Strategies for Enhanced Dual Failure Restorability with Static or Reconfigurable p-cycle Networks

Transcription

1 Reprnt of paper to appear at the 2004 Internatonal Conference on Communcatons (ICC 2004), Pars, France, June Strateges for Enhanced Dual Falure Restorablty wth Statc or Reconfgurable p-cycle Networs Domnc A. Schupe * Semens AG, Corporate Technology Informaton and Communcatons Otto-Hahn-Rng 6, Munch, Germany E-mal: <frstname>.<lastname>@semens.com Wayne D. Grover, Mattheu Clouqueur TRLabs and Unversty of Alberta 7 th Floor, St NW Edmonton, Alberta, Canada T6G 2V4 E-mal: {grover, clouqueur}@trlabs.ca Abstract We suggest methods for achevng hgh dual-falure restorablty n p-cycle networs that are optmally desgned only to wthstand all sngle falures, or have mnmzed amounts of addtonal capacty for dual falure consderatons. In one set of crcumstances we consder statc p-cycles (that cannot be rearranged once establshed) and propose strateges to enhance the dual-falure restorablty based on concepts of falure spreadng and lmtng the mamum number of protecton relatonshps of any p-cycle. We then consder the case of reconfgurable p-cycles, n whch the spare capacty can be reconfgured dynamcally, creatng a new set of p-cycles that are optmzed to wthstand possble second falures. Results show sgnfcant mprovements of the dual-falure restorablty obtaned by lmtng the number of protecton relatonshps n statc desgn. For 100% dual-falure restorablty, p-cycle reconfguraton reusng most of the estng p-cycles and addng some new ones appears to be the most promsng approach. Keywords- p-cycle, Protecton, Dual Falures, Reconfguraton. I. BACKGROUND The concept of p-cycle protecton fnds applcaton n many nds of networs [1]. Hgh capacty-effcency and fast protecton swtchng tmes are acheved for span-protecton n transport networs [1][2]. Le a shared-protecton rng, a p-cycle protects worng capacty on the spans t covers ( oncycle spans ) by provdng loop-bac protecton paths. But unle rngs, a p-cycle can also protect worng capacty of offcycle spans whch have ther end-ponts on the p-cycle ( straddlng spans ). Moreover, a p-cycle offers two protecton paths for each straddlng span, and, thus, protects twce as much worng capacty on these spans as on the on-cycle spans. For a detaled descrpton of the p-cycle concept see [1] or [3]. Most of the wor on p-cycle networ desgn to date has focused on effcently provdng a guarantee of 100% restorablty aganst any sngle span falure. Ths R 1 = 1 condton remans the most cost-effectve sngle step that networ operators can tae to enhance the avalablty of all servces n ther networs. Once ths s n place, however, t s reasonable to consder how the nvestment n redundant * Ths wor orgnated durng a research stay of D. Schupe at TRLabs, Edmonton, Canada and s part of the wor when he has been wth the Insttute of Communcaton Networs at Technsche Unverstät München, Munch, Germany. capacty to acheve R 1 = 1 mght be further leveraged or optmzed to also wthstand dual falures as well as possble. Especally n large networs [4][5], or where mantenance actvtes reduce the avalable spare capacty, dual falures may become the domnant factor n determnng the ultmate servce avalablty of those networs. We therefore nvestgate several deas for provdng networs, based on p-cycles, whch have assured survvablty to any sngle falure, but also have an enhanced or optmzed level of dual-falure restorablty, denoted R 2. These strateges may nvolve usng only the spare capacty already nvested for R 1 = 1, or addng addtonal capacty to acheve specfc goals for R 2. Restorablty s defned as the fracton of demands that are affected by a falure scenaro, but whch survve by vrtue of recovery mechansms usng spare capacty of the networ. For sngle span falure scenaros, p-cycles guarantee (by the usual desgn default condton) restorablty R 1 of 100%. Some ntal analyses of dual-falure restorablty wth statc p-cycles have already been publshed [6][7]. Ths paper further etends and refnes the methods for statc p-cycle networs wth enhanced dual-falure restorablty and ncludes completely new methods for reconfgurable p-cycle networs that acheve 100% dualfalure restorablty. In general, we can dentfy three reasons to trgger dynamc p-cycle reconfguraton [8]: 1. The demand changes: p-cycles adapt to protect newly arrved demands [9][10]. 2. A (frst) falure occurs: Immedate p-cycle reconfguraton can mamze the potental for protecton aganst a subsequent falure. 3. Operator/protocol-ntaton: p-cycles are reoptmzed, e.g., after falure-repar or n a mantenance wndow. The paper focuses on case 2; the related aspects of case 3, however, wll be dscussed. For smplcty, we assume that n a dual falure scenaro, the second falure always occurs after the frst falure has been fully responded to by p-cycle protecton. In other words any second falure s assumed to occur on a networ n whch the frst falure and any reconfguratons n response to t have been completed. The nvestgatons on statc p-cycles [6] show that the dual-falure restorablty can be mproved by ncreasng the 1/6

2 D. A. Schupe, W. D. Grover, M. Clouqueur, Strateges for enhanced dual falure restorablty wth statc or reconfgurable p-cycle networs dversty of cycles used n the desgn and by desgnng so as to manage the number of protecton relatonshps each p-cycle may have through a susceptblty measure. In ths paper, we nvestgate these and other methods for both statc and reconfgurable p-cycles, to acheve the hghest state of readness after a frst falure should a second falure occur. In partcular, statc and reconfgurable p-cycles are compared wth respect to capacty requrements and we consder whether the number of reconfguraton actons followng a frst falure can be mnmzed whle stll achevng goals for R 2. In Secton II we classfy the desgn cases arsng when dual falures are taen nto account and formulate correspondng optmzaton models. Secton III dscusses restorablty and capactyeffcency results for networ case studes desgned usng the proposals of Secton II. Secton IV concludes and gves an outloo on further research. II. P-CYCLE DESIGN MODELS TO ACHIEVE DUAL-FAILURE RESTORABILITY Fgure 1 gves a logcal classfcaton of desgn and operatonal approaches that can be consdered from a standpont of R 2 consderatons. The optons depend on the networ operaton (e.g., f reconfguraton s supported or not) and on desgn preferences (e.g., desgn for sngle or dual-falure restorablty). Dual-falure optmzaton (R 2 optmzaton) means that the desgn ams to acheve hgh dual-falure restorablty, wth respect to the average case (or, alternatvely the worst case). No R 2 optmzaton Statc p-cycles R 2 optmzaton Protect vulnerable worng capacty only p-cycle desgn Protect vulnerable protecton capacty only Reconfgurable p-cycles No R 2 optmzaton R 2 optmzaton Protect any vulnerable capacty After frst falure Fgure 1. Possbltes n the p-cycle networ desgn when dual falures are taen nto account. Let us now dscuss desgn formulatons correspondng to the dfferent optons n Fgure 1. A. Basc Models for p-cycles Wthout R 2 Optmzaton Ths formulaton, frst gven n [9] and employed n [1][2], serves as the benchmar for R 1 desgn (sngle falure restorable) at mnmum spare capacty cost and s also the startng pont for the new models developed here. It assumes WDM networs wth full (or suffcent) wavelength converson, or networs wth equvalent characterstcs. The topology s descrbed by the set of spans S, ndeed by for faled spans and by for non-faled spans. A set of elgble and smple 1 cycles X s pre-computed, ncludng the protecton capacty matr, ( π ), ndcatng on whch spans S each canddate p-cycle X les, and the worng capacty coverage matr, or matr of protecton relatonshps, ( ρ ), 1 A smple cycle passes through a node or a span at most once. ndcatng how many worng capacty unts on span S can be protected by a sngle p-cycle on cycle X (1 for on-cycle spans, 2 for straddlng spans, 0 otherwse). For a span S, the cost of one unt of capacty s denoted by C. The desgn soluton s descrbed by varables n for the number of unt copes of p-cycles selected on cycle X, and by aulary varables s for the number of unts of protecton (or spare ) capacty on span. We assume the worng capactes w are gven nput parameters. s w Mnmze: C s (1) = π n X X S ρ n n,, S (2), S (3) X The objectve functon (1) mnmzes the total costweghted protecton capacty, constrants (2) determne the protecton capacty needed to form the p-cycles and constrants (3) ensure that there are enough p-cycles to protect the worng capacty. B. Models for Statc p-cycles wth R 2 Optmzaton 1) Reduce Susceptblty to Dual Falures Reference [6] proposes the followng etenson to the basc model. It ams to reduce the susceptblty 2 by eepng the worng capacty coverage (or protecton relatonshps) of the cycles low. New varables t ndcate whether a cycle s used n the desgn,.e., t s 1 f n s greater than 0, otherwse t s 0: n t n N, { 0,1} t, X (4) X The constant N s a large number. The addtonal (mnma) varable ρ ma s mnmzed (ths can be ncluded n the objectve, see [6]): ρ ma ρ t, S X (5) By mnmzng ρ ma we lmt the mamum number of untcapacty protecton relatonshps provded by any sngle p- cycle. Ths mproves the dual-falure restorablty by reducng the propensty for any sngle p-cycle to be smultaneously affected by two falures. New results here, show that a dfferent susceptblty measure,, relatng to the number of protected spans nstead of capacty unts, allows an even better control of the dual-falure restorablty. 2 A p-cycle s sad to be susceptble to a specfc dual falure combnaton f t bears protecton relatonshps to worng capacty on two or more spans whch are both faled smultaneously n the falure scenaro. 2/6

3 D. A. Schupe, W. D. Grover, M. Clouqueur, Strateges for enhanced dual falure restorablty wth statc or reconfgurable p-cycle networs We defne as: And (5) s replaced by: ma ρ + π = (6) 2 S t, X (7) B C Fgure 2. Eample of the fault-sharng concept. A The matr entry ndcates that cycle can protect capacty on span,.e., = 1, f span bears an on-cycle or a straddlng span logcal relatonshp to cycle, 0 otherwse. Ths s wthout consderaton of the actual amount of worng capacty that cycle may protect on span. Hence, a cycle s susceptble to 1 dual falures. S S Now, nstead of ncludng the susceptblty n the objectve, we can tae t drectly nto account durng the selecton of the elgble p-cycles. We thereby ncorporate susceptblty as a restrcton,.e., a canddate p-cycle n X s responsble for at most ma spans. 2) Falure Dspersal or Fault Sharng Another prncple s to desgn so that the total protecton of worng capacty on any gven span s dversfed or dspersed over multple p-cycles. The ntent s to avod the stuaton where a dual falure taes down the entre worng demand gong through a span because they have all been protected by the same p-cycle whch may be fully utlzed by the frst falure or otherwse overwhelmed by the partcular dual falure combnaton. A frst approach s to requre that at least µ dverse cycles share the protecton relatonshps on a span (ths s smlar to the multplcty or sharng factor concept for a dfferent nd of shared bacup cycles n [11] where a multple protecton cycles per span are requred) 3 : t µ X, S (8) For nstance, wth reference to Fgure 2, f µ s 2, then the span 7-2 can be protected by p-cycle combnatons A/B, A/C, B/C, or A/B/C. Note that after a frst falure of ths span, the combnaton A/B s better than A/C, snce the cycles A and B are dsjont ecept for ths span. If the two correspondng cycles each cover half of the worng capacty of span 2-7, then the worst case second falure for combnaton A/B (e.g., 2-3) causes loss of at most half of the worng capacty, whle the worst case second falure for combnaton A/C (e.g., 2-3) causes the full worng capacty to be lost. 3 Ths constrant can only hold for spans that are members of at least µ cycles. Although we can reformulate to t µ X, S : X µ, for smplcty we assume µ to be low enough, such that every span s member of at least µ cycles. C. Models for Reconfgurable p-cycles wth R 2 Optmzaton After a frst falure, p-cycles that are used for restoraton of affected worng lns are no longer avalable to protect other unaffected worng lns that they normally protect. In addton, the protecton path(s) of p-cycles whch are n an actve restoral state are themselves eposed, should a second falure ht them drectly. For nstance, the falure of span 2-7 n Fgure 3 maes worng capacty on spans 2-3, 3-4, 4-5, 5-6, 6-7, and 3-6 vulnerable to a second falure, f p-cycle A protects some worng capacty on these spans. Also the protecton path (2-3, 3-4, 4-5, 5-6, and 6-7) s vulnerable D Fgure 3. Eample of the p-cycle reconfguraton concept. p-cycle reconfguraton can hedge aganst ths vulnerablty by ntroducng, for eample, cycle D after the falure. To fully protect the worng capacty and the protecton path, two copes of cycle D are needed. If the p-cycle s not perfectly loaded, t may not be necessary for t to be able to protect all spans wthn tself. In the eample, p-cycle A may be responsble only to protect span 2-7; then, p-cycle D n the reconfguraton needs (for dualfalure protecton) the same capacty as cycle A only, snce t protects the protecton path only. To avod mang the reconfguraton models that generalze these deas overly comple, we mae the followng assumptons for a survvng span after a frst falure on span : 1. Reconfguraton should protect an upper bound on the unprotected worng capacty followng the frst falure,.e., t should protect ' w, = ρ n A X, ρ > 0 unts on span. Ths term sums the total number of worng capacty unts on span that have potentally lost ther protecton because that protecton was guaranteed by p-cycles also protectng span. Ths number s therefore greater than or equal to the worng capacty that s actually left unprotected after falure of span. Equalty holds only for perfectly loaded p-cycles on span. Overall, ths assumpton can result n more capacty consumpton than needed, because reconfguraton protects the vulnerable 3/6

4 D. A. Schupe, W. D. Grover, M. Clouqueur, Strateges for enhanced dual falure restorablty wth statc or reconfgurable p-cycle networs worng capacty of perfectly loaded p-cycles. If, however, the p-cycles are well loaded, lttle addtonal capacty wll be epected. 2. Reconfguraton should protect an upper bound on the capacty of the unprotected protecton paths that are actvated by the frst falure,.e., t should protect ' s, = π n unts on span. Ths term sums X, ρ > 0 the total protecton capacty on span of all p-cycles affected by the frst falure on span. Note that, agan, ths assumpton can result n more capacty consumpton than needed, because reconfguraton wll protect a p-cycle s protecton capacty even f t s not actually used after a frst falure on one of ts oncycle or straddlng spans (.e., the frst faled span could have been but was not actually protected by the p-cycle). Note that a perfectly loaded p-cycle protects all ts on-cycle and straddlng spans (.e., the upper bound s reached). So far we have also consdered that both the eposed (but not faled) worng capacty followng the frst falure, and the protecton path(s) themselves should be consdered for reestablshng protecton coverage durng reconfguraton n response to a second falure. Ether of them, however, can n prncple also be consdered alone. If a worng path s guaranteed to survve a sngle falure only, t may be suffcent to regard the am of the post-falure reconfguraton as beng to acheve protecton of the remanng unfaled worng capacty only. In other words, the phlosophy would be to protect normal worng capacty from dual falures, but not to reprotect protecton paths themselves. Ths ponts out that there can actually be two slghtly dfferent meanngs to what one has n mnd when we say dual-falure survvablty. The central dstncton s whether: Every worng demand should be protected from both falures. Every worng demand should be protected from any one falure that hts t drectly, even f the networ as a whole has already sustaned a pror falure. These two contets are consdered n our results as R 2 = 100% and R 2 = 100% only for worng capacty, respectvely. As ponted out n [7], certan dual falures can be survved wth falure sgnallng on the cycle or under certan falure crcumstances, but for the reasons gven ths s not consdered n ths secton. In addton, we assume that estng worng capacty and the routes of worng demands that are unaffected by a frst (or second) falure are not re-arranged. 1) Complete Cycle Reconfguraton After Frst Falure Complete reconfguraton can be thought of as meanng that for each sngle falure state n the networ we have a predetermned optmzed set of alternate p-cycles to swtch to, to mamze the protecton aganst a subsequent falure. In addton the no-falure state has a specfc confguraton of p- cycles that s nown to support 100% restorablty to any frstfalure. To model ths we ntroduce an addtonal set of varables denotng whch p-cycles are to be formed after falure of span : n, X, S : π = 0 Note that for certan dual falures (e.g., n a rng-le p- cycle wthout straddlng spans) reconfguraton cannot acheve 100% dual-falure survvablty. Then, for these dual falures n s not defned by default (.e., by π = 0 ). Constrants (2) are replaced by constrants determnng the upper lmt of spare capacty requred for each falure case: s s X π n π n X, π = 0, S (9),, S, (10) These addtonal constrants determne the set of cycles after falure of span : w + π n ρ n X, ρ > 0 X, π = 0,, S, (11) The second term n (11) sums the capacty of the protecton paths after the frst falure. Overall, the soluton specfes the set of p-cycles to use to fully wthstand any second falure after each frst-falure state has been entered. It also generates enough spare capacty on each span so that all of the varous p- cycle confguratons are feasble to confgure. Reconfguraton can be done n practce usng crossconnect nodes (XC) wth pre-planned nformaton: once the falure-span dentty s dssemnated networ-wde, each XC ndependently mplements ts local set of spare capacty connectons that effects the net requred p-cycle set networ wde. Snce the p-cycle confguraton s pre-planned, ts computaton can be done offlne, and thus, s not a tme-crtcal factor. Alternately, networs wth the dstrbuted cycle preconfguraton (DCPC) protocol [9] can re-actvate the protocol to autonomously self-organze the requred new set of p-cycles after any sngle-falure state s entered (and dssolvng any p- cycles not used n the frst falure protecton response). 2) Incremental Confguraton After a Frst Falure Another approach s to ncrement the estng confguraton n each sngle-falure state. In other words, after a frst falure, we want the p-cycles, whch are ntally found and whch are partally actve n recoverng the frst falure, to stay. Therefore, to protect aganst second falures, we deploy addtonal p- cycles and, as an opton, assgn more (prevously unused) protecton relatonshps to the estng p-cycles. The most straghtforward ncremental approach nvolves ntroducng only an addtonal set of p-cycles. These addtonal varables denote how many addtonal cycles are employed n the reconfgured state after falure of span : n, X, S : π = 0 4/6

5 D. A. Schupe, W. D. Grover, M. Clouqueur, Strateges for enhanced dual falure restorablty wth statc or reconfgurable p-cycle networs The constrants (2) are replaced by constrants determnng spare capacty needed n the non-falure case plus the upper lmt of spare capacty for each falure case: s π n + π n X X, π = 0,, S, (12) The followng addtonal constrants determne the set of addtonal cycles after falure of span : ρ n + π n ρ n, X, ρ > 0 X, ρ > 0 X, π = 0, S, (13) Wth ths modfcaton of the model, addtonal p-cycles are found and the ntal p-cycles reman untouched n ther confguraton (they do not even get new protecton relatonshps). As the ntal p-cycle set and thus a large part of the networ confguraton s ept autonomous, ths may ease admnstraton of the protecton confguraton. However, to realze better capacty effcency, we can also eplot the ntal set of cycles to protect aganst secondary falures. In ths approach, besdes deployng new p-cycles, estng p-cycles get new protecton relatonshps. In ths case we can use the complete cycle reconfguraton approach of the prevous subsecton plus the followng constrants: n n, X, S : π = 0 (14) 3) Mnmzng Reconfguraton Actons The number of cross-connectons to effect these reconfguratons could be a concern. Theoretcally a complete update of protecton confguraton may be needed. Dependng on technology and control assumptons ths may or may not be an ssue. Reconfguraton s assumed to happen after the frst falure s fully protected and before a second falure has occurred. It can therefore be a relatvely slow process (mnutes) and stll be very short relatve to the repar tme of the frst falure or the mean tme to a second falure. If the nodes are able to do parallel operatons (e.g., parallel cross-connectng nstructons over the connecton control nterface), such concerns may also not be of mportance. If relevant, however, one approach to mnmze the reconfguraton worload s to eep the number of ntally confgured cycles n place to the β 0,1 ): etent possble (controlled by [ ] β, X, S : π = 0 (15) n n Whle we epress the problem as one of mnmzng the number of reconfgured p-cycle unts, the concept can easly be enhanced to epress the reconfguratons n terms of reconfgured nodes or reconfgured p-cycles (havng one or more unts). III. CASE STUDIES We now present results of a study networ (wth the COST239 topology as studed n [2]) wth 11 nodes, 26 spans, and 3531 possble cycles (the average nodal degree d s 4.7). COST239 s three-connected therefore t s topologcally feasble for double falures to be survved f adequate spare capacty s present. A. Statc p-cycles Fgure 4 shows how R 2 can be mproved usng the fault sharng approach and the susceptblty approach both of whch are based on statc p-cycles and shows the assocated capacty cost penalty. For the fault sharng approach, factor µ s vared from 1 to 10 (a µ of 1 has no effect, and a µ hgher than 10 means that the protecton of any span has to be shared by more than ten cycles, whch seems to be mpractcal). For the susceptblty restrcton approach, ma s vared from 4 to 26 (a ma of 3 s nfeasble, and a ma of 26 means no effect). Fgure 4 also ncludes the comparson value for R 2 reconfgurable networs. Addtonal Relatve Spare Capacty Cost Fault Sharng Susceptblty Reconfguraton (global) ma =9 0.1 ma= Dual-falure Restorablty ma=4 smaller ma Fgure 4. Cost over dual-falure restorablty R 2 for three p-cycle desgn approaches. The capacty optmal desgn wthout R 2 optmzaton results n a reference cost C ref of and an R 2 of 61 percent. As we can see, the fault sharng approach does not brng any sgnfcant mprovement of R 2. As t turns out, the fault sharng approach can only mprove worst cases, but not the average case. The susceptblty restrcton, however, s a tool to mprove R 2 : wth every decrement n the mamum number of allowed protecton relatonshps per cycle we get a sgnfcant gan n restorablty. Smlar to the results of [6], mnmzng the mamum susceptblty such that the cost C ref are preserved, however, yelds only n very small restorablty devatons (about 1% only). Both for the span sharng factor and the mamum susceptblty, the hgher the value, the more cycles wll be selected for the optmal soluton. But for all the correspondng nstances of Fgure 4, the number of cycles s at most 15 only. We get smlar results for three other test networs. B. Reconfgurable p-cycle Desgns for 100% Dual-Falure Restorablty Fgure 5 shows a comparson of the cost of the desgns for statc, non-r 2 -optmzed p-cycles (1 st bar) and reconfgurable p- cycles (bars 2 to 7). All desgns whch ensure R 2 = 100% (bars 3 to 5, and bar 7) requre at least 70% more capacty than the reference R 1 desgn. An ntermedate value s reached for the case where only the worng capacty s protected aganst dual falures (see Secton II.C), wth 60% addtonal cost (2 nd bar). Only a margnal cost ncrease for the global reconfguraton 5/6

6 D. A. Schupe, W. D. Grover, M. Clouqueur, Strateges for enhanced dual falure restorablty wth statc or reconfgurable p-cycle networs desgn s nvolved, f at most 5% of the p-cycles can change (4 th bar). Compared wth the global reconfguraton output (3 rd bar), 47% more cost s needed for eepng p-cycles untouched and addng new ones (7 th bar). If estng p-cycles are eploted, the cost ncrease s only 2% (5 th bar). Wth a cost ncrease of 27%, agan an ntermedate value s reached for the case where only the worng capacty s protected aganst dual falures and estng p-cycles are untouched (6 th bar). Relatve Spare Capacty Cost Statc, R1=100% (acheves R2=61%) Reconfguraton, R1=100%, R2=100% only for w orng capacty, global reconfguraton Reconfguraton, R1=R2=100%, global reconfguraton Reconfguraton, R1=R2=100%, reconfguraton w th at most 5% p-cycles changed Reconfguraton, R1=R2=100%, reconfguraton w th no p-cycles changed Reconfguraton, R1=100%, R2=100% only for w orng capacty, reconfguraton w th addtonal p-cycles Reconfguraton, R1=R2=100%, reconfguraton w th addtonal p-cycles Fgure 5. Cost comparson between desgns for statc and reconfgurable p-cycles (for bars 4 to 7 only appromate solutons, the ntervals ndcate the LP relaaton bounds). Although a capacty cost ncrease of 70% for an R 2 desgn relatve to the R 1 desgn sounds hgh, ths value may be epected. In addton, gven the low ntal redundancy of the R 1 reference p-cycle desgn, ths stll represents R 2 restorable networs that typcally wll have less capacty than rng or 1+1 APS-protected R 1 networs. From the capacty pont-of-vew, an effcent R 1 desgn for COST239 could acheve a spare to worng capacty rato of 1/(d-1) = 27% (see for eample [12] for the dervaton of ths formula). For an effcent R 2 desgn, however, the estmaton clmbs to 2/(d-2) = 74% (durng a double span falure ncdent at a node, the worng capacty of the 2 spans s best spread evenly on the node s survvng d-2 spans). Therefore, f we relate these capacty estmatons for the two desgns, we epect appromately 2.7 tmes more spare capacty needed for R 2 = 1 compared wth R 1 = 1. For the test networ, however, as we deal wth cost weghted spare capacty n (1) and we do not deal wth an estmator value for a pure-optmal R 1 desgn, the cost dfference between R 2 = 1 and R 1 = 1 can be lower. IV. OUTLOOK In ths paper, we presented capacty desgn methods for mproved or complete dual-falure restorablty usng ether statc p-cycles or p-cycles that can be reorganzed after the event of a frst falure. Test case studes show that usng reconfgurable p-cycles allows one to offer a dual-falure restorablty guarantee at a sgnfcantly lower capacty cost, although ths cost s stll 70% hgher than the cost of desgn for sngle falure restorablty only. Another approach that could be of nterest for a p-cycle networ that s optmally desgned to protect aganst sngle falures, would be to consder how much the ablty to reconfgure p-cycles can enhance the dual-falure restorablty (wthout addng any new capacty.) Although ths approach may not offer a hgh gan for the statc case [6], reconfgurable networs may tae advantage of the common pool of protecton capacty to acheve hgher average or mnmal dualfalure restorablty. The approach can be used n complete p- cycle reconfguraton and for statc p-cycles wth new protecton relatonshps n the reconfguraton and s for further study. Future wor on the present topc could also nvestgate the possblty of desgnng a p-cycle networ for multple protecton classes (R 1 and R 2 ) as presented n [13] n the contet of span-restorable networs. The authors than M. C. Scheffel for hs helpful comments on the paper. REFERENCES [1] W. D. Grover and D. Stamatelas, Brdgng the rng-mesh dchotomy wth p-cycles, n Proc. of DRCN 2000, pp [2] D. A. Schupe, C. Gruber, and A. Autenreth, Optmal confguraton of p-cycles n WDM networs, n Proc. of ICC 2002, pp [3] W. D. Grover, p-cycles, Chapter 10 n Mesh-based Survvable Networs: Optons for Optcal, MPLS, SONET and ATM Networng, Prentce-Hall, Aug [4] M. Clouqueur and W. D. Grover, Avalablty analyss of span restorable mesh networs, IEEE JSAC, May 2002, pp [5] D. A. Schupe, A. Autenreth, and T. Fscher, Survvablty of multple fber duct falures, n Proc. of DRCN 2001, pp [6] D. A. 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