Service Overlay Network Design with Reliability Constraints

Transcription

1 Serice Oerlay etwork Deign with Reliability Contraint gok Lam, Lorne G. Maon Department of Electrical and Computer Engineering, McGill Unierity, Montreal, Canada {ngok.lam, Zbigniew Dziong Department of Electrical Engineering, Ecole de Technologie Superieure, Montreal, Canada Abtract We tudied a cla of Serice Oerlay etwork (SO) deign problem with reliability contraint. It i aumed that a SO network could enter an inadmiible tatu for two reaon; firt when there i inufficient reource to accommodate new connection, econd when ome hardware deice malfunction. The deign problem i uually formulated a either a Maximum Profit (MP) contrained optimization problem or a Minimum Cot (MC) contrained optimization problem. In thi article we inetigate the relationhip between the two formulation in the context of enuring ytem operability. By uing the et of Lagrange multiplier from the MC formulation a a tool, we how the general condition that MP and MC gie exactly the ame network deign. The key contribution of thi paper i the proiion of inight into the olution nature of the MP and the MC formulation in deigning a reliable oerlay network, thereby giing guideline to the proper formulation the network deigner may conider in deigning a reliable yet economically optimal SO network. Autonomou Sytem (ASe), in the form of Serice Leel Agreement (SLA). The leaed bandwidth act a logical link that connect the SO gateway. Each logical link conit of exactly one autonomou ytem. Once all the logical link are in place, the SO i realized and the oerlay network formed i under the adminitration of a ingle authority. Becaue the SO i adminitrated by a ingle operator, it i capable of proiding end-to-end QoS guarantee to the alue-added erice proided by it (i.e. online gaming, Video conferencing, etc). A uer with acce to the Internet can acce the erice gateway to ue the alue-added erice, proided the hot holding the content are alo connected to ome SO erice gateway. In a SO, the connection are claified by the origin and the detination (OD) gateway. Uer pay the erice charge baed on the origin and Keyword- Reliability; Serice Guarantee; etwork Deign; Profit Maximization; Cot Minimization ITRODUCTIO The demand for end-to-end Quality of Serice (QoS) guarantee in the Internet ha increaed ignificantly due to the introduction of new application like VoIP, online gaming, and ideo conferencing. Thi poe a major challenge to the current Internet architecture. Owing to hitorical reaon, the Internet conit of a large collection of independent Autonomou Sytem (ASe). To enure end-to-end QoS guarantee for the data, one ha to build a multi-lateral buine relationhip with all the independent ASe hi data tranit. Thi make it unrealitic to obtain end-to-end QoS guarantee. A higher leel mechanim on the top of the Internet known a Serice Oerlay etwork (SO) i thu propoed to alleiate thi problem []. The SO network operate in a manner imilar to a irtual network. The SO operator own the SO gateway which are placed in trategic location. To realize the SO network, the SO operator leae bandwidth with QoS guarantee from the underlying Fig.. An example SO network. detination of their connection a well a their connection duration. Figure how an example of the SO network. It i quite common that in formulating network deign problem, one may be confronted with the choice of chooing to maximize the profit or to minimize the cot, a the deign criteria. An intereting quetion i thu raied: ince the maximum profit (MP) and the minimum cot (MC) approache optimize different objectie, how do their olution compare and how doe a network deigner decide which approach to employ. Thee quetion could be nontriial under the SO enironment. Thi i becaue the SO operator enjoy great degree of freedom in deploying their own inter-domain routing cheme a the network i olely under their adminitration. Owning to thi flexibility, different network adminitrator are likely to employ different

2 routing cheme to fit their buine objectie. The routing cheme employed would influence the network deign ignificantly. The concluion drawn regarding SO deign for a particular routing cheme may not be alid for another. Thi paper addree thi objectie choice problem faced by the operator for a cla of Serice Oerlay etwork (SO) capacity allocation problem. The reult doe not depend on a pecific routing cheme a long a certain condition are met. Thi paper i tructured a follow: Section i the decription of the problem aumption and formulation, Section 3 dicue the major reult, Section 4 how a imple example that illutrate the reult, Section 5 i the concluion ection that dicue the concluion obtained and poible future extenion. THE PROBLEM. Problem Formulation To deploy a SO, a major challenge faced by the operator would be the optimal amount of bandwidth to be leaed on the logical link. The allocated bandwidth hould fit the economic objectie of the operator yet meet the uer expectation regarding the Grade of Serice (GoS) leel. The Grade of Serice (GoS) leel are uually quantified by connection blocking probabilitie. The connection blocking include the blocking due to reource contraint and alo the blocking reulting from hardware failure. Thi blocking probability i iewed a a reliability meaure of the network by the uer. By conidering the SO a a lo network [5], the deign problem can be formulated a a contrained maximum profit problem (MP) a lited in (..a). The problem i aumed to be oled uing the Lagrangian relaxation method [3]. max λ w ( B ) C ( ) B L u 0 z (.. a) B c = ( w + u )( ),.t. > 0 (..b) λ h h n H = h h n nn B B nm λ + nn λ + nm n h = ( w u ), h = ( w u ) (..c) The correponding firt order optimality condition and the Heian matrix of the relaxed problem are lited in (..b) and (..c) repectiely. We hall denote formulation (..a) a MP in the remainder of thi article. In the formulation (..a), λ i the gien poionian connection arrial rate demanding the connection of the node pair (i,j) (i.e. origin gateway i i, detination gateway i j), w i the expected erice charge/expected reward generated by an admitted (i,j) connection. We hall ue the term reward and erice charge interchangeably in thi article. The notion B denote the analytical end-to-end blocking probability for connection of the node pair (i,j), and it i routing cheme dependent. A connection can be blocked for three reaon: a) there i no aailable reource, b) hardware failure on the logical link, c) a combination of (a) and (b). The GoS contraint for each OD pair (i,j) i gien by L, which pecifie the maximum allowed end-to-end blocking probability due to reource contraint or hardware failure. The capacity of a link i denoted by and it i a deciion ariable of thi problem. The function C (.) i the cot function that quantifie the cot of allocating unit of capacitie on link per unit time, baed on ome SLA. Thi function i aumed to be a linear function of the ariable. The ariable u and z are the Lagrange multiplier with repect to the contraint. The minimum cot formulation (MC) of the ame problem, the correponding firt order optimality condition and the Heian matrix are gien by (..a), (..b) and (..c) repectiely. The only new notation introduced in (..a) i. It i the Lagrange multiplier correpond to the GoS contraint. We hall denote formulation (..a) a MC in the ret of the article. It i aumed that both MP and MC formulation employ the ame (optimal) routing cheme in the routing layer. min C ( ) B L 0 z (.. a) B c = ( ),.t. > 0 (..b) h h n B B H =, hnm =, hnn = (..c) n m n hn h nn To analyze the two different formulation, a typical approach i to inetigate the correponding optimality condition ince they proide the richet information regarding the optimal olution. From (..b) and (..b), one can ee that proided B are the ame, the firt order neceary condition of the two formulation are equialent if the expreion and λ w + u are equal for all the OD pair (i,j). In thi cae the econd order ufficient condition of the MP and MC will alo be the ame, a the Heian matrice (..c) and (..c) are identical then. Therefore, regardle of their different objectie, MP and MC could delier the ame olution if one can et the multiplier u appropriately, ubject to the dual feaibility condition. A we hall ee, there exit a general condition that the MP and MC formulation can be made equialent. When the olution of MP and MC coincide, it indicate the cot function dominate the reward function o that the incorporation of additional reward information doe

3 not differentiate the olution of MP from the olution of MC. We hall how that for ome pecific range of reward parameter w, the cot dominate the deign. While in other range of the w parameter, the reward parameter are large enough to influence the network deign.. The End-to-End Blocking Probabilitie Before the analyi can be made, a proper analytical form of the blocking function, B, i needed. The functional form of B arie with routing cheme [4]. We employ technique from the reliability theory [9] to deie a general and functional form for B, regardle of the actual routing cheme employed. The B function deried below i primarily baed on the reduced load approximation model [0], which aumed tatitical link independence and Poion link arrial rate. We conider the collection of network path, that connect a particular origin node i with a particular detination node j, a a complete ytem. The tak of thi ytem i to ere the connection between the node pair (i,j). Aume the network link are independent of one another. The link in the collection of path are the independent component of the ytem. Denote thee link by and let R be a et that contain all uch link. Define an indicator ariable y for the link. Wherea y equal to zero if link i functioning and ha enough reource to admit at leat one connection, and y equal to one if link ha failed or doe not hae reource to ere a new connection. The expected alue of y i the blocking probability of link due to reource contraint or hardware problem. According to the reliability theory [9], a Boolean function φ ( Y ) that indicate whether the ytem i unaailable for a new (i,j) connection can be defined by taking the link tatu ector Y=[y ] a the input. The expected alue of φ ( Y ) i effectiely B, the end-to-end blocking probability for connection pair (i,j). Since y are independent zero-one random ariable and φ ( Y ) i a Boolean function, we can perform the Shannon decompoition on the function φ ( Y ). Uing link a the piot we hae (..): φ ( Y) = y φ (, Y) + ( y ) φ (0, Y) = φ (0, Y) + [ φ (, Y) φ (0, Y)] y (..) (0, Y ) and (, Y ) are the tatu ector that differ only in the th link. The function φ (0, Y ) and φ (, Y ) indicate that whether the ytem ha been blocked gien that the link i in admiible tatu/ blocking tatu. Thee two function do not depend on the link. Since the link are independent, taking expectation on the both ide of (..) yield (..). E[ φ ( Y )]= f ( E[ y ]) + f ( E[ y ]) E[ y ] (..), ', ' Where, are the link in the et R. In the SO architecture aumed [], each pair of SO gateway are eparated by one autonomou ytem, o a link alway conit of one-hop. By the definition of y, the expectation E[y ] i the probability for a ingle link to be in blocking tatu. We aume link failure due to hardware problem to be independent of the reource related blocking. We alo aume the link connection arrial rate to be independent Poion procee. Hence we hae E[y ]= E (λ, ) + h -E (λ, )h. Where λ i link connection arrial rate at equilibrium, where E (λ, ) i the Erlang-B formula, and h i a contant to denote the probability for link to fail owning to hardware iue (i.e. random hardware failure, lo of electricity, etc), thi probability can be etimated from the hitorical data of the AS. Therefore (..) can be written to (..3). B = f, ( λ', ', h' ) + f, ( λ', ', h' ) h + f, ( λ', ', h' )( h) E( λ, ) (..3) It hould be clear that B =E[ φ () Y ] i eentially a reduced-load approximation of the OD pair blocking probability, a link independence and Poion link arrial rate are aumed. ote that in (..3), f, [0,], a it i the blocking probability of the ytem gien that link i in admiible tate (refer to..). We alo need f, 0 if the end-to-end blocking probability B i non-increaing in the preence of additional aailable link capacitie (when f, 0, it i the Birnbaum importance meaure of link ). Thi i a monotonic property we impoed on the routing cheme and it hould be atified by any reaonable (centralized) routing cheme ince uch cheme hould not perform poorer with more aailable reource. We alo impoe the aumption that the routing objectie function being uni-modal with repect to the capacity aignment. Moreoer, we aume the continuou extenion of Erlang-B formula uggeted in [] i ued throughout thi article. Expreion (..3) i alid for any link. So we can repreent B in the form of f, + f, h + f, (-h )E (λ, ) for any link, where f,, f are independent of link. Thi general, expreion of B facilitate eay acce to the deriatie of B with repect to the capacity on an arbitrary link. In a imilar pirit, by applying the Shannon decompoition again on expreion (..3), B can be further decompoed in term of the blocking probabilitie of link and link, a in (..4). B = g + g E (.) + g E (.) + g E (.) E (.) 0 ' 3 ' g = f (0 ) + f (0 ) h + ( f ( ) f (0 )) h 0, ', ', ', ' ' + ( f, ( ' ) f, (0 ' )) hh ' = h f, ' + f, ' f, ' h' = h' f, ' f, ' + f, ' f, ' h 3 = h h' f, ' f, ' g ( )( (0 ) ( ( ) (0 )) ) g ( )(( ( ) (0 )) ( ( ) (0 )) ) g ( )( )( ( ) (0 )) (..4) The notion 0 and in (..4) indicate the tatu of the link. For intance, f, ( ) indicate the function f (.) gien, that link ha been blocked. Expreion (..4) can be employed to derie the econd order information of the MP

4 and MC formulation. The function g 0, g, g and g 3 in (..4) are independent of the link and. By uing expreion (..3), the optimality condition (..b) and (..b) can be rewritten to (..5) and (..6). E( λ, ) [ ( λ w + u ) f, ( λ ', ')]( h )( ),.t. > 0 (..5) E( λ, ) [ f, ( λ ', ')]( h )( ),.t. > 0 (..6) Aume the continuou Erlang-B formula in [] i employed. Since it i a C function [], o the econd deriatie exit. By uing (..3) and (..4) the element of the Heian matrice of (..c) and (..c) can be re-written to (..7) and (..8) repectiely. En(.) Em(.) hnm =[ ( w λ + u )( f, n( m)- f, n (0 m))](- hn )(- hm ) n m En (.) hnn =[ ( w λ + u ) f, n(.)](- hn ) (..7) n m hnm =[ ( f, n( m)- f, n( m n m n m En (.) hnn =[ f, n(.)](- hn ) (..8) n n E (.) E (.) 0 ))](- h )(- h ) 3 THE MP AD MC FORMULATIO 3. The General Condition for the Equialence of MP and MC Formulation ote that from the new optimal condition (..5) and (..6), if the um ( λ w + u ) f, (.) and f, (.) can be θ θ made identical for all the link at the optimal olution of MC, the firt order optimality condition of MP and MC will be identical. The econd order optimality condition will then alo be identical. To ee thi, we ubtitute (..7) and (..8) into the Heian matrice of (..c) and (..c). Recall that f, (0 ' ) and f, ( ' ) are the function f, gien that link i in admiible tatu/ha been blocked. So ( λ w + u ) f (.) = f (.) implie θ,, θ ( λ w + u ) f ( y ) = f ( y ) where y i either 0 θ, ', ' θ or. It i then eay to ee the Heian matrice of the two formulation are identical. Hence, the problem i now reduced to howing the exitence of a et of non-negatie (dual feaible) multiplier u, uch that the two aforeaid um can be made equal. We hall how the general condition for the multiplier to exit uing Farka lemma [8]. o aumption regarding the number of OD pair and network link i made. Without lo of generality, let aume the multiplier exit and the OD pair are indexed from to, where i number of ditinct OD pair. Let alo aume the link are indexed from to, where i the total number of link in the network. Define a matrix A and a ector B a in (3..). Then the non-negatie matrix A ha a dimenion of. The = k function f, (.) in the matrix A indicate it i a function correpond to the k th OD pair and it i the f function of the link. Thi function take the optimal olution of the MC formulation a the input in (3..) and (3..). The ector B ha a dimenion of. The problem of finding a et of nonnegatie u i equialent to finding a non-negatie ector U uch that AU=B. ( λ w ) f,(.) = = f, (.) f, (.) θ A= B = = = f, (.) f, (.) ( λ w ) f, (.) θ T T = = T T =,, (3..) Ap pf (.),, pf (.) = [ η,..., η ] 0 (3..) By the Farka lemma, the non-negatie ector U exit, if and only if for eery ector P uch that A T P 0, the inner product B T P i non-negatie. That mean for any ector P that atifie (3..), the expreion (3..3) mut be poitie. < B p >= p w f, ( λ ), (.) = ( λ w ) p f, (.) d η = (3..3) In (3..3), d =( -λ w ) are not retricted in ign. The ymbol η denote the element of (3..) which are alway nonnegatie. From the expreion (3..) and (3..), we note that it i poible to et ome η arbitrarily large by properly picking alue() for p for ome intance of A. Since d are not retricted in ign, the inner product of (3..3) can become negatie unle ome further contraint are impoed on the routing matrix A. The only cae that (3..3) i alway poitie, regardle of the A matrix, occur when all the d are nonnegatie. Thi implie condition (3..4). The reult can be alternatiely erified by the regularity condition. 0 w λ (3..4) Condition (3..4) i the ufficient condition uch that the MP and MC formulation can be made equialent, regardle of the routing cheme employed. So the following reult are alid for all routing cheme that are monotonic and unimodal. We firt need to etablih the uniquene of the allocated link capacity with repect to the link cot, a et of real alue, and the link connection intenity ector Λ (i.e. offered connection intenitie which i equal to AΛ, where Λ=[λ ]). Defineθ to be a et which contain indexe of all the

5 OD pair that utilize link in at leat one of their path. Let be the capacity of a link, which i aumed to be continuou. Let c be the cot of allocating one unit of capacity on link, and the contant h to be the hardware failure probability on link. Let E ( λ, ) denote the Erlang-B formula with offered connection intenity λ and link capacity of a parameter. Lemma For poitie contant, c, and fixed Λ, there exit at mot one on the link that atifie the following equation: E( λ, ) c = [ f, ( λ', ' )]( h)( ) (3..5) θ Proof: Aume the continuou extenion of Erlang-B formula uggeted in [] i being employed. ote that the function f = E[ φ (, Y ) - φ (0, Y )] i the ame a the one defined earlier and it i poitie (becaue we impoed the condition θ to enure it will not be zero). The alue λ are the element of Λ. We rearrange the term and get (3..6): c E( λ, ) = ( ) (3..6) [ f ( λ, )]( h ) θ, ' ' Since the function f, i independent of. It can be regarded a a poitie contant function with repect to. Denote E ( λ, ) ( ) by f (, ) 3 λ. It i known that the continuou Erlang B formula i a C function [], which i trictly conex in the capacity []. Therefore for a fixed λ, the function f 3 (.) i trictly decreaing and continuou in, and there i an oneto-one correpondence between the function alue and for the fixed λ. Moreoer it can be een that f 3 (.) i poitie, ince when i increaed by delta, the function -E (.) alway increae if λ i fixed. So for poitie contant, c, and fixed λ, there i a unique that atifie equation (3..6). There are two pecial cae for expreion (3..6), firt if the LHS of (3..6) i larger than max( f3( λ, )) then doe not exit, econd if the cot c equal to zero, then equal to infinite. The phyical interpretation of Lemma i that when the link connection intenity i fixed, then for each et of c and, there i at mot one alue that ole (3..5) and thereby atifying the firt order neceary condition. Lemma proide a tool for u to how the condition uch that MC and MP are equialent. The following theorem will etablih the relationhip between MC and MP under ome pecified condition. Theorem Conider a MP formulation (..a) and a MC formulation (..a) of the SO capacity allocation problem. If a et of Lagrange multiplier exit for the MC formulation, and if the condition (3..4) i atified, then an optimal olution of the MC formulation i alo an optimal olution of the MP formulation. Further to thi, if the et of i unique, then MP and MC formulation are equialent. Proof: The tationary condition for MP and MC are (3..7) and (3..8) repectiely. All the ymbol hae the ame definition a being defined earlier. ote that (3..7) and (3..8) both repreent n et of equation where n equal to the number of link in the network. E( λ, ) c = [ ( λ w + u) f, ( λ', ' )]( h)( ) (3..7) θ E ( λ, ) c = [ f ( λ, )]( h )( ) (3..8), ' ' θ Suppoe are the Lagrange multiplier at optimality for MC. If (3..4) i atified, then it i poible to find a non-negatie ector U=[ u ] uch that ( λ w + u ) f, (.) equal to θ f, (.) at the optimality of the MC formulation. Aume θ c a ector of link intenitie Λ i deignated by ome optimal routing rule at the optimal olution of the MC formulation. c c c c c Suppoe the tuple ( Λ, ( Λ )), where ( Λ ) i the ector of optimal capacitie, i a olution that atifie (3..8) and alo the complementary lackne condition. Then it i c c c obiou from Lemma that the olution ( Λ, ( Λ )) i c unique with repect to Λ. ow if we ubtitute the olution c ( c c Λ, ( Λ )) along with U into (3..7), the olution hould atify (3..7) if it atifie (3..8). Thi i becaue ( λ w + u ) f, (.) equal to f, (.) for all the n θ θ equation, and alo becaue the expreion E ( λ, ) ( ) depend only on c c c Λ and ( Λ ). The complementary lackne condition are alo atified ince the contraint of MP and MC are identical. In other word, when (3..4) i atified, the olution that atifie the firt order neceary condition of MC alo atifie the firt order neceary condition of the MP. With the ame ector U, the econd order ufficient condition check i triial, becaue the Heian matrice of the correponding Lagrangian dual are identical (ee (..7) and (..8)). Therefore we hae hown that the optimal olution of MC i alo an optimal olution of MP. Further to thi, if the ector U i unique and condition (3..4) i atified, then formulation MP and MC are equialent. Thi can be proed by contradiction. If there exit another et of multiplier X = [ x ] of MP, uch that ( λ w + x ) f, (.)) = y f, (.) f, (.), where θ θ θ

6 x 0 and y 0. Suppoe for any optimal olution p p Λ, ( Λ )) that correpond to X and atifie both the neceary and ufficient condition of the MP formulation. It i obiou that the multiplier y = ( λ w + x ) along with p p p ( Λ, ( Λ )) atify (3..8). The complementary lackne p ( condition are alo atified becaue (..a) and (..a) hae the ame contraint. Moreoer thi olution alo atifie the econd order necearily condition of MC a long a it atifie thee condition for MP. That implie Y=[ y ] i alo a et of multiplier for MC. But thi contradict with the claim that MC ha a unique et of multiplier. Therefore it i impoible for the ector X to exit. A a reult MP mut alo hae a unique et of multiplier (i.e. U=[ u ]). Thi how that both MP and MC gie the ame optimal olution if (3..4) i atified, and are therefore equialent. Aume there exit a unique et of multiplier for the MC formulation and the correponding Λ i unique; the aboe theorem indicate that if the erice charge are lower than ome alue (i.e. (3..4) i atified), then MP and MC gie the ame olution. Thi i the region uch that the cot function dominate the reward function. For all the other region, the erice charge are large enough to influence the deign and MC and MP are different. Owing to the pace limitation we hall dicu the reult for one uch region in Section The MP and MC Formulation a Differentiated by the Serice Charge If condition (3..) hold for the erice charge w, then it i obiou that condition (3..4) can not be atified, a condition (3..) implie the B ector to be trongly negatie. We hall how the MP formulation i different from the MC formulation in thi region in two way. Firt, MP proide olution with trictly better GoS guarantee than that of MC. Second, MP alo proide olution with trictly more profit than the MC formulation. The inetigation will proide u with inight into the relationhip between the erice charge and GoS guarantee (which will be dicued in another paper). 0 f ( λ, ) < w λ f ( λ, ) (3..) θ c c c c, ' ', ' ' θ Before we preent the main reult, again we firt need a lemma. Lemma etablihe the relation between the optimal capacitie allocated and the magnitude of erice charge. Lemma For any two real number,, where > > 0. If λ i a contant in (3..), then we hae >, where and are the alue of correpond to = and = repectiely in (3..). E( λ, ) c = ( ) (3..) Proof: It i known from the proof of lemma that for a fixed λ, the function expreion in E( λ, ) ( ) E( λ, ) ( ) i trictly decreaing and continuou. So the i continuou and trictly decreaing, when i a poitie number. For a contant c, the E( λ, ) larger the alue, the maller the expreion ( ) will be required to atify the equality condition of expreion (3..). Thi require a larger alue. A a reult if > > 0 and if and both exit then >. With Lemma, we hall now how that MP proide olution with trictly better GoS guarantee than MC in theorem. Theorem If (3..) hold, moreoer, if the routing cheme i monotonic in the allocated capacitie, and doe not reduce the importance of a link a it capacity increae. Then the olution of the MP formulation will delier trictly better GoS guarantee than that of the MC formulation. The optimal olution for (..) will be the ame a it uncontrained counterpart. Proof : c c c Conider the cae that the optimal olution ( Λ, ( Λ )) of the MC formulation i regarded a the initial olution of the MP formulation. Aume that the optimization proce attempt to ole (3..7) at each iteration and thi proce conerge to the optimality of MP [3]. Aume alo that the initial ector [ u ] to be 0 T, thi ector will increae iff ome GoS c c c contraint() i/are iolated. Subtitute ( Λ, ( Λ )) into the optimality condition of MP (3..7). Recall that ((3..) hold, by lemma the allocated capacitie on all the link trictly increae at the end of the firt MP iteration. Since it i aumed that the importance of a link doe not decreae a it capacity increae, therefore f, i not decreaed in the econd MP iteration. A a reult w λ f, (.) i non-decreaing in θ the econd MP iteration, by lemma the allocated capacitie are non-decreaing in the econd iteration. Thi augmentation p of capacitie continue until the tationary point at i p c p reached. So we hae >, where i the capacity ector at the optimality of the MP. ow becaue of the monotonic aumption of the routing cheme and alo p c p becaue >, the GoS guarantee offered by i

7 c trictly better than that being offered by. Moreoer, ince the optimization proce augment the capacitie, o [ u ]=0 T for all the iteration and we hae [ u ]=0 T. Becaue [ u ]=0 T, o the olution of MP remain the ame een if all the GoS contraint are relaxed. By definition the MP formulation i to maximize the profit o the profit generated by it olution mut not be wore than that of the MC. We hall how in the following theorem that MP proide olution with trictly more profit than the MC formulation if condition (3..) i atified. Theorem 3 If the routing cheme i monotonic in the allocated capacitie and if condition (3..) hold, the MP formulation delier olution with trictly more profit than the MC formulation. Proof: ote that from the proof of theorem, when condition (3..) hold, u in (3..7) are all zero at optimality, thi i due to the theorem and the complementary lackne condition. ow expreion (3..7) can be written a (3..3): E ( λ, ) c = [ λ w f ( λ, )]( h ) ) (3..3) p, ' ' θ To how the profit generated by the olution of MP formulation i trictly larger if condition (3..) hold, we firt rewrite the R.H.S of (3..) a w λ f ( λ, ) = f ( λ, ) + x f ( λ, ) c c c c c c, ' ', ' ', ' ' θ θ θ where x > 0 for all. We ubtitute the optimal olution c c c ( Λ, ( Λ )) of MC with thi expreion into the RHS of (3..3). Rearranging the term we hae (3..4): E( λ ) E( λ ) (, )( ) (, )( ) (3..4) c c c c c c f, λ ' ' h + x c f, λ ' ' h c θ θ Expreion (3..4) denote the marginal profit from the link by c allocating a capacity of when (3..) hold. By referring to (3..8), the firt term in (3..4) equal to c, o (3..4) can be reduced to (3..5). c c c E ( λ ) +, λ' ' c θ c E( λ ) c, f, c x[ f (, )]( h)( ) (3..5) We know that are poitie, and h i a probability, therefore the econd term of (3..5) i poitie. A a reult, from expreion (3..5) one can conclude that at the c optimality of MC formulation (i.e. ), the marginal reward i trictly larger than marginal cot c. Thi implie trictly more profit can be generated if extra capacitie are allocated, and the profit attend maximum at the olution of MP (i.e. (3..3) ), From the analyi in theorem 3, we can gain the inight that the olution of MC offer trictly wore profit than that of MP becaue the MC formulation merely allocate the minimum amount of capacitie to atify the GoS requirement. Additional profit that could hae been gained are lot. The main concluion from ub-section 3. and 3. i that the MP formulation offer no wore performance than the MC formulation in all cenario. When reward are low and the cot i the major concern, the MP formulation minimize the cot jut like MC. When the reward are high enough, the MP formulation witche to maximize the profit. In thi ene, deigning the SO network uing the MP formulation i uperior to the MC formulation. 4 A ILLUSTRATIVE EXAMPLE We conider a imple SO network a hown in figure for illutration. In thi example, SO Gateway A i required to end a Poion tream of connection per unit time, on the behalf of it client, to Gateway B. Gateway C i required to end a Poion tream of 4 connection per unit time, on it client behalf, to gateway B. We conider two typical routing cheme. amely (I) the direct-link cheme that ue only the direct link, and (II) the minimum cot cheme that ue the cheapet path. Without lo of generality we aume the mean holding time of the connection to be identically ditributed with unit mean. The cot of leaing one unit of bandwidth of capacity for one unit of time are unit, unit and 50 unit repectie for link,, and 3. The allocated capacitie are required to be integral alue. Aume the hardware failure probabilitie for the link are h = h = h 3 = 0.0. Let the GoS/reliability requirement for the two tream of connection be 0.9 and 0.95 repectiely (i.e. at mot 0./0.05 probability of failure on an aerage). The multiplier ector for the directlink cheme and the minimum cot cheme are found to be (994.9, 0.) and (6.7, 80.6), which tranlate to erice charge ector of (66.4, 8.4) and (5., 7.53). We et the erice charge to (66, 8) and (5, 7) repectiely for the routing cheme I and II. Thu condition (3..4) i atified. ote in table and, for both routing cheme, MP and MC formulation gie the ame olution. Which i exactly the reult being depicted in theorem. Figure. A imple SO network. We then et the erice-charge ector to (67,9) and (6,8). It i eay to erify that (3..) i atified. Table and again ummarize the reult obtained. Since the olution are rounded up to integer from the real-alued optimal olution,

8 the GoS contraint may not be tight een for the MC formulation. But note that in the table when the erice charge atify (3..), the olution of the MP formulation indeed offer trictly lower blocking probabilitie for the both routing cheme, which i the reult to be expected a indicated by theorem. The profit of MP deign alo urpa that of the MC deign for the two routing cheme, and thi illutrate the reult proen in theorem 3. Table. Capacity allocation reult for routing cheme I MP MP MC Direct Link Serice charge Serice charge Routing (67, 9) unit (66, 8) unit Reliability of (0.93, 0.96) (0.9, 0.95) (0.9, 0.95) the deign Allocated (0,3,6) (0,30,5) (0,30,5) Capacitie Cot Objectie alue Expected Profit /75.5 Table. Capacity allocation reult for routing cheme II MP MP MC Direct Link Serice charge Serice charge Routing (6, 8) unit (5, 7) unit Reliability of (0.9, 0.96) (0.90, 0.95) (0.90, 0.95) the deign Allocated (7,43,0) (7,4,0) (7,4,0) Capacitie Cot Objectie alue Expected Profit /3.0 5 COCLUSIOS AD FUTURE WORKS We tudied a cla of Serice Oerlay etwork (SO) capacity allocation problem with GoS/reliability conideration. The problem can be formulated a Maximum Profit (MP) or Minimum Cot (MC) optimization problem. We howed the condition uch that the MP and MC approache offer the ame olution. Intuitiely we can iew the region that the MP and MC are equialent to be the region that i not profitable to offer the required GoS guarantee. In thi region the MP approach minimize the cot in an exactly ame way a the MC approach. While in the profitable region, MP take adantage of that and allocate more reource wheneer neceary to aoid potential profit loe. Thi i an attractie feature becaue the MP approach automatically decide whether to minimize the cot or to maximize the profit in an optimal ene. While the MC approach, on the other hand, doe not hae thi capability. In cae that GoS requirement are baed on ome non-economic conideration (for example GoS could be decided by external entitie like the regulating authoritie), the MP approach will proide olution with better alue than that of the MC approach a the MC approach will be ticking blindly to the potential arbitrary GoS requirement. The computation complexitie for both the MP and the MC will be imilar in a ene that major effort for both approache will be on oling the optimal routing problem and calculating the gradient of B. Therefore the MP approach appear to be a better option for SO network deign. Owning to the pace limitation, the reult in thi paper do not coer the region uch that the ector B i poitie but the condition (3..4) i not atified, and region uch that condition (3..) i partially atified (i.e. it i atified on ome of the link, but not on the other link). The reult for thee region will be included in a more comprehenie extenion of the current paper. From the reult in ection 3., one can ee that the MC Lagrangian multiplier act a a form of threhold for pricing under the GoS contraint. They can be employed a a form reference metric for the erice charge. Thi reult will alo be dicued in an extenion of the current paper. REFERECES [] A. Jager and E. V. Doom, 986. On the continued Erlang lo function. Operation Reearch Letter, Vol. 5, o., pp [] J. S. Etee, J. Craeirinha, and D. M. Cardoo, 006. Second order condition on the oerflow traffic from the Erlang-B ytem, Caderno de Matemática, Unieridade de Aeiro, CM06/I-0. [3] A. Girard and B. Liau, 993. Dimenioning of adaptiely routed network, IEEE tranaction on networking, Vol., o. 4, pp [4] A. Girard, 993. Reenue optimization of telecommunication network, IEEE tranaction on communication, Vol. 4, o. 4, pp [5] F.P. Kelly, 988. Routing in circuit-witched network: Optimization, Shadow Price and Decentralization, Adanced in applied probability, Vol. 0, o., pp [6]. Lam, Z. Dziong and L.G. Maon, 007. etwork capacity allocation in erice oerlay network, Proceeding of 0th International Teletraffic Congre (ITC), Ottawa, pp [7] A. Girard, 990. Routing and dimenioning in circuit-witched network. Addion-Weley, USA. [8] D. P. Berteka, 999. onlinear Programming, nd edition, Athena Scientific, USA. [9] Wallace R. Blichke, D.. Prabhakar Murthy, 000. Reliability Modeling, Prediction, and Optimization. John Wiley & Son, USA. [0] W. Whitt, 985. Blocking when Serice i Required from Seeral Facilitie Simultaneouly, AT&T Technical Journal, ol. 64, o. 8, pp [] Z. Duan, Z. L. Zhang, and Y. T. Hou, 003. Serice Oerlay etwork: SLA, QoS, and bandwidth proiioning, IEEE/ACM Tranaction on networking, Vol., o. 6, pp