A New Optical Signal Routing Scheme for Linear. Lightwave Networks. Milan Kovacevic and Mario Gerla

Transcription

1 New Optcal Sgnal Rutng Scheme fr Lnear Lghtwave Netwrs Mlan vacevc and Mar Gerla bstract In ths paper we prpse a new scheme fr ptcal sgnal rutng wthn Lnear Lghtwave Netwr (LLN) subnets. LLN s a recently prpsed ber ptc netwr whch perfrms nly lnear peratns n ptcal sgnals: pwer cmbnng, splttng and pssbly lnear ptcal amplcatn. LLN can be parttned nt several subnets where each subnet s a tree prvdng full bradcast amng all statns cnnected t t. We study the synchrnzatn prblem that exsts n these subnets whch prevents ecent mplementatn f tme dvsn multple access schemes fr sharng a cmmn bradcast medum. slutn t ths prblem s prpsed, based n a new ptcal sgnal rutng scheme, called rted rutng. The mpact f rted rutng n pwer lsses s analyzed, and an apprach fr mnmzng pwer lss n LLNs wth rted rutng s presented. It s shwn that when ln and excess lsses are small, rted rutng pwer budget can be made clse t the pwer budget yelded by the rgnal, shrtest path rutng scheme. It s als shwn that rted rutng pwer budget can be sgncantly mprved usng a sngle ptcal ampler. Intrductn It s antcpated that ne f the mar cnsumers f bandwdth n future ptcal MNs wll be vde dstrbutn and telecnferencng. Ths type f trac requres large bandwdth, and t must be dstrbuted t several (multcast) r even all (bradcast) netwr users. Therefre, the best canddates fr future MNs are thse archtectures whch can supprt multcast trac ecently. Netwrs that t ths categry are manly thse based n a bradcast star [] r a tree [] tplgy. These netwrs, hwever, have lmtatns n supprtng large numbers f statns. Snce each sgnal s bradcast t all statns, pwer lss ncreases wth the number f statns due t splttng. There s als a capacty lmtatn. The number f avalable wavelengths s lmted by the recever's tunng range t values frm tens t a hundred wavelengths, thus creatng a bttlenec when the number f statns s large []. esdes, relablty f thse tplges s als a prblem. sngle ln r nde falure dscnnects ne r mre statns frm the rest f the netwr. mparng a star and a tree tplgy, the tree s preferable wth respect t ber layut csts. In rder t establsh a star, t s necessary t nstall a ln frm each Ths research was supprted n part by a grant frm Mtsubsh Electrc. The paper was presented n part at the IEEE INFOOM ' cnference, Flrence, Italy, May. M. vacevc was wth the mputer Scence epartment, Unversty f alfrna, Ls ngeles, 00. He s nw wth the enter fr Telecmmuncatns Research, lumba Unversty, New Yr, NY 007. M. Gerla s wth the mputer Scence epartment, Unversty f alfrna, Ls ngeles, 00. nde t the central nde (n bth drectns). It s shwn n [] that fr a gven nde placement, the nstallatn cst (.e. the amunt f ber used) can be reduced sgncantly f a tree tplgy s chsen nstead f a star tplgy. nvel ber ptcs archtecture, called Lnear Lghtwave Netwr (LLN) [0] was recently prpsed t vercme the lmtatns f star and tree tplges and at the same tme explt the cst benets f tree tplges. LLN s based n a mesh tplgy, but t allws parttnng such a tplgy nt several subnets. Each subnet s a tree prvdng full bradcast amng all statns cnnected t t. y parttnng the netwr nt clusters, the pwer splttng lsses are reduced, and the same wavelength can be reused n dfferent subnets leadng t an ncrease n aggregate netwr capacty. LLN s als very rbust snce t can be dynamcally recngured n rder t vercme ln and/r nde falures. Prevus research n bradcast netwrs shws that gd perfrmance can be btaned f bth wavelength and tme dvsn multple access are used t share a cmmn bradcast medum []. Thus, we wuld le t apply the same multaccess scheme n LLN subnetwrs. Hwever, we wll shw that a mar bstacle t the mplementatn f a tme dvsn multaccess scheme n LLNs s represented by the dculty n synchrnzng the transmtters. nsequently, ths leads t an necent utlzatn f the shared medum. In ths paper we prpse a slutn t ths prblem usng a specal rutng scheme wthn each LLN subnet. The paper s rganzed as fllws. Sectn gves an vervew f the LLN archtecture. In Sectn the synchrnzatn prblem n LLN s descrbed. Sectn prpses a slutn t ths prblem. In Sectn the mpact f the prpsed slutn n budget s analyzed and an apprach fr pwer budget ptmzatn s presented. Sectn 6 cncludes the paper. Overvew f LLN The Lnear Lghtwave Netwr (LLN) [0,, ] s a recently prpsed lghtwave netwr archtecture. s ts name mples, LLN perfrms nly lnear peratns n ptcal sgnals: pwer cmbnng, splttng, and lnear (nnregeneratve) ptcal amplcatn. LLN cnssts f statns and ndes ntercnnected amng themselves va bdrectnal ber ptc lns. The ey cmpnent f LLN s a devce called Lnear vder-mbner (L) nstalled at each netwr nde. L s a multprt devce that

2 perfrms swtchng, splttng and multplexng f ptcal sgnals. L s, n fact, an electrcally cntrllable ptcal swtch wth addtnal features such as multcastng (an nput sgnal can be dstrbuted t several utputs) and multplexng (several nput sgnals can be cmbned n a sngle utput ln). Fgure shws a L wth fur pars f ncmng and utgng ber lns. Input lnes represent ncmng ptcal bers and utput lnes utgng bers. pwer transfer matrx: M 6 a a a a a a a a a where matrx element a represents the fractn f ptcal sgnal frm nput prt sent t utput prt. The matrx elements can be expressed as 7 L Fgure : Lnear vder-mbner a ; ; f; ; : : :; g () where represents the prtn f pwer frm nput prt splt t utput prt (thrugh the dvder stages), and the prtn f pwer frm nput prt cmbned nt utput prt (thrugh the cmbner stages). Nte that and must satsfy the fllwng sets f cnstrants: ; f; ; : : :; g () ; f; ; : : :; g () Pwer splttng and cmbnng n a L s llustrated n Fgure. δ δ δ δ σ σ σ σ Lnear vder-mbner f arbtrary sze can be bult usng drectnal cuplers wth adustable cuplng rat mplemented n T:LNbO technlgy [8]. Fgure llustrates mplementatn f a L usng these cuplers. We nte that the rst tw stages are "dvder" stages, and the last tw are "cmbner" stages. δ δ δ δ δ σ σ σ σ σ δ δ δ σ σ σ δ σ δ δ δ σ σ σ Fgure : Pwer splttng and cmbnng n a L Fgure : Implementatn f a L usng drectnal cuplers The L's swtchng functn s dened by a pwer transfer matrx. L wll have the fllwng LLN can have an arbtrary mesh physcal tplgy. Fgure shws a LLN wth sx statns (,,,, E and F ) and ve ndes (g, h,, and ). Each edge n the graph represents a bdrectnal ln,.e. t may be vewed as tw undrectnal ber lns gng n ppste drectns. y adustng the and parameters, arbtrary cnnectvty clusters can be created. Statns that cmmuncate amng themselves belng t the same cluster. Suppse that we

3 g h Fgure : LLN n a mesh tplgy have tw clusters: cluster wth statns,, and F and cluster wth statns, and E. The statns that belng t the same cluster frm a subnet that has a tree tplgy. The subnet tplgy must be a tree n rder t avd sgnal nterference that wuld happen f the same sgnal reaches ts destnatn travelng n multple paths. ls, clusters are dsnt by dentn. Therefre, derent subnets cannt share lns. In Fgure the dtted edges represent lns that are used by subnet and the bld edges lns used by subnet. Suppse nw that ln gh faled. Subnet can be recngured by changng the L pwer transfer matrces n rder t add lns h and. Smlarly, n the case f falure f ln h, subnet can be recngured by addng ln h. Ths example shws the ptental f LLNs n mprvng netwr fault tlerance. In the abve scheme there s a prblem f statns frm derent subnets wsh t cmmuncate wth each ther. One slutn s t merge these subnets nt ne. Ths s acceptable f the ttal number f statns n these subnetwrs s small. nther apprach s t ntercnnect these subnets wth anther wavelength, usng wavelength selectve Ls [0]. In ths paper we fcus n cmmuncatns wthn a LLN subnet, and d nt cnsder the ntercnnectn prblem. Therefre, we assume that a sngle wavelength s used. Hwever, ur prpsed strategy can als be extended t the multple wavelength case. In the rgnal LLN prpsal, the rutng f ptcal sgnals n a subnet s perfrmed usng shrtest path rutng, as llustrated n Fgure. If transmts, the rutng pattern n Fgure.a s fllwed. If transmts, the pattern shwn n Fgure.b s fllwed. In general, a separate bradcast tree s asscated wth each surce. Ls are cngured t supprt such dstnct bradcast trees. Fr example, the pwer transfer matrces fr the Ls at ndes e and f n the netwr shwn n Fgure have the fllwng values f the pwer dstrbutn at the Ls s unfrm (.e. the pwer frm each nput prt s splt equally amng all selected utput prts, and each utput prt receves equal F E prtn f the pwer frm all selected nput prts): M e M f a a a f a a a f a f a f a ff a a a e a a a e a e a e a ee TM synchrnzatn prblem ecause f the hgh data rate (up t several Gb/s) generally avalable n ber ptc channels, ptcal MNs belng t the categry f netwrs characterzed by a lw transmssn tme t prpagatn delay rat. In such netwrs, sharng the cmmuncatn medum n tme dman usng randm access technques s nt eectve. Therefre, a reservatn technque such as TM shuld be used n rder t acheve better channel ecency. Ths technque has prven t be eectve n ther bradcast netwrs such as fr nstance the satellte netwrs [7] (whch are nt ptcal but have cmparable transmssn t prpagatn tme rat), and has n fact been appled n sme recently prpsed lghtwave netwrs []. In TM, tme s dvded nt slts,.e., tme ntervals n whch a sngle statn s allwed t transmt a pacet. perd f several cnsecutve slts frms a frame. In star tplgy netwrs all transmssns are drected tward a central nde, frm whch they are then bradcast t all ndes. Synchrnzatn s requred t crdnate transmssns s that n tw transmssns verlap at the central nde. Ths can be acheved by determnng fr each nde the tme nstants when t can begn ts transmssn. y measurng the prpagatn delays frm each nde t the center t s pssble t schedule transmssns s that they arrve at the central nde at the desred tme. Thus, t s pssble t fully utlze each slt n the frame, achevng maxmum channel capacty. Hwever, synchrnzatn s much mre dcult t acheve n a LLN netwr than n a star netwr. What maes the synchrnzatn prblem dcult s the fact that n LLN there s n sngle synchrnzatn pnt, s that transmssns must be synchrnzed wth respect t nt ust a sngle (central) nde but several ndes. Furthermre, these synchrnzatn requrements may be cnctng, mang t mpssble t acheve the ptmal utlzatn f a shared channel. Fgure 6 llustrates the prblem. The netwr n the Fgure 6.a cnssts f the statns,, and ntercnnected amng themselves va the netwr ndes e and f. These statns and ndes, alng wth the lns ncdent n them, frm a tree. In ths example we assgn / f the ttal channel capacty t statns and t bradcast ther trac t all ther statns. The requrement fr bradcast cnnectns s mre restrctve than fr pnt-t-pnt cnnectns. It dctates that an ptcal sgnal must nt cllde wth anther sgnal anywhere n the netwr. We assume that slt sze s

4 e f e f (a) (b) Fgure : Shrtest path rutng ne tme unt, and that prpagatn delays are nrmalzed t slt sze. Fgure 6.a shws the tplgy f a netwr where ln labels represent nrmalzed prpagatn delays. The TM frame carres tw slts, ne slt each fr statns and. Let us assume that statn transmt at tmes ( 0; ; ; : : :). Statn 's slts arrve at nde e at tmes + : and at nde f at tmes + :6. In rder t have the transmssn arrvals frm and synchrnzed at e, statn shuld start ts transmssns at tmes ( + ) + :? : + :. On the ther hand, n rder t synchrnze the transmssn arrvals at nde f, shuld transmt at tmes ( + ) + :6? 0: + :7. Obvusly, we are nt able t satsfy bth requrements at the same tme, and therefre cannt avd transmssn cllsns. Fgure 6.b and 6.c shw the cases where the synchrnzatn s perfrmed wth respect t ndes e and f, respectvely. In bth cases cllsns ccur due t msalgnment f transmssn arrvals. In the rst case statn receves cllded transmssns and n the secnd case statn. In rder t avd cllsns t s necessary t ntrduce gaps between slts. In ths example, we need t nsert a gap f length 0.6, as shwn n Fgure 6.d whch wuld reduce channel utlzatn t 77%. The synchrnzatn prblem can be slved by reducng the frame t a sze that s equal t a cmmn dvsr f prpagatn delays, as stated by Stern []. Hwever, the requrement fr havng the frame sze dependent n the prpagatn delays s nt practcal. Frst f all, ths requrement may mae the frame sze t small fr any practcal applcatn. esdes, prpagatn delays may vary wth temperature changes, thus requrng cntnuus adustments n the frame sze. Furthermre, the prpagatn delays wll als change when tplgcal recnguratns ccur. There are ther appraches that can be taen t slve the synchrnzatn prblem. The slt sze can be chsen t be equal t the sum f the transmssn tme and the maxmum prpagatn delay between tw ndes. Ths allws a transmssn t dran ut f the netwr befre the next transmssn starts. The dsadvantage f ths scheme s a lw channel ecency. The ecency depends n the transmssn tme t maxmum prpagatn delay rat, transmts transmts receves receves transmts transmts receves receves transmts transmts receves receves e f Fgure 6: Synchrnzatn prblem n a tree tplgy netwr and may becme very pr n metrpltan area hgh-speed netwrs. nther apprach that can be appled t netwrs that use bth tme and wavelength dvsn multplexng s t ptmze netwr thrughput by apprprate schedulng []. Ths requres a cmplex schedulng algrthm whch s expensve t mplement. Stern [] als prpsed a smple assgnment prcedure, called Pseudrandm schedulng. Each nde s gven the pprtunty t transmt n a partcular channel at tmes dened by a (a) (b) (c) (d)

5 e f e f (a) (b) Fgure 7: Rted rutng predened pseudrandm sequence. The advantage f ths apprach s that the synchrnzatn s smpled, requrng each surce t be synchrnzed nly wth ts destnatn. Hwever, ths apprach has a relatvely lw ecency, arund 0%. ls, ths methd s nt well suted t bradcast/multcast trac. In general, the prevusly mentned appraches d nt acheve ptmal channel utlzatn. The mre ecency we want t btan, the mre cmplexty n synchrnzatn and schedulng we must ntrduce. Yet, the result wll nt be ptmal. slutn fr the TM synchrnzatn prblem Frm the prevus dscussn n TM synchrnzatn we bserved that the dculty wth general LLN netwrs s that they requre synchrnzng the arrvals f transmssns at derent ndes - a cndtn whch cannt be easly satsed. In the case f a star r a rted tree (such as Tree-net []), we nly need t synchrnze transmssns wth respect t ne nde,.e. the rt. Ths s easy t acheve, and yelds the maxmum utlzatn f the shared channel. The same dea can be appled t LLN. nde s selected t be the rt, and all ptcal sgnals, nstead f travelng n the shrtest paths, must g thrugh that rt nde befre reachng the destnatn(s). Ths rutng scheme we call rted rutng. Fgure 7 llustrates rted rutng. We may arbtrarly chse nde f t be the rt nde. If transmts, the rutng pattern n Fgure 7.a s fllwed. If transmts, the pattern shwn n Fgure 7.b s fllwed. The sgnal s bradcast t all statns, ncludng the statn rgnatng the sgnal. The statns can use ths "ech" sgnal as an acnwledgment f a successful transmssn (.e. n cllsn ccurred), r as a pwer reference t tune ther transmt pwer level, r as a tme reference t measure the rund-trp prpagatn delay (fr transmssn synchrnzatn). Thus, rted rutng s a smple and straghtfrward slutn f the TM synchrnzatn prblem. The statns need nly t synchrnze ther transmssns t nde f. y synchrnzng ther transmssns t f, statns and autmatcally acheve synchrnzatn at nde e, fr example. Fgure 8 shws transmssn schedulng fr the example frm Fgure 6 when rted rutng s used wth the rt at nde f. We see that all statns receve the transmssns wthut cllsns and there s n need t ntrduce any gaps between slts. Thus, we can practcally acheve 00% ecency f the TM channel. transmts transmts receves receves receves receves e Fgure 8: Synchrnzatn wth rted rutng Frm ths example we see that a transmssn frm t has t pass thrugh lns e, ef, fe and e n ts way t the destnatn. Ths s a lnger path than n the case f shrtest path rutng where the sgnal ges nly thrugh nde e befre reachng statn. Therefre, the prpagatn delay ncreases. ls, the pwer lss may ncrease. In rder t mplement ths scheme, the Lnear vders- mbners must be prperly cngured. In the example, the pwer transfer matrces fr ndes e and f, wth the unfrm pwer dstrbutn at the Ls, shuld have the f (a) (b)

6 fllwng values: M e M f a a a f a a a f a f a f a ff a a a e a a a e a e a e a ee Nte that all the elements f matrx M f have value. If the L at nde f were replaced by a star cupler the matrx elements wuld have value. Thus, nstead f attenuatng each sgnal by a factr f (due t pwer splttng), the L attenuates each sgnal tmes. Ths s a cnsequence f the requrement that pwer dstrbutn n Ls has t satsfy cnstrants (), () and (). ascally, t s the penalty we pay n rder t eny the recnguratn exblty f LLN. We further dscuss ths lmtatn f Ls n the next sectn. If all statns are at the same dstance frm the rt, the maxmum prpagatn delay n a LLN wth rted rutng s the same as n a LLN wth shrtest path rutng. If ths s nt the case, the delay ncreases when rted rutng s used. The actual ncrease depends n the netwr tplgy and the chce f the rt nde. It s shwn n ppendx that the maxmum prpagatn delay f rted rutng s at mst twce the maxmum prpagatn delay f shrtest path rutng f the degree f the rt s at least tw (.e., the rt has at least tw bdrectnal lns cnnected t t). ny nde n the netwr can be chsen as the central nde. One f the crtera fr chsng the center s t mnmze the average r the maxmum prpagatn delay. Ths can be dne by chsng the nde whse average r maxmum dstance frm all statns s mnmal. nther crtern fr chsng the rt nde s t mnmze pwer lsses. s t wll be shwn n the next sectn, the ptmal pwer budget des nt depend n the chce f the rt nde when ln and excess lsses are neglgble. When ln lss becmes the dmnant factr (as s the case n large metrpltan netwrs), mnmzng prpagatn delays wll als mnmze pwer lsses, snce bth prpagatn delay and ln lss are prprtnal t ln lengths. Pwer budget analyss The man lmtatn n the number f statns that can be supprted n an ptcal netwr s due t pwer lsses. The maxmum rat f the sgnal sent by a transmtter and the sgnal receved by a recever whch ensures sucent sgnalt-nse rat at the recever fr successful sgnal detectn represents the pwer margn. Wth current technlgy, fr example, t s pssble t supprt a data rate f Gb/s wth 0? errrs per bt wth -0dm receved pwer usng a slcn avalanche phtdde [6]. ssumng a 0dm laser surce, ths yelds a pwer margn f 0d r We need t ensure that the maxmum pwer lss between any tw statns, dened as pwer budget, des nt exceed the pwer margn. Therefre, the man cncern n the pwer budget analyss s t evaluate the maxmum pwer lss n a netwr, and t mnmze t whenever pssble. In addtn t pwer budget, the dynamc range s als an mprtant factr. Recever dynamc range can be dened as the dfference n pwer levels f the largest and the smallest sgnal that the recever can handle. Thus, we must ensure that the derence n the maxmum and the mnmum pwer lss between transmtters and the recever s smaller than the recever's dynamc range. In rted rutng the dynamc range prblem s easly reslved by requrng that each transmtter calbrate ts pwer s that ts wn ech pwer s equal t the pwer receved frm a reference statn. Snce all the sgnals are ruted thrugh the same rt, the calbratn wth the reference statn ensures that every statn receves the same pwer frm all surces []. Next, we analyze the pwer budget f a LLN wth rted rutng. There are fur factrs that cntrbute t the pwer lsses n ths netwr:. splttng lsses due t the splttng f ptcal pwer at a nde t several utgng lns,. ln lsses due t the attenuatn f an ptcal sgnal n a ber ptc ln,. excess lsses that ccur at ndes due t the mperfect cuplng f ncmng and utgng lns wth cuplers and. cmbnng lsses. The cmbnng lsses are the result f the L's lmtatn n multplexng ptcal sgnals. Ths lmtatn was frmulated n Sectn by the set f cnstrants () whch state the fact that t s nt pssble t buld a pwer cupler that cmbnes tw r mre uncrrelated nputs and delvers them t ts utput lsslessly []. Fr example, f we have a L and we want t cmbne the sgnals frm all ncmng lns ust t a sngle utgng ln, say r, deally we wuld le t have: a r a ; f; : : :; g 0; f; : : :; g; f; : : :; r? ; r + ; : : :; g that gves a r r Snce generally >, ths assgnment f the matrx cef- cents des nt gve a feasble slutn. In rder t have a feasble slutn, the pwer cmbnng frm an nput prt t utput prt r must be reduced tmes. Therefre? f ttal ncmng pwer wll be lst n ths L due t pwer cmbnng. In ths pwer budget analyss we are manly nterested n the eect f splttng and cmbnng lsses n ttal pwer lss. When the number f statns s large, these lsses are dmnant. Whle the excess and ln lsses can be further reduced wth technlgy advancements, the splttng and cmbnng lsses cannt be elmnated. Thus, we begn ur analyss studyng the deal case where excess and ln lsses d nt exst. Let us cnsder the pwer budget n a LLN wth rted rutng mplemented n the tplgy shwn n Fgure. We chse nde h t be the rt nde. We cnsder rst the deal case when pwer dstrbutn and multplexng at 6

7 N/... g h... N/ + N/ + Fgure : tree tplgy wth N statns and tw ndes Ls s unfrm. Ths means that that at nde g an equal prtn f pwer s cmbned frm each ln g t ln gh and an equal prtn f pwer s dstrbuted frm ln hg t each ln g where ; : : :; N. Lewse, at nde h an equal prtn f pwer frm each nput prt s dstrbuted t each utput prt. The maxmum pwer lss n ths netwr ccurs between the statns and where ; f; : : :; N g. sgnal frm statn after passng nde g s attenuated N tmes due t the cmbnng lsses. fter passng nde h, the sgnal n ln hg s attenuated ( N + ) tmes due t the pwer splttng and cmbnng at nde h. Fnally, the sgnal n ln g s agan attenuated N tmes due t pwer splttng at nde g. The ttal pwer lss f the sgnal sent frm statn t statn s N P [d] 0 lg 0 (( N + ) ( N ) ) 0 lg 0 (( N ) ) () If the shrtest path rutng s used, the maxmum pwer lss fr a LLN wth unfrm pwer dstrbutn at Ls ccurs between the statns and, where f; : : :; N g and f N + ; : : :; Ng. The sgnal s attenuated at each L ( N ) tmes due t splttng and cmbnng lsses. Thus, the maxmum pwer lss s P [d] 0 lg 0 (( N ) ( N ) ) 0 lg 0 (( N ) ) () Pwer budget can be reduced n LLN wth bth types f rutng f pwer dstrbutn and multplexng at each L s ptmzed rather than beng chsen as unfrm. We need t slve the fllwng ptmzatn prblem: etermne the pwer transfer matrces fr all ndes n the netwr, such that the maxmum pwer lss between any par f ndes s mnmal. Let us rst ntrduce the fllwng ntatn: N Ttal number f statns n the netwr. R Rt nde. r (l) Ln ncdent t nn-rt nde l (l) Number f lns ncdent t nde l. and n a path frm that nde t the rt. m (l) Nde r statn t whch nde l s cnnected va ln. n (l) Number f statns that can be reached frm nde l va ln. S (l) Lpt n(l) Lpt ut(l) E (l) l Set f statns that can be reached frm nde l va ln. Optmal rat between pwer transmtted by statn s S (l) and pwer at nput prt f nde l ( 6 r (l) ). Optmal rat between pwer at utput prt f nde l ( 6 r (l) ) and pwer receved by statn s S (l). Excess lss at nde l expressed as rat between ttal pwer enterng the nde and ttal pwer leavng the nde. Pwer lss n ln (; l). a (l) Pwer transfer matrx element fr nde l. Fr example, the netwr shwn n Fgure 0 has the fllwng parameters: N R d r (c) r (e) (c) (d) (e) m (c) m (c) m (c) d m (d) c m (d) e m (d) H m (e) d m (e) F m (e) G n (c) n (c) n (c) n (d) n (d) n (d) n (e) n (e) n (e) S (c) fg S (c) fg S (c) S (d) f; g S (d) ff; Gg S (d) ff; G; Hg fhg S (e) f; ; Hg S (e) ff g S (e) fgg E (e). ef. F e.. c E (c). E (d). eg de. G cd. c d. dh c. Fgure 0: netwr wth ve statns and three ndes The slutn fr the pwer budget ptmzatn prblem n a LLN wth rted rutng whch taes nt accunt all types f lsses s presented n ppendx. If the lns have the same lsses n bth drectns (e.g., ), t s shwn n the appendx that Lpt n(l) Lpt ut(l) Lpt (l). In such a case we have the fllwng result fr the ptmal pwer budget: where P [d] 0 lg 0 (E ( p m (l) H Lpt ) ) (6) 7

8 Lpt (l) ( pl P f p s a statn ( (p) ;6r (p) Lpt(p) )E(p) pl f p s a nde It s shwn n ppendx that when pwer dstrbutn s ptmal, pwer lss between any par f statns s the same. Ths practcally means that a recever wll receve sgnals n the same pwer level frm any transmttng statn (assumng that all statns transmt the same pwer). nversely, all statns receve the same pwer frm the same transmtter. Thus, the dynamc range requred by the recever n a LLN wth rted rutng and ptmal pwer dstrbutn s practcally zer. s an example, let us calculate the ptmal pwer budget fr the netwr shwn n Fgure 0. There we have: Lpt (c) c : Lpt (c) c : Lpt (e) ef : Lpt (e) eg : Lpt (d) (Lpt (c) + Lpt (c) )E(c) cd : Lpt (d) (Lpt (e) + Lpt (e) )E(e) de :6 Lpt (d) dh : P [d] 0 lg 0 (E (d) (Lpt (d) + Lpt (d) + Lpt (d) )) 0:0 Frm ths general slutn we can get a slutn fr the deal case. There we have E (l) l Lpt (l) n (l) Lpt and the ptmal pwer budget s P [d] 0 lg 0 N n N 0 lg 0 N + 0 lg 0 N (7) Ths s an nterestng result that shws that n the deal case the ptmal pwer budget des nt depend n the netwr tplgy nr n the chce f the rt nde. The ptmal pwer budget s determned nly by the number f statns n the netwr. ls, we can see that the cmbnng lsses cntrbute t the ttal pwer lss as much as the splttng lsses d. It s shwn n ppendx that the ptmal pwer budget n the deal case when shrtest path rutng s used s P [d] 0 lg 0 (N? ) 0 lg 0 (N? ) + 0 lg 0 (N? ) (8) The results fr the ptmal pwer budget fr bth rutng schemes are very smlar. The derence s that when rted rutng s used, a sgnal s splt amng N statns, and when shrtest path rutng s used, t s splt amng N? statns. Let us cnsder agan the netwr shwn n Fgure. The ptmal pwer budget fr ths LLN when rted rutng and shrtest path rutng are used s gven n expressns (7) and (8), respectvely. If we cmpare these results wth the pwer budget when pwer dstrbutn at Ls s unfrm, gven n expressns () and (), we can see that the pwer budget s reduced sgncantly by ptmzng the pwer dstrbutn. If we tae nt accunt the excess and ln lsses, and assume that all ndes have the same lss E and all lns have the same lss, we can shw frm expressn (6) that the maxmum pwer lss n the ptmzed LLN wth rted rutng s P [d] 0 lg 0 (E( N E + N ) ) 0 lg 0 (( N ) E ) f E Let us estmate nw the maxmum pwer lss n the LLN wth shrtest path rutng and ptmal pwer dstrbutn. We can determne a smple upper bund fr ths lss f we use the same pwer dstrbutn whch was ptmal fr the deal case (.e., subptmal pwer dstrbutn). In such a case the maxmum pwer lss s equal t the splttng and cmbnng lsses (whch are the same as n the deal case) ncreased by ln and excess lsses encuntered n the path between statns and where f; : : :; N g, f N + ; : : :; Ng. Thus, we have that the maxmum pwer lss n LLN wth shrtest path rutng and ptmal pwer dstrbutn s P [d] 0 lg 0 ((N? ) E ) Frm ths example we can mae the fllwng bservatns. Shrtest path rutng s generally better than rted rutng when the ln and/r excess lsses are dmnant. When the ln and excess lsses are small cmpared t the splttng and cmbnng lsses, the perfrmances f bth rutng schemes are almst the same. Wth a pwer margn f 0d we nd frm expressn (7) that t s pssble t supprt up t 00 statns n a LLN subnet wth rted rutng n the deal case. If t s necessary t supprt mre statns r f the excess and/r ln lsses are sgncant, ptcal amplcatn shuld be used. One pssblty s t nstall an ptcal ampler at each utput prt f each nde. In rder t cmpensate fr excess and ln lsses, the amplers wth a cnstant amplcatn gan can be used. Each ampler shuld have an amplcatn gan equal t the excess lsses wthn ts nde and t the lsses n ts utgng ln. These lsses are cnstant because they depend nly n the sze f a L (.e. n the number f drectnal cuplers wthn the L a sgnal has t pass gng frm an nput t an utput prt) and the ln length. If such amplers are used the pwer lss analyss fr LLN wth rted rutng n the deal case gves the exact slutns. ue t the smplcty f these slutns, the cmplexty f the netwr ptmzatn and management can be greatly reduced. If we want t use the amplcatn 8

9 t vercme the splttng and cmbnng lsses, the ampl- ers whse gan can be dynamcally cntrlled wuld be preferred because these lsses depend n the tplgy f a LLN subnet that can be changed dynamcally. Hwever, the nstallatn f an ptcal ampler per each ln wuld drastcally ncrease the netwr cst. nther, mre ecnmcal slutn that can be appled t LLNs wth rted rutng requres addng a sngle ptcal ampler. In ths scheme a sngle undrectnal ber ln s added t the rt nde mang a self-lp. Thus, ths scheme requres an addtnal nput and utput prt at the rt nde. The ptcal amplcatn s perfrmed nly at that ln usng the Erbum-dped ber ampler []. Fgure llustrates the scheme. Optcal sgnals cmng t the rt e Erbum ped Fber Fgure : Lp ln at the rt nde nde frm all ts nput lns (except frm the lp ln) are multplexed t the lp ln, where ptcal amplcatn s perfrmed. The ampled sgnal s then sent bac t the rt nde where t s dstrbuted t all ts utput lns (except t the lp ln). Ths rutng scheme s dened by the fllwng pwer transfer matrx fr the nde f when pwer dstrbutn and multplexng s unfrm: M f 6 a a a e a f a a a e a f a e a e a ee a ef a f a f a fe a ff 7 f The pwer budget ptmzatn fr ths scheme s cnsdered n ppendx. If the amplcatn gan at the lp ln s G[d], the ptmal pwer budget fr such a netwr s P [d] 0 lg 0 (E ( Lpt ))? G[d] () ;6r Prvded that G s hgh enugh t cmpensate the pwer lsses frm statns t the lp ln, we have the fllwng expressn fr the ptmal pwer budget P [d] 0 lg 0 (E ( Lpt )) (0) ;6r and fr the deal case P [d] 0 lg 0 N () Thus, nly splttng lsses are ncurred and pwer budget becmes equal t the pwer budget f the star netwr wth N statns. The number f statns can be ncreased up t 0000 whch shuld be sucent n mst cases. 6 nclusn In ths paper we prpsed a new scheme fr ptcal sgnal rutng n Lnear Lghtwave Netwrs, called rted rutng. Ths new scheme vercmes the TM synchrnzatn prblem that exsts n LLNs when shrtest path rutng s used. The rted rutng scheme, hwever, ncreases prpagatn delays and pwer lsses. The pwer budget ptmzatn prblem fr LLN wth rted rutng s presented and slved, and t s shwn that, when the excess and ln lsses are small cmpared t the splttng lsses, and when pwer dstrbutn s ptmzed, bth rutng schemes perfrm almst the same. Pssbltes f usng ptcal amplcatn n LLNs were als dscussed and t was shwn that the pwer budget f a LLN subnet that uses rted rutng can be sgncantly mprved usng a sngle ptcal ampler. ppendx Fact: The maxmum prpagatn delay f a LLN wth rted rutng s at mst twce the maxmum prpagatn delay f the same LLN wth shrtest path rutng prvded that the degree f the rt nde s at least tw. Prf: If the degree f the rt nde s at least tw, and the tplgy s a tree, at least ne par f statns has the shrtest path that ges thrugh the rt nde. In such a case the maxmum prpagatn delay frm a statn t the rt (r vce versa) cannt be larger than the maxmum prpagatn delay between statns cnnected by the shrtest paths. The maxmum prpagatn delay n a LLN wth rted rutng s twce the maxmum prpagatn delay between a statn and the rt. Therefre, the maxmum prpagatn delay n a LLN wth rted rutng s at mst twce the maxmum prpagatn delay between statns that are cnnected by the shrtest paths. ppendx In ths ppendx, we present a slutn fr the pwer budget ptmzatn prblem n a LLN wth rted rutng tang nt accunt all types f lsses. In a LLN wth rted rutng an ptcal sgnal sent by a statn s attenuated n ts way t the rt due t the cmbnng and excess lsses at the ndes and the ln lsses at the lns n the path frm the statn t the rt. ls, an ptcal sgnal cmng ut f the rt nde s attenuated due t the splttng and excess lsses at the ndes and the ln lsses at the lns n the path frm the rt t a statn. In rder t mnmze the maxmum pwer lss between any tw statns,.e. the pwer budget, we can parttn ths prblem nt three subprblems and slve them separately. We need t

10 . mnmze the maxmum pwer lss frm a statn n ts path t the rt nde,. mnmze the maxmum pwer lss frm the rt t a statn, and. gven the mnmzed lsses between statns and the rt, ptmze pwer dstrbutn /multplexng at the rt nde such that the maxmum pwer lss between any tw statns s mnmal. In rder t nd the pwer transfer matrx fr a nn-rt nde, we need t slve the rst tw prblems. In nn-rt nde l sgnals frm all ts nput lns (except frm ln r (l) ) are multplexed t utput ln r (l). ue t the lmtatn n cmbnng the pwer t utput prt r (l), nly a prtn f the pwer frm each f the nput lns s transferred t r (l). The sgnals cmng frm nput ln r (l) t nde l are dstrbuted t all ts utput lns except r (l). The maxmum lss f any sgnal transmtted by statn s mnmal at nput prt f nde l f the lss s the same fr each f the statns n S (l). The maxmum attenuatn f a sgnal frm utput prt f nde l at statn s S (l) s mnmal f splttng s perfrmed n such a way that each f the statns n S (l) receves an equal prtn f pwer. Usng these facts, we can derve the s S (l) expressns fr the ptmal pwer lss frm statn s S (l) t nput prt f nde l and frm utput prt f nde l t statn s S (l). Let p m (l) Lpt n(l) ( pl P f p s a statn ( (p) ;6r (p) Lptn(p) )E (p) () pl f p s a nde Lpt ut(l) ( lp P f p s a statn ( (p) ;6r (p) Lptut(p) )E (p) () lp f p s a nde In rder t acheve these pwer lsses, elements f the pwer transfer matrx fr each nn-rt nde l shuld have the fllwng values: a (l) a (l) Lpt ut(l) P ; r (l) ; 6 r (l) () (l) ; 6 Lptut(l) (l) Lpt n(l) P ; r (l) ; 6 r (l) () (l) ; 6 Lptn(l) (l) Fr the rt nde we have a (l) 0; r (l) (6) a (l) 0; ; 6 r(l) (7) Lpt ut P Lptut and a Lpt n P Lptn ; ; f; : : :; g (8) Let be the pwer lss between statns and expressed as rat between pwer transmtted by statn and pwer receved by statn. We have that Lpt n ( ( (a Lpt n )E )? E Lpt ut Lpt ut ); ; f; : : :; Ng () We see that s ndependent f and. Therefre, the pwer lss between any par f statns has the same value whch s n fact the pwer budget. The ptmal pwer budget (expressed n decbels) s thus P [d] 0 lg 0 P P 0 lg 0 (( Lptn )E ( Lptut )) (0) If the lns have the same lsses n bth drectns (e.g., ) we have frm () and () that Lpt n(l) Lpt ut(l) Lpt (l) () whch allws us t smplfy the frmula fr the ptmal pwer budget. In such a case we have P [d] 0 lg 0 (E ( ppendx Lpt ) ) () Fact: The ptmal pwer budget n a LLN wth shrtest path rutng n the deal case (wthut the ln and excess lsses) s P [d] 0 lg 0 (N? ) Prf: Let us cnsder the pwer lss n the shrtest path frm statn t statn as shwn n Fgure. The ndes n the path frm t are labeled g ; g ; : : :; g p where p s the number f ndes n the path. Fr each f these ndes we label the ncmng and the utgng prt n the path as and, respectvely. Let be the pwer lss n the path frm t. We have ( py l a (gl) )? When pwer dstrbutn s ptmal we have that (gl) n (gl) P (g l ) ;6 n(gl) n(gl) N? n (gl) 0

11 g g g p Fgure : Path between statns and and whch gves and (gl) N? n(g) n (g) Snce n (gl) P (g l ) ;6 n(gl) a (gl) n(gl) N? n(g) n (g) n (gl) N? n (gl) N? n (g) n (g) N? n (g) n (g) n(gl) N? n (gl) n (gl) N? n (gl) N? n (gl+) N? n(gp) n (gp) the expressn fr can be smpled. Thus, ecause n (g) N? n(g) n (g) n (gp), we have (N? ) N? n (gp) n (gp) N? n (gp) n (gp) Snce the pwer lsses are the same fr all pars f statns, we have the fllwng result fr the pwer budget P [d] 0 lg 0 0 lg 0 (N? ) () ppendx In ths appendx we cnsder the pwer ptmzatn n a LLN wth rted rutng that has a self-lp ln at the rt nde. n ptcal ampler wth amplcatn gan G[d] s nstalled at ths ln. Intrductn f the lp ln des nt change the ptmal pwer dstrbutn and multplexng at the nn-rt ndes. nsequently, the pwer transfer matrces fr the nn-rt ndes are als dened by the expressns (), (), (6) and (7). The ptmal pwer dstrbutn and multplexng at the rt nde s nw dne n the same manner as fr the nn-rt ndes. Thus, the elements f the pwer transfer matrx f the rt nde have the fllwng values: a a Lpt ut P ; r ; 6 r ; 6 Lptut () Lpt n P ; r ; 6 r ; 6 Lptn () a 0; r (6) a 0; ; 6 r (7) The ptmal pwer budget s P [d] 0 lg 0 ( E Lpt n E RR Lpt ut ) 0 lg 0 ((E Lpt ) RR ) (8) where RR represents the pwer lss at the lp ln. Snce amplcatn s perfrmed n the ln, ths lss s negatve (.e., G[d]?0 lg 0 RR ). Thus, we have P [d] 0 lg 0 (E References Lpt )? G[d] () [] rshna ala, Thmas E. Stern, and avta ala. lgrthms fr rutng n a lnear lghtwave netwr. In Prceedngs f IEEE INFOOM ', vlume, pages {, al Harbur, Flrda, prl.

12 [] Jseph. annster, Lug Fratta, and Mar Gerla. Tplgcal desgn f the wavelength-dvsn ptcal netwr. In Prceedngs f IEEE INFOOM '0, vlume, pages 00{0, San Francsc, alfrna, June 0. [] harles. racett. ense wavelength dvsn multplexng netwrs: Prncples and applcatns. IEEE Jurnal n Selected reas n mmuncatns, 8(6):8{6, ugust 0. [] Mar Gerla and Lug Fratta. Tree structured ber ptc MN's. IEEE Jurnal n Selected reas n mmuncatns, S-6(6):{, July 88. [] P.S. Henry. Hgh-capacty lghtwave lcal area netwrs. IEEE mmuncatns Magazne, 7(0):0{ 6, Octber 8. [6] Rbert J. Hss. Fber Optc mmuncatn esgn Handb. Prentce Hall, Englewd ls, New Jersey, 0. [7] Y. It, Y. Uran, T. Muratan, and M. Yamaguch. nalyss f a swtch matrx fr an SS/TM system. Prceedngs f the IEEE, 6(), March 77. [8] S.. rty and R.. lferness. Wavegude electrptc devces fr ptcal ber cmmuncatn. In S.E. Mller and I. amnw, edtrs, h. f Optcal Fber Telecmmuncatns II. cademc Press, 88. []. Naagawa, S. Nsh,. da, and E. Yneda. Trun and dstrbutn netwr applcatn f erbum-dped ber ampler. Jurnal f Lghtwave Technlgy, (), February. [0] Thmas E. Stern. Lnear lghtwave netwrs: Hw far they can g? In Prceedngs f GLOEOM '0, pages 866{87, San eg, alfrna, ecember 0. [] Thmas E. Stern. lnear lghtwave MN archtecture. In Guy Pulle, edtr, Hgh-apacty Lcal and Metrpltan rea Netwrs, pages 6{7. Sprnger- Verlag, erln, Germany,. He ned the Faculty f the UL mputer Scence epartment n 77. Hs research nterests cver the perfrmance evaluatn, desgn and cntrl f dstrbuted cmputer cmmuncatn systems and hgh speed cmputer netwrs (TM and ptcal netwrs). Mlan vacevc (M ') receved hs.s.e.e. and M.S.E.E. degrees frm the Schl f Electrcal Engnnerng, Unversty f elgrade, Yugslava n 8 and 87, respectvely, and M.S. and Ph.. degrees n mputer Scence frm the Unversty f alfrna, Ls ngeles n. Frm 8 t 87 he was a Research Engneer at the epartment f Electrncs and mputer Engneerng at Unversty f elgrade. In he was a Member f Techncal Sta at the dvanced mputng Systems mpany n Ls ngeles. In r. vacevc ned the enter fr Telecmmuncatns Research and the epartment f Electrcal Engneerng at lumba Unversty as sscate Research Scentst. Hs current research nterests nclude bradband (TM), wreless, and ptcal netwrs, and perfrmance analyss. Mar Gerla (M '7) was brn n Mlan, Italy. He receved a graduate degree n engneerng frm the Pltecnc d Mlan, n 66, and the M.S. and Ph.. degrees n engneerng frm UL n 70 and 7, respectvely.