Heuristics for dimensioning large-scale MPLS networks

Heurisics for dimensioning large-scale MPLS newors Carlos Borges 1, Amaro de Sousa 1, Rui Valadas 1 Depar. of Elecronics and Telecommunicaions Universiy of Aveiro, Insiue of Telecommunicaions pole of Aveiro ABSTRACT MuliProocol Label Swiching (MPLS) echnology allows he suppor of muliple services wih differen Qualiy of Service (QoS) requiremens in classical IP newors. In an MPLS domain, pace flows belonging o a paricular class are classified in he same Forward Equivalence Class (FEC). Based on differen FECs, each service can be se up in he newor hrough logical newors. Each logical newor is a se of Label Swiched Pahs (LSPs), one for each service raffic run. The newor-dimensioning problem is formulaed as he deerminaion of roues for all LSPs o achieve he leas cos physical newor. To solve his problem some widely nown heurisics are used and wo enhancemen algorihms are proposed ha allow for significan gains when compared wih he basic heurisics. The heurisics esed include a geneic algorihm, a greedy based heurisic and a lagrangean relaxaion based heurisic. The enhancemens are proposed for applicaion o he greedy based heurisic and o he lagrangean heurisic. The resuls show ha he enhanced lagrangean heurisic is he bes overall echnique for he case sudies presened. This echnique yields significan average gains when compared o he basic lagrangean heurisic. Keywords: MPLS, newor dimensioning, opimisaion, heurisics, lagrangean relaxaion 1. INTRODUCTION In oday s newor echnologies, he successful inegraion of services wih differen QoS requiremens is one of he mos criical issues. Nowadays, here is an unprecedened demand of IP based services ha require higher level of conrol and predicabiliy, such as Voice Over IP, Virual Privae Newors, and ohers. As such, effors have been made o provide some QoS and rouing policy capable mechanisms wihin he ypically bes-effor based Inerne. One such mechanism is he newly developed MPLS forwarding scheme 1,2. MPLS echnology combines some of he benefis of previously exising ATM echnology wih he advanages of IP based soluions. In paricular, MPLS provides connecion-oriened swiching ha allows for beer raffic engineering and managemen in IP newors. An MPLS domain is a newor composed by a se of Label Swiched Rouers (LSRs) and a se of connecions beween LSRs 3. The rouers on he fronier of he MPLS domain are called edge rouers ha ac as ingress rouers for incoming IP pace flows and egress rouers for oucoming IP pace flows. LSRs ha inerconnec edge rouers are named core rouers. An LSR can wor boh as an edge rouer and as a core rouer. A he ingress rouer of an MPLS domain, paces are classified ino a Forward Equivalence Class (FEC) based on IP header informaion such as source and desinaion addresses, por numbers, ype-of-service bis, ec. Then, a label is added o each pace, which is forwarded hrough one of he Label Swiched Pahs (LSPs). An LSP is a se of physical connecions ha form a pah beween one ingress LSR and one egress LSR. When an LSP is se up, suiable label ranslaion enries are configured in all inermediae LSRs along is pah. LSPs can be configured explicily hrough adminisraion acion or may be se up by an underlying consrained-based rouing proocol. A raffic run is an aggregaion of raffic flows of he same class ha is characerized by is ingress and egress rouers and he FEC ha is associaed wih 4. Each ime a new raffic flow reservaion is requesed or released in an ingress LSR, he bandwidh reservaion for he corresponden raffic run along is assigned LSP is updaed. In his paper, we address wo slighly differen newor-dimensioning problems. In boh problems, we assume ha for all suppored services, he maximum bandwidhs of all raffic runs are nown based on expeced raffic flow demands and QoS requiremens. The physical newor-dimensioning problem considers he deerminaion of a physical newor wih enough ransmission capaciies beween rouers o accommodae all raffic runs of all services a heir maximum bandwidh. We assume also ha each raffic run mus be conveyed hrough a single LSP. Rouer locaions are 1 Email: clopes@av.i.p, asou@ua.p, rv@ua.p Phone: +351 234377900; fax: +351 234377901; Insiue of Telecommunicaions, Universiy of Aveiro, Campus de Saniago, 3810-193 Aveiro, Porugal

represened by nodes in an undireced graph and available ransmission faciliies beween rouer locaions are represened by edges in he same graph. The newor-dimensioning problems addressed here belong o a class of problems nown in he Operaions Research field as muli-commodiy capaciaed newor design problems. These problems are hard o solve. Easier versions of hese problems were solved up o he opimaliy only for graphs up o 16 nodes 5,6. For large-scale newors, one soluion is he uilisaion of heurisics ha can calculae good feasible soluions in an efficien way. For ATM newors, similar dimensioning models were invesigaed and an heurisic based on Lagrangean relaxaion wih sub-gradien opimisaion has been used o generae feasible soluions in very low compuing imes 7,8. Taing his heurisic as he saring poin, we explore some oher well-nown heurisic approaches and propose wo enhancemen algorihms in order o achieve beer dimensioning resuls. This paper is organised as follows. Firs, he newor dimensioning models are presened in secion 2. Then, secion 3 describes he differen heurisics adoped for he soluion of he dimensioning models. Finally, secion 4 presens compuaional resuls and he conclusions on he performance comparison of he proposed heurisics. 2. NETWORK DIMENSIONING MODELS Consider ha a newor is represened by an undireced graph (N, A). The nodes represen LSR locaions and he edges represen available ransmission faciliies beween LSR locaions. Consider ha T is he se of ypes of ransmission faciliies available. The newor-dimensioning problem is modelled as an opimisaion problem. This problem aims o minimise he operaional and mainenance cos of he ransmission faciliies in a physical newor. The cos of each ransmission faciliy is given by a fixed swiching cos a he edge end-nodes and by a ransmission cos, which is proporional o he lengh of he edge. Therefore, he cos of a ransmission faciliy a edge {i, is equal o: C { i, = 2 Swiching_cos_of_ + Edge_{i, _lengh Transmission_cos_per_lengh_uni Consider he following noaion: N Se of rouer locaions A Se of edges, i, j N where ransmission faciliies can be se up T Se of possible ypes of ransmission faciliies defined by: α Bandwidh capaciy of ransmission faciliy of ype, α > 0 K i j i j i, j > C {, } Cos of a ransmission faciliy of ype in edge, C { } 0 Y, Maximum number of ransmission faciliies of ype on edge { { } Se of raffic runs o be suppored by he newor defined by: o Ingress rouer locaion, o N d Egress rouer locaion, d N b Maximum bandwidh, b > 0 i, j i,, Y { } 0 y { i, Ineger variables ha define he number of ransmission faciliies of ype a edge x i, j Binary variables: if se o 1, define ha he edge { { } run is conveyed i, is in he pah of he LSP hrough which raffic In his noaion, we assume ha raffic runs are symmerical, e.g., hey require he same maximum bandwidh in boh direcions. However, he newor dimensioning problems and heurisic soluions presened in his paper can be easily exended for he asymmerical case. Variables y represen he physical newor soluion given by he ype and number of ransmission faciliies o be insalled beween rouer locaions. Binary variables x represen, for each raffic run, an LSP pah soluion in he physical newor defined by variables y. The firs dimensioning problem considered (named as P1) is defined as follows:

Minimize: C{ i, y (1) T Subjec o: { x { i, j } : x { i, =1} is a pah from o o d, K (2) b x α y, A (3) K T y Y, A, T (4) x ij {0,1}, y { i, non negaive ineger (5) The objecive funcion (1) is he newor cos funcion, which is he sum of he coss of all ransmission faciliies. Consrains (2) sae ha variables x mus define a pah for each LSP. Consrains (3) guaranee ha he capaciy of ransmission faciliies is enough o suppor he maximum bandwidh demand of all raffic runs ha cross i. Consrains (4) limi he maximum number of ransmission faciliies of each ype in each edge. P1 assumes LSRs can be conneced wih more han one ransmission faciliy in parallel. Consrains (3) assume ha i is possible o spli he raffic run bandwidh beween differen ransmission faciliies. This means ha he se of ransmission faciliies insalled beween wo connecing LSRs is reaed as one ransmission faciliy wih he oal bandwidh. This soluion is only possible if MPLS newor suppors load disribuion across parallel raffic runs 4. The second problem addressed assumes ha load disribuion across parallel raffic runs is no available on rouers and herefore, one only ransmission faciliy can be deployed beween wo connecing LSRs. This second dimensioning problem (named P2) is defined by subsiuion of consrains (4) in P1 wih he following consrains: T y, { i A 1, (6) These consrains guaranee ha he number of ransmission faciliies in each edge is a mos one, whaever ype of ransmission faciliy is seleced. 3. HEURISTICS DESCRIPTION 3.1 Geneic Algorihm This heurisic assumes ha, for each raffic run, we have a se of candidae pahs o chose he associaed LSP. These pahs are generaed hrough an N-shores pahs compuaion algorihm 9. This adds anoher degree of freedom o our problem: he number of candidae pahs. A small number limis he se of soluions, possibly neglecing some of he bes ones. A large number will increase he complexiy of he algorihm, maing i harder o converge o good soluions. Each soluion is represened as a chromosome composed of N genes, where N is equal o he number of raffic runs. Each gene conains a number ha indicaes one of he candidae pahs, as depiced in Figure 1. Figure 1: A chromosome represening a soluion for 8 raffic runs. The informaion conained in a chromosome along wih he specificaion of he candidae pahs defines he x variables. To deermine he y variables, he bandwidh required a each edge is calculaed as he sum of he bandwidh demand of all raffic runs, whose LSPs use he edge. Then, he se of ransmission faciliies ha provides his bandwidh a he lowes cos is chosen from a previously ordered lis of ses. This lis is ordered by increasing amoun of bandwidh and does no include he ses ha provide an already exising bandwidh capaciy a a greaer cos. For problem P2, such lis consiss of ses wih only one ransmission faciliy. The evoluion of he populaion is based on a muaion operaor and a wo poins crossover operaor 10. The selecion of individuals for he crossover operaion is based on roulee wheel selecion. The meri funcion is given by:

f ( x) = c c x i X i min c c min 1 + 1 + α α ( c c ) (7) max ( c c ) max min min where f(x) is he probabiliy of soluion x of being seleced; c x is he cos of soluion x; c min is he cos of he bes soluion in he populaion; c max is he cos of he wors soluion in he populaion; α is a parameer ha defines he grade of seleciviy (a smaller value of α implies ha he soluions wih he higher coss have less chances of being chosen for crossover). A he beginning of he geneic algorihm all genes in he populaion are assigned random values uniformly chosen beween [0; NKP[, where NKP is he number of candidae pahs. This means ha we begin wih soluions of random candidae pahs for all he LSPs. Then, for he nex ieraions, new soluions are generaed hrough selecion and crossover and he applicaion of he muaion operaor o a given percenage of he genes of he new populaion. The new populaions generaed replace all he soluions of he old populaions. The soluion of he geneic algorihm is given by he soluion wih lower cos among all soluions of all generaed populaions. 3.2 Greedy based heurisic This heurisic is based on running a simple greedy algorihm several imes. The algorihm is defined as follows: Repea for a given amoun of ime: Randomly sor he LSPs For each LSP do: Esablish he LSP hrough he pah ha requires he leas cos in addiional ransmission faciliies Deploy he addiional ransmission faciliies Reserve he associaed LSP raffic run bandwidh along he chosen pah Evaluae he soluion and if i is he bes found so far, save i as he final soluion. Remove all he ransmission faciliies The esablishmen of each LSP hrough he leas cos pah aes ino accoun he already deployed ransmission faciliies and he exceeding bandwidh available a each edge. If he bandwidh of he raffic runs assigned o an LSP can be esablished hrough an edge wih exceeding bandwidh, hen he cos of he edge is se o zero; if no, he edge cos is equal o he difference in cos beween he new se and he already insalled se of ransmission faciliies. 3.3 Lagrangean heurisic The lagrangean heurisic implemenaion is based on previous published wor 7,8. In each of he ieraions, a feasible soluion is deermined based on he values of he x variables of he lower bound soluion provided by he lagrangean relaxaion echnique 11. Given a se of lagrangean mulipliers λ, he lagrangean lower bound program (LLBP) is defined as: ( ) { } ( { } { } ) { } Minimize: λ b x i, j + C i, j λ i, j α y i, j (8) K T subjec o consrains (2), (4) and (5) for problem P1 or (2), (5) and (6) for problem P2. The firs par of (8) (he member ha depends on he x variables) is calculaed hrough he use of a shores pah algorihm for each LSP. For problem P1, he second par of (8) is calculaed by replacing he y variables wih: 0, C λ{ i, α y = { } i, j Y, C < λ α (9) For problem P2 he second par of (8) is given by: ( ) min min C λ α, 0 (10) T

In his case and for each edge {i,, variables y { i, are all zero if all erms C { i, j } λ { i, j } α are posiive or he variable y { i, is se o one for he index whose erm C { i, j } λ { i, j } α is minimum. Based on he variables x of he soluion of LLBP, a feasible soluion (variables y) is deermined in he same way as described for he geneic algorihm: i is based on a previously ordered lis of ses of ransmission faciliies. Given a se of lagrangean mulipliers λ, a new se is compued hrough sub-gradien opimisaion 12. Le Z LB be he curren lower bound o he problem, Z UB a feasible soluion and π a relaxaion parameer in he range ]0; 2]. The procedure is as follows: Se he lagrangean mulipliers equal o zero Se π equal o 2 Repea for a given amoun of ime Solve LLBP wih he curren se of lagrangean mulipliers o obain Z LB Deermine a soluion based on he values of he x variables of Z LB o obain y { i, Evaluae he soluion: if i is he bes found so far, se Z UB equal o is cos and save i as he final soluion Calculae he subgradiens as: G = b x α y (11) K T Calculae T as: π ( ZUB Z LB ) T = (12) 2 ( G ) Updae he lagrangean mulipliers as: λ { i, = max ( 0, λ{ i, + TG { i, ) (13) If Z LB has no improved in he las 40 ieraions halve he value of π 3.4 Heurisic Enhancemens Boh greedy based (secion 3.2) and lagrangean algorihms (secion 3.3) are ieraive processes ha ry o generae a feasible soluion in each ieraion. In his secion, wo heurisic enhancemen algorihms are proposed ha use each feasible soluion found by previous heurisics as he saring basis o find beer feasible soluions. The firs proposed heurisic enhancemen (named E1) is defined as follows: Repea for a given amoun of ime: Perform an ieraion of he greedy/lagrangean heurisic o obain a soluion Evaluae he soluion and if i is he bes found so far, save i as he final soluion If he soluion is feasible, perform for a given number of ieraions NSI: Deermine a random order for he LSPs Repea unil all LSPs have been removed and re-esablished Remove he ransmission faciliies required for N LSPs according o he random order For each of he N removed LSPs do: Esablish he LSP hrough he leas cos pah in addiional ransmission faciliies Insall he required newor resources and reserve he associaed LSP raffic run bandwidh along is pah Evaluae he soluion and if i is he bes found so far, save i as he final soluion The procedure of esablishing he LSP hrough he leas cos pah is done in he same way as described in he greedy based heurisic. The evaluaion of a soluion is performed by adding he bandwidh of he associaed LSPs raffic run ha cross each edge, and by choosing he leas cos se of ransmission faciliies ha provides ha bandwidh a he edge. Wih his enhancemen algorihm, a oal of NSI NL/N new soluions are generaed for each ieraion of he

main heurisic, where NL is he oal number of LSPs, N is he number of LSPs o be removed simulaneously and NSI is he number of ieraions of he enhancemen heurisic. The second proposed heurisic enhancemen (named E2) has a somewha similar behaviour. In his case however, we remove and re-esablish all LSPs ha cross a randomly chosen edge. By doing so, we hope o decrease he amoun of bandwidh required a ha edge by rying o esablish some LSPs hrough oher edges wih exceeding bandwidh available. The enhancemen algorihm E2 is defined as follows: Repea for a given amoun of ime: Perform an ieraion of he greedy/lagrangean heurisic o obain a soluion Evaluae he soluion and if i is he bes found so far, save i as he final soluion If he soluion is feasible, perform for a given number of ieraions NSI: Chose a random edge of he newor graph Remove he allocaed ransmission faciliies for all he LSPs esablished hrough his edge For all of he LSPs whose resources were removed do: Esablish he LSP hrough he leas cos pah in addiional ransmission faciliies Insall he required newor resources and reserve he associaed LSP raffic run bandwidh along is pah Evaluae he soluion and if i is he bes found so far, save i as he final soluion Wih his enhancemen a oal of NSI new soluions are generaed for each of he main heurisic ieraions. 4. COMPUTATIONAL RESULTS For compuaional resuls, we have considered wo ficional newors wih differen number of nodes and edges. The number of nodes and edges chosen was (50; 100) and (100; 200). The newor opologies are displayed in Figure 2. The available ransmission faciliy ypes and swiching coss are: (34Mbps, 10); (155 Mbps, 35); (622Mbps, 130); (2488 Mbps, 450). The ransmission cos per uni of lengh is 1 for all ransmission faciliies. Figure 2: The newor graphs used (50 nodes newor and 100 nodes newor) For each newor, we have considered he suppor of hree services: service 1 wih a bandwidh of 64 Kbps for each pace flow reservaion, service 2 wih a bandwidh of 128 Kbps for each pace flow reservaion and service 3 wih a bandwidh of 2 Mbps for each pace flow reservaion. Ten differen dimensioning problems were randomly generaed for each newor. In each problem, 80% of nodes were randomly seleced o be edge rouer locaions for each service. Then, raffic runs beween all pairs of edge rouers were considered. The maximum number of pace flows of each raffic run was randomly seleced beween 1-50 for service 1, 1-30 for service 2 and 1-5 for service 3. The maximum bandwidh of each raffic run is given by he produc of is maximum number of pace flows and he bandwidh reservaion of each individual pace flow.

The compuaional resuls were obained on a 300MHz Penium II wih 64Mb RAM running MS Windows2000 OS for he 50 node newor dimensioning problems and a 350MHz Penium II wih 64Mb RAM running MS Windows98 OS for he 100 node newor dimensioning problems. The heurisics were used wih he following parameers (chosen based on previous experience 13 ): Geneic algorihm: populaion size of 100 individuals; muaion probabiliy of 1%, α=0.01; NKP=5 Greedy based heurisic wih E1: NSI=1000; N=5 Greedy based heurisic wih E2: NSI=1000 Lagrangean heurisic wih E1: NSI=5; N=5 Lagrangean heurisic wih E2: NSI=100 for he 50 node dimensioning problems NSI=10 for he 100 node dimensioning problems All 20 dimensioning problems were solved considering he wo dimensioning models P1 and P2. For all heurisics, a maximum compuing ime of 3 hours was given and he he algorihm would sop if he bes soluion did no improve wihin 30 minues. Table 1 shows he average gains of each group of 10 problems relaive o lagrangean heurisic ((LH Heurisic)/LH) for he oher heurisics: geneic algorihm (GA); greedy based heurisic (GH); greedy based heurisic wih E1 (GHE1); greedy based heurisic wih E2 (GHE2); lagrangean heurisic wih E1 (LHE1); lagrangean heurisic wih E2 (GHE2). A cell wih N/F indicaes ha a feasible soluion was no found wihin he ime limi. Whenever a number n is shown beween braces, i means ha he heurisic has found a feasible soluion only for n of he 10 dimensioning problems. Table 2 shows he average elapsed imes (in seconds) beween he sar of he algorihm and he finding of he bes soluion. GA GH GHE1 GHE2 LHE1 LHE2 P1 50 nodes -12,11% -25,40% -9,64% -5,87% 2,25% 8,63% P1 100 nodes -33,95% -121,10% -86,64% -82,11% -15,15% 3,32% P2 50 nodes -31,93% -49,21% -10,54% 7,93% 4,73% 16,20% P2 100 nodes N/F -47,37% (1) -9,71% (4) -1,44% (2) 7,00% 12,23% Table 1: Average of gains compared wih lagrangean heurisic resuls LH GA GH GHE1 GHE2 LHE1 LHE2 P1 50 nodes 4,9 3514,7 1077,8 1721,0 1009,2 1294,8 2390,0 P1 100 nodes 135,2 7526,3 1586,1 9982,0 988,7 4593,1 2334,4 P2 50 nodes 17,4 2578,5 1743,3 2860,5 2358,6 2656,5 2552,5 P2 100 nodes 224,1 N/F 330,0 2428,3 980,5 3318,3 1970,0 Table 2: Average of compuing imes (seconds) The compuaional resuls show ha heurisics based on Lagrangean relaxaion wih sub-gradien opimizaion achieve he bes resuls. In paricular, he combinaion of his heurisic wih he enhancemen algorihm E2 has reached significan gains specially for dimensioning model P2. These gains are obained hrough an increase on compuing imes. However, we consider ha he maximum compuing ime allowed for he heurisics (3 hours) is reasonable for a newor-planning as and, herefore, his combined heurisic represens an improvemen on he abiliy of calculaing good feasible soluions for large-scale MPLS newors. Concerning he performance of he wo proposed enhancemen algorihms, i is clear ha he second one is beer since he average gains are always beer when E2 is applied, independenly of he problem class and of he basic heurisic. Concerning he oher approaches, he geneic algorihm proposed has very poor resuls. The greedy based heurisics perform significanly worse han lagrangean based heurisics.

In shor, a guideline for solving he aforemenioned problems would be o use he lagrangean heurisic for ime criical siuaions or exremely large insances and he lagrangean heurisic wih he second enhancemen o obain more refined lower cos soluions. 5. CONCLUSIONS In his paper, we have presened differen heurisic approaches for dimensioning large-scale MPLS newors. We have addressed wo slighly differen dimensioning problems ha arise in MPLS echnology. The firs one has been previously addressed where a heurisic based on Lagrangean relaxaion wih sub-gradien opimisaion was proposed. In his paper, we have shown he meri of his heurisic in solving MPLS large-scale newor dimensioning problems by comparing is resuls wih oher well-nown heurisic approaches (geneic algorihms and greedy based algorihms). Moreover, we have improved his heurisic hrough he inroducion of enhancemen algorihms, which significanly decrease he cos of he compued feasible soluions a he cos of an increase of compuing ime. AKNOWLEDGEMENTS This wor was par of projec PRAXIS/P/EEI/12135/98 DOTE "Dimensioning and Resource Managemen Opimizaion Algorihms for Telecommunicaions Newors", funded by Fundação para a Ciência e Tecnologia, Porugal. We also acnowledge he suppor of Porugal Telecom Inovação. C. Borges also wishes o han Fundação para a Ciência e a Tecnologia for suppor under gran BM/21073/99. REFERENCES 1. Xipeng Xiao, Alan Hannan, Broo Bailey, Traffic Engineering wih MPLS in he Inerne, IEEE Newor, March/April 2000. 2. Special issue on Muliproocol Label Swiching, IEEE Communicaions Magazine, 37(12), pp. 36-68, December 1999. 3. E. Rosen, A. Viswanahan, R. Callon, Muliproocol Label Swiching Archiecure, IETF RFC 3031, January 2001. 4. D. Awduche, J. Malcolm, J. Agogbua, M. O Dell, J. McManus, Requiremens for Traffic Engineering Over MPLS, IETF RFC 2702, Sepember 1999. 5. T. Magnani, P. Mirchandani, R. Vachani, Modeling and Solving he Two Faciliy Capaciaed Newor Loading Problem, Operaions Research, 43(1), pp.142-156, 1995. 6. D. Biensoc, O. Günlü, Capaciaed Newor Design Polyhedral Srucure and Compuaion, INFORMS Journal on Compuing, 8(3), pp. 243-257, 1996. 7. A. Sousa, R. Valadas, L. Cardoso, A. Duare, ATM Newor Dimensioning for Mixed Symmerical and Assymerical Services wih Dynamic Reconfiguraion in a Muli-Newor Provider Environmen, Third IFIP Worshop on Traffic Managemen and Design of ATM Newors, pp. 5/1-5/15, England, 1999. 8. D. Medhi, Muli-Hour, Muli-Traffic Class Newor Design for Virual Pah-Based Dynamically Reconfigurable Wide-Area ATM Newors, IEEE/ACM Transacions on Neworing, 3(6), December 1995. 9. Erneso Marins, Mara Pascoal, José dos Sanos, A New Algorihm for Raning Loopless Pahs, Research Repor, CISUC, May 1997. 10. Misuo Gen, Runwei Cheng, Geneic Algorihms and Engineering Opimizaion, Wiley-Inerscience, 2000. 11. J. Beasley, Lagrangean Relaxaion, Modern Heurisic Techniques for Combinaorial problem, Ed. by Colin Reeves, Blacwell Scienific publicaions, 1993. 12. M. Held, P. Wolfe, H. Crowder, Validaion of subgradien opimisaion, Mahemaical Programming, 6, pp. 62-88, 1974. 13. Carlos Borges, Heurisics for he Dimensioning Problem of Muli-Service Telecommunicaion Newors (in Poruguese), M.Sc. Disseraion, pp. 92-93, Aveiro, submied April 2001.