3878 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 9, SEPTEMBER Optimal Algorithms for Near-Hitless Network Restoration via Diversity Coding

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1 3878 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 Optiml Algorithms for Ner-Hitless Network Restortion vi Diversity Coing Serht Nzim Avi, Stuent Memer, IEEE, n Ener Aynoglu, Fellow, IEEE Astrt Coing-se restortion tehniques hve protive restortion whih results in time svings over other stte-ofthe-rt restortion tehniques. Diversity oing is oingse reovery tehnique whih offers ner-hitless restortion with ompetitive spre pity requirement with respet to other tehniques. In this pper, we show tht iversity oing n hieve su-ms restortion time. In ition, we evelop two optiml lgorithms for pre-provisioning of the stti trffi n one for the ynmi provisioning of the trffi onemn. There is one lgorithm for systemti n one for non-systemti iversity oing in pre-provisioning. An MIP formultion n n ILP formultion re evelope for systemti n non-systemti ses, respetively. The MIP formultion of the systemti iversity oing requires muh fewer integer vriles n onstrints thn similr optiml oing-se formultions. In ynmi provisioning, n ILP-se lgorithm overs oth of the systemti n non-systemti iversity oing. In ll senrios, iversity oing results in smller restortion time, higher trnsmission integrity, n muh reue signling omplexity thn the existing tehniques in the literture. Simultion results inite tht iversity oing hs signifintly higher restortion spee thn Shre Pth Protetion (SPP) n p-yle tehniques from the literture s well s Synhronous Optil Network (SONET) rings, whih re ommonly eploye y servie proviers toy. In terms of pity effiieny, it outperforms SONET rings n + APS, wheres it my require more extr pity thn the p-yle tehnique n SPP. Diversity oing offers preferle treoff whih offers two orers of mgnitue inrese in restortion spee t the expense of less thn 26% extr spre pity. Inex Terms Network fult-tolerne, network oing, liner progrmming. I. INTRODUCTION CABLE uts in wie re networks re ommon. They hppen pproximtely 4.39 times yer per sheth miles []. In this pper, we fous on reovery from single link filures whih onsist of 7% of ll the filures [2], lthough our tehniques n e generlize to multiple link filures. The prominent restortion tehniques n e liste s ring-se restortion, mesh-se restortion, n the p-yle tehnique [3], [4], [5]. Eh tehnique offers ifferent treoff in terms Mnusript reeive Otoer 28, 22; revise Mrh 2 n June 4, 23. The eitor oorinting the review of this pper n pproving it for pulition ws P. Popovski. The uthors re with the Center for Pervsive Communitions n Computing, Deprtment of Eletril Engineering n Computer Siene, University of Cliforni, Irvine (e-mil: {svi, ynoglu}@ui.eu). This work ws prtilly supporte y the Ntionl Siene Fountion uner Grnt No Any opinions, finings, n onlusions or reommentions expresse in this mteril re those of the uthors n o not neessrily reflet the view of the Ntionl Siene Fountion. This work ws presente in prt uring the IEEE Glol Communitions Conferene, Anheim, USA, Deemer 22. Digitl Ojet Ientifier.9/TCOMM /3$3. 23 IEEE of pity effiieny n restortion spee. For voie trffi, telephone network inustry set the restortion time gol to 5 ms, whih is the lower oun of the filure pereption time y users. For IP trffi, it is lwys esirle to erese the restortion time to even smller vlues ue to the omplexities introue y the ifferent lyers of the networking hierrhy. In networking inustry, hitless swithing is onsiere to e the ultimte restortion tehnique, in whih en noes o not experiene the filure even s it [3]. Mesh-se restortion tehniques re groupe into two, nmely pth-se restortion n link-se restortion. The simplest form of pth-se restortion tehniques re + n : Automti Protetion Swithing (APS). They reserve link-isjoint eite kup pth for every primry pth. In this pper, link-isjointness tully refers to spn-isjointness, where eh spn onsists of two opposite iretionl links with ritrry pity. In + APS, the kup pth trnsmits the sme t in the primry pth t ll times. The :APS sheme n e extene to N : M APS, whih requires M link-isjoint kup pths to protet the N link-isjointprimrypths fromny M link filures [3]. Mesh-se protetion shemes employ shring of the spre pity mong ifferent primry pths in orer to offer high pity effiieny t the expense of lower restortion spee n higher signling omplexity. Shre-pth protetion (SPP) is pth-se restortion tehnique tht hs een stuie extensively in oth ll-optil [6] n opque [7] networks. The ompnying restortion tehnique of the stnr Synhronous Optil Network (SONET) is se on protetion swithing over reserve pity of multi-noe ring strutures, known s self-heling rings or SONET rings. More thn % pity, in terms of fier miles, is eploye over self-heling rings to mth n protet ll of the ffete trffi over the file links. However, ue to geogrphil eployment of self-heling rings, require reunnt pity in fier miles exees %. A tehnique tht omines the spee vntge of SONET rings n the pity effiieny of mesh-se restortion is known s p-yle protetion [8]. P-yle protetion is fster thn mesh-se tehniques euse it elimintes most of the ross-onnet onfigurtions tht re require to reroute the trffi fter the filure. The sme ie is use in mesh-se tehniques y [9] n [], nme s hot-stny n preross-onnete trils (PXT), respetively. Coing se link filure reovery ws introue in [], [2] n ws lle iversity oing. In single filure iversity oing, N primry links re protete using seprte N + st protetion link whih rries the moulo-2 sum of the t

2 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 3879 signls in eh of the primry links. Assuming ll N + links re isjoint, in other wors physilly iverse, then ny single link filure n impir only one of them n the file t n e extrte y pplying moulo-2 sum to the reeive t. The iggest vntge of this tehnique is the fst utomti reovery from single link filures y eliminting the omplex n time-onsuming signling n rerouting opertions. The ft tht single protetion link rries the oe t of N primry links les to pity svings. As result, oth the restortion spee n the pity effiieny gols n e hieve, leit within ertin limits. Its protive restortion simplifies the network mngement, minimizes the signling overhe, n elimintes the instility thret tht n e use y the ynmi onfigurtion of the optil rossonnets [3]. Diversity oing, like APS, n e generlize to multiple link filures y eploying M protetion links to protet N primry links from M link filures. In [4], iversity oing is pplie to ritrry network topologies, using heuristi lgorithm. There, it is shown tht iversity oing is muh fster thn typil SPP tehnique known s soure rerouting, n the p-yle tehnique. In [4], oth primry pths n protetion pths re inorporte into oing opertions whih results in further pity svings over typil iversity oing strutures. The optiml lgorithms of iversity oing for oth pre-provisioning n ynmi provisioning re presente in [5]. It is shown tht, iversity oing n hieve su-ms restortion time if proper synhroniztion n uffering re implemente. We introue tehnique tht onverts ny shring-se solution of SPP into oing-se solution in [6], lle s Coe Pth Protetion (CPP). The onversion mkes the restortion utomti, fster, n simpler with some slight extr pity. In tht pper, it is shown tht, oing-se restortion tehniques preserve the trnsmission integrity fter link filure. CPP provies, in ition to enoing insie the network, eoing insie the network s hs een sought for within the ontext of network oing. We woul like to note tht lthough the pulition of [] n [2] prete the topi of network oing, iversity oing is form of network oing. For the purposes of this pper, it hs the gol of minimizing istne metri, in ition to optimum ersure oing in network. Designing survivle network ginst single link filures onsists of two min steps, nmely restortion tehnique n pity plement lgorithm. The performne of the esign epens on oth of these strutures n it is evlute y numer of ifferent riteri. The restortion spee n the pity effiieny re the most importnt metris. Sine the gol is to implement the restortion tehniques in ritrry networks, the omplexity of the pity plement lgorithm plys vitl role for esign purposes espeilly in ig networks n ense trffi senrios. Therefore, the hllenge is to evelop suffiiently simple n effiient restortion tehnique jointly with esign lgorithm tht n optimlly provision the trffi with low omplexity. Both stti n ynmi trffi onsist of multiple unists etween ifferent noes. A smll improvement in the restortion tehnique n use n exponentil inrese in the omplexity of the optiml pity plement lgorithm. A theoretilly superior restortion tehnique n result in inferior results if the ompnying esign lgorithm is not simple enough to fin the ner-optimum solutions with the limite resoures. In this pper, optiml esign tehniques with low omplexity re presente for iversity oing uner stti n ynmi trffi senrios, respetively. Coing-se reovery tehniques hve higher restortion spee thn the rerouting-se tehniques sine they re protive. In this pper, for the first time, the restortion spee of oing-se reovery tehnique is quntifie within su-ms. In ition, the synhroniztion mehnism require for this opertion is simpler thn the ompetitive tehniques in [5] n [7] sine it only requires N uffers for eh oing group with N onnetion emns. Seon, in this pper, we present n optiml esign lgorithm for iversity oing tht n hieve ompetitive pity effiieny ompre to stte-of-the-rt reovery tehniques. The optiml lgorithm is se on Mixe Integer Progrmming (MIP) n is muh simpler thn the oing-se optiml lgorithms in the literture e.g., [5], [7], n [8]. Thir, we present the first oing-se ynmi provisioning lgorithm for single link filure reovery. This lgorithm is lso optiml uner set of ssumptions. This pper onsists of two prts. The first prt resses the pre-provisioning prolem of the stti trffi wheres the seon prt els with the ynmi trffi. II. DIVERSITY CODING TREE In this setion, we present how to hieve reovery within su-ms in the se of single link filures n how to preprovision the stti trffi with n optiml esign lgorithm. The reovery tehnique we opt is form of iversity oing in whih the oing opertions re rrie out mong the onnetions with the sme estintion noe. In this pper, there re three oservtions tht leverge the simpliity of this oing struture. First, this oing struture hieves su-ms restortion time in optil networks with simple synhroniztion struture. Besies the restortion time n synhroniztion omplexity, it lso simplifies the signling omplexity. Seon, it enles eomposition of the trffi mtrix into smller groups whih ereses the esign lgorithm omplexity without loss of optimlity. The prtitione trffi sugroups n e input to prllel MIP formultions. Thir, the nture of the oing struture helps eliminte some of the vriles n onstrints in the MIP formultion. This enles pplition of iversity oing on relisti networks. The optiml esign lgorithm is relize with n MIP formultion n lle Diversity Coing Tree lgorithm. In this lgorithm, primry tree struture serves s the primry pths n protetion tree serves s the protetion pths of set of onnetions. Moreover, the iversity oing tree inputs uniiretionl onnetions. This flexiility s strength to iversity oing tree tehnique in offering solutions for nonsymmetri trffi senrios. An exmple is provie in Fig. (). In this figure, noes S, S2, ns3 trnsmit their t, given s,, n, respetively, to the ommon estintion noe D. The primry tree is shown with soli lk lines n rrows. The protetion

3 388 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 () () Fig.. An exmple of the iversity oing tree struture () There re three link-isjoint primry pths spnne y the primry tree n there is link-isjoint protetion tree, () The synhroniztion uffers for the tree struture. tree is isjoint to ll these links use y the primry tree n it enoes the t to e protete on the tree struture, shown in this se with she lk lines n rrows. In the sequel, we ll the set of onnetions protete y the sme set of primry n protetion trees Coing Group. A. Synhroniztion n Buffering In orer to hieve ner-hitless reovery, the rrivl instnes of the signls in the sme oing group must e the sme. If so, restortion time will e upper-oune y very smll intervl whih inlues filure etetion, protetion swithing, n single XOR opertion. This is ensure vi synhroniztion n uffering. We present simpler synhroniztion mehnism thn the one in [5]. In ition, the uffering elys re reue. The exmple Fig. () is replite in Fig. () with some itionl fetures. The oxes B, B2, n B3 re uffers tht synhronize the primry tree n the protetion tree for ner-hitless swithing. These uffers equlize the rrivl instnes of the primry signls n the protetion (prity) signl. The ely vlues of the uffers n e lulte with the help of vrile i x,y,z,v whih is equl to totl time when signl i trverses from noe x to noe v over intermeite noes y n z. Then uffer elys re lulte oring to the exmple in Fig. (), ssuming,2,3,6 is the longest pth in the oing group, 2,3,6 2,7,6,n 4,3,6 4,,,6 B =,2,3,6,6 () B2 = 2,3,6 2,7,6 (2) B3 = 4,3,6 4,,,6. (3) Totl numer of uffers is N, reue y N ompre to the synhroniztion mehnism in [5]. They re ple t the inoming links of the estintion noe exept the link whih rries the ltest rriving signl of the eoing opertion. The uffers t the intermeite noes in [5] re eliminte. Assuming the protetion tree is the longest pth in the oing group, eh signl on the primry tree is elye s long s the propgtion ely ifferene of the sme signl etween the primry n protetion tree. The uffering elys re reue y eliminting the synhroniztion mehnism over the protetion tree, whih oes not ompromise the eoing struture. To hieve fine synhroniztion, either pointer proessing or the Glol Positioning System (GPS) n e use. In simpler implementtion, in pket networks, pket heers with sequene numers n e employe for the sme purpose. Besies the synhroniztion n uffering requirements, the noes shoul e le to rry out XOR opertions in high spee. One wy is to rry out those opertions in the optil omin s shown in [9]. The seon lterntive is to onvert those signls into eletril omin n rry out these opertions in the eletril omin. The estintion noe lso hs smll memory requirement for eoing purposes. A frme struture enles the orretion of it errors vi two-imensionl mehnism. When one employs CRC heks on the its of the primry pths, then one knows the existene of errors on those pths, then one n tret the frme with the CRC hek s erse n reover it using the t in the protetion pth n the primry pths. Assuming N is the numer of primry pths, there is one protetion pth, p is the rw Bit Error Rte (BER), n L is the size of frme, the proility of the frme eing in error is ( p) L.The lultion of the proilities of error is omplite, ut they re ominte y polynomil in the form of ( p) N. Then, the proility of error fter this mehnism is ominte y NLp 2, whih is muh smller thn the rw BER p for prtil vlues of N, L, np. B. Diversity Coing Tree The esign lgorithm for systemti iversity oing is given in this setion. We ll this lgorithm when employe for pre-provisioning, iversity oing tree. In systemti iversity oing, there re link-isjoint primry pths n oe protetion pths whih re link-isjoint to primry pths. The optiml iversity oing tree lgorithm uses ies from p- yle pproh tht is se on yle exlusion tehnique [2]. The iversity oing tree lgorithm is lso simpler thn the optiml lgorithms of similr oing-se reovery tehniques in [5], [7], n [8].

4 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 388 One of the novelties in the iversity oing tree lgorithm is the wy the primry n protetion pths re forme. Inste of uiling the primry n protetion pths of eh onnetion with seprte vrile in the MIP [7], the iversity oing tree lgorithm uils trees tht reple primry n protetion pths for the onnetions tht re protete together. Therefore, it rries out the sme tsk with signifintly smller numer of vriles n onstrints. More etils out the omplexity nlysis n e foun in Setion II-B6. There re two seprte trees forming the primry n protetion pths of the onnetions, respetively. In primry tree, there is link-isjoint pth from eh soure noe to the estintion noe, efining the primry pth of tht soure noe. The primry tree onsists of multiple link-isjoint rnhes originting from the soure noes tht merge t the ommon estintion noe. Even though these rnhes o not shre link, the resulting struture is tree in the ontext of grph theory. The protetion tree serves s the ommon protetion pth for ll of the onnetions protete y the sme tree. The rnhes of this tree n merge until they reh the root of the tree, whih is the estintion noe. The primry tree n the protetion tree of the sme iversity oing tree struture re link-isjoint. The optiml MIP formultion of iversity oing tree tehnique is provie elow. The input prmeters re G(V,E) : Network grph, S : The set of spns in the network, spn onsists of two links in opposite iretions, N : Enumerte list of ll unit-emn onnetions whih hve the estintion noe, e : Cost ssoite with link e, T : Mximum numer of iversity oing groups llowe, out hlf of the numer of onnetions in eh suprolem, Γ i (v) : The set of inoming links of eh noe v, Γ o (v) : The set of outgoing links of eh noe v, s i : Soure noe of the onnetion emn i, : The ommon estintion noe, ND : The nol egree of the estintion noe, α : A onstnt employe in the lgorithm where V α, β : A onstnt employe in the lgorithm, β 2 mx( V, mx i (ND i )). Next we provie the vriles. Exept the lst two, they re inry n tke the vlue of or. n(i, t) : Equls iff onnetion i is route n protete y the iversity oing group t, e (t) : Equls iff the primry tree of oing group t psses through link e, e (t) : Equls iff the protetion tree of oing group t psses through link e, p v (t) : A ontinuous vrile etween n, resulting in n MIP formultion. It keeps the voltge vlue of noe v in the protetion tree of t. It is possile to set this vrile s n integer lrger thn ut tht mkes the simultion run slower, g v (t) :Smesp v (t) exept it is use for the primry tree of t. The ojetive funtion is to minimize the totl ost inurre y primry n protetion pths of eh oing group min e E T e ( e (t)+ e (t)). (4) t= ) Coing Group Formtion: T n(i, t) = i, (5) t= N n(i, t) ND t. (6) i= The first onstrint ensures tht onnetion n e route n protete y only one iversity oing group. In oing group with N onnetions, there re N link-isjoint pths insie the primry tree n t lest link-isjoint pth s the protetion tree. The require numer of link-isjoint pths re t lest N + for oing group with size N. However,the nol egree of the estintion noe is limite. Therefore, the mximum size of oing group is limite y ND. 2) Builing Primry Trees: N e (t) = n(i, t)+ e Γ o(v) i=,s i=v e Γ i(v),v= e (t) = e Γ o(v),v= e Γ i(v) e (t) v, t. (7) N n(i, t) t, (8) i= e (t) = t, (9) The onstrints ove efine the struture of the primry trees epening on the noe they trverse. A primry tree must hve link-isjoint pth from eh soure noe of the protete onnetions to the estintion noe. If noe v is not estintion noe, it n e n intermeite noe or soure noe or oth. However, the ehvior of the primry tree is the sme on these noes. There re two rules to onsier while uiling primry trees on non-estintion noes. First, there must e rnh of the primry tree t on the outgoing links of non-estintion noe for eh protete onnetion originting from this noe. Seon, non-estintion noe must forwr the primry tree rnhes tht re input using its outgoing links. In eqution (7), on non-estintion noe, the numer of outgoing rnhes elonging to the primry tree t is equl to the numer of inoming links rrying rnh of primry tree t plus the numer of onnetions protete y oing group t tht re originte t tht noe. Eqution (7) stisfies oth of the rules. If noe v is estintion noe, then the totl numer of inoming links rrying the primry tree of t must e equl to the numer of onnetions protete y tht oing group t s mthemtilly stte y eqution (8). In eqution (9), it is ensure tht there is no primry tree on the outgoing links of the estintion noe sine they re suppose to terminte t tht noe.

5 3882 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 u e v..3.3 p u (t) e (t) p v (t). Contrition Fig. 2. A typil link in the protetion tree t. 3) Builing Protetion Trees: e Γ o(v) e (t) N i=,s i=v e Γ i(v),v= β n(i, t) e (t) e Γ o(v),v= + e Γ i(v) β N n(i, t) i= β e (t) v, t. () t, () e (t) t. (2) The struture of the protetion tree lso epens on the nture of the noe it is trversing over. Inequlity () is similr to eqution (7) exept with one funmentl ifferene. Primry pths re link-isjoint n re not oe so they nnot merge. On the other hn, protetion pths, whih re link-isjoint to primry pths, re oe when they merge t the enoing noes. Inequlity () mkes sure tht if noe inputs one or more protetion signls then it enoes those signls n uses t lest one of its outgoing links to trnsmit the enoe signl to the estintion noe. The ojetive funtion mkes sure tht the enoe signls re trnsmitte over only single outgoing link of the enoing noe. The onstrints regring the estintion noe re ifferent thn the onstrints regring other noes. In inequlity (), if oing group t protets t lest one onnetion then there must e t lest one rnh of the protetion tree of t rrying the protetion signls on the inoming links of the estintion noe. In some ses, some of the rnhes of the protetion tree, rrying ifferent signls, my not merge n input to the estintion noe seprtely. This oes not ompromise the eoing struture. Inequlity (2) ensures tht there re no rnhes of the protetion tree on the outgoing links of the estintion noe sine they re suppose the terminte t tht noe. 4) Link-Disjointness: e (t)+ f (t)+ e (t)+ f (t) e, f g, g S, t. (3) The primry tree n the protetion tree must e link-isjoint whih is stisfie y (3). The vriles e n f re the links of spn g in the opposite iretions euse filure over this spn ffets oth of these links t the sme time. The link-isjointness riterion etween the primry pths of the onnetions in the sme oing group is ensure impliitly while the MIP formultion uils the primry trees. 5) Cyle Prevention: In orer to prevent getting yli (or loop) strutures insie the trees, we hoose to ssign two voltge vlues to eh noe in the tree, s in [2], for the primry n protetion trees, respetively. These re not tul voltge vlues, inste they re eh metri use s Fig Voltge vlue ontrition in loop struture. vrile in the formultion. We woul like to emphsize tht this voltge vlue is only use in the sense of resemlne to the fmilir Kirhoff s voltge lw. It is n ssigne vrile to prevent loops. g v (t) g u (t) α e (t) ( e (t)) e = u v, t. (4) p v (t) p u (t) α e (t) ( e (t)) e = u v, t. (5) In inequlities (4) n (5), the voltge vlue t the he noe shoul e higher thn the voltge vlue t the til noe of the links whih re prt of the primry or protetion trees, respetively. Fig. 2 shows typil link in the network. The voltge vlue of noe v must e higher thn the voltge vlue of noe u. This voltge reltionship prevents the yli strutures to e prt of the iversity oing trees suh s in Fig. 3. These vriles re ruil to ensure tht the inequlities (7)-(2) proue vli primry n protetion trees. As n exmple, in Fig. 4(), ssume tht there re 8 onnetions originte from S i stod. The nol egree of the estintion noe is 5 whih mens the mximum size of oing group is 4. Therefore, the onnetions re prtitione into two ifferent oing groups, the first group inlues S D, S 2 D, S 3 D, ns 4 D. The other group inlues the rest of the onnetions. In Fig 4(), there is n exmple of the primry tree elonging to the first oing group forme in Fig. 4(). In tht exmple, there re four onnetions originting from noes S, S 2, S 3,nS 4 to noe D rrying signls,,, n, respetively. The primry tree of this oing group is epite in Fig. 4() with thik stright rrows. As seen on noe D, there re four rnhes of the primry tree on the inoming links of the estintion noe, eh rrying the signl of ifferent onnetion. Noes, 2, n4 re pure soure noes of the onnetions protete y this oing group. It is seen tht new primry tree rnh origintes from eh soure noe. Noe is oth n intermeite n soure noe. When eqution (7) is pplie to this noe, there re two rnhes of primry tree on its outgoing links. One of the primry tree rnhes rry the signl of the onnetion S 4 D n the other rnh is forme y forwring the tree rnh on the inoming link 4 to outgoing link. As seen in Fig. 4(), equtions (7)-(9) ensure tht there is linkisjoint primry pth in the primry tree for eh protete onnetion. The protetion tree of the exmple in Fig. 4 is epite in Fig. 4(). Noes, 4, n re pure soure noes wheres noe 3 is pure intermeite noe n noe 2 is oth. There is only single rnh of protetion tree over the inoming

6 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 3883 () () () () Fig. 4. A iversity oing tree exmple with 8 onnetion emns, () The soure noes of the 8 onnetions n the ommon estintion noe, () In the primry tree, on eh non-estintion noe, the numer of inoming signls is equl to tht of outgoing signls, () In the protetion tree, nonestintion noe merges the inoming signl flows into single outgoing link, () In oing group, the primry n the protetion trees re link-isjoint. links of the estintion noe. At noe 2, the signls n re oe sine noe 2 inputs two signls n merges them over single rnh of the protetion tree. At noe 3, three inoming rnhes of the protetion tree re merge into single outgoing rnh. The signls of those rnhes re oe s well. As seen in Fig. 4(), inequlities ()-(2) ensure tht the protetion tree origintes from the soure noes n its rnhes merge until they rrive t the estintion (root) noe of the tree. In Fig. 4(), it is shown tht the primry n protetion trees of oing group re link-isjoint to ensure eoility. 6) Design Complexity Comprison: We leverge the nture of the opte iversity oing tehnique to simplify the esign lgorithm in two wys. First, the novel iversity oing tree lgorithm requires rmtilly fewer numer of vriles n onstrints thn the similr optiml lgorithms of oingse reovery tehniques e.g., [7]. Seon, the trffi mtrix n e prtitione into smller groups n eh sugroup n e input to prllel simultion without loss of optimlity. The iversity oing tree lgorithm reples the iniviul primry n protetion pths with trees whih les to svings in the numer of integer vriles in the MIP formultion. Tle I ompres the omplexities of ifferent LP formultions of optiml oing-se lgorithms. The tehniques in [5], [7], n [8] re ompre with the novel tehnique in terms of the totl numer of integer vriles n onstrints. We ssume T = N /2. In MIP, the omplexity inurre y the ontinuous vriles re negligile ompre to tht of the integer vriles. As seen in Tle I, the novel lgorithm requires signifintly fewer numer of integer vriles n onstrints ompre to the other tehniques inluing the preeing work in [5]. It is lso more slle thn the other tehniques with lrger network size n lrger trffi mtrix. The opte iversity oing struture implements oing on onnetions with the sme estintion noe. In Fig. 5(), there is n exmple of typil trffi mtrix of network with noes. The rows re soure noes n the olumns re estintion noes. The inies re the totl onnetions etween two noes, whih mkes N = 76. However, in our lgorithm, the onnetions with ifferent estintion noes nnot intert with eh other. Therefore, it is possile to prtition tht mtrix into vetors of onnetions epening on their estintion noes. In Fig. 5(), the set of onnetions with estintion noe equl to re enirle, whih mkes N =2for this group. This oservtion les to signifint simplifition of esign omplexity sine N hs signifint effet on the omplexity s shown in Tle I. The whole trffi mtrix n e prtitione into vetors whih n e input to prllel simultions without loss of optimlity. On the other hn, in generl iversity oing, ny onnetion n e enoe with other onnetions in the trffi mtrix. Therefore, it is impossile to prtition the trffi mtrix into smller groups without loss of optimlity. The only insight is to prtition onnetions whih hve loser estintion noes [4]. Even in tht se, the size of the non-optiml prtitions in generl iversity oing nnot e s smll s the size of the optiml prtitions in the opte iversity oing. C. Non-systemti Coing In this setion, we present the optiml esign lgorithm for non-systemti iversity oing, where eh onnetion hs

7 3884 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 TABLE I COMPLEXITY COMPARISONS OF THE LP FORMULATIONS OF DIFFERENT TECHNIQUES Tehnique Numer of integer vriles Numer of onstrints Diversity Coing Tree N E + N 2 /2 3 N E /2+ N V +7 N /2 Diversity Coing Tree in [5] 3 N E /2+ N 2 /2 N 3 E /2+ N 2 E /2+... ILP formultion in [7] N 2 /2( E +)+3 N E N 4 /8+... ILP formultion in [8] N E ( V +2)+ N ( N +2 V )+... N V (3 E + N + V )+... () Fig. 5. () An exmpler trffi mtrix. The rows re soure noes n the olumns re estintion noes. The inies re totl numer of unitemn onnetions etween these noes, () The onnetions re prtitione epening on their estintion noe. The onnetions with estintion noe re enirle s sugroup n so on. two pths n eh pth n e oe with others uner some rules. There re N onnetion emns in oing group. Eh onnetion emn hs two link-isjoint pths rrying the sme signl, whih is istint from other onnetion emns. The pths tht re to e oe together re ssigne to the sme sugroup of oing group. The totl numer of sugroups vries etween N + n 2N. The numer of pths in sugroup tkes vlues from zero to N. In the reeive vetor of the estintion noe, eh onnetion emn is represente s vrile n eh sugroup is represente s n eqution. If there re smller thn or equl to N sugroups, some t nnot e reovere in some filure senrios euse tht leves N equtions for N unknowns. In the opposite extreme, there will e mximum 2N sugroups if eh pth is trnsmitte seprtely, whih is the se in + APS. For exmple, in systemti iversity oing, there re totl of N +sugroups, N of them re the primry pths n one of them is the omintion of protetion pths. The ommon estintion noe rries out the eoing opertion over the reeive vetor. A non-systemti oe n e uilt y ssigning pths to the sugroups ritrrily. However, the ritil point in the onstrution of non-systemti oe is the eoility of ll N trnsmitte signls. The N t signls n e eoe uner ny single link filure senrio s long s ny N equtions of the reeive vetor re linerly inepenent. It is ler tht ny suset of liner equtions with size N of the reeive vetor re inepenent n N + sugroups re suffiient in systemti iversity oing. In non-systemti oing, the pths in eh sugroup must e speifie. In [4], onnetion emns re rnomly hosen n pths re ssigne to sugroups of the existing oing group one y one. However, generl rule is neee to optimlly uil non-systemti oes. In [2], it is reporte y Lemm tht the estintion noe n reover N t signls from non-systemti oe s long s ny suset of the t () signls with size k re trnsmitte over t lest k + pths. In our tehnique, Lemm n e prphrse s ) Lemm : The non-systemti oe will e vli s long s ny suset of t signls with size k re memers of t lest k + sugroups in oing group. The proof n e followe from [2] y ssuming U s s the set of onnetion emn signls n L s s the set of sugroups in oing group. We uil vli non-systemti oes with the ojetive of minimizing totl pity. Therefore, we evelop n optimiztion lgorithm to fin the oe tht requires lowest totl pity while eliminting the oes tht violte Lemm. The following exemplifies how n invli non-systemti oe n e etete. Assume we hve four onnetion emns, rrying signls,,, n in oing group n eh onnetion emn hs two link-isjoint pths. Assume the first three sugroups of this oing group re given s + +, (6) + whih inites tht one pth of n, n, n n re oe together. Tht les to oing reltionship mp shown in Fig. 6(). In this mp, there re two symols for eh onnetion emn, referring to their two link-isjoint pths. In Fig. 6(), iiretionl rrow etween two pths mens they re in the sme sugroup n therefore oe together. If pthof is oe together with pth of n pth of is oe together with pth, then onnetion emn is iniretly relte to onnetion emn, whih is shown with she rrow in Fig. 6(). In ition, pirs n re iniretly relte s well. If the fourth sugroup onsists of + then four onnetion emns re oune within four sugroups, whih is violtion of Lemm. InFig.6(),the reltionship mp is upte to inlue iiretionl rrow etween pth of n pth of. As result, onnetion emns n re oe together n iniretly relte t the sme time, whih uses irle shown in Fig. 6(). We ll this oing irle, whih is n inition of the violtion of Lemm. Therefore, in the ILP formultion, we seek to prevent oing irles y ensuring two ifferent onnetion emns n either e oe together or iniretly relte. The resulting non-systemti oe will e vli s long s oing irles re prevente. An ILP formultion is evelope to implement the propose tehnique with n ojetive to minimize totl pity on ritrry networks. The ILP uses the sme prmeter s in Setion II-B. The vriles relte to non-systemti iversity oing prolem re given elow. All re inry n tke the vlue

8 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 3885 () () () Fig. 6. Formtion of oing irle. A oing irle violtes Lemm. of or. x e (i) : Equls iff the pth i psses through link e, n(i, t, s) : Equls iff pth i is in sugroup s of oing group t, m(i, j) : Equls iff pth i n pth j re oe together, r(i, f) : Equls iff pth i n onnetion emn f re iniretly relte, otherwise, θ e (t, s) : Equls iff the topology of sugroup s of oing group t inlues link e. Note tht i, j, s 2N, f N, n t T.The ojetive funtion is T 2N min e θ e (t, s). (7) t= s= e E The following inequlity fins two pths for eh onnetion emn x e (i) - if v = s i, x e (i) = if v =, (8) e Γ i(v) e Γ o(v) otherwise. Note tht we require mo(i, 2) = s i = s i for i 2N. Eh pth must e ssigne to single sugroup of single oing group whih is ensure y T 2N n(i, t, s) = i, (9) t= s= n(i, t, s)+n(i,t,s) i, s, t :mo(i, 2) =, (2) 2N s= n(i, t, s) = 2N s= () n(i,t,s) i, t :mo(i, 2) =, (2) m(i, j) n(i, t, s)+n(j, t, s) i j, s, t. (22) Inequlity (2) ensures tht pths of the sme onnetion nnot e in the sme sugroup. However, eqution (2) ensures tht they must e in the sme oing group. If two pths re in the sme sugroup then they re ssume to e oe together, whih is stisfie y inequlity (22). r(i, f) m(i, j)+m(j, 2f)+m(j, 2f ) m(i, 2f) m(i, 2f ) i, j, f, : i j (23) suh tht j = j if mo(j, 2) = n j = j + otherwise. r(i, f) r(i, g)+m(2g, 2f)+m(2g, 2f ) +m(2g, 2f)+m(2g, 2f ) i, f g : i 2f,i 2f,i 2g, i 2g. (24) In inequlity (23), if pth i eomes iniretly relte to emn f if there exists pth j tht is oe with oth pth i n one of the pths rrying emn f. Moreover, pth i must not e oe with either pths of emn f. Inequlity (24) ensures tht pth i eomes relte to emn f if pth i is relte to emn g n one of the pths rrying emn g is oe with one of the pths rrying emn f. Inequlity (25) ensures tht only one of the pths rrying emn f n e either oe with one of the pths rrying emn g or e iniretly relte to emn g. This inequlity ensures the vliity of the non-systemti oe y preventing the oing irles. r(2f,g)+r(2f,g)+m(2f,2g)+m(2f, 2g) +m(2f,2g ) + m(2f, 2g ) g, f : g f, (25) θ e (t, s) x e (i)+n(i, t, s) e, i, s, t (26) θ e (t, s )+θ e (t, s 2 )+θ f (t, s )+θ f (t, s 2 ) e, f g, g S, t, s,s 2 (27) Inequlity (26) fins the topologies of the sugroups. The topology of sugroup is the union of the protetion pths of the onnetions in tht sugroup. Inequlity (27) ensures tht two sugroups hve link-isjoint topologies. In Setion II-D n III-D, we will present our simultion results. Although our tehniques re pplile to eletril n optil networks, in our simultions we fouse on optil networks. D. Pre-Provisioning Results In this setion, we present the simultion results of systemti n non-systemti iversity oing on test networks. Those results re ompre with the simultion results of SPP, p-yle protetion, SONET rings, n + APS in terms of pity effiieny n restortion spee. Cpity effiieny is mesure s the totl pity to route n protet onnetion emns n restortion spee is mesure s the worst-se restortion time. We hve three test networks to nlyze the omprtive performne of these five tehniques. These networks re the COST 239 network [3], the NSFNET network [22] n the Smllnet network [2]. Their topologies re given in Fig. 7(), Fig. 7() n Fig. 7(), respetively. In Fig. 7()-7(), the numers next to the noes re noe inies, wheres in Fig. 7()-7(), the numers next to the links re osts (lengths) of using tht link. In the Smllnet network, link lengths re set to kilometers. The trffi mtrix of the COST 239 network is tken from [2].The COST 239 network hrterizes the network etween mjor Europen metropolises. The trffi mtrix of the NSFNET network onsists of 3 rnom unit-size emns, whih re hosen using relisti

9 3886 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 () () () Fig. 7. Test networks, () COST 239 network, () NSFNET network, () Smllnet network. grvity-se moel [23]. Eh noe in the NSFNET network represents U.S. metropolitn re n their popultion is proportionl to the weight of eh noe in the onnetion emn seletion proess. The trffi mtrix of the Smllnet network onsists of rnomly selete 25 rnom unit-size emns oring to the grvity-se moel. It is ssume tht noe 3 n noe hve 5 units of popultion n rest of the noes hve unit popultion to highlight the omprtive performne of iversity oing in trffi senrio similr to U.S. ost-to-ost trffi. For p-yle protetion, optiml yle-exlusion se ILP for joint pity plement (JCP) n spre pity plement (SCP) lgorithms from [2] re employe epening on the network senrio. The SCP esign lgorithm for SPP is tken from [3, p. 46]. The lgorithm of SPP is suoptiml ue to the extremely high omplexity of the optiml ILP formultion of SPP. The esign lgorithm for SONET 4- fier iiretionl line swithe rings is evelope using the esign lgorithm of p-yle protetion. The esign lgorithm for + APS is simply fining the shortest pir of isjoint primry n protetion pths for eh onnetion. It n e implemente using ifferent, reltively simple esign lgorithms. The esign lgorithms for SPP, p-yle protetion, n SONET rings re evelope with the ssumption of symmetri trffi. Diversity oing tree n + APS tret nonsymmetri trffi the sme s the symmetri trffi. Therefore, they n mintin their performne uner nonsymmetri trffi wheres the other tehniques re expete to ehve worse. In COST 239 n Smllnet networks, JCP is rrie out with the exeption of the esign lgorithm of SPP. In the NSFNET network, sine oth the network size n the trffi mtrix re reltively lrge, SCP is rrie out inste of JCP for ll. It will eome ler from simultion results tht this hnge oes not ffet our overll onlusion in this pper. In SCP, the primry pths re route using the shortest pths n the lgorithms minimize the totl spre pity. The trffi mtrix is prtitione into smllest numer of groups when the omputtionl resoures re not le to ompute the prolem for the whole trffi mtrix t one. Simultion results for the COST 239 network, the Smllnet network, n the NSFNET network re given in Tle II, Tle III, n Tle IV, respetively. TC mens totl pity to route n protet the trffi n RT is the worst-se restortion time. The restortion times of systemti n non-systemti iversity oing re simply RT SDC = F +M +S, RT N SDC =2 F +2 M +S +T, where F is the filure etetion time, M is the noe proessing time, n S is the noe swithing time. The mximum vlues for F, M, n S re tken s μs [6]. RT N SDC is lrger thn RT SDC ue to e omplexity in the eoing struture. In non-systemti oing, ssume two primry pths rrying t n re enoe n + is forme. If the link rrying t fils, the reeiver will still tret the enoe t s +, not +, unless the intermeite noe etets this filure n issues feeforwr error signl. In orer to gurntee orret eoing, there is n extr etetion n noe proessing opertion in the intermeite noe so tht the reeiver n just its eoing struture. T represents the trnsmission time of the feeforwr signl in non-systemti iversity oing, whih is lso tken s μs. There is lso itionl etetion n noe proessing times in RT N SDC. In relity, these vlues epen on the nture of the network, the opte network protool, n the trnsmission tehnology. However, in toy s networks, these numers re in the sums rnge. The propgtion ely ifferene in the restortion time formultion of iversity oing is eliminte euse of the uffers, whih equlize the rrivl time to the estintion noe of eh pth in the sme oing group. The restortion time formultion of SPP is tken from [24] s RT SPP = F +2 s +(h si +) M + X +(h +) M. The symol X refers to the onfigurtion time of n optil ross-onnet (OXC), h is the numer of hops in the protetion pth etween estintion noe n soure noe s, h is is numer of noes etween noe i, whih etets the filure, n

10 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 3887 TABLE II SIMULATION RESULTS OF COST 239 NETWORK COST 239 Network, noes, 26 spns Sheme TC RT for ifferent X vlues (ms).5ms ms 5ms ms N-S. Div. Co F +2 M + S + T (6μs) S. Div. Co F + M + S (3μs) +APS F + S (2μs) SPP p-yle SONET TABLE IV SIMULATION RESULTS OF NSFNET NETWORK NSFNET Network, 4 noes, 2 spns Sheme TC RT for ifferent X vlues (ms).5ms ms 5ms ms N-S. Div. Co F +2 M + S + T (6μs) S. Div. Co F + M + S (3μs) + APS 8888 F + S (2μs) SPP p-yle SONET TABLE III SIMULATION RESULTS OF SMALLNET NETWORK Smllnet Network, noes, 22 spns Sheme TC RT for ifferent X vlues (ms).5ms ms 5ms ms N-S. Div. Co F +2 M + S + T (6μs) S. Div. Co. 78 F + M + S (3μs) +APS 37 F + S (2μs) SPP p-yle SONET noe s. Finlly, s represents the propgtion ely etween noe s n noe. It is optimistilly ssume tht the OXC onfigurtions over the protetion pth n e rrie out simultneously, whih is oppose y some reserhers in [8], [], n [3]. The restortion time formultion of p-yle protetion is tken from [6], whih is RT p yle = F + X + h M +, where the prmeter is the longest propgtion ely etween ny two noes in p-yle n h is the numer of noes in yle. The restortion time formultion of SONET rings is thesmesp-yle protetion exept tht, in simultions, the longest rings re usully shorter thn longest p-yles resulting in shorter restortion time. In + APS, restortion is silly eteting the filure n swithing the trffi from the primry pth to the protetion pth ssuming they re synhronize. Then, RT + = F + S. As stte erlier, we onservtively ssume tht F, M, ns hve vlues out μs eh. This mkes RT pproximtely 3μs for systemti iversity oing, 6μs for non-systemti iversity oing n 2μsfor+ APS in Tles II-IV, where S. n N-S. men systemti n non-systemti, respetively. It is note tht the restortion time results re evlute se on the ssumptions ove n the simultion results. In this pper, CPLEX 2.2 is use to run LP formultions. As seen from the results, iversity oing is muh fster thn oth SPP n the p-yle tehnique in eh network in eh onfigurtion. In ll networks, non-systemti iversity oing is more pity effiient thn the systemti version s expete. However, systemti iversity oing is fster thn the non-systemti version. For pre-provisioning of the stti trffi, they re less pity effiient thn SPP in eh network. The restortion spee inreses more thn hunre times over SPP. Therefore, our lgorithms offer treoff for network esigners where the spee inrese is t lest two orers of mgnitue t the expense of less thn 26% extr pity. We elieve with the existene of unnt fier on toy s networks, our tehniques offer esirle treoff. The restortion time of SPP inreses s the expete time of OXC onfigurtion inreses. Relistilly, in some ses OXC onfigurtion my tke seons [9]. It shoul e note tht, if the noes in the kup pths re not le to rry out ynmi OXC onfigurtions simultneously, then the restortion time of SPP inreses signifintly. The p- yle tehnique lso results in higher pity effiieny thn iversity oing in eh network. The pity effiieny of the iversity oing gets loser to the pity effiienies of the p-yle tehnique n the SPP, when the JCP simultions re rrie out in the COST 239 network. On the other hn, iversity oing is lso more thn hunre times fster thn the p-yle tehnique. It is importnt to relize tht, the restortion time of the p-yle tehnique my inrese if link is protete vi multiple p-yles. In this se, en noes of the file link hve to onfigure multiple OXCs simultneously. Some noes my not e le to rry out these onfigurtions in prllel. Therefore, restortion time of p-yles n signifintly inrese in some ses. Furthermore, it is oserve tht pity effiieny of the p-yle tehnique vnishes while going towrs more sprse networks. On the other hn, oth versions of iversity oing re signifintly more pity effiient thn + APS n SONET rings in eh network. They re lso muh fster thn the SONET rings, whih is slightly fster thn the p-yle tehnique. SONET rings n offer less thn 5 ms restortion time if there is length limit over the nite rings with the expense of higher totl pity. The ft tht + APS is fster thn iversity oing eomes negligile onsiering the extr ely oming from imperfetions insie the optil network. In iversity oing, the mximum synhroniztion ely of the primry pths in the COST 239 network is 8.7 ms. This vlue is equl to ms n 2 ms for the NSFNET n the Smllnet networks, respetively. As rekown of the iversity oing results, the pity effiieny is investigte epening on eh estintion noe for three networks in Tle V. The spre pity perentge

11 3888 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 (SCP) is lulte s Totl Cpity Shortest Working Cpity SCP =. Shortest Working Cpity Shortest W orking Cpity is the totl pity when there is no protetion n the trffi is route over the shortest pths. As it is seen from the results, iversity oing results in lower SCP when the estintion noe hs higher nol egree or it is loser to the ege of the network. The SCP results of non-systemti iversity oing re lower thn or equl to tht of systemti iversity oing s expete. III. DYNAMIC PROVISIONING The stti trffi ssumptions re not lwys vli. In some ses, the future network emns nnot e known or preite. The onnetions n e set up y the nwithon-emn prigm. Therefore, there is nee for ynmi provisioning tehnique tht will ynmilly provision the new onnetion emns without ny future knowlege of the trffi. To our knowlege, this pper presents the first oingse ynmi provisioning tehnique exept its preeing works in [5] n [25]. Some of the hllenges in the ynmi provisioning prolem re the tight timing onstrints, the lk of knowlege out the future trffi n preservtion of the integrity of the existing onnetions. In this pper, those hllenges re mitigte with simple n optiml esign lgorithm. The simpliity of the esign lgorithm is ue to the ft tht provisioning of eh onnetion is one one-y-one inste of optimizing the whole set of onnetions t one. One-yone provisioning of onnetion emns ws use for oing in [4] n [26] s heuristi lgorithms for stti trffi. The optimlity of the esign lgorithm epens on the ssumptions liste s. The existing onnetions nnot e rerrnge ue to QoS requirements, 2. At the eginning, the emn mtrix is n empty set, 3. Centrlize informtion out the stte of the network is upte n onveye to the noes every time there is hnge, 4. Every noe is le to run the lgorithm n lulte the routes, 5. Connetions n e set up when emns pper n terminte when they no longer exist, 6. The ojetive funtion is to minimize the totl ost. The opte iversity oing shemes re the sme s the previous setion. We evelope ILP formultions for oth systemti n non-systemti iversity oing with single estintion noe. These shemes result in lower omplexity, lower restortion time, lower signling, n higher oing flexiility. A novelty of this pper is to show tht non-systemti oing n e implemente optimlly without ing ny extr omplexity to tht of systemti oing. The only ifferene etween the two shemes is the wy the ost prmeters re efine in eh ILP formultion. In this setion, we will first present the lgorithm for nonsystemti iversity oing. Lter, systemti iversity oing will e expline s speil se of non-systemti oing. New Connetion Demn Ientify Coing Groups Coing Group Coing Group 2 Coing Group N ILP ILP ILP ESC ESC2 ESC N Choose Smllest Fig. 8. Extr spre pity is lulte for eh oing senrio n the minimum is hosen. The ynmi provisioning lgorithm is se on ing new onnetion emns to the estlishe oing groups suh tht the new oing group preserves the eoility uner ny single link filure. When new emn rrives, its estintion noe is etete. Then, the existing oing groups whih shre the sme estintion noe with the new onnetion re liste. One of the oing groups is n empty set whih llows the new emn to estlish new oing group. After tht, the new onnetion is hypothetilly e to eh liste oing group. We evelope n ILP formultion tht oes the ing opertions with minimum extr pity. It fins link-isjoint primry n protetion pths for the new emn epening on eh oing senrio. Finlly, the ost of eh ition senrio is evlute n the new onnetion emn is e to the group requiring the lowest extr pity. The lgorithm is epite in Fig. 8. In this figure, ESC i mens totl extr pity require to new onnetion emn to oing group i. The ost vetor of the links is juste for every oing senrio, epening on the topology of the oing group. One the est ville oing group is selete, the oing group topology is upte with the new onnetion n the lgorithm is rey to inorporte new onnetion emn. Before ttempting to optimize the extr ost, the eoility of the oing groups must e preserve ue to the ition of new emn. In our lgorithm, we efine set of rules to preserve the vliity of oing groups fter ing new onnetion emn. These rules re. One of the pths of the new onnetion must e linkisjoint to ny pth in the oing group, 2. The other pth of the new onnetion must e oe with only one pth in the oing group, 3. No pth in the oing group n iverge fter ny noe. The eoility of the ugmente oing groups n e proven using inution if the rules ove re followe. In non-systemti iversity oing, the reeive vetor t the estintion noe of oing group looks like in Fig. 9(), where,,, n re the oe signls n x ij re the inry oing prmeters, whih n tke vlue of or. In relity, there my e more thn N + inoming pths ut some of the pths n e enoe n merge just efore they enter the estintion noe so tht the struture ove is estlishe. It is ssume tht the signls n e extrte from this oing vetor uner ny single link filure senrio. Therefore, the reeive mtrix X =[x ij ] 5 4 is full rnk even if ny one of its rows is elete. Deleting row is the nlog ESCj

12 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 3889 TABLE V SCAP RESULTS FOR EACH DESTINATION NODE Dest. Noe Non-systemti Diversity Coing Systemti Diversity Coing Smllnet COST 239 NSFNET Smllnet COST 239 NSFNET Noe Noe Noe Noe Noe Noe Noe Noe Noe Noe Noe NA NA Noe 2 NA NA 96.5 NA NA 96.5 Noe 3 NA NA 89.9 NA NA 89.9 Noe 4 NA NA 2.9 NA NA 2.9 Averge x + x x2 + x x3 + x x4 + x x5 + x x + x + x + x + x () x 4 + x + x + x + x e x2 + x22 + x23 + x24 + e x x32 x33 x34 e x4 + x42 + x43 + x44 + e x + x + x + x + e e () x + x2 + x3 + x4 + e x2 + x22 + x23 + x24 + e x x32 x33 x34 e x4 + x42 + x43 + x44 + e x + x + x + x + e e () x + x2 + x3 + x4 + e x2 + x22 + x23 + x24 + e x x32 x33 x34 e x4 + x42 + x43 + x44 + e x + x + x + x + e e Fig. 9. The reeive vetor of nonsystemti iversity oing, () X = [x ij ] 5 4 is full rnk + mtrix, () Aition of new onnetion emn rrying e, () Signls re eole if the first pth fils, () Signls re eole if the sixth pth fils. of single link filure wheres the full rnk property ssures the eoility of the signls. We ll these mtries to hve full rnk + property. The gol is to preserve the full rnk + property of the ugmente oing mtries fter ing new emn. Assume tht we hve new onnetion emn tht shres the sme estintion noe with the onnetions in mtrix X, rrying signl e. Following the rules, the reeive mtrix n e trnsforme to the formt in Fig. 9(). The link isjoint pth of the new onnetion is represente s the sixth row of the new oing mtrix. The other opy of signl e is oe with other signls in the oing group s ritrrily enote in the fifth row. In relity, the eision to hoose the oing row is me y the ILP formultion epening on the unerlying topology with n ojetive of minimum extr pity. For eoility, the new signl shoul e oe with t most one pth in the oing group. The full rnk + () property of the new oing mtrix n e shown y heking the eoility of the new oing vetor uner ny single link filure. For the no link filure se, we n erive,,, n y solving first four rows n we n erive e using the lst row. If we elete one of the first four rows, sy the first row, then the reeive mtrix is epite in Fig. 9(). We erive e from the lst row n sutrt it from the fifth row. Then the mtrix generte from rows 2 to 5 tht multiplies the vetor (,,, ) T hs full rnk. Therefore, ll of,,, n n e extrte. In Fig. 9(), the sixth row is elete. Then, the first four rows n e use to extrt signls,,, n. These signls n then e use to fin the vlue of e from the fifth row. To preserve the full rnk + property of the oing mtrix, none of the pths is llowe to iverge fter ny noe. If new onnetion is merge with pth in the existing oing group, they will sty together until the estintion noe is rehe. Otherwise, pth my spn multiple rows in the oing mtrix, whih my impir eoility of the new oing group. If the new onnetion emn nnot e e to ny of the possile oing groups following the mentione rules, then new oing group is estlishe y the new onnetion emn itself. The new oing group will onsist of two linkisjoint pths elonging to the new onnetion. The reeive vetor of the new group eomes [ ] f (28) f where f is the signl rrie y the new onnetion. The proof of eoility of oing groups vi inution is omplete sine the initil stte of oing mtrix X stisfies the full rnk + property. Coing mtries with full rnk + property n e uilt with oeffiients other thn n. However, we opt to hoose this pproh euse of its simpliity n the ft tht it shoul result in minimum overll pity.

13 389 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 () e e e () Fig.. emn. X = () () e e Trnsformtion of the X mtrix fter ing new onnetion The reeive vetor of the oing group looks like [ ] T (29) Fig.. Aition of new onnetion to n existing oing group () The existing oing group, () The new ugmente oing group fter the ition. An exmple is provie to highlight how new onnetion emn is e to n existing oing group. The oing group is shown in Fig. (). There re four onnetion emns from S, S2, S3, n S4 to D whose signls re represente s,,, n, respetively. The she links elong to the oing group. There is new onnetion emn from S5 to D to e e to the existing oing group, enote y e. As it is seen from Fig. (), there exists pth from S5 to D over 9 3, whihis link-isjoint to the oing group topology. The other opy of the new signl will e oe with one of the pths of the existing oing group. After the ition lgorithm, the new ugmente oing group topology is shown in Fig. (). As it is seen, the seonry pth of the new onnetion emn inurs no extr ost sine it is oe over the lrey estlishe portions of the oing group. The trnsformtion in the X mtrix is epite in Fig. (). The she retngle shows the oing mtrix efore ition. The orresponing trnsformtion of the reeive oing vetor is shown in Fig. (). The X mtrix preserves the full rnk + property fter the trnsformtion. Systemti iversity oing is speil se of the nonsystemti iversity oing. In this se, there is speifi istintion etween primry n protetion pths of eh onnetion emn. When new onnetion emn rrives, its protetion pth n only e enoe with the protetion pths of other onnetions in the existing oing group. Therefore, we nee to reefine the seon rule of ition s 2. The other (protetion) pth of the new onnetion must e oe with only protetion pths in the oing group. where the enoe pths spn only single row. The first row is the union of protetion pths n the rows 2 5 re primry pths. Assume new onnetion emn is to e e to this oing group. The primry pth of the new emn hs to e link-isjoint to the pths in the oing group. The protetion pth of it n only e enoe with other protetion pths ple in the first row of the oing vetor. The ugmente oing vetor looks like [ e e ] T (3) We evelope n integer liner progrmming (ILP) se lgorithm to estlish n enlrge oing groups s new emns rrive. The lgorithm is optiml given the ssumptions mentione ove. It mps oth iversity oing strutures into ritrry topologies in orer to protet the onnetions ginst single link filures in ost effiient wy. The ILP ore of the lgorithm inputs the new emn n oing group. It serhes for possile routing n oing senrios with lowest ost. The ost of the links re juste regring the topology of the oing group. When the new emn is oe over n existing link of the oing group, it inurs no extr pity. The ILP formultion leverges the unerlying oing group topology to fin pir of pths for the new onnetion t lowest extr ost. The prmeters of the ILP formultion to fin pir of link isjoint primry n seonry pths re s follows. G(V,E) : Network grph, S : The set of spns in the network, spn onsists of two links in the opposite iretions, N : Enumerte list of ll onnetions, e : Cost ssoite with link e for the primry pth, sme for oth non-systemti n systemti iversity oing, 2 e : Cost ssoite with link e for the seonry pth, epens on the nture of iversity oing,

14 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 389 Γ i (v) : The set of inoming links of eh noe v, Γ o (v) : The set of outgoing links of eh noe v. The inry ILP vriles whih tke vlues or re x e : Equls iff the primry pth of the new onnetion psses through link e, y e : Equls iff the seonry pth of the new onnetion psses through link e. The ojetive funtion is e Γ i(v) x e min e E x e e + y e e 2. (3) = e Γ o(v) x e = e Γ i(v) - if v = s, if v =, otherwise, y e e Γ o(v) y e v, (32) x e + x f + y e + y f e, f g, g S. (33) The origintion, flow, n termintion of the primry pth (x e ) n the seonry pth (y e ) re etermine y eqution (32), where s n re the soure n estintion noes of the new onnetion, respetively. The link isjointness etween the primry n seonry pths is stisfie y inequlity (33). A. Cost Ajustment Exmple In Fig. 2(), there re two uniiretionl links etween eh noe n the numers next to them re their osts. As n exmple, we hve oing group tht is shown in Fig. 2(). There re two soure noes S n S2 n one estintion noe D. The signls trnsmitte from S n S2 re n respetively. The thir pth lso rries the oe version of these signls to the estintion noe. The she lines show links tht re inorporte in the oing group. Assume tht we hve new onnetion request from ny ritrry noe (exept noe 3) to noe 3. We wnt to lulte the require spre pity to this onnetion to the oing group. When we run the ILP formultion, there will e two ifferent topologies in terms of the osts of the links. The topology for the primry pth iffers from the topology for the seonry pth epening on the existing oing group. The topology for the primry pth in oth non-systemti n systemti oing re the sme. However, the topology for the protetion pth epens on the oing senrio. We onsier the existing oing group shown in Fig. 2(). Consier rules -3 esrie erlier. This ft is visulize in Fig. 2() n Fig. 2() for the primry n seonry pths of nonsystemti oing, respetively. Following rule, in Fig. 2(), the links whih rry the signls in the existing oing group re remove from the network. Following rule 2, in Fig. 2(), the links whih elong to the existing oing groups hve zero ost. On the other hn, the ost of the links for the seonry pth of systemti oing is epite in Fig. 2(e) following the moifie rule 2. Only the links elonging to seonry pths of the existing oing group hve zero ost. Rest of the links in tht oing group re remove. The links whih re not ssoite with the oing group hve the sme regulr ost for oth of the pths s shown in Fig. 2(), 2(), n 2(e). B. Limite Cpity Cse We ssume tht there is no limit on the pity of the links. Therefore, ny new emn n e route n protete. In the opposite se, some of the new emns my not hve suffiient pity over the tril of their nite pths. In limite pity se, the links whih hve zero pity re remove from the network tht is input to the ILP formultion. If new emn nnot e route n protete within the existing topology, then it is loke. C. Connetion Terown We ssume tht onnetions leve the network fter urtion. The terown proess of these leving onnetions re First, the onnetion is roppe from its oing group n the topology of tht oing group is upte. This is one y sutrting the links whih purely rry the signl of interest from tht oing group topology. The links tht rry the oe version of this signl re kept in the oing group topology. In the seon step, the reeive mtrix of the oing group is upte y exluing the signl ssoite with the leving onnetion. In the lst step, the pity of the links tht re sutrte from the respetive oing group re inrese y. It shoul e note tht the lst step is require only if the pity of the links re limite. D. Dynmi Provisioning Results In this setion, the performne of the ILP-se ynmi provisioning lgorithm is ompre ginst p-yle protetion lgorithm given in [27], the optiml + APS lgorithm, n n ILP-se lgorithm for iverse routing given in [28], whih is nother form of SPP. Both systemti n nonsystemti oing re employe. Unlike the simultion setup in [5], the pities of links re not limite. We present the simultion results on test networks onsisting of worst-se restortion time n totl pity require to route n protet onnetions. The sme NSFNET n Smllnet networks re use for simultions. In [27], the Most-Free-Routing lgorithm is hosen for p-yle protetion mong similr tehniques ue to its superior performne. In the simultions, onnetion emns re provisione rnomly one-y-one from finite trffi emn mtrix, without ny future informtion out the onnetion emns. The trffi mtries re the sme s in the pre-provisioning simultions. The ojetive is to minimize totl pity without hnging the pths of the existing onnetion emns. The reovery time formultions of the tehniques re the sme s the se in stti preprovisioning. The omprtive results in terms of worst-se restortion time n totl pity re presente in Tle VI n Tle VII. Evlution of the simultion results of the ynmi provisioning onfirms tht the restortion time n pity effiieny nlysis of ifferent tehniques o not hnge muh with the esign lgorithm n the nture of trffi. It is euse the nture of these tehniques is preserve in oth stti n ynmi provisioning. Therefore, oth versions of iversity oing re fster thn p-yle protetion n iverse routing in ynmi provisioning y three orers of mgnitue. The

15 3892 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 23 () () () () (e) Fig. 2. () The topology of the network ssoite with the regulr link osts, () An existing oing group, () The network seen y the primry pth, () The network seen y the seonry pth in non-systemti oing, (e) The network seen y the seonry pth in systemti oing. TABLE VI SIMULATION RESULTS OF SMALLNET NETWORK Smllnet Network, noes, 22 spns Sheme TC RT for ifferent X vlues (ms).5ms ms 5ms ms N-S. Div. Co F +2 M + S + T (6μs) S. Div. Co. 885 F + M + S (3μs) +APS 37 F + S (2μs) SPP p-yle TABLE VII SIMULATION RESULTS OF NSFNET NETWORK NSFNET Network, 4 noes, 2 spns Sheme TC RT for ifferent X vlues (ms).5ms ms 5ms ms N-S. Div. Co F +2 M + S + T (6μs) S. Div. Co F + M + S (3μs) + APS 8888 F + S (2μs) SPP p-yle requirement of feeforwr signling when oth primry n protetion pths re oe mrginlly inreses the restortion time of nonsystemti iversity oing ompre to the se of systemti oing. On the other hn, oth versions of iversity oing re more pity effiient thn + APS. Both versions of iversity oing hve slightly lower pity effiieny thn iverse routing n p-yle protetion in oth networks. However, the ifferene in terms of pity effiieny my eome negligile ompre to the spee vntge of iversity oing for the network esigners in pursuit of higher restortion spee without strit restritions on the fier pity. It is oserve tht inrese oing flexiility in non-systemti iversity oing retes igger vntge in highly onnete networks thn the reltively sprse networks. IV. CONCLUSION This pper presents optiml esign lgorithms of iversity oing for pre-provisioning n ynmi provisioning ginst single link filures. The opte iversity oing tehnique hs two vritions, employing non-systemti n systemti oing, respetively. Only the onnetions with the sme estintion noe re enoe together in oth of the vritions of iversity oing. We were le to hieve su-ms restortion time with this sheme. In ynmi provisioning, optimlity of the esign lgorithm is supporte with set of ssumptions. We evelope novel MIP formultion tht routes n protets set of stti trffi emns optimlly. The signifine of this lgorithm is lower omplexity ompre to the similr tehniques in the literture. The MIP formultion forms the oing groups, first. Then, it retes primry tree struture for eh oing group whih serves s the primry pths of the onnetions in tht oing group. The primry tree is ompnie y link-isjoint protetion tree whih reples the protetion pths in tht oing group. The oing opertions in the protetion tree require no extr vrile. As result, the new formultion proves to e simpler in terms of the numer of integer vriles n onstrints. Evlution of the simultion results inite tht oth versions of iversity oing n hieve su-ms restortion time. Diversity oing is muh fster thn the other tehniques exept + APS in eh senrio. On the other hn, SPP n the p-yle protetion re more pity effiient thn iversity oing for these senrios. However, it offers esirle treoff where one n hieve spee gin of more thn hunre times with less thn 26% extr pity over SPP n the p-yle protetion. In the seon prt of the pper, n ILP-se ynmi

16 AVCI n AYANOGLU: OPTIMAL ALGORITHMS FOR NEAR-HITLESS NETWORK RESTORATION VIA DIVERSITY CODING 3893 provisioning lgorithm is evelope for ynmi trffi. New onnetions re route n protete one-y-one y using oth versions of iversity oing. The esign lgorithm is oth optiml n fst enough to provision the quikly hnging trffi. The ie ehin the lgorithm is to new onnetion emns to the existing suitle oing groups without impiring the integrity of the existing onnetions. A new emn is e to the oing group whih requires the lowest extr pity. The evlutions of the simultion results re onsistent with the results in stti pre-provisioning. Both versions of iversity oing re still signifintly fster thn p-yle protetion n iverse routing. However, the pity effiieny of iversity oing gets loser to those of p-yle protetion n iverse routing ompre to the stti trffi senrio. V. ACKNOWLEDGEMENTS The uthors woul like to thnk the nonymous reviewers whose omments improve the qulity of the presenttion in the pper. 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Moufth, On hieving optiml survivle routing for shre protetion in survivle next-genertion Internet, IEEE Trns. Rel., vol. 53, no. 2, pp , June 24. Serht Nzim Avi (S 2) reeive his B.S. egree in Eletril n Eletronis Engineering from Bilkent University, Ankr, Turkey in June 29. During the sme yer, he ws epte into the Deprtment of Eletril Engineering n Computer Siene of the University of Cliforni, Irvine with eprtmentl fellowship. He reeive his M.S. egree from this eprtment in Deemer 2. Currently, he is Ph.D. nite in the sme eprtment. His reserh interests re in pplitions of oing theory into prolems in networking, in vritions of liner progrmming, espeilly in the genertion of fst lgorithms for liner progrmming pplitions suh s utting stok or olumn genertion solutions. Ener Aynoglu (S 82-M 85-SM 9-F 98) reeive the B.S. egree from the Mile Est Tehnil University, Ankr, Turkey, in 98, n the M.S. n Ph.D. egrees from Stnfor University, Stnfor, CA, in 982 n 986, respetively, ll in eletril engineering. He ws with the Communitions Systems Reserh Lortory, AT&T Bell Lortories, Holmel, NJ (Bell Ls, Luent Tehnologies fter 996) until 999 n ws with Ciso Systems until 22. Sine 22, he hs een Professor in the Deprtment of Eletril Engineering n Computer Siene, University of Cliforni, Irvine, where he serve s the Diretor of the Center for Pervsive Communitions n Computing n hel the Conexnt-Broom Enowe Chir uring Dr. Aynoglu is the reipient of the IEEE Communitions Soiety Stephen O. Rie Prize Pper Awr in 995 n the IEEE Communitions Soiety Best Tutoril Pper Awr in 997. Sine 993, he hs een n Eitor of the IEEE TRANSACTIONS ON COMMUNICATIONS n serve s its Eitor-in-Chief from 24 to 28. From 99 to 22, he serve on the Exeutive Committee of the IEEE Communitions Soiety Communition Theory Committee, n from 999 to 2, ws its Chir.