Zbigniew Dziong Department of Electrical Engineering, Ecole de Technologie Superieure 1100 Notre-Dame Street West, Montreal, Quebec, Canada H3C 1k3

Transcription

1 Capacity Allocation in Serice Oerlay Network - Maximum Profit Minimum Cot Formulation Ngok Lam Department of Electrical and Computer Engineering, McGill Unierity 3480 Unierity Street, Montreal, Quebec, Canada H3A A7 Zbigniew Dziong Department of Electrical Engineering, Ecole de Technologie Superieure 00 Notre-Dame Street Wet, Montreal, Quebec, Canada H3C k3 Lorne G. Maon Department of Electrical and Computer Engineering, McGill Unierity 3480 Unierity Stree Montreal, Quebec, Canada H3A A7 ngok.lam@mcgill.ca zdziong@ele.etmtl.ca lorne.maon@mcgill.ca ABSTRACT We tudied a cla of Serice Oerlay Network (SON) capacity allocation problem with Grade of Serice (GoS) contraint. Similar problem in the literature are typically formulated a either a Maximum Profit (MP) optimization problem or a Minimum Cot (MC) optimization problem. In thi article we inetigate the relationhip between the MP and MC formulation. When the erice charge are zero, the MP and MC formulation are obiouly equialent. By uing the et of Lagrange multiplier from the MC formulation a a tool, we inetigate the extent that thi equialence hold with repect to the erice charge. The ame et of multiplier alo act a threhold for erice charge o that the SON operator will be happy to proide adequate erice leel een if he/he i not obligated to do o. A key contribution of thi paper i the proiion of inight into the olution nature of the MP and the MC formulation under different erice charge parameter, thereby giing guideline to the proper formulation hould be choen. The econd contribution i the ue of the Lagrange multiplier in proiding pricing information to the network operator. Keyword Serice oerlay network deign, GoS, Capacity allocation, Profit maximization, Cot minimization, Optimization method, Pricing of network erice. INTRODUCTION It i not uncommon that in formulating network deign problem, one may be confronted with the choice of chooing whether to minimize the inetment or to maximize the profit from the network. While thi choice of the criterion of optimization could hae profound influence on the final olution obtained, we note that in the literature the choice are uually made in a omewhat undicloed manner without giing the detailed background rationale to the reader. A natural quetion i: Since the MC and MP formulation optimize different objectie, how do their olution compare? Or to be pecific, for a gien et of parameter, which formulation hould be picked and how do the olution compare. Thi paper addree the objectie function choice problem faced by the operator for a cla of Serice Oerlay Network (SON) capacity allocation problem. We identify the region uch that Maximum Profit (MP) and Minimum Cot (MC) formulation offer the ame olution. We alo identify the region uch that MP gie different olution than the MC formulation. It i hown that the et of Lagrange multiplier from the MC formulation play an important role in eparating the aforeaid region. When the erice charge are lower than the et of multiplier, the MC and MP formulation will gie the ame olution. When the erice charge are higher than the multiplier, the olution will be different. The ame et of Lagrange multiplier alo act a threhold for the erice charge o that the SON operator would proide adequate erice leel under the MP formulation. All thee oberation will be elaborated hortly. Thi paper i tructured a follow: ection i the decription of the problem aumption and the employed analytic model, ection 3 dicue the major reult, ection 4 how the relation between thi tudy and the pricing of erice with GoS guarantee, a imple example illutrating the obtained reult will be proided in ection 5, and ection 6 i the concluion ection that dicue the reult obtained and poible future extenion.. PROBLEM DESCRIPTION The SON network operate in a manner imilar to a irtual network. The SON operator own the SON gateway which are placed in trategic location. To realize the SON network, the SON operator leae bandwidth with QoS guarantee from the underlying Autonomou Sytem, (ASe) in the form of Serice Leel Agreement (SLA). The leaed bandwidth proide logical connection to the oerlay network. Once the bandwidth are in place, the SON i realized and i ready to offer end-to-end QoS guarantee for the alue-added erice it proide (i.e. VoIP, Video On Demand erice, etc). Uer with acce to the Internet can acce the erice gateway and ue the alue-added erice. The SON connection are claified by the ource and the detination gateway. Uer pay the erice charge baed on the ource and detination of their connection a well a their connection duration. To deploy a SON, a major challenge faced by the SON operator would be the optimal amount of bandwidth to be leaed on each of the logical link. The allocated bandwidth hould proide the operator with maximum economic benefit yet meet the uer expectation regarding the connection acceptance ratio. Thi acceptance ratio i alo known a the Grade of Serice requirement (GoS) which are uually repreented in the form of connection blocking probabilitie. The problem confronting the SON operator i: gien a et of SON gateway, decide the optimal bandwidth to be leaed on the logical link when the traffic intenitie and the gateway location are known, and the erice charge a well a GoS requirement are gien.

2 The maximum profit formulation of thi problem i gien by the expreion in (). max w ( B ) C ( N ) N B L u N 0 z () Where i the poionian connection arrial rate for the node pair (i.e. ource node i i, detination node i j), w i the expected reward generated by an admitted connection, both and w are aumed to be gien parameter. The ariable B i the blocking probability for traffic pair, which i a alue returned by the routing layer. The GoS contraint for each OD pair i gien by L which pecifie the maximum allowed connection blocking probability, it i aumed that the Erlang B equation i employed to quantify the blocking probabilitie. The capacity of a link i denoted by N and it i a deciion ariable of thi problem. The function C (.) i the cot function that quantifie the cot rate of allocating N unit of capacitie on link and it i aumed to be a linear function of the ariable N. The ariable u and z are the Lagrange multiplier with repect to the contraint. The minimum cot formulation of the ame problem i gien by (). min N C ( N ) B L N 0 z () It i important to mention that the blocking ariable B donate the theoretical blocking probabilitie of the OD pair and thee probabilitie are dependent on the routing cheme employed and it performance model. For the ame network configuration (ame allocated capacitie), different routing cheme may gie different B alue. A routing cheme that utilize the network reource efficiently may return lower B alue than a imple and primitie routing cheme een with the ame et of allocated capacitie. 3. THE MP AND MC FORMULATIONS 3. The Equialence of MP and MC Formulation From () and () the Maximum Profit (MP) and the Minimum Cot (MC) formulation are identical if the w are all zero. Intuitiely when the w gradually increae, the cot would till dominate the objectie until a threhold i hit. In other word, the MP and MC formulation hould be equialent when the w are maller than ome threhold. In thi ection we formally inetigate thi and how the condition that the MP and MC formulation are equialent. The reult rely on the non-negatie nature of a et of multiplier. The trictly conex property of the Erlang B equation i alo employed to make the proof clearer. Only the reult for the cae where the MC formulation ha a unique olution are hown, and thi i aumed throughout the whole article. The reult for the multiple-olution cae will be included in an extended erion of the paper. Before howing the main reult, we need to proe a lemma that etablihe the uniquene of the allocated capacity with repect to the cot, a et of real alue (which mimic the Lagrange multiplier correponding to the GoS contraint) and the link traffic intenity (i.e. offered traffic intenity) ector Λ. Lemma : For poitie contant, c, and fixed Λ, there exit at mot one N on the link that atifie the following equation: B c = ( ) (3) N θ Where θ i a et containing the indexe of all Origin-Detination pair that hae link in their route, N i the capacity of link and it i relaxed to be a continuou ariable, c i the cot of allocating one unit of capacity on link, are ome poitie real number, Λ i the ector of traffic intenitie on the link a deignated by ome optimal routing rule. Aume the continuou extenion of Erlang B equation uggeted in [] i being employed. B can be written a (4), where R i a et that contain all the link being ued by the route of traffic, the function f (.) and f (.) are to characterize the blocking of traffic due to link other than link in the route, therefore they are both non-negatie. The function E (.) i the Erlang B equation that quantifie the blocking due to lack of reource on link. The alue are the element of Λ which denote the offered traffic intenity on link. The idea for (4) i that the blocking probability of the traffic pair inole term that are independent of link and alo term that are dependent on link. Thi idea i rather general and the expreion (4) i alid for ariou routing cheme like the alternate routing cheme [3], [4], the load haring routing cheme [5], and extended erion of the load haring/alternate routing cheme [6]. Note that for expreion (4) to be alid, the condition R i needed. If thi condition i not atified, (4) can be till employed but f (.) become zero.. B = f (, N ) + f (, N ) E (, N ) (4) ' R ' R ' R ' R ' ' ' ' By ubtituting (4) into (3) and by auming link independence we hae: θ c E (, N ) = (5) f N N ( ' R, ' ) R ' ' Since the function f (.) are independent of N and, they can be regarded a a non-negatie contant function with alue in the interal (0,], note that 0 i not included ince θ. Denote E (, N ) by f ( 3, ) N. It i known that the continuou Erlang B formula i a C function [], which i trictly conex in the capacity []. Therefore for a fixed, the function f 3 (.) i trictly decreaing and continuou in N, and there i an one-to-one correpondence between the function alue and N for the fixed. Moreoer it can be een that f 3 (.) i poitie (ince when N i increaed by delta, the function -E (.) alway increae if i fixed ), o for poitie contant, c, and fixed offered traffic intenity on the link, there i a unique N that atifie equation (5). There are two pecial cae for expreion (5), firt if the LHS

3 of (5) i larger than max( f (, N )) then N doe not exit, econd N 3 if the cot c equal to zero (which i unlikely), then N equal infinity. Q.E.D. Expreion (3) together with the complementary lackne and the non-negatiity requirement of the multiplier correpond to the t order neceary condition of the MC formulation. The phyical interpretation of lemma i that when the link traffic intenitie are fixed, then for each et of c and, there i at mot one N alue that ole (3) and thereby atifying the t order neceary condition. The reult implie that we may write N a N (c,, Λ). Thi lemma proide a tool for u to how the condition uch that the MC and MP formulation are equialent. The proof of the following theorem relie on the non-negatiity nature of the multiplier correponding to the GoS. Theorem (relationhip between the minimum cot and maximum profit formulation): Conider a maximum profit formulation () and a minimum cot formulation () of the SON capacity allocation problem. If a et of Lagrange multiplier exit for (), and if the connection reward w atify condition (6), then an optimal olution to () i alo an optimal olution to (). Further if there i a unique et of for (), then () and () are equialent. 0 w (6) The tationary condition for () and () are lited in (7) and (8) repectiely, where c i a contant cot for allocation a unit bandwidth on link. Note that (7) and (8) both repreent n et of equation where n equal to the number of link in the network. Again θ in (7) and (8) i a et that contain the indexe of all OD c w u = ( + ) (7) p p θ Λ c = (8) c c θ Λ pair that conit of link in their route. Note that the capacitie c N and p N are repreented a function of the link traffic intenitie to explicitly indicate the dependence on the olution of the routing layer. Suppoe are the Lagrange multiplier at optimality for (). It i eay to ee that if (6) i atified, letting u = w 0 for all the OD pair in all the n equation of (7) would make the alue ( w + u ) equal to for all the n equation. If (6) θ θ i not atified, thi will not be alway poible ince u mut be c non-negatie for all. Aume a ector of link intenitie Λ i deignated by ome optimal routing rule, and uppoe the tuple c c c ( Λ, N ( Λ ) ) i a olution that atifie (8) and alo the complementary lackne condition. Then it i obiou from c c c c lemma that the olution ( Λ, N ( Λ ) ) i unique for the Λ. Now if we ubtitute the olution along with the u into (7), the olution hould atify (7) if it atifie (8), thi i becaue that ( w + u ) = for all, and alo becaue that the expreion Λ depend only on Λ and N ( Λ ). The complementary lackne condition are alo atified ince the contraint of () and () are identical. In other word, when (6) i atified, a olution that atifie the firt order necearily condition of the formulation () mut alo atify the t order necearily condition of the formulation (). With the ame et of u (i.e. u = w ), the econd order ufficient condition check i triial, becaue the Heian matrice of the correponding c c c Lagrangian dual are identical. If the olution ( Λ, N ( Λ ) ) atifie the nd order ufficient condition of formulation () then it mut alo atify the ufficient condition of formulation (). Therefore we hae hown that the optimal olution of MC i alo an optimal olution of MP. Further to thi, if the et of i unique and condition (6) i atified, then formulation () and () are equialent. Thi can be proed by contradiction. If there exit another et of multiplier x uch that ( w + x ) = y for the formulation (), where x 0 and 0. Suppoe for c c c any optimal olution ( Λ, N ( Λ ) ) that correpond to x and atifie both the neceary and ufficient condition of the formulation (). It i obiou that the multiplier y = ( w + x ) c c c along with ( Λ, N ( Λ ) ) atify (8). Moreoer thi olution mut alo atify the econd order necearily condition of () a long a it atifie thee condition for (), ince formulation () and () hae the identical firt order and econd order optimality condition. That implie y i alo a et of multiplier for (). Thi contradict with the claim that () ha a unique et of multiplier. Therefore if (6) hold and the et of multiplier i unique for (), then () alo hae a unique et of multiplier u uch that ( w + u ) =. And thi how the equialence of the MP and the MC formulation. Q.E.D. For the dicuion in thi article, the ector Λ i aumed to be unique with repect to a gien et of multiplier. Yet thi requirement i not needed for the aboe proof to be alid (but then we may hae multiple optimal olution een if the et of multiplier i unique). 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. condition (6)), the MC and the MP formulation both gie the ame olution to the capacity allocation problem. In other word, if the erice charge are lower than a et of threhold, formulating the SON capacity allocation problem a either the MC or the MP problem doe not matter a thi i the region uch that MC and MP are equialent. A quetion 3

4 one may ak i: what if the erice charge are higher than the threhold? The reult for the quetion i preented in ection The MP and MC Formulation a Differentiated by the Serice Charge If condition (9) hold for the erice charge w, then we hall how in thi ection that the MP formulation proide olution with trictly better performance than the MC formulation in term of profit generation. Although ome of the reult obtained may look omewhat triial, the formal inetigation of the problem proide u with aluable inight into the relationhip between the erice charge and GoS guarantee which will be dicued in the ection 4. 0 < w (9) Note that though condition (6) and (9) are omewhat complementary, they are in fact rather looe a there are grey region which are not coered. Thee grey region are the region where 0 w < for ome, and 0 w for the remaining. To tackle thee region, the tudy will inole the comparion of the term R ( w ) and R = θ =. θ Our reult indicate that if there exit at leat one R that i larger than R for ome link, then the MP formulation will gie olution that generate more profit, otherwie the MP and MC formulation gie olution that generate the ame profit. The detailed analyi of thee cae require more careful treatment which need ignificantly more pace, thee reult will be included in a future paper a mentioned earlier. The reult proided here i a implified erion baed on the condition (9). Before we preent the main reult, again we firt need a lemma. Thi lemma etablihe the relation between the optimal capacitie allocated and the magnitude of erice charge. Lemma : Conider equation (0), where c i a poitie contant. Aume that the link traffic intenitie, Λ, are fixed. If, are two ector whoe element are indexed by (i.e. =[ ] T, =[ ] T ), then if hae N > > > 0 for all, and N N and N both exit then we, where N and N are the alue of N in (0) that correpond to the ector and repectiely. θ B c = ( ) (0) N Subtituting (4) into (0) we hae: E(, N ) c = f (, N ) () ' R ' R θ ' ' From the proof of lemma, It i known that E(, N ) i trictly decreaing and continuou for fixed, moreoer it i alo known that f (.) are non-negatie contant function with repect to N. So the expreion θ E(, N ) f (, N ) ' R ' R ' ' continuou and trictly decreaing in N. A a reult for a contant c, the larger the function f (, N ) the θ ' R ' R ' ' maller the expreion E(, N ) will be required to atify the equality condition of expreion (), thi therefore require a larger N alue. It i eay to ee that if > > 0 for all, the function f (, N ) θ ' R ' R ' ' f (, N ), o if N and N ' R ' R θ ' ' N > N. Q.E.D. y i will be larger than both exit then With lemma, we can now continue with our main theorem of thi ection. Theorem : Aume the link traffic intenitie at optimality, Λ, are the ame for the MC and MP formulation. When condition (9) hold, the MP formulation delier olution with trictly better performance than the MC formulation in term of profit. Note that when condition (9) hold, u in (7) are all zero at optimality, thi i due to the theorem 3 in ection 4 and the complementary lackne condition. Now expreion (7) can be written a () c = θ ( w ) () p From the expreion (8) and () and alo lemma, it i eay to erify that when condition (9) hold the optimal capacity allocated by MP i trictly larger than that of MC on all link. To how the profit generated by the olution of MP formulation i trictly larger if condition (9) hold, we rewrite (9) a 0 + x = w, where x > 0 for all. We ubtitute thi into () and rearrange the term we hae (3) c = ( ) + ( x ) (3) p p θ θ c we ubtitute the optimal olution of MC (i.e. N ) into the RHS of (3) and we get (4), 4

5 θ ( ) + ( x ) (4) c c θ expreion (4) denote the marginal profit from the link at the c allocated capacity of N, and it can be further modified to (5) + ( x ) (5) c θ c from the proof of lemma, we know that the nd term of the expreion (5) i trictly poitie, therefore expreion (5) c ugget that at the optimality of MC formulation (i.e. N ), marginal reward i larger than marginal cot on the link, which implie that additional profit can be generated if extra capacitie are allocated. Therefore when (9) hold, the MP formulation proide trictly better performance in term of profit generation. Q.E.D. The reult of theorem i omewhat triial, becaue after all, the MP formulation i to maximize the profit, but from the analyi in theorem, we can gain the addition inight that the olution of the MC formulation offer wore profit than that of the MP formulation becaue the MC formulation merely allocate the minimum amount of capacitie to atify the GoS requirement. Additional profit that could hae been gained from the high reward connection i lot. The main concluion from ubection 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 are the major concern, the MP formulation minimize the cot, when the reward are high enough, the MP formulation witch to maximize the profit. Therefore it i enible to deign the SON network uing the MP formulation. If the MP formulation i choen, it can be further hown that the multiplier in (9) act a a et of threhold price for the erice o that the operator would be happy to delier adequate leel of GoS een if he i not contrained to do o. We hall elaborate thi in the following ection. 4. THE PRICE OF OFFERING GOS: AN OPERATOR S PERSPECTIVE In thi ection we aume that the SON operator regard the profit a the primary metric for the network performance. The following theorem formally how that the multiplier act a a form of threhold for erice charge in the MP formulation. Theorem 3: Aume that OD-pair blocking probabilitie are trictly decreaing function of allocated capacitie. If (9) hold then the olution of MP formulation will delier lower blocking than that of the MC formulation. Moreoer all the GoS contraint in () will be nonbinding at optimality, and the optimal olution for () will be the ame a it uncontrained counterpart. if (9) hold then ( w ) >, by lemma, we hae p N > c θ θ N, where p N i the capacity ector that atify (7). Since p N > c p N, the GoS offered by N mut be better than that c being offered by N. It i known that all the GoS contraint are c p atified by the allocation N, o it mut be atified by N. Since all the GoS requirement are atified at the optimal olution of () without the need of haing the GoS contraint if (9) hold, o the olution of () remain the ame een if all the GoS contraint are relaxed. Q.E.D. Aume that the MC formulation ha a unique et of multiplier. For the ake of dicuion we define a et of threhold alue in (6): w% = w% (6) If the erice charge w are le than the w% for all, then from theorem, the MP and MC formulation are equialent, and the multiplier u in (7) will be poitie. From the complementary lackne condition, the MP formulation will delier a olution with tight GoS contraint. It can alo be een from theorem 3 that if the erice charge w are larger than w%, the MP formulation will delier GoS erice leel better than that are being required by the GoS contraint. The et of w% act a a threhold et for the GoS erice leel. In other word, the alue w% can be interpreted a the minimum erice price that drie the network operator to offer the required leel of GoS when they are not contrained to offer any GoS guarantee. Thi et of w% proide intereting cot information in the context of proiding GoS guarantee. The threhold in (6) are deried from the multiplier, which are well known metric that quantify the price of the GoS contraint: the cot objectie in () can be improed by unit if the correponding GoS contraint i relaxed by one unit. So, intuitiely thi i alo the amount of reward the uer of the OD pair hould bring to the network o a to enjoy the GoS. 5. AN ILLUSTRATION EXAMPLE Conider a imple three-node SON network a hown in figure. Aume that there are three Poion tream of traffic in the network, and the offered traffic intenitie for the three independent tream are AB =0 per unit time, CB =5 per unit time, AC =0 per unit time. To make the dicuion imple, all the traffic tream are routed through direct link (i.e. direct link AB, CB and AC). Without lo of generality the mean holding time of the traffic are aume to be identically ditributed with unit mean. The cot for leaing one unit of bandwidth for one unit of time are 5 unit, 6 unit and 7 unit repectie for the link AB, CB, and AC, and the allocated capacitie are aumed to be 5

6 integral alue. Aume that GoS requirement for all the tream are 0. (i..e.0% probability of blocking)), thi alue i deliberately made large o a to facilitate eay comparion. The multiplier, and are found to be 8, 67 and 37 AB CB AC repectiely (which tranlate to erice charge of 8., 7.8 and 8.6). To illutrate theorem, we deliberately et the erice charge to be (0, 0, 0) for the tream AB,CB and AC. Condition (6) i atified under thi et of erice charge. Table ummarize the reult obtained. We can ee that, under thee erice charge, the MP and MC formulation do gie the ame allocated capacitie een though the objectie alue are different, which confirm the reult in theorem. Figure. A imple SON network. Table. Capacity allocation reult for low erice charge Serice charge (0,0,0) GoS ( AB, CB, AC) Allocated capacitie on link (AB, CB,AC) MP formulation MC formulation (0.084, 0.086, 0.085) (0.084, 0.086, 0.085) (3, 8, 3) (3, 8, 3) Objectie alue To illutrate theorem, we et the erice-charge ector to be (00, 00, 00). Table ummarize the reult obtained. The profit rate improement of the MP oer the MC i The MP formulation delier better performance oer the MC formulation in thi cae. Thi i conitent with the reult proed in theorem. Table. Capacity allocation reult for high erice charge Serice charge (00,00,00) GoS ( AB, CB, AC) Allocated capacitie on link (AB, CB,AC) MP formulation MC formulation (0.0037,0.009,0.0034) (0.084,0.086,0.085) (9, 6, 3) (3, 8, 3) Expected Profit rate Since the olution are rounded up from the real-alued optimal olution, the GoS contraint are not tight een for the MC formulation, but note that in table when the erice charge atify (9), the olution of the MP formulation indeed offer lower blocking probabilitie than that of the MC formulation and thi illutrate the reult in theorem CONCLUSIONS AND FUTURE WORKS We tudied a cla of Serice Oerlay Network (SON) capacity allocation problem with GoS contraint. By uing the et of Lagrange multiplier in the MC formulation, we howed the condition uch that the MP and MC formulation are equialent. Moreoer we alo howed the condition uch that the two formulation are different. It i particular intereting to note that when the erice charge are low the MP formulation minimize the cot/inetment in realizing the SON network by proiding minimum leel of reource to meet the GoS contraint. When the erice charge are high enough, howeer, the MP formulation automatically take adantage of that and witche to maximize the profit in realizing the SON network. Thi phenomenon ugget that MP formulation adapt well to the different range of parameter in the SON deign problem and i attractie to be employed a the dimenioning formulation. By employing the MP formulation in the SON capacity allocation problem, we are able to quantify the price of offering GoS in the SON. An intereting fact i that regardle that the MP formulation i being employed; the aforeaid GoS price are related to the et of Lagrange multiplier of the MC formulation. Major effort are being pent to refine the reult obtained regarding the relationhip of the MP and MC formulation when the MC formulation ha multiple optimal olution, the poible extenion to the cae where heterogeneou blocking are preent i alo being tudied. Moreoer we are alo inetigating the extenion of the reult to the more general multi-rate cae. Mot of the new reult will be included in a more comprehenie future paper. 7. REFERENCES [] A. Jager and E. V. Doom On the continued Erlang lo function, Operation Reearch Letter, Vol. 5, No., pp , 986. [] J. S. Etee, J. Craeirinha, and D. M. Cardoo, Second order condition on the oerflow traffic from the Erlang-B ytem, Caderno de Matemática, Unieridade de Aeiro, CM06/I-0, 006. [3] A. Girard and B. Liau, Dimenioning of adaptiely routed network, IEEE tranaction on networking, Vol., No. 4, pp , 993. [4] A. Girard, Reenue optimization of telecommunication network, IEEE tranaction on communication, Vol. 4, No. 4, pp , 993. [5] F.P. Kelly, Routing in circuit-witched network: Optimization, Shadow Price and Decentralization, Adanced in applied probability, Vol. 0, No., pp. -44, 988. [6] N. Lam, Z. Dziong and L.G. Maon, Network capacity allocation in erice oerlay network, in Proc. 0th International Teletraffic Congre (ITC), Ottawa, 007, pp [7] A. Girard, Routing and dimenioning in circuit-witched network, Addion-Weley, 990. [8] D. P. Berteka, Nonlinear Programming, nd edition, Athena Scientific,