VNE-AC: Virtual Network Embedding Algorithm based on Ant Colony Metaheuristic

Transcription

1 : Vrtual Network Embeddng Algorthm based on Ant Colony Metaheurstc Ilhem Faar, Nadb Atsaad, Guy Puolle and Hubert Zmmermann Gnkgo Networks: 2 A Rue Danton, Montrouge, France HIPERCOM INRIA: Domane de Voluceau - Rocquencourt - B.P. 105, Le Chesnay Cede - France UPMC - Unversty of Pars 6: 4 Place Jusseu, Pars, France lhem.fear@gnkgo-networks.com, nadb.atsaad@nra.fr, guy.puolle@lp6.fr, hubert.zmmermann@gnkgo-networks.com Abstract In ths paper, we address vrtual network embeddng problem. Indeed, our obectve s to map vrtual networks n the substrate network wth mnmum physcal resources whle satsfyng ts requred QoS n terms of bandwdth, power processng and memory. In dong so, we mnmze the reect rate of requests and mamze returns for the substrate network provder. Snce the problem s NP-hard and to deal wth ts computatonal hardness, we propound a new scalable embeddng strategy named based on the Ant Colony metaheurstc. The ntensve smulatons and evaluaton results show that our proposal enhances the substrate provder s revenue and outperforms the related strateges found n current lterature. Keywords: Network vrtualzaton, Embeddng problem, Ant colony metaheurstc. I. INTRODUCTION An ambtous vson of future Internet networks would nclude Network Vrtualzaton. Ths new paradgm conssts of leasng and sharng the Substrate Network (SN nfrastructure between several Vrtual Network (VN provders n the am of enhancng the proftablty of the physcal resources. However, few studes have dealt wth practcal challenges such as vrtual network embeddng, resource schedulng, moblty, nteroperablty, prvacy, etc. [1], [2]. In ths paper, we nvestgate the ssue of VN embeddng n whch network performance and the SN provder s economc returns depend strongly on ts success. Each vrtual node must be assgned to only one substrate node. As well, each vrtual lnk must be mapped nto a substrate path. Specfcally, our obectve s to ncrease the VN provder s revenue by reducng the reect rate ofvn requests, whle also satsfyng the requred physcal resources n terms of bandwdth, power processng, and memory. The mappng of vrtual nodes s proved to be NP-hard n [3]. Addtonally, the vrtual lnk assgnment can be formulated as an unsplttable flow or a mult-way separator problem and s thus also NP-hard [4], [5]. Snce the VN embeddng problem s computatonally ntractable, the optmal soluton could only be generated n smallszed nstances of VN and SN. In ths respect, we propose a new scalable VN mappng strategy based on a Ma-Mn Ant System metaheurstc (Ant Colony algorthm [6], denoted by. The man dea behnd our proposal s to take nspraton from the behavour of collectve ants n fndng the best path between ther nest and a food source. To do so, frst dvdes the VN request nto a set of sub-problems, denoted by soluton components whch are sorted and then sequentally assgned to form the sequence of mappng transtons. Afterward, launches the artfcal ant colony, where ants perform a parallel search. To ths end, each ant teratvely bulds a pece of the soluton (.e. transton where each soluton component s embedded accordng to the avalable resources and the artfcal pheromone tral n the SN. It s worth notng that each substrate node s assocated wth a pheromone tral value for each transton. Once all the canddate solutons (.e. full VN are mapped, updates the pheromone tral n the SN. The pheromone tral s evaporated n all substrate nodes. Nevertheless, t wll also be renforced n the contrbutng substrate nodes to buld the best mappng topology. Indeed, the soluton component s lkely to be embedded n the substrate node wth the hghest pheromone tral value. Fnally, the ants repeat the same process, durng a predefned number of teratons, to mprove the soluton accordng to the new tral values. To gauge the effectveness of our proposal, we compared the algorthm to some promnent estng proposals found n related lterature: [7], [7], VNE-Subdvdng [7], and [8]. The smulaton results obtaned show that outperforms the related algorthms n terms of VN reect rate, mappng cost, request revenue, and resource utlzaton. The remnder of ths paper s organzed as follows. The net Secton wll summarze related work addressng the VN embeddng problem. In Secton III, we wll formulate both the network model and the VN mappng problem mathematcally. Then, we wll descrbe our mappng algorthm,, and the smulaton results obtaned n Secton IV and Secton V respectvely. Fnally, Secton VI concludes ths paper. II. RELATED WORK Few research papers have studed the VN embeddng problem. In ths secton, we wll outlne the man proposals found n estng lterature. In [7], the authors propose three VN assgnment algorthms denoted by,, and VNE-Subdvdng. The frst method,, treats vrtual nodes and lnks mappng separately. Thus, substrate and vrtual nodes are sorted accordng to ther stress and connectvty degree respectvely. Then, assgns the vrtual node wth hghest connectvty degree to the least stressed substrate node recursvely untl all the vrtual nodes are embedded. Thereafter, makes use of the shortest dstance algorthm to connect the mapped vrtual nodes. On the other hand, the and VNE-Subdvdng algorthms take nto account the substrate lnk load when selectng the substrate nodes. assgns the vrtual nodes n a smlar manner to, ecept that nodes stress and path dstance are replaced by the newly defned metrcs. The fnal proposal, VNE-Subdvdng, subdvdes the VN request nto star topologes so as not to deal wth the request as one whole unt. Net, the stars are mapped sequentally usng. However, the authors assume that

2 the resources are unlmted n the SN whch s not a realstc assumpton. Thus, the VN request reect rate s not evaluated n the paper. In [8], the authors proposed the vrtual network embeddng algorthm. In ths system, the substrate and vrtual nodes are sorted accordng to the avalable and requested resources respectvely. Then, the vrtual node wth the hghest resource request s assgned to the substrate node contanng the largest avalable resource metrc value recursvely untl all the vrtual nodes are mapped. Net, the nodes are connected usng the K-shortest paths algorthm. s man drawback s the substrate path buldng algorthm. Indeed, the shortest path algorthm does not consder the congested SN lnks, whch mples an ncrease of hot-spots n the SN and an ncrease n request reect rate. Moreover, the vrtual nodes and lnks mappng are not coordnated. The authors of [9] model thevn as a drected graph wth two types of nodes: access and core. The requredvn resources are defned n terms of the epected traffc, whch s epressed as an upper lmt on allowed traffc between all access node pars. The authors obectve s to calculate the mnmum request bandwdth for the VN accordng to the access nodes upper lmt of traffc. Nonetheless, the weakness n the proposal conssts n s the lack of consderaton of SN capactes (.e. unlmted and the use of statc routng tables n the network. In addton, the proposal requres the star VN topology, whch s strongly bndng. In [10], the authors propound two VN embeddng algorthms, named Determnstc-VNE and Randomzed-VNE. Here, the substrate graph s augmented wth meta-nodes and meta-edges to form a meta-graph. Each meta-node corresponds to one vrtual node and meta-edge s a lnks between a metanode and the substrate nodes located n ts requred geographc area. Note that each vrtual node s assocated wth a specfed regon where t could be hosted. Nevertheless, t s not realstc to epect end-users to specfy all vrtual nodes locatons. In fact, only the locatons of access nodes can be fed. The man drawback here s the nodes locatons constrants. Indeed, when these are not defned, D-VNE and R-VNE cannot be eecuted snce the meta-graph cannot be bult. In ths paper, we propose a new VN embeddng algorthm based on the ant colony metaheurstc, named. It s worth notng that we do not restran the VN embeddng problem by assumng unlmted substrate resources, or specfc VN topologes or restrctng geographc locatons of the substrate core node. To the best of our knowledge, ths paper s the frst proposal that takes nto consderaton the above assumptons. III. PROBLEM STATEMENT A. Network Model We model the SN as an undrected graph denoted by G s (N s,e s, where N s and E s are the sets of physcal nodes and ther connected lnks respectvely. Each physcal node, n s Ns, s characterzed by ts resdual processng power (C n s, resdual memory (M n s, type: access or core (X n s, and v geographc locaton (L n s. Note that f X n s = 1, then n s s an access node. Otherwse, X n s = 0. Lkewse, each physcal lnk, e s Es, s typfed by ts avalable resdual bandwdth, denoted by B e s. Smlarly, the VN request s modeled as an undrected graph, denoted by G v (N v,e v, where N v and E v are the sets of vrtual nodes and ther vrtual lnks respectvely. Wthn the VN request, each vrtual node, n v Nv, s assocated wth the requred processng power (C n v and memory (M n v, ts type (X n v, and geographc locaton (L n v whether s an access node. Moreover, each vrtual lnk, e v Ev, requests B e v n terms of bandwdth. B. Vrtual Network Embeddng Problem In ths secton, we wll model the embeddng problem of the VN (G v n the SN (G s. Frst, we assume that all the physcal resources (.e. bandwdth, processng power, and memory ng s are lmted. In fact, G s s not able to host an nfnty number of VN requests. Consequently, an ntellgent and udcous mappng of the VN n G s s necessary n order to mamze the acceptance rate and the substrate provder s proft. Let N denote the bnary node-mappng matr wth m rows and n columns. Each row s assocated wth one vrtual node andms equal to N v. As well, each column s assocated wth one substrate node and n s equal to N s. Then, N (, = 1 when the vrtual node n v s mapped n the substrate node ns. Otherwse, N (, = 0. Node mappng s constraned so that for each VN request, G v, two vrtual nodes cannot be assgned to the same substrate node. Formally, m {1,2,...,n}, N (, 1 (1 In addton, each vrtual node must be assgned to only one physcal node. Formally, n {1,2,...,m}, N (, = 1 (2 The vrtual node, n v, can be mapped n the substrate node, n s, f the avalable resdual resources (.e. C n s and M n s are at least equal to those requred (.e. C n v, M n v. Formally, N (, (C N C s N 0 v N (, (M N M s N 0 (3 v {1,2,...,n}, {1,2,...,m} Besdes, f n v s an access node then t can only be embedded n substrate access nodes located n the geographc area delmted by a crcle wth a center L n v and a radus D. Formally, N (,X n v L n v L n s D (X n v X n s 1 ( N(,=1 = 0 {1,2,...,n}, {1,2,...,m} where 1 cond s equal to 1 f cond s true, otherwse s 0. Each vrtual lnk, e v between nv and n v s assgned to a unsplttable substrate path denoted by P e v between n s and ns. Note that P e v s a set of substrate lnks. In fact SN s manly make use of shortest-path-based routng protocols such as Open Shortest Path Frst (OSPF. Indeed, to employ sophstcated splttable routng algorthms, already deployed IP routers must be upgraded. Ths would sgnfcantly ncrease captal ependtures. Let E denote the bnary edge assgnment matr wth p rows and q columns. Each row s related to one vrtual lnk and p s equal to E v. Furthermore, each column s related to one substrate lnk and q s equal to E s. E (,y s equal to 1 f the substrate lnk e s y P e v. Moreover, n the matr E, all the allocated substrate lnks e s y n each row ev must form a path between n s and ns. Otherwse, the confguraton s not allowed. Moreover, the avalable resdual bandwdth of all the physcal lnks n P e v must be at least equal to B e v. Formally, (4

3 E (,y (B e B sy e 0 v {1,2,...,p}, y {1,2,...,q} Our obectve s to generate, for each VN request, the best possble mappng (.e. N and E whle also mnmzng the embeddng cost n terms of allocated resources n the SN. Note that for a specfc VN request, whatever the mappng the resources allocated by the vrtual nodes are dentcal. However, the assgned resources for the vrtual lnks depend on the substrate path length. For ths reason, we defne the cost of the mappng request by the total bandwdth used n all the substrate paths. In ths respect, our obectve s to mnmze the mappng request cost as epressed below. p q E (,yb e v (6 =1y=1 We wll now outlne our VN embeddng optmzaton problem. Problem 1 VN embeddng optmzaton problem p q mnmze =1 y=1 E(,yB e v subect to: m N (, 1 n N(, = 1 N (, (C N C s N 0 v N (, (M N M s N 0 v N (,X n v L n v L n s D (X n v X n s 1 ( N(,=1 = 0 {e s y } bulds a path for ev E (,y (B e B sy e 0 v (5 N (, {0,1} E (,y {0,1} {1,2,...,n}, {1,2,...,m} {1,2,...,p}, y {1,2,...,q} Our problem s a bnary combnatoral optmzaton problem. As stated earler, t has been proved to be NP-hard [3] [5]. In the net secton, we wll propose a new scalable VN embeddng algorthm based on the ant colony metaheurstc, denoted by. IV. PROPOSAL: ALGORITHM As stated above, the VN embeddng problem s resolved manly by usng greedy algorthms or standard solvers (e.g., Cple, Glpk, etc. Unfortunately, a greedy approach usually converges to a local optmum and solvers cannot tackle largesze problems. In ths respect, we propose to make use of metaheurstcs. These can be appled to several optmzaton problems, but there are no detals eplanng how they can be appled. Indeed, the man challenge s to adapt the metaheurstc to address the problem n hand. In estng lterature, several metaheurtscs can be found such as Ant Colony, Tabu Search, Genetc Algorthm, etc. In ths paper, we propose a new VN embeddng algorthm, denoted by. It s based on a Ma-Mn Ant System [6] metaheurstc whch operates as follows. Frst, the problem s dvded nto a set of soluton components that are sorted and then solved sequentally. Note that solvng a soluton component s only equvalent to buldng a small part of the overall soluton. Net, A ma parallel artfcal ants are launched and teratvely eplore the search space untl a predetermned number of teratons N ma s reached. Durng each teraton, each ant ncrementally constructs the soluton by transtng (.e. movng from one soluton component to another. To do so, the ant localzes potental canddates n the SN for each soluton component n the search space then selects ust one by applyng a stochastc local decson. It s worth notng that the decson s based on heurstc nformaton and artfcal pheromone trals, whch respectvely quantfy the desrablty of a pror and posteror transton. Once each ant bulds ts full soluton, the best one (.e. the one that most enhances the obectve functon among all solutons generated by all ants s selected. Furthermore, the artfcal pheromone s slghtly evaporated n all the envronment. Ths helps the ants to dscover new traectores n the envronment and not to be trapped n local optmums. Nevertheless, the artfcal pheromone tral of each soluton component s renforced at the vsted ponts n the envronment accordng to the best traectory traveled by ants to buld the whole soluton. Ths helps the ants to mprove and contnually refne the best soluton obtaned. The process s repeated durng N ma teratons and the global best soluton generated by the A ma ants s consdered to be the output soluton. The fundamental stages of our proposal are: Formaton of soluton components, Localzaton of potental canddates, Stochastc selecton of the canddate, v Selecton of the best soluton and v Updatng the pheromone tral. These wll now be descrbed n greater detal. A. Formaton of soluton components In ths stage, the VN request, G v, s splt up nto many subrequests, known as soluton components, denoted bysc a. To do ths, we frst map all access nodes n G v snce ther geographc locatons are fed. Indeed, the vrtual access node n v (.e. X n v = 1 s embedded n the substrate access node n s (.e. X n s = 1 offerng the most resdual resources (gvng prorty to processng power wthn the crcular geographc area defned by the center L n v and the radus D. Recall that the resdual resources of n s (.e. C n s and M n s must be at least equal to those requested (.e. C n v and M n v. Then, estng vrtual lnks between the vrtual access nodes wll be mapped as eplaned n the net secton, IV-C. The remanng vrtual nodes and lnks are denoted by G v. Note that n G v, we found many vrtual lnks mssng one of ther outermost vrtual nodes, whch had already been mapped. Henceforth, we wll refer to ths type of lnk as hangng lnk. Thus, after mappng vrtual access nodes and the estng vrtual lnks between them, we can now sequentally generate the sequence of soluton components {SC a },a = 1,2,,ξ. To do so, we select the vrtual node wth the hghest number of hangng lnks n G v. Then,SC a s consttuted usng the selected vrtual node and all ts attached hangng lnks only. Afterward, we subtract SC a from G v (.e. Gv G v \SC a and we then repeat the same process recursvely, to form the rest of soluton components, untl G v becomes empty (.e. Gv =. As a result, the {SC a } seres, a = 1,2,,ξ, s generated. Thus the artfcal ants can start mappng {SC a } sequentally n the same order durng each teraton. In ths respect, to embed the soluton component SC a, each ant frst of all localzes

4 potental canddate substrate nodes that could host SC a, as detaled hereafter. B. Localzaton of potental canddates The potental substrate nodes {PN b a } for hostng SC a are those suppled wth suffcent resdual resources (.e. processng power and memory and localzed wthn H hops from the nearest substrate core node to a pont P. We defne P as the barycentre of the set of substrate nodes, denoted by N a, hostng the vrtual nodes of hangng lnks n SC a. By dong so, we am to embed the vrtual node of SC a n the substrate node located at the same dstance to ts drectly connected nodes, N a, that are already mapped. Consequently, the path lengths wll be balanced and consumed resources wll be mnmzed. Once the potental set of substrate canddates s determned PN a = {PN b a}, one s selected accordng to a stochastc decson outlned n the followng secton. C. Stochastc selecton of the canddate To select the best potental substrate node PN b a, we defne the probablty of desrablty, denoted by p ab, to embed SC a n PN b a. Ths probablty, p ab, depends on the heurstc nformaton,η ab, and the artfcal pheromone tral,τ ab, ofsc a npn b a, as recommended n Ma-Mn Ant System [6]. Formally, τab α ηβ ab p ab = PNa c=1 [ τ α ac ηβ ac ] (7 where α and β respectvely determne the relatve mportance of the pheromone tral and the heurstc nformaton. Remember that PN a s the set of potental canddates for hostng SC a determned n the prevous stage. Wth ths n mnd, for each substrate node n s b we can defne the heurstc nformaton η ab as: η ab = C n s +M n s + B Pbk (8 b b n s k Na where P bk s the best path between n s b and ns k n terms of our new metrc (see equaton (9. Thanks to η ab, substrate nodes wth hgh avalable resources are assocated wth a hgh value of p ab and thus, are more lkely to be selected as PN b a. Furthermore, the artfcal pheromone tral τ ab s defned n each teraton as epressed n equatons (10 and (11. In dong so, we assgn SC a to one canddate n PN a accordng to the probablty values {p ab } calculated above. Net, we generate the substrate paths assocated wth the vrtual hangng lnks n SC a snce both etremty nodes are now assgned. Our man obectve s to reduce the allocated bandwdth n the SN. So, the Dkstra shortest path n terms of hops would be the best approach. Unfortunately, we notced many VN request reects due to the hot spots n the SN and the requred bandwdth s not taken nto consderaton. For ths reason, we propose to generate the shortest path accordng to our own new metrc, whch takes nto account the resdual bandwdth and the path hop-length. In fact, we am to choose the shortest paths offerng the best bandwdths n order to reduce resources allocated to substrate lnks. The dstance of a sngle path, P, s defned as: d(p = length(p (9 mn e s P B e s Fnally, once all the ants map {SC a },a = 1,2,,ξ, and generate the full VN mappng soluton SL 1,SL 2,,SL Ama, we can select the best one, SL best, as detaled n the net secton. It s worth notng that each ant converges to a soluton that s dfferent to the ones generated by the other ants due to the probablty-based nature of the embeddng decson. D. Selecton of the best soluton The selecton of one soluton among those generated by all ants s based on the obectve functon value defned by equaton (6. Thus, we select the solutonsl best that mnmzes VN request s cost whch s defned as the total allocated bandwdth n the SN. E. Updatng the pheromone tral Before reteratng the process, movng from teraton t to t + 1, the pheromone tral s evaporated for all {SC a },a = 1,2,,ξ, n the whole SN. To acheve ths, the pheromone tral s weghted by ρ wth 0 < ρ < 1. Formally, τ ab (t+1 = ρ τ ab (t (10 Conversely, we renforce the pheromone tral of {SC a }, a = 1,2,,ξ, n the substrate canddate nodes {PN best 1,PN best 2,...,PN best ξ } partcpatng n the constructon of the best soluton SL best. To do so, we ncrease the pheromone tral by the nverse of SL best s mappng cost n the SN weghted by φ as suggested n [6]. τ ab (t+1 = τ ab (t+1+ p =1 φ q y=1 E(,yB e v (11 where p and q are equal to E v and E s respectvely, as mentoned n equaton (5 and φ a constant parameter. It s worth notng that the lowest and the largest bounds of τ ab are fed n relaton to the Ma-Mn Ant System as descrbed n [6]. Moreover, at t = 0, τ ab (0 s set to some arbtrarly hgh value n order to ncrease the eploraton of solutons durng the frst teratons. V. PERFORMANCE EVALUATION In ths secton, we wll study the effcency of our proposal,. To acheve ths, frst we descrbe our dscrete event VN embeddng smulator. Then, we wll defne several metrcs n the am of showng the advantages of compared wth the promnent estng schemes mentoned above: [7], [7], VNE-Subdvdng [7], and v [8]. Fnally, buldng on the outputs of the above smulatons, we wll assess our proposal and comment the results obtaned. A. Smulaton envronment We mplement a dscrete event VN embeddng smulator. In ths respect, we use the GT-ITM tool to generate random SN and VN topologes. Note that we model the arrval of VN requests by a Posson Process wth rate λ A. We also model VN lfetme by eponental dstrbuton wth mean µ L. As stated n [8], we set the SN sze to 100. In ths case, the rato of access and core nodes s fed to 20% and 80%, respectvely. Furthermore, we set the VN sze accordng to a dscrete unform dstrbuton, usng the values gven n [2, 10]. Snce vrtual access nodes are defned by customers, we assume that each vrtual node could be access or core wth a probablty of 0.5. It s worth notng that n both cases (VN and SN, each par of nodes s randomly connected wth a probablty of 0.5. The arrval rate λ A and the average lfetme µ L of

5 VN s are fed to 4 requests per 100 tme unt and 1000 tme unts respectvely. We calbrate the capacty of substrate nodes and lnks (.e. C n s, M n s, and B e s accordng to a contnuous unform dstrbuton takng values n [50, 100]. As well, we set the requred vrtual resources (.e. C n v, M n v, and B e v accordng to a contnuous unform dstrbuton, usng the values gven n [10,20]. In our smulator, we mplement our proposal alongsde the related strateges:,, and. Based on etensve smulatons, we calbrate our heurstc s parameters. We set the number of ants A ma and the number of teratons N ma to 10 and to 20, respectvely. We f the hop-range wthn local search area H, the mportance of heurstc nformaton and pheromone tral α and β to 2, 1 and 2 respectvely. We set the number of VN requests to All smulaton results of pseudorandom strateges are calculated wth a confdence level equal to 99.70%. Note that tny confdence ntervals are not shown n the followng fgures. B. Performance metrcs In ths secton, we wll defne the performance metrcs used to assess our proposal. 1 Q(t: s the reect rate of VN requests durng the tme perod [0, t]. 2 C tot (t: s the cumulatve provsonal cost of all accepted requests n the SN durng the tme perod [0, t]. Formally, C tot (t = G v AR t C(G v (12 where C(G v s defned as n equaton (6 and AR t s the set of accepted requests durng [0, t]. 3 C(t: s the average provsonal cost of requests n the SN durng the tme perod [0, t]. Formally, G C(t = v AR t C(G v (13 AR t 4 R tot (t: evaluates the SN provder s proftablty by calculatng the total beneft of all accepted requests durng the tme perod [0, t] (.e. cumulatve total revenue. Formally t can be epressed as: R tot (t = G v AR t R(G v (14 where R(G v s the request revenue defned as n [7] [8] [10]: R(G v = C n v + M n v + B e v (15 n v Nv n v Nv B e v E v 5 R(t: measures the average revenue of VN requests. It s worth notng that when R(t s hgh, the embeddng strategy maps requests generatng a larger revenue. It can be epressed as: G R(t = v AR t R(G v (16 AR t 6 U(t: measures average usage rate of substrate lnks at tme t. It can be epressed as: U(t = 1 E s ( Be s Be ma (17 s e s Es In the followng secton, we wll present the results of our smulatons and summarze the key observatons. C. Evaluaton Results In Table I, we compare the reect rate and total revenue of the whole (.e VN requests. It shows that our proposal,, sgnfcantly outperforms the related strateges. We can see that denes only 4.56 ± 0.40% of requests. It s worth notng that,, and VNE-Subdvdng declne the most of requests (69%, 73.90%, and respectvely. Ths s manly because the latter strateges do not consder the resdual resources n the SN. In addton, our proposal mproves by roughly three tmes (4.56%± %. As well, we can see that mamzes the substrate provder s revenue. Note that request s revenue s proportonal to the amount of allocated resources. Ths can be eplaned due to the fact that the larger number of accepted requests leads to the ncrease of allocated resources. TABLE I PERFORMANCE OF VN REQUESTS Strategy reect rate (% provder s revenue ( ± ± VNE-Subdvdng Reect rate of requests (logarthmc scale VNE-Subdvdng Tme unt (logarthmc scale Fg. 1. Reect rate - Q Fg. 1 llustrates the reect rate of requests Q(t durng the smulaton. It shows that notably mnmzes the reect rate compared wth the related strateges, throughout the smulatons. The ratonale behnd ths can be eplaned by the fact that our scheme ams to mamze the resdual resources n the SN. Thus, avods more crtcal resources and consequently prevents congeston n the SN. As llustrated n Fg. 1, t s worth notng that the requests s reect process starts late wth (at 1420 tme unt compared to the related strateges. Moreover, the reect rate s peak s the lowest and then decreases durng the smulaton whch proves that the resources are allocated effcently. In fact, unlke other algorthms, our soluton s based on pseudo-randomzed node selecton. Ths approach, as proved n algorthm desgn randomzaton, leads to effectve performance n computatonally ntractable problems. Fg. 3 shows the cumulatve and average requests provsonal cost obtaned. As depcted n Fg. 3.(a, we can see that enhances the total provsonal cost (.e. lowest compared wth and. In ths case, our proposal allocates fewer resources (.e. bandwdth than and whle also acceptng more requests, as llustrated n Fg. 1. As a result, our proposal,

6 , effcently embeds the requests. As shown n Fg. 2, thanks to our dstance path metrc defned n equaton (9, we succeed n makng a trade-off between mnmzng the length of substrate paths and mamzng resdual bandwdth n the SN. We can see that allocates appromately the same bandwdth n the SN but our proposal assgns more requests. It s worth notng that and VNE-Subdvdng consume less resources than our proposal. Ths can be eplaned by the fact that s based on shortest path algorthm and VNE-Subdvdng favors the mappng of small VN requests n terms of requred resources (as llustrated n Fg. 4(b whch ratonally decreases the provson cost. However, both strateges have the hghest reect rate snce they do not take nto account the bottleneck effect n the SN. Average usage rate of lnks VNE-Subdvdng revenue potentals. Ths s clearly shown n the Fg. 4(b whch plots the average revenue of requests. We can see that the average revenue of s roughly constant, n spte of ncomng requests. Total revenue of requests Average revenue of requests VNE-Subdvdng Tme unt (a Cumulatve - R tot VNE-Subdvdng Total cost of requests Average cost of requests (logarthmc scale 0 1.2e+06 1e Fg. 2. VNE-Subdvdng Tme unt Lnk s average usage rate - U Tme unt (a Cumulatve - C tot Tme unt (b Average - C VNE-Subdvdng Fg. 3. Comparson of provson cost It s clear to see n Fg. 3.(b that produces the best average provsonal cost thanks to our path metrc and the large number of accepted requests. We can also see that s average provsonal cost s constant throughout the smulaton, even though more VN requests are embedded n the SN. Fg. 4(a shows that our proposal,, performs better than the other approaches tested, snce t offers hgher revenue to the SN provder. We can see that returns generated by rse lnearly together wth ncomng requests. Ths can be eplaned by the fact that accepts many more requests than related schemes. Addtonally, coordnated nodes and lnks mappng stages leads to effcent mappng and thus generates a hgher ncome. Moreover, VNM-AC not only mproves the total revenue but also embeds requests wth larger 160 Fg. 4. Tme unt (b Average - R Comparson of revenue VI. CONCLUSION Ths paper studed the vrtual network embeddng problem. Note that t s NP-hard and computatonally ntractable. In response, we proposed a new effcent and scalable strategy named. It s based on a Ma-Mn Ant System metaheurstc. Therefore, our am s to mnmze the allocated resources n the substrate network for each vrtual network request n order to mnmze the reect rate and mamze the provder s revenue. Based on etensve smulatons, we have shown that our proposal makes embeddng more effectve and acheves better resource utlzaton. Moreover, outperforms related strateges proposed n estng lterature:,, VNE-Subdvdng and. REFERENCES [1] N. M. M. K. Chowdhury and R. Boutaba, Network vrtualzaton: State of the art and research challenges, IEEE Communcatons Magazne, vol. 47, [2], A survey of network vrtualzaton, Computer Networks, vol. 54, [3] D. G. Andersen, Theoretcal approaches to node assgnment, 2002, [4] J. Klenberg, Appromaton algorthms for dsont paths problems, PhD thess - MIT, [5] S. G. Kollopoulos and C. Sten, Improved appromaton algorthms for unsplttable flow problems, In Proc. IEEE Symposum on Foundatons of Computer Scence, [6] T. Stützle and H. H. Hoos, MAX-MIN ant system, Future Gener. Comput. Syst., vol. 16, pp , [7] Y. Zhu and M. Ammar, Algorthms for assgnng substrate network resources to vrtual network components, IEEE INFOCOM, pp. 1 12, [8] M. Yu, Y. Y, J. Reford, and M. Chang, Rethnkng vrtual network embeddng: substrate support for path splttng and mgraton, SIGCOMM Comput. Commun. Rev., vol. 38, pp , [9] J. Lu and J. Turner, Effcent mappng of vrtual networks onto a shared substrate, Washngton Unversty n St. Lous, Tech. Rep., [Onlne]. Avalable: l1/research/tech report 2006.pdf [10] N. Chowdhury, M. Rahman, and R. Boutaba, Vrtual network embeddng wth coordnated node and lnk mappng, IEEE INFOCOM, pp , 2009.