Virtual Network Embedding with Coordinated Node and Link Mapping

Transcription

1 Vrtual Network Embeddng wth Coordnated Node and Lnk Mappng N. M. Mosharaf Kabr Chowdhury Cherton School of Computer Scence Unversty of Waterloo Waterloo, Canada Emal: Muntasr Rahan Rahman Cherton School of Computer Scence Unversty of Waterloo Waterloo, Canada Emal: Raouf Boutaba Cherton School of Computer Scence Unversty of Waterloo Waterloo, Canada Emal: Abstract Recently network vrtualzaton has been proposed as a promsng way to overcome the current ossfcaton of the Internet by allowng multple heterogeneous vrtual networks VNs) to coexst on a shared nfrastructure. A major challenge n ths respect s the VN embeddng problem that deals wth effcent mappng of vrtual nodes and vrtual lnks onto the substrate network resources. Snce ths problem s known to be NP-hard, prevous research focused on desgnng heurstc-based algorthms whch had clear separaton between the node mappng and the lnk mappng phases. Ths paper proposes VN embeddng algorthms wth better coordnaton between the two phases. We formulate the VN embeddng problem as a mxed nteger program through substrate network augmentaton. We then relax the nteger constrants to obtan a lnear program, and devse two VN embeddng algorthms D-VNE and R-VNE usng determnstc and randomzed roundng technques, respectvely. Smulaton experments show that the proposed algorthms ncrease the acceptance rato and the revenue whle decreasng the cost ncurred by the substrate network n the long run. I. INTRODUCTION Snce ts ncepton, the Internet has proven ts worth by supportng myrads of dstrbuted applcatons and heterogeneous networkng technologes. However, due to the exstence of multple stakeholders wth conflctng goals and polces, alteratons, even necessary ones, to the present archtecture have now become almost mpossble to acheve. To fend off the rgdty of the exstng Internet once and for all, network vrtualzaton has been propounded as a fundamental dversfyng attrbute of the future nter-networkng paradgm that wll allow multple heterogeneous network archtectures to coexst on a shared substrate [1], [2]. In a network vrtualzaton envronment NVE), multple servce provders SPs) wll be able to create heterogeneous vrtual networks VNs) to offer customzed end-to-end servces to the end users by leasng shared resources from one or more nfrastructure provders InPs) wthout sgnfcant nvestment n physcal nfrastructure [2] [4]. Each VN n the NVE s a collecton of vrtual nodes e.g., vrtual routers) that are hosted on dfferent substrate nodes and nterconnected by physcal paths n the substrate network correspondng to vrtual lnks. Snce multple VNs can share the same underlyng physcal resources, effcent and effectve embeddng/mappng/assgnment 1 of each of the onlne VN requests s of utmost mportance n order to ncrease utlzaton of the substrate network resources and consequently revenues of the InPs. However, the VN embeddng problem, wth constrants on vrtual nodes and vrtual lnks, can be reduced to the NPhard mult-way separator problem [5], even f all the VN requests are known n advance. Even when all the vrtual nodes are already mapped, embeddng the vrtual lnks wth bandwdth constrants onto substrate paths s stll NP-hard n the unsplttable flow scenaro. As a result, a number of heurstc-based algorthms have appeared n the relevant lterature [6] [9]. Most of these proposals focused prmarly on edge mappng usng, for example, shortest path, k-shortest paths, and mult-commodty flow algorthms) after employng greedy methods to preselect the node mappngs. However, preselectng node mappngs wthout consderng ts relaton to the lnk mappng stage restrcts the soluton space, and may result n poor performance. In ths paper, we ntroduce better correlaton between the node mappng and the lnk mappng phases by proposng two new VN embeddng algorthms D-VNE Determnstc VN Embeddng) and R-VNE Randomzed VN Embeddng). In these algorthms, we map the vrtual nodes to substrate nodes n a way that facltates the mappng of vrtual lnks to physcal paths n the subsequent phase. To ths end, we extend the physcal network graph by ntroducng meta-nodes for each vrtual node and connect the meta-nodes to a selected subset of physcal nodes Secton IV-A). We then treat each vrtual lnk wth bandwdth constrants as a commodty consstng of a par of meta-nodes. As a result, fndng an optmal flow for the commodty s equvalent to mappng the vrtual lnk n an optmal way. If we ntroduce addtonal bnary constrants that force only one meta-edge to be selected for each metanode, we can effectvely select exactly one substrate node for each meta-node correspondng to a partcular vrtual node. We use mxed nteger programmng MIP) formulaton [] to solve the embeddng problem wth bnary constrants on the meta-edges and lnear constrants on actual substrate network 1 The words embeddng, mappng, and assgnment are used nterchangeably throughout ths paper /09/$ IEEE 783

2 edges. Snce solvng an MIP s known to be NP-hard [], fndng an optmal VN embeddng usng MIP becomes NPhard as well. As a result, we relax the nteger program to obtan a lnear programmng formulaton whch can be solved n polynomal tme. We then use determnstc and randomzed roundng technques on the soluton of the lnear program to approxmate the values of the bnary varables n the orgnal MIP. Once all the vrtual nodes have been mapped, we use the mult-commodty flow algorthm to map the vrtual lnks onto the substrate network between the mapped vrtual nodes [9], [11]. Ths can also be solved n polynomal-tme snce we assume that path splttng s supported by the substrate network [9]. The rest of ths paper s organzed as follows. Secton II provdes an overvew of the related work. Followng that, Secton III formalzes the network model and the VN embeddng problem tself. In Secton IV, we provde the optmal MIP formulaton for the VN embeddng problem usng substrate network augmentaton. Secton V relaxes the MIP formulaton to obtan a lnear program and presents D-VNE and R-VNE usng determnstc and randomzed roundng technques. Secton VI presents smulaton results that evaluate the proposed algorthms, and we conclude n Secton VII dentfyng future research drectons. II. RELATED WORK The VN assgnment problem s smlar to the prevous works on embeddng Vrtual Prvate Networks VPN) n a shared provder topology and the network testbed mappng problem [12], [13]. However, a typcal VPN request conssts only of bandwdth requrements, specfed n terms of a traffc matrx, wthout any constrant on ts nodes. As a result, most VPN desgn algorthms come down to fndng paths for source/destnaton pars. On the other hand, the Assgn algorthm [13] used n the Emulab testbed consders bandwdth constrants alongsde constrants on exclusve use of nodes,.e., dfferent VNs cannot share a substrate node. But n network vrtualzaton, there are capacty and placement requrements on both the vrtual nodes and the vrtual lnks; n addton, substrate nodes and lnks can be shared by multple VNs. In order to reduce the hardness of the VN assgnment problem and to enable effcent heurstcs, exstng research has been restrctng the problem space n dfferent dmensons, whch nclude: 1) consderng offlne verson of the problem.e., all the VN requests are known n advance) [7], [8], 2) gnorng ether node requrements or lnk requrements [6], [7], 3) assumng nfnte capacty of the substrate nodes and lnks to obvate admsson control [6] [8], and 4) focusng on specfc VN topologes [7]. The authors n [9] consder all these ssues, except for the locaton constrants on the vrtual nodes, by envsonng support from the substrate network through node and lnk mgraton as well as mult-path routng. We do not restrct the problem space by assumng nfnte capacty of the substrate network resources, nor do we assume any specalzed VN topologes. Contrary to the algorthms proposed n ths paper, all the exstng algorthms can clearly be separated nto two basc phases: 1) assgnng vrtual nodes usng some greedy heurstcs, e.g., assgn vrtual nodes wth hgher processng requrements to substrate nodes wth more avalable resources [8], [9], and 2) embeddng vrtual lnks onto substrate paths usng shortest path algorthms [8] n case of unsplttable flows, or usng mult-commodty flow algorthms n case of splttable flows [9], [11]. The authors n [14] have proposed a dstrbuted algorthm that smultaneously maps vrtual nodes and vrtual lnks wthout any centralzed controller, but the scalablty and performance of ther algorthm s stll not comparable wth the centralzed ones. The nteger and mxed nteger programmng approaches have been appled to a number of resource allocaton and optmzaton problems n the networkng area. In [15], the authors have proposed an nteger programmng model to solve the VPN tree computaton problem for bandwdth provsonng n VPNs. Technques of randomzed roundng for lnear programmng relaxatons to obtan approxmaton algorthms was frst ntroduced n [16]. In ths paper, we take a formal approach to solve the onlne VN embeddng problem usng a mxed nteger programmng formulaton. To the best of our knowledge, ths s the frst attempt to apply mathematcal programmng to ths problem. III. NETWORK MODEL AND PROBLEM DESCRIPTION A. Substrate Network We model the substrate network as a weghted undrected graph and denote t by G S = N S,E S), where N S s the set of substrate nodes and E S s the set of substrate lnks. Each substrate node n S N S s assocated wth the CPU capacty weght value c n S) and geographc locaton loc n S). Each substrate lnk e S, j) E S between two substrate nodes and j s assocated wth the bandwdth capacty weght value b e S) denotng the total amount of bandwdth. We denote the set of all substrate paths by P S, and the set of substrate paths from the source node s to the destnaton node t by P S s, t). Fg. 1 shows a substrate network, where the numbers over the lnks represent avalable bandwdths and the numbers n rectangles represent avalable CPU resources 2. B. VN Request Smlar to the substrate network, we model VN requests as weghted undrected graphs and denote a VN request by G V = N V,E ) V. We express the requrements on vrtual nodes and vrtual lnks n terms of the attrbutes of the nodes and lnks of the substrate network. Each VN request has an 2 We use notatons smlar to [9] to denote capactes and requrements. 784

3 d 20 b a 20 e 12 c VN Request f 20 VN Request 2 Fg. 1. a C A B 22 d D E e f 17 G H b Substrate Network Mappng of VN requests onto a shared substrate network. assocated non-negatve value D V expressng how far a vrtual node n V N V can be placed from the locaton specfed by loc n ) V. D V can be expressed n terms of physcal dstance or n terms of permssble delay e.g., RTT) from loc n ) V. Fg. 1 shows two VN requests wth node and lnk constrants. C. Measurement of Substrate Network Resources The resdual or the avalable capacty of a substrate node, R ) N n S s defned as the avalable CPU capacty of the substrate node n S N S. R N n S ) = c n S) cn V ) n V n S where x y denotes that the vrtual node x s hosted on the substrate node y. Smlarly, the resdual capacty of a substrate lnk, R ) E e S s defned as the total amount of bandwdth avalable on the substrate lnk e S E S. R E e S ) = b e S) be V ) e V e S where x y denotes that the substrate path of the vrtual lnk x passes through the substrate lnk y. The avalable bandwdth capacty of a substrate path P P S s gven by R E P ) = mn R E e S ) e S P D. VN Assgnment When a VN request arrves, the substrate network has to determne whether to accept the request or not. If the request s accepted, the substrate network then determnes a sutable assgnment for the VN and allocates network resources on the substrate nodes and paths selected by that assgnment. The allocated resources are released once the VN expres. The assgnment of the VN request V to the substrate network can be decomposed nto two major components: c F 1) Node assgnment: Each vrtual node from the same VN request 3 s assgned to a dfferent substrate node by a mappng M N : N V N S from vrtual nodes to substrate nodes such that for all n V,m V N V, subject to M N n V ) N S M N m V ) = M N n V ), ff m V = n V c n V ) R N MN n V )) ds loc n V ),loc M N n V )) ) D V 1a) 1b) where ds, j) measures the dstance between the locaton of two substrate nodes and j. In Fg. 1, the frst VN request has the node mappng {a C, b H, c B} and the second VN request has {d A, e D, f H}. Note that two vrtual nodes b and f from dfferent VN requests are mapped onto the same substrate node H. 2) Lnk assgnment: Each vrtual lnk s mapped to a substrate path unsplttable flow) or a set of substrate paths splttable flow) between the correspondng substrate nodes that host the end vrtual nodes of that vrtual lnk. It s defned by a mappng M E : E V P S from vrtual lnks to substrate paths such that for all e V = m V,n V ) E V, M E m V,n V ) P S M N m V ), M N n V ) ) subject to R E P ) b e V ), P M E e V ) 2) The frst VN { request n Fg. 1 has been assgned the lnk mappng a, b) {C, D), D, G), G, H)}, a, c) } {C, A), A, B)}, b, c) {H, F), F, E), E,B)}, and the second VN request has the lnk mappng } {A, C), C, D)}, e, f) {D, G), G, H)}. { d, e) E. Objectves Our man nterest n ths paper s to propose onlne VN embeddng algorthms that map multple VN requests wth node and lnk constrants. We also want to ncrease revenue and decrease cost of the InP n the long run, n addton to balancng load of the substrate network resources. Smlar to the prevous work n [8], [9], we defne the revenue of a VN request as: RG V )= e V ) + n V ) 3) e V E V b n V N V c Whle revenue gves an nsght nto how much an InP wll gan by acceptng a VN request, t s not very useful wthout knowng the cost the InP wll ncur for embeddng that request. 3 Even though multple vrtual nodes from dfferent VN requests can be mapped to the same substrate node. 785

4 a A B c An example of augmented graph constructon s shown n Fg. 2. B. MIP Formulaton C Meta-edge Meta-node Cluster D G Fg. 2. Constructon of an augmented substrate graph wth meta-nodes and meta-edges for a VN request. We defne the cost of embeddng a VN request as the sum of total substrate resources allocated to that VN. CG V )= b f e V e V E V e S E S E H e S + F n V N V cn V ) 4) where f ev e denotes the total amount of bandwdth allocated on S the substrate lnk e S for vrtual lnk e V. We use a modfed verson of 4) as the objectve functon of our MIP formulaton. IV. MIXED INTEGER PROGRAMMING FORMULATION FOR OPTIMAL VN EMBEDDING A. Augmented Substrate Graph Constructon In order to coordnate the node mappng phase wth ts lnk mappng counterpart, the base substrate network must be extended to create an augmented substrate graph usng the locaton requrement of the vrtual nodes as the bass for the extenson. Snce each n V N V has an assocated constrant loc n ) V on ts possble placement, we can create one cluster for each vrtual node N V n total) n the substrate network wth radus D V. We denote such a cluster by Ωn V ) and call t the Ω set of the vrtual node n V. Ωn V )= {n S N S ds loc n V ),loc n S)) } D V In Fg. 2, substrate nodes B, E, and F are wthn D V dstance of the vrtual node c, hence Ωc) ={B,E,F}. For each n V N V we create a correspondng meta-node μn V ), and we connect μn V ) wth all the substrate nodes belongng to Ωn V ) usng meta-edges wth nfnte bandwdth. We wll sometmes wrte the Ω set as Ωm) nstead of Ωn V ), where m = μn V ). We combne all the meta-nodes and meta-edges wth the substrate graph to create the augmented substrate graph G S =N S,E S ), where N S = N S {μn V ) n V N V } E S = E S {μn V ),n S ) n V N V,n S Ωn V )} The VN embeddng problem can now be formulated as a mxed nteger E V -commodty flow problem. We consder each vrtual edge e V 1 E V ) as a commodty wth source and destnaton ) nodes s and t, respectvely, s,t N S \N S. Each flow, n ths settng, starts from a meta-node and ends n another meta-node. By ntroducng restrctons on the meta-edges, each meta-node μn V ) can be forced to choose only one meta-edge to connect tself to an actual substrate node n Ωn V ). Ths effectvely selects a substrate node for each meta-node,.e., maps the vrtual node correspondng to that meta-node to a substrate node. At the same tme, all the vrtual edges.e., flows) are also mapped effcently nsde the substrate network wth the help of path splttng. We present the MIP formulaton n the followng manner. VNE MIP Varables: f uv: A flow varable denotng the total amount of flow from u to v on the substrate edge u, v) for the th vrtual edge. x uv : A bnary varable, whch has the value 1 f f uv + f vu) > 0; otherwse, t s set to 0. Objectve: mnmze + w N S α uv fuv R E u, v)+δ β w x mw cm) 5) R N w)+δ m N S \N S uv E S Constrants: - Capacty Constrants: fuv + fvu) R E u, v) x u,v, u, v N S 6) R N w) x mw c m), m N S \ N S, w N S 7) - Flow Related Constrants: fuw fwu =0,, u N S \{s,t } 8) w N S w N S fs w fws = be V ), 9) w N S w N S ft w fwt = be V ), ) w N S w N S 786

5 - Meta and Bnary Constrants: x mw =1, m N S \ N S 11) w Ωm) - Doman Constrants: Remarks: m N S \N S x mw 1, w N S 12) x uv R E u, v), u, v N S 13) x uv = x vu, u, v N S 14) f uv 0, u, v N S 15) x uv {0, 1}, u, v N S 16) The objectve functon 5) of the MIP tres to mnmze the cost of embeddng the VN request as well as balance the load. By dvdng the cost wth the resdual capacty, t s ensured that the resources wth more resdual capactes are preferred over the resources wth less resdual capactes. 1 α uv R E u, v) and 1 β w R N w) are parameters to control the mportance of load balancng whle embeddng a request. δ 0 s a small postve constant to avod dvdng by zero n computng the objectve functon. Constrant set 6) and 7) contans the node and edge capacty bounds. Summng up f uv and f vu n 6) ensures that the summaton of flows on both drectons of the undrected edge u, v) remans wthn ts avalable bandwdth. Constrant set 8), 9), and ) refer to the flow conservaton condtons, whch denote that the net flow to a node s zero, except for the source node s and the snk node t. Constrant sets 11) and 12) are related to the augmented porton of the substrate graph. Constrant set 11) makes sure that only one substrate node s selected for each meta-node, whereas constrant set 12) ensures that no more than one meta-node s placed on a substrate node. Constrant sets 13) and 14) together wth 2) ensure that x uv s set whenever there s any flow n ether drecton of the substrate edge u, v). Fnally, constrant sets 15) and 16) denote the real and bnary doman constrants on the varables f uv and x uv, respectvely. V. LP RELAXATION AND ROUNDING-BASED ALGORITHMS Snce solvng an MIP s known to be computatonally ntractable [], smultaneous node and lnk embeddng usng VNE MIP s practcally nfeasble. Hence we relax the nteger constrants 16) of the MIP, and obtan the followng lnear program VNE LP RELAX). Once we have the LP soluton, we use determnstc and randomzed roundng technques to obtan nteger values for the varable x and embed VN requests. VNE LP RELAX Objectve: α uv mnmze fuv R E u, v)+δ uv E S + β w x mw cm) 17) R N w)+δ w N S m N S \N S Constrants: - Capacty Constrants: fuv + fvu) R E u, v) x u,v, u, v N S R N w) x mw c m), m N S \ N S, w N S - Flow Related Constrants: fuw fwu =0,, u N S \{s,t } w N S w N S fs w fws = be V ), w N S w N S ft w fwt = be V ), w N S w N S - Meta and Relaxed Bnary Constrants: x mw =1, m N S \ N S w Ωm) m N S \N S x mw 1, w N S x uv R E u, v), u, v N S - Doman Constrants: Remarks: x uv = x vu, u, v N S f uv 0, u, v N S x uv 0, u, v N S 18) The doman constrant set 18) on the x uv varables has been relaxed. The rest of the constrants are smlar to the ones n VNE MIP. A. Determnstc Roundng Based Vrtual Network Embeddng Algorthm D-VNE) D-VNE Fg. 3) takes onlne VN requests as nputs and maps them onto the substrate network one at a tme. It takes decsons based only on the past VN requests that t has already seen,.e., D-VNE uses no look-ahead. Snce the nteger doman constrants 16) on the x varables have already been relaxed, we no longer get nteger values for the x varables. Instead, we employ determnstc roundng technque to get nteger values for x. We ntroduce ϕ : N S {0, 1}, whch s ntally set to zero for all n S N S sgnfyng that all the substrate nodes are ntally unused. Whenever a vrtual node 787

6 1: procedure D-VINEG V =N V,E V )) 2: Create augmented substrate graph G S = N S,E S ) 3: Solve VNE LP RELAX 4: for all n S N S do 5: ϕn S ) 0 6: end for 7: for all n N V do 8: f Ωn) {n S N S ϕn S )=1} = then 9: VN request cannot be satsfed : return 11: end f 12: for all z Ωn) do 13: p z f µn)z + f zµn) )x µn)z 14: end for 15: Let z max = arg max z Ωn) {p z ϕz) =0} 16: set M N n) z max 17: ϕz max ) 1 18: end for 19: Solve MCF to map vrtual edges. 20: Update resdual capactes of the network resources. 21: end procedure Fg. 3. D-VNE: Determnstc roundng based Vrtual Network Embeddng algorthm s mapped to a partcular physcal node n S,wesetϕn S ) to 1 to ensure that no substrate node s used twce for the same VN request. 1) Descrpton and Dscusson: The procedure begns by creatng an augmented substrate graph, G S = N S,E S ) for the VN request G V = N V,E ) V usng the augmentaton method descrbed n Secton IV-A. Next t solves VNE LP RELAX to get a fractonal soluton whch s at least as good as the nteger soluton of VNE MIP. For each vrtual node, D-VNE frst checks whether there are any unmapped substrate nodes wthn ts feasble regon the substrate nodes n the vrtual nodes Ω set). If the correspondng Ω set s empty, D-VNE stops the embeddng process mmedately and rejects the VN request. Otherwse the determnstc roundng procedure s ntated n lne 12. For each vrtual node n, D-VNE calculates a value p z for each substrate node z Ωn) n ts cluster. p z s calculated as the product of the value x µn)z and the total flow passng through the meta-edge μn)z n both drectons. The reason behnd usng ths multplcaton nstead of just x µn)z s as follows. In the MIP soluton x uv s set to bnary values based on the presence of flows n ether drecton n the edge u, v). When the bnary constrant x s relaxed, one mght expect that the fractonal values of x uv would also be proportonal to the total flow n the edge u, v). But durng the LP relaxaton process, the correlaton between the flow varable f and the bnary varable x s lost. It s because a lnear program tres to optmze the objectve functon wthout volatng the constrants; t does not care about the values as long as they are wthn ther permtted domans. As a result, n 1: procedure R-VINEG V =N V,E V )) 2: Create augmented substrate graph G S = N S,E S ) 3: Solve VNE LP RELAX 4: for all n S N S do 5: ϕn S ) 0 6: end for 7: for all n N V do 8: f Ωn) {n S N S ϕn S )=1} = then 9: VN request cannot be satsfed : return 11: end f 12: for all z Ωn) do 13: p z f µn)z + f zµn) )x µn)z 14: end for 15: p sum z Ωn) p z 16: for all z Ωn) do 17: p z p z /p sum 18: end for 19: set M N n) z wth probablty p z 20: ϕz) 1 wth probablty p z 21: end for 22: solve MCF to map vrtual edges. 23: Update resdual capactes of the network resources. 24: end procedure Fg. 4. R-VNE: Randomzed roundng based Vrtual Network Embeddng algorthm the relaxed lnear program, t s possble that the f values are very hgh and the correspondng x values are very low or vce versa. Multplyng the f and x values thwarts the possblty of selectng a substrate node based on hgh x value but very low f value on ts correspondng meta-edge and vce versa. The ones that have better values for both the varables f and x are more lkely to be n the soluton of the MIP than others. D-VNE maps the vrtual node n onto the unmapped substrate node z.e., ϕz) =0) wth the hghest p z value, breakng tes arbtrarly. Once all the vrtual nodes have been mapped to dfferent substrate nodes, D-VNE apples the mult-commodty flow algorthm to map the vrtual edges n E V onto the substrate paths. One can also use shortest path algorthms when path splttng s not supported by the substrate network. Fnally, D-VNE updates the resdual capactes of the substrate nodes and lnks to prepare for the next request. 2) Tme Complexty: An mportant aspect of D-VNE s that the mult-commodty flow algorthm s executed twce; frst, durng the node mappng phase snce VNE LP RELAX s a lnear programmng relaxaton of the orgnal mxed nteger mult-commodty flow problem), and second, durng the edge mappng phase. Snce the mult-commodty flow algorthm can be solved n polynomal-tme usng ether the ellpsod algorthm or Karmarkar s nteror pont algorthm for lnear programmng []; hence, D-VNE s a polynomal tme algorthm. 788

7 B. Randomzed Roundng Based Vrtual Network Embeddng Algorthm R-VNE) R-VNE Fg. 4) s qute smlar to D-VNE except that t uses randomzed roundng nstead of determnstc roundng. Once the p z values are calculated as n D-VNE, R-VNE normalzes those values wthn 0 to 1 range. The normalzed values for each z Ωn) correspond to the probabltes of n beng mapped to z by the optmal MIP. R-VNE selects a substrate node z Ωn) to map a vrtual node n wth probablty p z. The remander of ths algorthm s smlar to ts determnstc counterpart, and t s clear that ths algorthm also runs n polynomal-tme. VI. PERFORMANCE EVALUATION In ths secton, we frst descrbe the evaluaton envronment, and then present our man evaluaton results. Our evaluaton focuses prmarly on quantfyng the advantage of coordnatng node mappng and lnk mappng phases n terms of acceptance rato, revenue and cost. We also compare D-VNE and R-VNE wth exstng algorthms modfed to ft nto our model. A. Smulaton Settngs We have mplemented a dscrete event smulator to evaluate the performance of our algorthms whch s freely avalable at [17]. The substrate network topologes n our experments are randomly generated wth 50 nodes usng the GT-ITM tool [18] n 25 25) grds. Each par of substrate nodes s randomly connected wth probablty 0.5. The cpu and bandwdth resources of the substrate nodes and lnks are real numbers unformly dstrbuted between 50 and 0. We assume that VN requests arrve n a Posson process wth an average rate of 4 VNs per 0 tme unts, and each one has an exponentally dstrbuted lfetme wth an average of μ = 00 tme unts. In each VN request, the number of vrtual nodes s randomly determned by a unform dstrbuton between 2 and followng smlar setups to prevous works [8], [9]. The average VN connectvty s fxed at 50%. The cpu requrements of the vrtual nodes are real numbers unformly dstrbuted between 0 to 20 and the bandwdth requrements of the vrtual lnks are unformly dstrbuted between 0 to 50. Wehaveused the open source mxed nteger programmng lbrary glpk [19] to solve VNE LP RELAX. B. Comparson Method In our evaluaton, we have compared sx algorthms that combne dfferent node mappng and lnk mappng strateges ncludng our contrbutons and algorthms from prevous research [8], [9] modfed to ft nto our model.e., no reconfguraton). The notatons that we use to refer to dfferent algorthms are enumerated n Table I. C. Evaluaton Results We use several performance metrcs for evaluaton purposes n our experments. We measure the average acceptance rato, revenue R), and provsonng cost C) for VN requests over tme. We also measure the average node utlzaton and average VN Request Acceptance Rato Average Revenue Notaton D-VNE R-VNE [8] [9] TABLE I COMPARED ALGORITHMS Algorthm Descrpton Determnstc Node Mappng wth Splttable Lnk Mappng usng MCF Randomzed Node Mappng wth Splttable Lnk Mappng usng MCF Greedy Node Mappng wth Shortest Path Based Lnk Mappng Greedy Node Mappng wth Splttable Lnk Mappng usng MCF Determnstc Node Mappng wth Shortest Path Based Lnk Mappng Determnstc Node Mappng wth Splttable Lnk Mappng usng MCF, where α uv = β w = 1, u, v, w N S D-VNE R-VNE 0.55 Tme Fg. 5. VN request acceptance rato over tme D-VNE R-VNE 1.5 Tme Fg. 6. Tme average of generated revenue 789

8 Average Cost Node Utlzaton Lnk Utlzaton Fg. 7. Tme D-VNE R-VNE Average cost of acceptng VN requests over tme D-VNE 0.05 R-VNE Fg. 8. Tme Average node utlzaton D-VNE R-VNE 0 Fg. 9. Tme Average lnk utlzaton lnk utlzaton of the substrate network. In all these cases we plot the performance metrcs aganst tme to show how each of these algorthms Table I) actually perform n the long run. We summarze our key observatons n the followng. 1) Coordnated node and lnk mappng leads to hgher acceptance rato and larger revenue. Fg. 5 and Fg. 6 depct that the proposed algorthms, D-VNE and R-VNE, lead to better acceptance rato as well as hgher revenue than the exstng algorthms and ) through coordnated node and lnk mappng. Havng hgher revenue along wth better acceptance rato mples that our proposed algorthms actually embed VN requests that generate more revenue, nstead of embeddng smaller VN requests just to ncrease the acceptance rato. 2) Load balancng further ncreases the acceptance rato and the revenue. From Fg. 5 and Fg. 6, t s evdent that generates more revenue and accepts more VN requests than the basc D-VNE algorthm. In, the value of the objectve functon 17) of VNE LP RELAX depends on the resdual capacty of the network resources n addton to the provsonng cost α and β values are set to 1 here). The lower the resdual capacty of a partcular node or lnk, the hgher the value of the objectve functon. As a result, tres to avod hghly utlzed nodes and lnks as long as t can, leavng those crtcal resources avalable for the VN requests that absolutely need them. 3) Randomzaton can lead to better performance. It s well establshed n the algorthm desgn lterature that randomzaton allows effcent solutons to many ntractable problems n polynomal tme wth low probablty of error. Our experments show that the randomzed verson of our VN embeddng algorthm R-VNE) performs better than ts determnstc counterpart D-VNE) n terms of acceptance rato and revenue generaton Fg. 5 and Fg. 6). In addton to that, for networks wth large numbers of nodes randomzaton has been shown to be effectve for load balancng [20]. Ths phenomena s also vsble n our experments, snce R-VNE performs smlar to n most scenaros. 4) Load balancng slghtly ncreases the average provsonng cost. Whle load balancng ncreases revenue and acceptance rato by avodng hghly utlzed resources, t runs the rsk of ncreasng the average provsonng cost as shown n Fg 7. Snce tres to avod hghly utlzed resources, sometmes t ends up suggestng a longer path to map a partcular vrtual edge whch eventually sums up to slghtly hgher average provsonng cost n the long run. 5) Coordnaton ncreases resource utlzaton. Fg. 8 and Fg. 9 depct the average utlzaton of substrate nodes and substrate lnks for dfferent VN embeddng algorthms. Snce has the hghest acceptance rato, t also has the hghest node and lnk utlzaton. However, acheves a relatvely hgher gan n lnk utlzaton over ts counterparts than n node utlzaton. 790

9 We beleve that the reason behnd ths s the dstrbutve nature of algorthm. In order to avod lnks wth lower resdual capactes,.e., n order to mnmze 5), uses longer paths contanng more substrate lnks wth hgher resdual capactes to embed vrtual lnks. VII. CONCLUSION To make network vrtualzaton an ntegral part of the future Internet archtecture, effcent and practcal algorthms for VN embeddng are specally requred. In ths paper, we proposed algorthms for VN embeddng that dffer from the prevous algorthms by ntroducng coordnaton between node and lnk mappng phases. We argued that ths coordnaton greatly ncreases the soluton space and the qualty of the heurstc algorthms. To ths end we frst formulated the embeddng problem as a mxed nteger program. We then relaxed the nteger constrants and used determnstc and randomzed roundng technques to obtan polynomal-tme solvable algorthms for node mappng. The node mappng phase combned wth the mult-commodty flow based lnk mappng phase n our algorthms outperformed the exstng approaches n terms of acceptance rato, revenue, and provsonng cost, as shown through smulaton. However, there are a number of ssues that reman unresolved n ths work and can be good startng ponts for further research n ths drecton. Frst and foremost s the analyss of theoretcal approxmaton factors of the proposed algorthms n the worst case. To ths end, we are currently workng on developng a prmal-dual based analyss framework to obtan good lower bounds on the performance of D-VNE and R- VNE. Fndng out advanced economc models, nstead of the smple revenue model used n the exstng lterature, for VN prcng s another mportant research topc that needs further attenton. Fnally, avalable approaches to drectly solve nteger and mxed nteger programs e.g., column generaton) can be employed to develop effcent algorthms to obtan optmal or near-optmal solutons for the orgnal mxed nteger formulaton VNE MIP) wthout any relaxaton. ACKNOWLEDGMENT Ths research was partally supported by the Natural Scence and Engneerng Councl of Canada NSERC) and partally by WCU World Class Unversty) program through the Korea Scence and Engneerng Foundaton funded by the Mnstry of Educaton, Scence and Technology Project No. R ). REFERENCES [1] T. Anderson, L. Peterson, S. Shenker, and J. Turner, Overcomng the Internet mpasse through vrtualzaton, Computer, vol. 38, no. 4, pp , [2] J. Turner and D. Taylor, Dversfyng the Internet, n IEEE GLOBE- COM, vol. 2, [3] A. Baver, N. Feamster, M. Huang, L. Peterson, and J. Rexford, In VINI vertas: Realstc and controlled network expermentaton, n Proceedngs of SIGCOMM, 2006, pp [4] N. Feamster, L. Gao, and J. Rexford, How to lease the Internet n your spare tme, ACM SIGCOMM Computer Communcaton Revew, vol. 37, no. 1, pp , [5] D. Andersen, Theoretcal approaches to node assgnment, Unpublshed Manuscrpt, dga/papers/andersen-assgn. ps, [6] J. Fan and M. Ammar, Dynamc topology confguraton n servce overlay networks - a study of reconfguraton polces, n Proceedngs of IEEE INFOCOM, [7] J. Lu and J. Turner, Effcent mappng of vrtual networks onto a shared substrate, Washngton Unversty, Tech. Rep. WUCSE , [8] Y. Zhu and M. Ammar, Algorthms for assgnng substrate network resources to vrtual network components, n Proceedngs of IEEE INFOCOM, [9] M. Yu, Y. Y, J. Rexford, and M. Chang, Rethnkng vrtual network embeddng: Substrate support for path splttng and mgraton, ACM SIGCOMM Computer Communcaton Revew, vol. 38, no. 2, pp , Aprl [] A. Schrjver, Theory of lnear and nteger programmng. New York, NY, USA: John Wley & Sons, Inc., [11] W. Szeto, Y. Iraq, and R. Boutaba, A mult-commodty flow based approach to vrtual network resource allocaton, n Proceedngs of the IEEE Global Telecommuncatons Conference GLOBECOM 03), 2003, pp [12] A. Gupta, J. M. Klenberg, A. Kumar, R. Rastog, and B. Yener, Provsonng a vrtual prvate network: A network desgn problem for multcommodty flow, n Proceedngs of ACM STOC, 2001, pp [13] R. Rcc, C. Alfeld, and J. Lepreau, A solver for the network testbed mappng problem, ACM SIGCOMM Computer Communcaton Revew, vol. 33, no. 2, pp , Aprl [14] I. Houd, W. Louat, and D. Zeghlache, A dstrbuted vrtual network mappng algorthm, n Proceedngs of IEEE ICC, 2008, pp [15] A. Kumar, R. Rastog, A. Slberschatz, and B. Yener, Algorthms for provsonng vrtual prvate networks n the hose model, IEEE/ACM Transactons on Networkng, vol., no. 4, pp , [16] P. Raghavan and C. D. Tompson, Randomzed roundng: a technque for provably good algorthms and algorthmc proofs, Combnatorca, vol. 7, no. 4, pp , [17] VNE-Yard, [18] E. Zegura, K. Calvert, and S. Bhattacharjee, How to model an Internetwork, n Proceedngs of IEEE INFOCOM, 1996, pp [19] GNU Lnear Programmng Kt, [20] M. Mtzenmacher, A. W. Rcha, and R. Staraman, The power of two random choces: A survey of technques and results, n Handbook of Randomzed Computng. Kluwer Academc Press, 2001, pp