APOC #232 Capacity Planning for Fault-Tolerant All-Optical Network

Transcription

1 APOC #232 Capacy Plannng for Faul-Toleran All-Opcal Nework Mchael Kwok-Shng Ho and Kwok-wa Cheung Deparmen of Informaon ngneerng The Chnese Unversy of Hong Kong Shan, N.T., Hong Kong SAR, Chna -mal: Absrac Fuure communcaon neworks wll carry many WDM channels a very hgh b rae. Thus, s desrable o avod elecronc swchng a he core. The all-opcal backbone nework can be nerconneced by opcal cross-connecs a sraegc locaons o allow for flexble capacy provsonng and faul-oleran reroung. Such an all-opcal core layer ncely decouples he long-erm capacy plannng problem from he shor-erm dynamc bandwdh allocaon problem whch can be beer ackled n he elecronc doman. An essenal requremen for he all-opcal core layer s ha mus be fully faul-oleran, oherwse, a sngle faled lnk can cause a dsaser for he enre nework. We consder he problem of how o allocae he requred capaces and spare capaces on a gven all-opcal core nework so as o make he nework fully sngle-faul (or mul-fauls) oleran. The obecve funcon s he oal cos of he spare fbers. Based on a gven raffc requremen on all source-desnaon pars, he opmal bandwdh requremen for each lnk n he gven opology s frs compued. We hen consder lnk falures one-by-one for he enre nework. For each lnk falure, we show how spare capacy can be added n oher lnks so as o ake advanage of exsng spare capaces ha have already been added. The algorhm s based on he shores pah roung algorhm and has a polynomal me complexy. Prelmnary nvesgaons sugges ha he algorhm can gve resuls comparable o hose obaned by neger programmng. 1. Inroducon The need for relable communcaon servce has become exremely mporan for hgh capacy neworks. In order o provson for nework survvably, spare capacy mus be provded on he exsng bul-up nework under a chosen resoraon sraegy. There are wo man resoraon sraeges: Lnk resoraon and Pah resoraon. In lnk resoraon, he broken raffc s reroued beween he end nodes of he faled lnk. In pah resoraon, he orgn desnaon node pars (OD pars) whose raffc raversed he faled devce are responsble for resoraon and reroue over he enre pah se beween each affeced orgn desnaon par. The wo resoraon sraeges are shown n Fgure 1 and Fgure 2. Fgure 1: Lnk Resoraon Fgure 2: Pah Resoraon

2 If he spare capacy plannng s no opmally done, wll cause a waseful and neffcen use of resources. However, opmal spare capacy placemen n a mesh resorable nework has shown o be an NP-hard problem [1]. Based on hs reason, many researchers have aemped oher approaches o solve hs problem [2-8]. Grover suggesed he Spare Lnk Placemen Algorhm (SLPA) heursc for solvng he spare capacy placemen problem n a mesh nework wh lnk resoraon [3]. The SLPA s a heursc approach wh polynomal me complexy, and has been mplemened n some elecom nework for spare capacy plannng. In hs approach, he spare capacy allocaon wh full resorably s acheved by erave lnk addon n he phase of forward synhess and hen he spare capacy wll be reduced whle mananng he nework resorably n he phase of desgn ghenng. Sakauch, Herzburg and Venables aemped an neger programmng (IP) approach based on max-flow mn-cu consderaons o solve he spare capacy placemen problem [2,4,5]. Ths approach uses cu ses n he IP formulaon and populaes he consran ses wh cu ses eravely unl he soluon provdes a spare capacy placemen whch s 100% lnk resorable and mnmzes he oal spare capacy of he nework. Ths approach can lead o a beer spare capacy plannng han SLPA whle he complexy s hgher. However, boh of hese approaches can be used for lnk resoraon only. A more recen and advanageous approach s sll an IP approach bu wh flow consrans based on a suable se of predefned roues over whch resoraon pah ses may be mplemened [6]. Ths s referred o as he flow-based approach. In hs approach, he IP s based on elgble resoraon roues beween each par of nodes ermnang a lnk. Ths IP flowbased approach s preferred over he IP cu se approach because only a sngle IP execuon s needed o oban he spare capacy assgnmen. The same process also yelds he exac roung used o resore each lnk falure. Ths approach has been exended o pah resoraon and a good resul was acheved [7]. However, he IP-based approaches canno provde any nsgh n how o pck he pah se whn a general seng n order o acheve good resuls whle mananng a compuaonal feasble problem ha scales. An addonal lmaon of he IP-based approaches s ha here s sll no effcen algorhm for fndng he opmal soluon when he obecve funcon or consrans are nonlnear [12]. All of hese sudes were neresed only n deermnng he mnmum capacy requremens of a gven mesh ype nework opology wh he capacy allocaon already gven o mee he normal raffc demand, under he falure scenaro of a sngle lnk break. ven hough hey specfed he cos parameer n he obecve funcon, hey dd no ake he dsance and lnk cos merc no accoun. However, n realy here wll always be some neworks ha have more oal spare capacy bu are less expensve han some ohers. Whou akng no consderaon of he dsance and lnk cos merc, he problem-solvng model s no complee and canno represen he real suaon. So far, none of he publshed researches evaluae he effec of cos and dsance merc. In hs paper, he dsance and lnk cos merc wll be aken no accoun n mnmzng he oal spare capacy cos for he capacy plannng wh full resoraon for a gven workng mesh nework. Here, an effcen polynomal me complexy heursc for solvng hs problem under boh lnk resoraon and pah resoraon schemes s provded. I s called he Spare Capacy Allocaon and Plannng smaor (SCAP). The remander of hs paper s organzed as follows: In Secon II, he mahemacal model of he spare capacy problem wll be descrbed. In Secon III, he developed heursc SCAP wll be explaned. The expermenal resuls and dscusson wll be presened In Secon IV. In Secon V, conclusons are drawn. 2. Mahemacal model Frs, he problem s defned n he followng mahemacal model. Gven a mesh ype nework opology, he normal raffc demand, and he capacy allocaon o mee he raffc demand, how much spare capacy should be assgned for each lnk so ha he nework can olerae a sngle lnk falure? The goal of he opmzaon s o selec he bes se of resoraon

3 roues under lnk resoraon scheme or pah resoraon scheme, and assgn each roue a specfc flow o survve a sngle lnk falure, such ha he leas expensve nework can be found whn a reasonable me. In hs problem, s assumed ha he nal gven opology s a leas wo-conneced. The normal raffc demand beween each node par s assumed o be symmercal and he same roue s used for boh drecons. 2.1 Varable defnons The mahemacal model for hs problem and noaons based on [7] are as follows: N number of nodes n a nework; number of lnks n a nework; C cos funcon of a capacy un assgned o lnk ; β cos of a un capacy assgned o lnk per un lengh; L lengh of lnk ; T oal number of nonzero raffc demand pars n he normal raffc demand marx; T oal number of raffc demand pars affeced by lnk cu ; d normal raffc demand beween orgn-desnaon(o-d) par ; Q oal number of workng roues avalable o sasfy he normal raffc demand beween O-D par ; q g, he workng flow requred on he q h workng roue for sasfyng he demand beween node par ; ζ akes he value of 1 f he q h workng roue for demand par uses lnk ; 0 oherwse;, q w workng capacy on lnk ; X normal raffc demand affeced by O-D par upon he falure of lnk ; P oal number of possble elgble resoraon roues for reroung he normal raffc demand beween O-D par upon he falure of lnk ; p f, he resoraon flow hrough he p h resoraon roue for O-D par r upon he falure of lnk ; δ akes he value of 1 f he p h resoraon roue for O-D par afer he falure of lnk uses lnk, and 0 oherwse;, p, s spare capacy on lnk. 2.2 The obecve funcon and consrans When he problem s ransformed o a mahemacal model, he obecve funcon s: Mn C s = 1 where = β For lnk resoraon: The consrans o be sasfed are: 1) Resoraon flow mees 100% resoraon level for each O-D par : C L

4 P p=1, p f w = 1,2,..., T. = 1,2,..., 2) Lnk s spare capacy s suffcen o mee he smulaneous demands of all node pars affeced by any one lnk falure: P p=1 δ f s = 1,2,.., T. (, ) = 1,2,...,,, p, p f,, p 3) The flows on resoraon pahs,, are non-negave negers. 4) Spare capaces, s, and workng capaces, w, are non-negave negers. For, pah resoraon: The consrans o be sasfed are: 1) resoraon flow mees 100% resoraon level for each O-D par : P p=1, p f X = 1,2,..., T. = 1,2,..., 2) The oal demand los by OD-par upon he falure of lnk s he sum of he flows over all workng roues raversng lnk : Q, q, q ζ g = X = 1,2,..., T. = 1,2,..., q=1 3) Lnk s spare capacy s suffcen o mee he smulaneous demands of all node pars affeced by any one lnk falure: T P = 1 p= 1 T Q, p, p, q, q, f s + ζ ζ = 1 q= 1, q δ g (, ) = 1,2,...,, p f, g, q 4) The flows on resoraon pahs,, and workng pahs,, are non-negave negers. 5) Spare capaces, s, and workng capaces, w, are non-negave negers. 2.3 Complexy The problem under sudy requres T = 1 = 1 P + 2 varables and T 2 consrans for lnk resoraon scheme. For pah resoraon, he mahemacal model consss of P + 2 varables and consrans. The varable = 1 = 1 2 wll be 1 for lnk resoraon and scales wh he number of nodes of O N n he nework. I can be seen ha he number of varables and consrans scales wh he number of lnks and nodes n he nework. Furhermore, he number of varables also scales wh he number of resoraon roues consdered. Therefore he number of dsnc roues n a nework of lnks O 2. s ( ) ( ) Fnally, he complexy of he whole mahemacal model s as follows: O consrans. For lnk resoraon, he problem requres O( 2 ) varables and ( ) =1 T + T

5 For pah resoraon, he problem requres O( N ) varables and ( N ) O consrans. Ths shows ha he problem s NP-hard. In order o solve hs NP-hard problem, we have developed an effcen nsghful heursc. The heursc wll be descrbed n he nex secon. 3. A Greedy Algorhm SCAP As he spare capacy plannng problem s NP-hard, many researchers have aemped dfferen heurscs for solvng he problem. Among hese heurscs, he mos common one s o ransform he spare capacy plannng problem no an IP formulaon and solve wh an Ineger Programmng echnque such as he Branch and Bound (B & B ) algorhm. Theorecally, n order o oban he global opmal soluon, all dsnc resoraon roues beween he end-nodes of a lnk (for lnk resoraon) or OD pars (for pah resoraon) mus be presen n he consran sysem of he IP model. However, by dong hese, he complexy wll be exponenal wh he oal number of lnk n he nework as shown n he prevous subsecon. In order o reduce he complexy and apply he neger programmng heursc o praccal problems, he general approach s o resrc he number of dsnc roues enered as consrans wh he use of hop-lmed approach or smlar echnques. In hs way, can be expeced ha he Ineger Programmng heursc no longer ensure opmaly. A he same me, does no provde any nsgh for choosng he pah se. Wha makes neger programmng worse o apply n solvng he spare capacy plannng problem s ha unl now, here s no effcen algorhm for solvng he nonlnear neger programmng problem opmally [12]. Tha means ha he IP heursc can be effecve n opmzaon only when he obecve funcon and he consran sysem of he problem are resrced o he lnear funcons. However, can be expeced ha some nonlnear varables and consrans, such as QoS consrans, may be necessary for he spare capacy plannng problem. Therefore, we need o develop some nsghful heursc whch can ackle he above drawbacks whle achevng nearopmal or even opmal soluon n a reasonable me. Wh hs movaon, we nvesgae hs problem and develop an effcen heursc SCAP o acheve a near-opmal or opmal soluon for hs NP-hard problem. In hs subsecon, he workng prncple and he mplemenaon of SCAP wll be nroduced. 3.1 Workng Prncple of SCAP In order o undersand he basc workng prncple of he SCAP, le s frs look a he followng smple example. Fgure 2.4 shows an example nework wh 6 nodes and 10 lnks. Fgure 3: An example nework for workng prncple of SCAP. In hs example nework, he dsance and cos parameers of each lnk are assumed o be 1 un. The numbers n he parenheses represen he spare capacy assgned o he lnk a some me. Assume ha for lnk (3,5), whch has a normal raffc demand of 2 uns passng hrough, s broken. By he lnk resoraon sraegy, 2 raffc demand uns need o be reroued hrough oher resoraon pahs beween he end-node par (3,5). Among hese resoraon pahs, 3 pahs are shown n he fgure as examples. They are Rpah_A, Rpah_B and Rpah_C respecvely. Among hese 3 resoraon pahs,

6 Rpah_A and Rpah_B are he shores one. If we choose any sngle one of hem for reroung, hen eher of hem should provde he leas cos. However, under he condon ha some spare capaces already exsed on he lnks, he resoraon pah se for reroung wll be changed. If Rpah_A s chosen, 4 spare capacy uns wll need o be added. For Rpah_B, 2 spare capacy uns need o be added. Bu for Rpah_C, us 1 spare capacy uns need o be added. Thus, Rpah_C whch s he longes pah among hese 3 pahs urns ou o be he leas cosly roue because of he spare capacy ha already exsed. Ths example gves us he frs key dea of he SCAP ha he placemen of new spare capacy on he lnks o acheve full resorably based on he prevous spare capacy placemen wll lead o a more economcal spare capacy plannng. We call hs he ncremenal assgnmen effec. Le s look a he prevous example agan. If we use us Rpah_C for resoraon only, 1 more spare un wll need o be added. However, f we spl he raffc ha need o be reroued no wo porons, each of whch s one raffc demand un. Then f we reroue one un hrough Rpah_C and one un hrough Rpah_B, we can see ha no new spare capacy s needed. Ths fac gves us anoher key dea ha reroued raffc splng can help o lower he spare capacy requremen. Snce pah resoraon s smlar o lnk resoraon bu wh more OD pars o be reroued each me a lnk s broken, can be expeced ha he ncremenal assgnmen effec and reroue raffc splng effec wll help boh resoraon sraeges. Based on he above reasonng, we develop he SCAP o solve he spare capacy plannng problem wh he goal of achevng he mos economcal spare capacy placemen n he nework. 3.2 The mplemenaon of SCAP Two procedures are used n he heursc: he Dksra and Updae procedure. Procedure Dksra(G,A,B,n,P) akes he nework srucure G, and wo nodes A and B of he faled lnk and he affeced raffc n as npu and hen compue he se of shores pah P as well as he flow for hese pahs as soluon. The Updae procedure wll replace he spare capacy n he nework for he correspondng lnk flow hrough by he se of pahs P. The Procedure Dksra(G,A,B,n,P) n fac s a modfed Dksra algorhm for our heursc. As we know ha afer he frs eraon of he algorhm, here wll be some spare capacy n some lnks. For he second eraon of he algorhm, he nework has been changed, and hose lnks wh spare capacy n he prevous falure cases wll be a zero cos lnk n he curren eraon. Hence he Dksra procedure s based on hs prncple o fnd ou he shores pah unl he spare capacy assgned n he prevous cases canno sasfy he affeced raffc n he curren eraon. The nework srucure G wll be updaed. Then a new se of pahs wll be chosen for assgnng he spare capacy of he affec raffc no ye handled n he curren eraon. Ths s he raffc splng prncple and he ncremenal assgnmen effec n our heursc. In he case of pahs lengh ed, he pah wll be chosen accordng o he Dksra algorhm. The pseudo code s as follows: Begn { G=Load_Neowrk_Informaon(); n=all_the_traffc_informaon_for_ach_lnk(); for each faled lnk case { Modfed Dksra(G,A,B,n,P); Updae_he_Nework(G); } } nd; 2 I can be seen ha as he procedure manly depends on he Dksra algorhm whch s of he complexy O( N ) where N s he number of node n he nework. Assume ha he edge number of he nework s. The maxmum number of shores 2 pahs spl for each case s k, whch can be expeced o be a small consan. Then he complexy wll be O( k N ). Ths s a very effcen mehod when compare o he neger programmng approach whch s of he complexy n exponenal form of and N. 3.3 Improved SCAP

7 Alhough he orgnal SCAP can acheve a good soluon wh low complexy, can be expeced ha such knd of greedy algorhm canno acheve he opmal soluon n he forward synhess phase. Ths s because he neracve effec beween he pah ses, he spare capacy assgnmen on each lnks and he raffc demand has been gnored. The spare capacy plannng problem s such a complex combnaoral opmzaon problem ha he neracon beween he varables needs o be consdered n order o acheve a beer resul. To make he SCAP an opmal or a near-opmal soluon for he spare capacy plannng problem, we develop he followng mprovemens. A. Run he Algorhm wh sored order: Based on he raffc splng and ncremenal assgnmen effec, we form he basc SCAP. However, here mus be a sarng pon for he algorhm o run hrough. If he algorhm sars wh a bad choce, wll lead o a low qualy soluon. I s mporan o choose a good sarng pon ha leads o a hgh qualy soluon. As he SCAP s based on he ncremenal assgnmen effec, s reasonable o expec ha he sarng pon should be he OD par ha lead o he leas ncremenal assgnmen effec. By hs assumpon, f he algorhm runs wh a sored ncremenal assgnmen effec, SCAP should lead o a hgh qualy soluon. However he ncremenal assgnmen effec depends on he oal pah cos and he raffc demand. Bu he pah we need o choose s no known ye for all OD-par and f we ry o search ou all possble pahs for all OD-par and hen do sorng, wll be a complex problem. Hence, we ry o sor he raffc demand ha s already known for all OD par and hen sar he SCAP from he leas-load OD-par. Ths s he frs enhancemen. B. Afer he basc SCAP execuon, perform backrackng: Backrackng has long been a sandard mehod n compuer search problems when everyhng else fals. Afer he backrackng procedure, he soluon wll usually be mproved [10]. The backrackng procedure s very smple n he SCAP. Afer geng he soluon from he basc SCAP, we can reduce he spare capacy un of each lnk n he nework and see wheher s sll a feasble soluon. If yes, hen updae he nework and hen use ha nework for anoher eraon of backrackng unl here s no furher mprovemen. The pseudo code of he backrackng s as follows: do { se mprovemen (mp) o zero For each lnk (A,B) n nework (G) { f spare capacy exss n lnk (A,B) { decremen he spare capacy of lnk (A,B) by one f nework (G) s sll a feasble confguraon.(*) { ncremen (mp) by one updae he nework (G) } else { resore spare capacy of lnk (A,B) } } } } whle (mp) s non-zero Agan, for boh lnk and pah resoraons, he backrackng procedures are he same. From he pseudo code, can be seen ha mos of he processng me n he above procedure s consumed n lne (*), he par of check feasbly. For lnk

8 resoraon he check feasbly par s even smpler as s us a sngle commody maxmum flow (SCMF) problem whch can be solved by he maxmal-flow algorhm wh complexy O( k ) where k s he maxmum flow value and s he edge number [11]. However, for he smplcy of he lnk resoraon, we decde o use shores pah for checkng he feasbly. The check feasbly par for pah resoraon s no so smple, because each lnk falure wll cause several OD-par raffc o be reroued, and each OD-par wll have several possble pahs for supporng s raffc demand. Tha means we canno choose any sngle pah of any OD-par for checkng so easly, as we need o consder all he oher raffcs whch are affeced by he check feasbly procedure. However, hs s no an easy ask, as hs wll nvolve anoher neger programmng problem, he neger mulcommody maxmum flow (MCMF) problem. So we ry o use he nerference heursc proposed by Iraschko whch s an effcen heursc for solvng MCMF problem [8]. Ths heursc s of complexy 4 O( N ) where N s he node number. Wh hs heursc, we solve he check feasbly procedure of pah resoraon que successfully and make backrackng procedure work for he pah resoraon scheme. 4. Numercal xpermens and Dscusson Ths secon repors expermenal resuls explorng he suably of he SCAP for spare capacy plannng desgn. Alhough SCAP res o mnmze he nework cos, we perform many expermens o mnmze he oal spare capacy n order o compare wh he resuls hose n he leraure. In anoher se of expermen, we wll show ha our algorhm can mnmze he nework cos, whch s no shown by oher researchers n her own expermens. Fnally, from hese wo ses of expermen, we hope o draw a concluson ha SCAP s a generc and robus heursc for spare capacy plannng. The SCAP was wren wh C++ language and execued on Penum II 733 MHz PC wh 128 MB RAM runnng Wndows 2000 Professonal. 4.1 xpermens o mnmze he spare capacy In hs se of expermens, smulaons are done for he four medum-szed neworks proposed by A. Al-Rumah [9]. The opologes of he four neworks are shown n Fgure 4-7. In each nework, s assumed ha here are wo raffc demand uns beween each node par n he nework. Table 1 summarzes he parameers of he four neworks. The workng capacy for each nework was deermned usng he shores pah roung for each raffc demand OD-par, ha means Q =1 for all cases n our mahemacal model. Fgure 4: Nework 1 Fgure 5: Nework 2 Fgure 6: Nework 3 Fgure 7: Nework 4 Nework Number of nodes Number of lnks Average node Number of O-D Toal nework degree pars load Table 1: General nformaon of he four es neworks.

9 Al-Rumah has done expermens on hese four neworks wh he use of hree mehods from he leraure: (1) he Spare Lnk Placemen Algorhm (SLPA) [3]; (2) Lnk resoraon usng Ineger Programmng (Lnk IP) [6]; and (3) Pah resoraon wh lnk dson roues usng Ineger Programmng (Pah IP) [7]. All of hese mehods are used wh he obecve o mnmze he oal spare capacy n he nework and wh hop coun lm a 7. The resul of smulaon for he above four neworks by he 3 heurscs us nroduced s adoped from Al-Rumah as he clamed ha he mplemenaon of he hree mehods above s vald by reproducng publshed resuls from he leraure [9]. We wll hen use SCAP wh lnk resoraon o compare wh he resul from mehod 1, 2 and use SCAP wh pah resoraon o compare wh he resul from mehod 3. For SCAP, n order o mnmze he spare capacy of each nework, he lnk cos parameer and dsance parameer for each lnk n each nework s aken o be equal o 1 un. The resuls are shown n Table 2 and Table 3. The runnng me of he SCAP for spare capacy plannng of hese four nework s shown n Table 4. For lnk resoraon: Nework Toal Spare Capacy Workng Improved SCAP Improved SCAP Capacy SLPA Lnk IP Basc SCAP w/o backrackng wh backrackng Table 2: Toal Spare Capacy for lnk resoraon For pah resoraon: Nework Workng Toal Spare Capacy Capacy Pah IP Basc SCAP Improved SCAP wh backrackng Table 3: Toal Spare Capacy for Pah Resoraon Nework Runnng me of Lnk Resoraon (ms) Runnng me of Pah Resoraon (ms) Basc SCAP Improved SCAP Basc SCAP Improved SCAP Table 4: The runnng me of he SCAP In Table 2, he oal spare nework capacy for 100% resoraon for any sngle lnk falure as deermned by dfferen algorhms s gven. From he resul shown n Table 2, Table 3 and Table 4, we observe he followng: 1. When we compare he resul obaned n Basc SCAP and Improved SCAP, we see ha Improved SCAP can ouperform Basc SCAP from 15% o 30% n all he cases. I provdes he srong base ha he ncremenal assgnmen sorng and backrackng procedure can brng a hgh qualy soluon wh grea mprovemen.

10 2. From he runnng me able, we can see ha, he backrackng procedure s abou 5 mes as complex as he basc SCAP. I can be seen ha even wh he backrackng procedure, he runnng me s very shor for a 20-node nework, ha means he SCAP s overall an algorhm wh very low complexy. 3. For lnk resoraon sraegy, when compared o he SLPA resuls, can be seen ha even he Improved SCAP whou any backrackng procedure performed, can brng up o 6% mprovemen. I should be noed ha he SLPA s based on he cu-based procedure, whch canno provde any roung and flow nformaon whle SCAP can. And SLPA s self a forward synhess and backward ghenng heursc whch wll be more complex han Improved SCAP whou backrackng. 4. When compared wh Lnk IP mehod for lnk resoraon sraegy, f wh he backrackng procedure, SCAP can perform whn 1.5% of he resul obaned for Lnk IP. However, we can fnd ha for he nework sze o become larger and larger, SCAP can even ouperform he Lnk IP up o 1.3% for he larges nework. I s because n order o reduce he complexy and runnng me of he Lnk IP procedure, here s a lm on he sze of he resoraon pahs wh hop lm, and some long pahs whch may be helpful n he opmzaon process are excluded. Whle for SCAP, res o assgn he flow and he roung nformaon based on he prevous eraon unl all he raffc demand s roued or reroued, so he problem n lmng he pah sze n he execuon does no exs. Hence, SCAP can cover a large search space and produce a beer resul for larger neworks. 5. For he pah resoraon sraegy, when compared o he Pah IP resuls, SCAP sll can perform well, whn 5.5% of he resul obaned by Pah IP. In he smaller nework, SCAP performs beer. For he smalles nework, SCAP even ouperform he Pah IP by 5%. I s because for he pah resoraon sraegy, a sngle lnk falure wll cause several OD par raffc demand o be reroued, ha means more neracon effor needs o be consdered beween hese OD par raffc n order o acheve he opmal soluon, whch s no he case for lnk resoraon and pah sze was no a domnan facor hen. In hs way, Pah IP wll perform beer han SCAP, because SCAP s no so srong n handlng he neracve effec beween he assgnmen. However, SCAP can provde he nsgh of choosng he Pah Se and wh furher sofware developmen, can acheve real-me processng wh good resul. Through he expermen resuls, can be concluded ha he prncples of raffc splng effec and ncremenal assgnmen effec work and form he basc SCAP. Wh he sorng of ncremenal assgnmen effec and a smple backrackng procedure, mproved SCAP can ouperform he SLPA and have a smlar performance wh neger programmng bu wh a much lower complexy. Wha SCAP conrbue s he nsgh for choosng he pah se whle provdng he flow and roung nformaon drecly. Ths s sll an open queson and canno be obaned by he neger programmng approach. The SCAP s also useful n developng real me processng sofware for spare capacy plannng. 4.2 xpermens o mnmze he cos of he nework As sad before, SCAP s developed o opmze he selecon of he bes roues accordng o he dmensonng n order o fnd he leas cos nework whn a reasonable me. In he frs se of expermens, has been shown ha SCAP can acheve opmzaon of a sngle dmenson,.e. he spare capacy. Here we show ha SCAP can also handle he dsance and cos mercs and we perform he second se of expermens o es he SCAP performance. Unlke he frs se of expermen, unl now and o he bes of our knowledge, no research work exss ha handles hese wo mercs and produce a gudelne for reference even hough he researchers ry o nclude he cos parameer n he obecve funcon. Therefore, we develop our own expermens and gudelnes o show ha SCAP can be used for cos mnmzaon as well as he spare capacy assgnmen. The expermen s smple. We apply a se of dsance and cos mercs o he same nework opologes and hen use he SCAP heursc o fnd he capacy assgnmen and hen compare wh he assgnmen based on he oher heursc o see wheher he SCAP heursc can lead o a more economcal resul. The nework we use for he second se of expermen s from Hasegawa [6]. The nework opology s shown n Fgure 8. We assgn a dsance merc ha scales wh he drawng dsance of he nework and a cos merc o he nework. They are shown n Fgure 9 and Fgure 10 respecvely. The spare capacy assgnmen (S.C.A.) by Herzburg [4], Grover [3] and

11 SCAP are shown n Fgure 11, Fgure 12 and Fgure 13. Table 5 gves he oal cos for he spare capacy plannng by hese dfferen ses of assgnmen. Fgure 8: Topology of Hasegawa Ne. Fgure 9: Dsance Merc. Fgure 10: Cos Merc. Fgure 11: S.C.A. by Hergburg. Fgure 12: S.C.A. by Grover. Fgure 13: S.C.A. by SCAP. Herzburg Grover SCAP Toal Spare Capacy Toal Spare Cos Table 5 Resul of he Second xpermen From he resuls shown n Table 5, we can fnd ha even hough he oal spare capacy found by SCAP s more han hose found by he Herzburg and Grover, he oal cos s less han eher of hem. Ths shows ha SCAP s beer han oher heurscs n he way ha can handle he cos merc and dsance merc successfully for obanng he lowes cos nework. Based on he adapve propery durng he execuon of SCAP, can be expeced ha can handle an adapve cos funcon oo. Ths s beer han Lnk IP and Pah IP whch need o keep he cos parameer consan durng he eraon o ensure lneary due o her nably o handle nonlnear neger programmng problem. 5. Concluson I can be seen ha SCAP s a generc and robus heursc for he spare capacy plannng wh boh lnk resoraon and pah resoraon sraeges. No only can acheve he opmzaon of he oal spare capacy, also mnmzes he oal cos n a reasonable me. From he expermenal resuls, can be seen ha he SCAP performance s almos as good as all exsng heurscs n he leraure whle SCAP conrbues o he nsgh of choosng he pah se as well as s capably n handlng nonlnear cos funcon based on s adapve propery.

12 References [1] B. D. Venables, Algorhms for he spare capacy desgn of mesh resorable neworks, Maser of Scence Thess, Unversy of Albera, Fall, [2] H. Sakauch, Y. Nshmura, and S. Hasegawa, A self-healng nework wh an economcal spare-channel assgnmen, GLOBCOM '90, pp vol.1, [3] W. D. Grover, T. D. Blodeau, and B. D. Venables, Near opmal spare capacy plannng n a mesh resorable nework GLOBCOM '91, pp vol.3, [4] M. Herzberg, A decomposon approach o assgn spare channels n self-healng neworks, GLOBCOM '93, pp vol.3, [5] B. D. Venables, W. D. Grover, and M. H. MacGregor, Two sraeges for spare capacy placemen n mesh resorable neworks, I Global Conference on Communcaons, pp vol.1, [6] M. Herzberg, S. J. Bye, and A. Uano, The hop-lm approach for spare-capacy assgnmen n survvable neworks, I/ACM Transacons on Neworkng, pp vol.3, [7] R. R. Iraschko, M. H. MacGregor, and W. D. Grover, Opmal capacy placemen for pah resoraon n STM or ATM mesh-survvable neworks, I/ACM Transacons on Neworkng, pp vol. 6, [8] R. R. Iraschko, and W. D. Grover, A hghly effcen pah-resoraon proocol for managemen of opcal nework ranspor negry, I Journal on Seleced Areas n Communcaons, pp vol. 18, [9] A. Al-Rumah, D. Tpper, Y. Lu, and B. A. Norman, Spare Capacy Plannng for Survvable Mesh Neworks, NTWORKING, pp , [10] H. S. Wlf, Algorhms and Complexy, Prence-Hall, [11]. Lawler, Combnaoral Opmzaon: Neworks and Marods, Dover, [12] S. Leyffer, Deermnsc Mehods for Mxed Ineger Nonlnear Programmng, Ph.D hess, Deparmen of Mahemacs and Compuer Scence, Unversy of Dundee, 1993.