considers ll the four chllenges outlined in Section 1. In prticulr, we improve the link mpping lgorithm through sustrte support for pth splitting nd migrtion. 7. CONCLUSION A key prolem in the current study of network virtuliztion, the VN emedding prolem, hs vrious constrints tht mke it computtionlly intrctle. In this pper, rther thn significntly restrict the prolem spce to mke the prolem trctle, we rethink the VN emedding prolem y proposing more flexile sustrte network to etter support virtul network emedding. This flexiility includes pth splitting nd migrtion. Pth splitting (i.e. multipth) hs een recurring theme in mny network reserch topics, nd we demonstrte the power of multipth in sustrte network for more cost-effective virtul network emedding. From oth the theoreticl nd engineering perspective, we show tht llowing sustrte pth splitting nd migrtion would help us to ttin etter resource utiliztion. Through our pulicly ville simultor, we demonstrte the enefits of these pproches in mking the emedding prolem computtionlly esier, nd the resulting emeddings more efficient. 8. REFERENCES [1] Akmi content distriution network. [2] GENI. [3] Plnetl. https://www.plnet-l.org/. [4] Virtul Network Emedding Simultor. [5] R.K.Ahuj,T.L.Mgnnti,ndJ.B.Orlin.Network Flows: Theory, Algorithms, nd Applictions. Prentice Hll, [6] D. G. Andersen. Theoreticl pproches to node ssignment. Unpulished Mnuscript, http: //www.cs.cmu.edu/~dg/ppers/ndersen-ssign.ps, 02. [7] T. Anderson, L. Peterson, S. Shenker, nd J. Turner. Overcoming the Internet impsse through virtuliztion. IEEE Computer Mgzine, 38(4):34 41, 05. [8] B. Augustin, X. Cuvellier, B. Orgogozo, F. Viger, T. Friedmn, M. Ltpy, C. Mgnien, nd R. Teixeir. Avoiding trceroute nomlies with Pris trceroute. In Proc. Internet Mesurement Conference, 06. [9] I. Avrmopoulos, D. Syrivelis, J. Rexford, nd S. Llis. Secure vilility monitoring using stelth proes. Technicl Report TR , Princeton University, 06. [] A. Bvier, N. Femster, M. Hung, L. Peterson, nd J. Rexford. In VINI verits: Relistic nd controlled network experimenttion. In Proc. ACM SIGCOMM, Septemer 06. [11] N.G.Duffield,P.Goyl,A.Greenerg,P.Mishr,K.K. Rmkrishnn, nd J. E. vn der Merwe. Resource mngement with hoses: Point-to-cloud services for virtul privte networks. IEEE/ACM Trns. Networking, 02. [12] D. Eppstein. Finding the k shortest pths. In Proc. IEEE Symposium on Foundtions of Computer Science, [13] J. Fn nd M. Ammr. Dynmic topology configurtion in service overly networks: A study of reconfigurtion policies. In Proc. IEEE INFOCOM, 06. [14] N. Femster, L. Go, nd J. Rexford. How to lese the Internet in your spre time. ACM Computer Communiction Review, 37(1):61 64, 07. [] A. Feldmnn, A. Greenerg, C. Lund, N. Reingold, nd J. Rexford. NetScope: Trffic engineering for IP networks. IEEE Network Mgzine, 14(2):11 19, Mrch 00. [16] A. Gupt, J. Kleinerg, A. Kumr, R. Rstogi, nd B. Yener. Provisioning virtul privte network: A network design prolem for multicommodity flow. In Proc. ACM Symposium on Theory of Computing, 01. [17] M. Hrchol-Blter nd A. B. Downey. Exploiting process lifetime distriutions for dynmic lod lncing. ACM Trnsctions on Computer Systems, (3): , [18] J. He nd J. Rexford. Towrds Internet-wide multipth routing. In IEEE Network Mgzine Specil Issue on Sclility, Mrch 08. [19] P. Key, L. Mssoulie, nd D. Towsley. Pth selection nd multipth congestion control. In Proc. IEEE INFOCOM, 07. [] S. Kirkptrick, C. D. Geltt, nd M. P. Vecchi. Optimiztion y simulted nneling. Science, Numer 4598, 13 My 1983, 2, 4598: , [21] J. Kleinerg. Approximtion lgorithms for disjoint pths prolems. PhD thesis, MIT, [22] S. G. Kolliopoulos nd C. Stein. Improved pproximtion lgorithms for unsplittle flow prolems. In Proc. IEEE Symposium on Foundtions of Computer Science, [23] J. Lu nd J. Turner. Efficient mpping of virtul networks onto shred sustrte. Technicl Report WUCSE-06-35, Wshington University, 06. [24] M. Mitzenmcher. The power of two choices in rndomized lod lncing. IEEE Trnsctions on Prllel nd Distriuted Systems, 12():94 14, 01. [25] S. Rghunth, K. K. Rmkrishnn, S. Klynrmn, nd C. Chse. Mesurement sed chrcteriztion nd provisioning of IP VPNs. In Proc. Internet Mesurement Conference, 04. [26] R. Ricci, C. Alfeld, nd J. Lepreu. A solver for the network tested mpping prolem. ACM Computer Communiction Review, 33(2):65 81, 03. [27] J. S. Turner nd D. E. Tylor. Diversifying the Internet. In Proc. IEEE GLOBECOM, 05. [28] K.We,M.Hiler,R.Ricci,A.Clements,nd J. Lepreu. Implementing the Emul-Plnetl portl: Experience nd lessons lerned. In Workshop on Rel, Lrge Distriuted Systems (WORLDS), 04. [29] T. Wood, P. Shenoy, A. Venktrmni, nd M. Yousif. Blck-ox nd gry-ox strtegies for virtul mchine migrtion. In Proc. Networked Systems Design nd Implementtion, 07. [30] E. W. Zegur, K. L. Clvert, nd S. Bhttchrjee. How to model n internetwork. In Proc. IEEE INFOCOM, [31] Y. Zhu nd M. Ammr. Algorithms for ssigning sustrte network resources to virtul network components. In Proc. IEEE INFOCOM, 06. ACM SIGCOMM Computer Communiction Review 29 Volume 38, Numer 2, April 08

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