Network Capacity Allocation in Service Overlay Networks

Transcription

1 Netwo Capacity Allocation in Sevice Ovelay Netwo Ngo Lam 1, Zbigniew Dziong 2, Lone G. Maon 1 1 Depatment of Electical & Compute Engineeing, McGill Univeity, 3480 Univeity Steet, Monteal, Quebec, Canada H3A 2A7 2 Depatment of Electical Engineeing, Ecole de technologie upeieue,1100 Note-Dame Steet Wet, Monteal, Quebec, Canada H3C 1K3 ngo.lam@mcgill.ca, zdziong@ele.etmtl.ca, lone.maon@mcgill.ca Abtact. We tudy the capacity allocation poblem in evice ovelay netwo (SON) with tate-dependent connection outing baed on evenue maximization. We fomulate the dimenioning poblem a one in pofit maximization and popoe a novel model with eveal new featue. In paticula the popoed methodology employ an efficient appoximation fo tate dependent outing that educe the cadinality of the poblem. Moeove, the new fomulation alo tae into account the concept of netwo hadow pice in the capacity allocation poce to impove the efficacy of the olution cheme. Keywod: Sevice Ovelay netwo, SON dimenioning, capacity allocation. Section 1: Intoduction: The ey component of Sevice Ovelay Netwo, (SON), ae the SON gateway and the inteconnecting logical lin that lie on the top of one o moe phyical lin. SON gateway can be teated a oute that elay evice pecific data and pefom contol function. SON logical lin povide connectivity to the SON netwo though exiting phyical lin. The SON gateway ae connected to adjacent gateway by the logical lin. To povide SON evice, the SON povide ha to puchae bandwidth and QoS guaantee fom the coeponding netwo infatuctue owne via Sevice Level Ageement (SLA). It i clea that the optimum amount of capacity to be puchaed fom the infatuctue owne, o a to maximize the net evenue, i an impotant iue to be faced by the SON povide. The liteatue fo the netwo dimenioning poblem i uually elated to cicuit witched netwo uch a the telephone netwo. We hall intoduce ome of the wo that ae elated to ou. In [21], Gavih and Neuman uggeted a method baed on Lagangian elaxation to allocate netwo capacity and aign taffic in pacet witching netwo, but thei model aumed that the taffic i outed though a ingle path. Medhi and Tippe did compaion of fou diffeent appoache in [20] to a combinatoial optimization poblem that decibe a multi-hou netwo dimenioning poblem fo ATM netwo, but thei tudy wa alo baed on the aumption that taffic i outed though a ingle path. Intead of maximizing the net income geneated fom the netwo, both of the pape choe to compute the

2 minimum capacity allocation cot fo the netwo. Duan et al [22] invetigated the capacity allocation poblem of the SON netwo in ode to maximize the net income gained by the SON netwo. Thei model wa alo confined to netwo with ingle fixed oute fo the taffic. Giad popoed in [6] an optimization famewo fo dimenioning cicuit-witched netwo employing a moe flexible load haing altenative outing cheme. Thi famewo wa applied in [5] fo the dimenioning of telephone netwo. The fomulation in [5] and [6] wee poblem pecific in that they dimenion cicuit-witched netwo coniting of only one-lin and two-lin path. Shi and Tune peented in [7] a heuitic appoach to ize SON multicat netwo. Thei main focu wa on the outing algoithm that optimize the delay and the bandwidth uage on the multicat evice node. The dimenioning ue a imple algoithm that equalize the eidual capacitie aco the multicat netwo. In ou tudy, we dimenion the SON netwo baed on evenue maximization. In thi apect we ae not only conideing the net income in the objective function, but we ae alo incopoating the notion of aveage netwo hadow pice in the dimenioning poce in ode to eflect the enitivity of net evenue to the dimenion of the lin. We conide the SON netwo a a geneic netwo and povide a famewo fo dimenioning baed on the taffic ewad. The dimenioning poblem i fomulated a a contained optimization poblem fo two ditinct outing model. Fom the KKT condition of the optimization fomulation, we devie an iteative method that lead to nea-optimal olution. Compaed with the peviou tudie epoted in the liteatue, ou model allow moe flexible outing cheme wheea each path can be compied of an abitay numbe of lin. We alo incopoate two ophiticated outing cheme to bette appoximate the tate dependent outing cheme aumed in the SON envionment. We peent analytical optimization model, and include detailed dicuion of the implementation iue, a well a numeical tudie that veify the model accuacy. A novelty of ou tudy i that we povide an economic integation of the contol laye and the dimenioning laye though the ue of aveage hadow pice concept. Thi aticle i tuctued a follow: ection 2 will be devoted to the deciption of outing algoithm ued in late ection. The mathematical fomulation i included in ection 3 togethe with the detail of the analytical model fo the netwo dimenioning poblem. We dicu implementation iue and peent numeical eult in ection 4. The concluion i given in ection 5. Section 2: Routing Algoithm A mentioned above, in ou SON famewo we apply the tate dependent ewad maximization outing tategy; uch a the MDPD tategy [4], in ode to achieve integated economic famewo. Nevethele, to implify analytical pefomance evaluation, in ou dimenioning model we appoximate MDPD outing tategy by a outing baed on a load haing concept. The pue load haing outing tategy i inefficient a call can be lot even when valid available path ae peent which i not the cae with MDPD appoach. To ovecome thi iue we employ two elatively imple yet efficient load haing outing tategie to povide conevative appoximation to the MDPD tategy. The blocing pefomance of thee two

3 tategie povide uppe bound fo the MDPD tategy. The dimenioning olution baed on them ae theefoe conevative. The fit outing tategy ued hee i nown a the combined load haing and altenate outing tategy [11]. We denote thi tategy a outing tategy I and the coeponding optimization model a model I thoughout the aticle. In thi outing tategy, the potential path fo a taffic flow ae odeed to fom a et of outing equence. Each of thee outing equence conit of all the potential path fo the taffic flow. The path ae aanged in diffeent ode in diffeent outing equence. Evey outing equence bea a load haing coefficient; the taffic flow i aigned to a outing equence with pobability popotional to the load haing coefficient of that equence. The taffic flow mut attempt all the path in it aigned equence befoe declaing connection failue. A connection would fail if and only if all the path in the aigned equence ae bloced. Figue 1 how an intance of uch a cheme fo the taffic flow between the node S and D, f SD. In that example, the fit outing equence caie a faction a 1 /(a 1 +a 2 +a 3 ) of the total taffic between node S and D, and the flow f SD mut attempt path in the ode P1, P2, P3. The econd equence in the example caie a 2 /(a 1 +a 2 +a 3 ) of the taffic and the path mut be attempted in the ode P2, P3, P1. The thid equence caie a 3 /(a 1 +a 2 +a 3 ) of the taffic flow f SD, and the path mut be attempted in the ode P3, P1, P2. In the econd tategy conideed fo appoximation of MDPD outing, each potential path fo a paticula taffic flow i aigned a outing coefficient. Fit, the taffic flow i aigned to a path with pobability popotional to the outing coefficient. If thi path i bloced, the cheme will attempt the emaining n-1 path with pobabilitie popotional to the path oiginal outing coefficient. If the new path choen by the cheme alo tun out to be bloced, the taffic will attempt the emaining n-2 path with pobabilitie popotional to thei oiginal outing coefficient. Thi poce continue until eithe the taffic flow i outed o until the cheme dicove that all n path ae bloced. Figue 2 depict a cae of uch a outing cheme. In that cenaio, path P1 i dicoveed bloced by a taffic flow aigned to it. The taffic theefoe oveflow to the emaining path P2 and P3, with pobabilitie diectly popotional to thei outing coefficient a 2 and a 3. In the emainde of thi pape the econd tategy i efeed to a outing tategy II and the coeponding optimization model a model II.

4 Section 3: The Optimization Model We teat the SON netwo a a geneic netwo and the SON gateway a geneic netwoing node that hip data to geneate evenue. In the following dicuion we hall ue SON gateway and node inte-changeably. The taffic conideed hee ae homogenou taffic with the ame bandwidth equiement following the exponential ditibution fo both thei inte-aival time and evice time. We aume, at thi tage, that the netwo topologie, the taffic intenitie, the GoS equiement, and taffic evenue (o evice pice) ae all given paamete. We alo aume that thee exit at mot one phyical lin between any pai of node in the undelying netwo, although it i poible that thee exit moe than one phyical lin. The GoS equiement ae pecified in the fom of blocing pobabilitie. We fomulate the poblem a a dimenioning poblem fo the lin in the netwo. Fo the ae of implementation, we leave mot of the patial deivative in the equation o a to enable numeical method uch a the finite diffeence method to be employed. Without lo of geneality, let aume the path in the equence et q ae indexed by the ode they will be attempted in outing cheme I. It i eay to ee that becaue we aumed evey outing equence fo a paticula flow f. contain all the coeponding end to end path, the blocing pobability fo a paticula taffic flow f. unde tategy I can be witten a: q R q q q q i = 1 = 2 α ( P ( bloced bloced, i 1, 2,... 1)) P( bloced ) (1) whee α q i the pobability of electing the equence q fo taffic pai f., q i a paticula path indexed by in the outing equence q of the taffic pai f., R ae all the end-to-end path fo the flow f.. To futhe implify (1), we can aume independence of the path q, =1,2, R. A a conequence of thi aumption, all the conditional pobabilitie of (1) ae educed to the unconditional pobabilitie, equation (1) i now given by: R B whee B i the blocing pobability fo a paticula path. The appoximation of (2) i exact when none of the path hae a common lin. Fo the ae of implicity we ue (2) in the fomulation of ou model. A ideba note i that the calculation of (1) (2)

5 can be equivalently viewed a finding the pobability uch that at leat one of the cut et with epect to R ha all it element failed, in ou cae it i poible to ue a ecuive technique to tacle that without the need of finding the et explicitly, we hall have a hot dicuion about thi in ection 3. Now let go to deive the oveflow taffic geneated by outing tategy I. We can ee the amount of taffic oveflowing to a paticula lin, a a eult of outing cheme I, i a follow: (( ( B )(1 B ))( ) /(1 B )) (3) λ α q m δ q Q 1, q > m, m q whee m ae all the path peceding in the equence q. We define 0 to be a dummy path that doe not conit of any phyical lin, and we atificially defined B =1 fo conitency. Note that we ue the ame independence aumption a that of (2) in fomulating the blocing pobabilitie. In the expeion (3), δ A = 1 if event A i tue, and δ A = 0 if event A i fale, q efe to a paticula equence of path, and Q i the et containing all the equence of path fo the flow f. A paticula taffic flow f. will oveflow to the lin if thi lin i ued by a path which i contained in one of the equence, q, inide the et Q, and all the path m befoe the path in the equence q ae bloced. Thi eult in the expeion (3) which will be ueful in calculating the lin blocing pobabilitie fo ou model. We peent the optimization model fo outing tategy I fit, and denote it a model I thoughout thi aticle. Optimization model II fo the outing tategy II i imila although the expeion fo oveflow taffic i moe complicated. Let fit define the vaiable being ued in model I in the table below: Table 1. The et of expeion ued. C (N ) = the cot function fo having a capacity of N on lin. w = the evenue geneated by taffic flow f. (i.e. taffic fom node i to node j) though path. B = the blocing pobability of path. λ = the offeed taffic in tem of numbe of connection fo the flow f. λ = the caied taffic fo flow f. on a path, it i equal to λ Σ q (α q P(C q )), whee α q i the load haing coefficient coeponding to a equence q, and P(C q ) i the pobability that the taffic i being admitted at oute of the equence q, whee q Q. L = the uppe bound fo the end-to-end blocing pobability of the flow f.. α q = the pobability of electing equence q fo the flow f.. E(a,N )= the Elang-B equation fo the lin, with offeed taffic a and capacity N. R = the et containing all the poible end-to-end path fo the flow f.. R = the aveage hadow pice fo the lin, thi i a enitivity meauement of the total evenue with epect to the lin capacity of lin. The optimization fomulation i hown in equation (4), whee x, v, u q, y, z ae the KKT multiplie. The Lagange equation fo (4) i hown in equation (5).The fit ode KKT condition of (5) ae lited in equation (6). Equation (6.III) involve the tem R, which i the enitivity of the total evenue with epect to the lin capacity 0

6 of lin and i deived a a patial deivative of the evenue with epect to the lin capacity. Thi tem tend to be ignoed in ome of the liteatue, but we dicoveed that the addition of thi tem enable ou methodology to yield bette eult, ince it tae into account of the impact of the lin capacitie on the total evenue geneated and eflect the noc-on effect of dimenioning lin ove the total evenue. Thi tem i alo nown a the aveage netwo hadow pice [4] fo the lin. Lin hadow pice i being ued extenively in the outing liteatue a a contol paamete to impove netwo eouce utilization and theefoe the incopoation of the aveage lin hadow pice in the dimenioning poce fom an economic famewo that integate the dimenioning model with the contol model of the SON netwo. M in( C ( N ) w λ ). t. Π R q Q q i, j, B L ( x ) q E ( a, N ) = B ( y ) α = 1 ( v ) α 0 ( u ) N 0 ( z ) q (4) L = C ( N ) w λ α (( B )(1 B )) + x ( B L ) q m, j, q,, > m q i q R m + v ( α 1) u α + y ( E ( a, N ) B ) q q q q Q q Q z N Π (5) Note that the KKT multiplie v at left hand ide of equation (6.I) i independent of the path taen,. A a conequence, the expeion at the ight hand ide of the equation hould have the ame value fo all the path caying a non-zeo taffic potion of the flow f.. Thi equation can be teated a the optimality equation fo the optimal outing poblem. Thi implie that all the path with a poitive hae of the taffic f. hould have the ame maginal cot fo the flow they cay. Thi i a well nown fact fo ytem optimality in the liteatue. Becaue of the complementay lacne condition, we can futhe conclude fom (4) that all path with poitive hae of taffic f. hould have the multiplie u q being 0. Togethe with the multiplie y fom the peviou iteation (o fom the initial value) we can olve (6.I) fo α q. With the outing coefficient α q and the multiplie x (fom the peviou iteation o fom the initial value) we can calculate the multiplie y fom (6.II). With the multiplie y we can olve the optimal dimenion ub-poblem in (6.III).

7 ( y E( a, N )) v u w ( B )(1 B ) ( I) = q + λ Π m q > m, m q αq ( B )(1 B ) ( B ) (, ) y w ( ) x ( ) y ( II) m > m, m q R E a ' N' = λ α q + + ' i, j, q, q B B ' B C ( N ) = R ' ( x ( B )) N Π Π R E( a, N ) N Π y + z ( III ) Then with all the multiplie and the optimization vaiable, we can olve the dual of (5) fo the multiplie x which i: max( L( x )), whee L(x) i the function min L x N, α with x a the vaiable, L(x) i continuou and concave fo any pimal, but if the pimal poblem ha non-unique olution. L(x) i non-diffeentiable. To get aound with thi, we employ the ub-gadient method to maximize L(x). With the new multiplie x, we can go bac to equation (6.I) and etat the whole poce again until the olution convege. When the olution convege, that mean equation (6.I), (6.II) and (6.III) will all be atified, which implie the fit ode KKT condition of the optimization poblem ae atified and the olution of the dimenioning poblem aive at a tationay point. It i alway poible to pefom a econd ode optimality condition chec to tet fo local optimality, although the computation of the Heian matix fo the Lagange equation (5) can be expenive a the ize of the Heian i of ode S 2, whee S i the et containing all the lin of the netwo. The main olution given by thi model i the dimenion of the individual lin. The iteative olution cheme employed hee i imila to that of [6], but the fomulation of [6] would be exceedingly complicated if it i etuctued to uit the multi-lin path in the SON envionment. Ou fomulation, on the othe hand, can tacle path coniting of an abitay numbe of lin without maing any modification. Moeove, we tae into account the notion of aveage hadow pice in dimenioning each of the individual lin. Thi i omething miing in othe tudie. Additionally, the moe ophiticated outing cheme ued hee alo impove the netwo eouce utilization, which in tun help to alleviate the poblem of ove-dimenioning in the final olution. Though outing tategy II appea to be moe complicated in the ene that it attempt the path to chooe a oute, it can be poven that the blocing pefomance expeion ae the ame a tategy I. Both tategie would declae connection failue fo a paticula taffic pai f., if and only if all the poible path ae bloced. In othe wod, we can ue the ame appoximation (2) to epeent the blocing pefomance fo tategy II. The expeion fo oveflow taffic of taffic f., howeve tun out to be moe complicated fo tategy II. We aume that tategy II only oveflow unbloced path, which i the eult of maintaining an up-to-date path tatu table. We can how that the expeion fo oveflow taffic to a lin, i epeented by (7). Note that in thi equation, R i the cadinality of the et R. Θ i a et that contain ome paticula et a element - each of the element i itelf a et (6)

8 that contain diffeent path fo the taffic f., we denote thee element by b, and Θ hold all the poible b. R 1 α P A P B A α '' λ δ = 1 b, b Θ '' b α ' ' R, ' b [[ ( ) ( )( )( )] ] (7) A in the above equation denote the event path i not bloced, and B denote the event only the path in b ae bloced. α R 1 ' ', ' b, ' R v = u + w λ P ( A ) + w P ( A ) P ( B A ) 2 '' 1, ( ' ) λ α = b b α Θ '' b + i, j, R R 1 α λ [ ( ) ( ) = 1 b, b α Θ ' ' b, ' R w P A P B A E ( a N ) y α, ' b, ' R ] δ b ( I ) y = ( P ( A )) w λ α i, j, R B + R 1 ( P ( A ) P ( B A )) α ( ) w λ α '' i, j, R 1 = b, b Θ B α ' '' b ' b, ' R x B R E a, N y B B Π ( x ( B )) ( ) ( II ) ' R E ( a, N ) ( ) = N + N ( ) C N R y z III The optimization model fo outing tategy II i imila to that of tategy I although the equation ae moe complicated becaue of the oveflow patten. To ave pace, we hall only lit the et of fit ode KKT condition in (8). Again R i the enitivity of the total evenue with epect to the lin capacity of lin. The above KKT condition can be olved by uing an iteative appoach a wa done in model I. Section 4: Numeical Reult and Dicuion We conducted a eie of numeical tudie with the mathematical model. We feel it i ueful to give a bief dicuion of ome of implementation iue of both model. In ou implementation we ue the Fan-Wolfe method to compute the load haing coefficient. A the Fan-Wolfe method may convege vey lowly when it i cloe to the optimal olution; we atificially upply an uppe bound fo the numbe of iteation. Thi lightly deceae the accuacy of the olution, but in geneal the efficiency of Fan-Wolfe method i impoved. We ue the Elang B fomula extenively in the model hee. Diect implementation of the Elang B fomula uffe fom two majo poblem. Fit, the magnitude of it component explode with the capacity and the offeed taffic. Second, diect (8)

9 calculation of the Elang fomula would equie a time complexity of O(n 2 ) whee n i the capacity of the lin. Both poblem can be cicumvented by the method mentioned in [18], and the time complexity i educed to O(n) in ou implementation. One futhe difficulty elated to the oiginal Elang fomula i that it i a dicete function in the capacity. A continuou veion of Elang B equation available in the liteatue[8] involve complicated component that mae the computation inefficient. We tae advantage of the fact that the Elang B fomula i a tictly deceaing function in the capacity, and ue the linea intepolation method to appoximate the continuou valued capacity. Fo model II, a geat deal of the difficulty lie in the computation of the expeion P(B A ). A ecuive tyle algoithm can be ued to calculate the oveflow taffic elegantly. We employ a ecuive DFS (Depth Fit Seach) algoithm [19] to each though all combination of path failue that can eult in path failue o a to avoid the complexitie involved in calculating the et Θ explicitly. The ame code can alo be exploited to find all the cut et between node i and node j. Peliminay numeical tudie wee conducted on two elatively mall ample netwo fo both of the model. The fit ample netwo i illutated in figue 5. Fo thi netwo, we aume thee ae two pai of taffic, one i fom node A to node B, with an aveage connection ate of 6 unit and evenue of 7 unit fo each caied connection, and the othe taffic i fom node A to node C, with an aveage connection ate of 5 unit and evenue of 8 unit pe connection. The GoS equiement i 0.1 fo both taffic demand. The following table ummaize eult obtained by the two model, the eult ae ounded to intege: Table 2. Reult fo the ample netwo in figue 5. Lin index Cot Net ewad Model I Model II The GoS containt ae atified by both aignment. Model II geneate lightly moe net ewad than model I. A plauible explanation to thi phenomenon i that in geneal the outing tategy I can geneate diffeent load ditibution on the conideed path when compaed with the outing tategy II. Thi fact, combined with ignificantly lage numbe of vaiable to optimize in cae of tategy I, may lead to a moe uboptimal olution in model I. The convegence gaph fo both model ae hown in figue 6 below, the y-axi coepond to the net evenue, while the x-axi

10 coepond to the iteation numbe. A we can ee, both of them convege in appoximately 5 iteation, and each iteation tae aound 6 econd of time on a 1.4Ghz P4 machine fo the both model. We alo conideed a lage poblem a hown in figue 7. Thi poblem ha 5 pai of taffic flow, the taffic detail ae lited in table 3 and the dimenioning eult ae depicted in table 4. Table 3. Taffic matix fo the netwo in figue 7. Aveage Rate Revenue pe connection Poible oute GoS (indexe of the lin) Taffic A-> B 25 unit 18 unit >5->2 Taffic B->A 15 unit 12 unit Taffic A->C 18 unit 25 unit 1-> >5 4->7->6 Taffic B->C 30 unit 17 unit >4->5 Taffic E->D 12 unit 18 unit 7-> >3 7->6->3 Table 4. Dimenioning eult fo the netwo in figue 7. Lin index Cot Net ewad Model I Model II

11 It tae appoximately 30 iteation fo both model to convege in thi lage netwo, with each iteation taing appoximately 8 econd. Model II again give highe net evenue while atifying all the GoS equiement. Thi i to be expected a accoding to ou pefomance model, model II geneate le oveflow taffic. A a eult, le eouce i needed fo model II to meet the GoS Containt, which become moe tivial in thi example. Depending on the implementation, model I might futhe uffe fom the poblem of lage cadinality in geneating equence, a the numbe of equence gow a a factoial function of the poible path. One may have to limit the numbe of equence geneated by model I in lage example and thi could be anothe diadvantage of model I fo lage-ize eal-wold netwo, Section 5: Concluion We tudied the poblem of SON dimenioning poblem by employing an iteative poce baed on two diffeent outing model. A majo contibution of thi tudy i that we povide an appoach to dimenion the SON netwo by conideing the SON netwo a a geneic netwo baed on the taffic evenue. Moeove the concept of aveage lin hadow pice i alo incopoated in the SON dimenioning model. We alo povided numeical eult to offe inight into the efficacy of the theoetical model. The numeical eult ae pomiing on the mall ample netwo teted, and convegence uually occu within a few iteation. The cuent effot i to captue the ey featue of the olution cheme o a to impove computational pefomance. The veification of the olution quality though tate dependent outing imulation and the convegence tudie ae both in poge. Oveall the tudy epoted hee povide, unde the new pepective of pofit maximization, an economic integation of the contol and dimenioning laye when allocating capacitie in the SON netwo. Refeence: 1. Zbigniew Dziong, ATM netwo eouce management. McGaw-Hill, New Yo, Michal Pioo and Deepana Medhi, Routing flow and capacity deign in communication and compute netwo. Mogan Kaufmann Publihe, Fan P. Kelly, Routing in cicuit-witched netwo: Optimization, Shadow Pice and Decentalization, Advanced In Applied Pobability, Vol. 20, No. 1, 1988, Zbigniew Dziong and Lone G. Maon, Call admiion and outing in multi-evice lo netwo, IEEE Tanaction On Communication, Vol. 42, No.2/3/4, 1994,

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