Wavelength Scheduling in Time-domain Wavelength Interleaved Networks

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1 Wavlgth Schdulig i Tim-dmai Wavlgth Itrlavd twrks Ya Li, Sajay Raka ad Sartaj Sahi Dpartmt f Cmputr ad Ifrmati Scic ad Egirig Uivrsity f lrida, Gaisvill, lrida {yali, raka, sahi}@cis.ufl.du Abstract Tim-dmai Wavlgth Itrlavd twrkig (TWI) is a w ptical twrk architctur which achivs a gd balac btw schdulig flxibility ad dplymt cst. I this papr, w slv th wavlgth assigmt prblm fr TWI twrks usig tplgy sharig apprach. W shw that dtrmiig th wavlgth assigmt that us th miimum umbr f wavlgths is a P-Cmplt prblm. ur grdy huristics ar prstd t cmput th apprximatd sluti withi rasabl tim. Th valuati rsults shw that prfrmig srtig dstiati trs ad wavlgths imprvs th assigmt rsults, spcially udr lw traffic lads. Hwvr, prfrmig srtig brigs sm xtra vrhads t th srt huristics ruig tim, but vrall cmputati csts ar still accptabl. I. ITRODUCTIO Tim-dmai Wavlgth Itrlavd twrkig (TWI) is a ptics-basd trasprt twrk architctur that aims t prvid cst ffctiv ptical grmig [] [3]. Traditial ptical twrks wrk i f th fllwig tw mds: ptical circuit switchig (OCS) r ptical packt switchig (OPS). I OCS twrks, th fist badwidth graularity ffrd by a ptical switch is at a wavlgth lvl, i.. sigl wavlgth a fibr ca b usd by ly d-t-d traffic ad cat b shard with thr traffic. This is t ffctiv wh th traffic dmad is much lwr tha th wavlgth capacity. At th thr xtrm, OPS twrks prmit sharig f ptical liks by traffic with diffrt surcs ad dstiatis. Ths twrks, which ar abld by ptical-lctric-ptical (OEO) cvrsi at ach d i th twrk, td t icur a rlativly high systm cst ad trasmissi dlay, as OEO cvrtrs ar grally xpsiv ad th cvrsi prcss is tim-csumig cmparig t dirct circuit switchig. Sm tchiqus hav b itrducd t imprv th utilizati f ptical liks by simulatig OPS vr OCS, such as ptical burst switchig (OBS) [4]. Hwvr, OBS still ds high-spd ptical switchs ad a ctti algrithm at ach switch. Widjja t al. tc. prpsd TWI t vrcm lik utilizati prblms i OCS but avid th high cst ad dlays rsultig frm OEO cvrtrs big dplyd at all th ptical switchs []. TWI prfrms ptical grmig ly at its dg switchs ad th twrk cr is purly basd passiv wavlgthslctiv switchs (WSS) that rut th wavlgths frm thir igrss prts t th apprpriat grss prts [5]. I a TWI twrk, th dg ds ca b ithr surcs r dstiatis. A trasmittr with a multi-frqucy lasr is lcatd at ach surc d. With this trasmittr, surc ds ca chag th wavlgth f thir ptical sigal i sub-ascds [6]. Surc ds cllct data uits frm varius clits ad assmbl data uits fr th sam dstiati it burst. Wh sdig th burst, th surc chags its fast-tuabl lasr t th wavlgth uiquly assigd t that dstiati. Th itrmdiat ds rut ptical bursts basd purly th wavlgth f th burst. Wh th burst is rcivd at its dstiati, it is disassmbld ad frwardd t th crrspdig clits. As currt ptical switchs cat sparat th bursts that shar th sam wavlgth, ly traffic with th sam dstiati may shar a wavlgth i th tim-dmai. This cstrait lads t tr-lik ruts i th twrk fr vry dstiati, whr th dstiati is th rt ad th surcs ar th lavs. I this papr, w discuss th wavlgth assigmt prblm fr TWI twrks (TWI-WA). W slv this prblm usig a tw-phas prcss: Tr Cstructi ad Tr-Wavlgth Assigmt. Tr-Cstructi grups th traffic dmads with th sam dstiati tgthr ad cstructs th crrspdig dstiati trs. Tr-Wavlgth Assigmt prcss assigs wavlgths t th dstiati trs cstructd i th prvius stp. Th gal f th Tr-Wavlgth assigmt phas is t miimiz th ttal umbr f wavlgths dd t accmmdat th traffic dmads. I this papr, w shw that th miimum umbr f dstiati trs ca b cstructd usig a grdy apprach i th Tr Cstructi phas. r th Tr-Wavlgth Assigmt prblm, w prv its P-Cmpltss by rducig th Graph-Clrig prblm t it. W prps a grdy stratgy that matchs dstiati trs ad wavlgths by. W als prpsd tw tr srtig mthds ad tw wavlgth srtig mthds t rgulat th rdr f tr-wavlgth matchig. Wh diffrt tr srtig ad wavlgth srtig mthds ar applid t th tr-wavlgth assigmt schm, fur huristics ar prstd: C-B, C-, P-B ad P-. Extsiv simulatis ar cductd t valuat th prfrmacs f ths huristics. Th rsults shw that prfrmig srtig dstiati trs ad wavlgths imprvs th assigmt rsults, spcially udr lw traffic lads. Hwvr, prfrmig srtig brigs sm xtra vrhad t th srt huristics ruig tim, but vrall cmputati csts rmai accptabl. I larg tplgis with havy wrklad, th huristic withut ay srtig bcms cmptitiv as it ca prvid similar schdulig prfrmac with much lss cmputatial cst. Th rst f this papr is rgaizd as fllws. I Scti II, w discuss rlatd wrk. I Scti III, w xplai th TWI

2 2 architctur i dtail ad dfi th TWI-WA prblm frmally. I Scti IV, a grdy algrithm fr Tr-Cstructi is prstd. I Scti V, w prv th P-Cmpltss f Tr-Wavlgth assigmt prblm ad fur huristics ar discussd. Scti VI prsts a xprimtal valuati f th fur huristics fr Tr-Wavlgth assigmt. Scti VII givs th cclusis. II. RELATED WORK Optical Circuit Switchig (OCS) with wavlgth-dimsi multiplxig (WD) [7] prvids th mst cmical sluti fr high spd ptical twrks. Hwvr, th iflxibl rutig schm ad cars multiplxig graularity mak it ly suitabl fr th lg-livd larg bulk data trasfrs. O th thr had, Optical Packt Switchig (OPS) [8] ad Optical Burst Switchig (OBS) [9] hav b prpsd t prvid subwavlgth schdulig graularity ad th capability f dyamic rutig. Hwvr, th ultra-high spd ptical-lctricptical switchs that ar rquird i th OPS/OBS twrks ar rmally xpsiv ad difficult t maitai. Th high cst i dplymt ad maitac ihibit th us f OPS ad OBS i mdr twrks. Tim-dmai Wavlgth Itrlavd twrkig (TWI) has b prpsd t fill th gaps btw OCS ad OPS/OBS. Th architctur f TWI is itrducd i []. Th gal f TWI is t prvid sub-wavlgth graularity fr traffic schdulig withut usig xpsiv high spd ptical switchs i th twrks. TWI achivs this by ly allwig light paths with th sam dstiati t shar a wavlgth. As th traffic th sam wavlgth will t b split agai, th cmical switchs usd i OCS twrks ar abl t rut th bursts i TWI twrks. TWI brigs w challgs t traditial ptical schdulig apprachs. [] prsts sm basic idas i th rutig th burst schdulig i TWI twrks ad prpsd a prfrmac masurmt framwrk. [3] ivstigatd th ptical burst schdulig prblm i TWI twrks. Thy shw that achivig th maximum thrughput with zr prpagati dlay is quivalt t th ptimal matchig prblm i bipartit graphs. Thy als dmstrat that v wh prpagati dlay is -gligibl, a factr- 2 apprximat schdulig algrithm xists t maximiz th thrughput. awhil, [] fcusd th prvidig bttr QS i TWI twrks. Thy itrducd a Itgr Liar Prgrammig frmulati that miimiz th quuig dlay f th ptical bursts. Thy als prpsd th Dstiati Slt St (DSS) algrithm t apprximatly slv th prblm withi rasabl tim. I this papr, w fcus th wavlgth assigmt prblm fr TWI twrks. Traditial wavlgth assigmt stratgis tak th availabl wavlgth umbr as th mai cstrait. Hwvr, as fractial wavlgth is allwd, ad th gral traffic flw is assumd i sub-wavlgth lvl, TWI wavlgth assigmt (TWI-WA) is rlaxd frm th itgr capacity cstrait i traditial twrks. Th mai ccr i TWI-WA is th cflict tplgis amg multipl dstiati trs wh thy shar wavlgth. rvr, istad f assigig wavlgth t ach light path, TWI assigs wavlgths t a dstiati tr. Traditial wavlgth assigmt prblm ar rmally quivalt t th Bi-Packig prblm [2]. Hwvr, TWI twrks wavlgth assigmt prblm is a variati f th Graph- Clrig prblm [3], as shw i Scti V-A. [4] [6] tgthr prvid a summary th xistig wavlgth assigmt stratgis. Th mst ppular wavlgth assigmt stratgis ar irst-it ad Bst-it, whr th wavlgths ar matchd with th rqust accrdig t a radm rdr r t thir rmaiig capacitis. [4] prpsd a dfrrd wavlgth assigmt stratgy fr ptical twrks with wavlgth cvrtr prvids.this stratgy imprvs th rqust accptig rat by dfrrig th wavlgth assigmt frm schdulig tim t actual jb start tim. [7] prpsd th last-cvrsi assigmt schm that attmpts t rduc th wavlgth cvrsi vrhad i spars wavlgth cvrtr twrks. I TWI twrks, th simplst wavlgth assigmt stratgy is t assig ach dstiati tr a idividual wavlgth. Hwvr, this stratgy rquirs that th umbr f availabl wavlgths b qual t th umbr f dstiati trs, th itral switchs ar uabl t sparat th traffics t diffrt dstiati trs th sam wavlgth. Wh th umbr f dstiati trs ar mr tha th availabl wavlgths, w d t stablish sm mchaism s that multipl dstiati trs ca shar sigl wavlgth withut cllisi. This prcss is dtd as wavlgth rus. st xistig wavlgth rus algrithms ar Tim Dimsi ultiplxig basd [2], [8]. Hwvr, th traffic flws i thir scaris cat b trasmittd at th max rat as th lik capacity is shard by diffrt dstiati tr i diffrt tim slics. This cstrait may t b cssary i may scitific ad cmmrcial applicatis, whr th jb fiish tim is critical fr th wrkflw. rvr, TD basd wavlgth rus rquirs cmprhsiv ptical burst schdulig algrithms, spcially wh th twrk dlay is csidrd. Our algrithm csidrs th wavlgth rus prblm i a diffrt dircti: multipl dstiati trs shar wavlgth if thir tplgis ar cmpatibl. I ur apprach, c th wavlgth is assigd fr th dstiati trs, th surc ds f all assigd traffic flws ca wrk i a full lad t trasmit thir bursts withut wrryig abut th flw cllisis th ptical liks that shard by diffrt dstiati trs, which gratly simplifis th burst schdulig. III. ETWORK ODEL AD PROBLE DEIITIO O th spctrum f ptical twrks, TWI twrks rsid btw th OCS ad OPS twrks. Cmpard t ths traditial ptical twrks, th TWI pss fllwig w faturs: ) Similar t traditial OCS twrks, TWI s data bursts travl alg th light path usig a pr-assigd wavlgth. Hwvr, a wavlgth ca b shard by multipl traffic flws frm diffrt surcs, ly if thir ttal flw siz ds t xcd th wavlgth capacity. 2) As currt ptical switchs cat sparat th bursts that shar th sam wavlgth, traffic ptical lik has t b rutd t th sam dstiati if thy ar usig th sam wavlgth. S TWI light paths with

3 a D 3 sam dstiati ar grupd tgthr as a tr structur whr th dstiati is th rt ad th surcs ar th lavs. Wavlgths ar assigd t ach f ths trs, rathr tha t a sigl light path. igur shws a simpl TWI twrk. Tw surc ds, S ad S 2 ar sdig traffic t tw dstiati ds, D ad D 2. A 5-d cmmuicati twrk ccts th surcs ad th dstiatis. I th twrk, thr ar 4 diffrt light paths: (S, D ), (S 2, D ), (S, D 2 ) ad (S 2, D 2 ). Bfr th wavlgths ar assigd, ths light paths ar grupd it 2 tr structurs accrdig t thir dstiati, dtd as T =< (S, S 2 ), D > ad T 2 =< (S, S 2 ), D 2 >, rspctivly. D is th rt f T whil D 2 is th rt f T 2. Th twrk ctais tw wavlgths: W ad W 2. Each dstiati tr has t b assigd t a wavlgth bfr th trasmissi ca start. I th simplst cas, T is assigd wavlgth W ad T 2 is assigd wavlgth W 2. Durig th trasmissi, S ad S 2 itrlav thir traffic t D ad D 2 by tuig th clr f thir lasr t th crrspdig wavlgth. r ach d i th cmmuicati twrk, a rutig tabl is maitaid t idicat th utgig prt fr diffrt wavlgths. Wh th traffic arriv at th itral switchs, rutig is prfrmd usig ly th ruls i th rutig tabl ad th clr f th icmig bursts. This guarats that ptical bursts f a giv wavlgth will b rutd t th itdd dstiati. r xampl, i igur, d a must cmbi th traffic frm d S ad d wavlgth W ad frward is t th lik that ccts t d b, accrdig t th rutig tabl. d b, aftr rcivig th bursts wavlgth W, will frward thm t d D, which is thir dstiati. phas, w prfrm th wavlgth assigmt. Durig th Tr-Cstructi phas, th traffic dmads i R ar grupd tgthr by thir dstiati. I ach grup, th crrspdig light paths ar mrgd tgthr t frm a dstiati tr. As fractial jb assigmt is allwd, a simpl grdy algrithm will grat th dstiati tr st with miimum siz. Tr- Wavlgth assigmt algrithms assig ach dstiati tr a wavlgth. W shw that fidig a ptimal assigmt that uss th fwst umbr f wavlgth is P-Hard. Svral huristics ar th prpsd fr wavlgth assigmt. IV. TREE COSTRUCTIO I a TWI twrk G < V, E >, a dstiati tr fr d D i is dtd as T(D i ) = (< S >, D i ), whr < S > is th st f all surc ds i T(D i ). Giv th st f traffic dmads R, w d t first cstruct dstiati trs frm th light paths bfr w ca actually assig th wavlgth. This prcss is calld Tr-Cstructi. Th gal f this prcss is t miimiz th ttal umbr f th dstiati trs i th rsult st T(D). S S 2 S 3. 6 S S 2. 2 a D S 3. 6 ig. 2. A xampl f TWI Tr Cstructi. ig.. A xampl f TWI twrk. r traffic whs rquird badwidth is fracti t th wavlgth capacity, TWI twrks will gratly facilitat thir schdulig by prvidig mr flxibl ad fir-graid rutig ad wavlgth assigmt schm. st ptical twrks still us static rutig i th high spd md as chagig th ruts -th-fly icurs vry high vrhads. Thrfr, i this papr, w als assum that th path fr ach surc/dstiati pair as pr-cmputd, ad fcus ur rsarch th wavlgth assigmt prblm fr TWI twrks (TWI-WA). Giv a TWI twrk G =< V, E >, a traffic dmad is dfid as r = (s, d, bw), whr s, d V is th surc ad dstiati d f th traffic flw, ad bw (, ] is th fracti f th wavlgth capacity rquird. TWI-WA taks a st f traffic dmads R as iput. Th gal is t accmmdat th all dmads r R usig a miimum umbr f wavlgths. As dscribd i Scti I, TWI-WA is slvd usig a 2- stp prcss. I th Tr-Cstructi phas, w cstruct th dstiati trs ad i th Tr-Wavlgth assigmt I this papr, w allw th traffic rqust (s, d, bw) t b partitid it multipl sub-rqusts that ca b assigd t diffrt dstiati trs. This is rasabl as mst mdr ptical switchs ar capabl f trasmittig/rcivig data bursts diffrt wavlgths simultausly. As lg as th dstiati ds ar capabl f packag rdrig ad r-assmbly, fulfillig rqust with multipl data flws is ttally fasibl. O th thr had, if w simply mrgig all th light paths with th sam dstiati, th rsultig dstiati tr may t b admissibl t th twrk, as th ttal flw siz fr dstiati may xcd th wavlgth capacity. igur 2 shws a xampl f tr cstructi. Thr surc ds, S, S 2 ad S 3, ar t sd data t d D simultausly. Th data rat at ach surc d is.6. If w mrg all 3 light paths it dstiati tr, th ttal traffic lik (a, D) wuld xcd th wavlgth capacity. S th dmads hav t b split it tw sparat dstiati trs, i.. T (D) = (< S, S2 >, D) ad T (D) = (< S2, S3 >, D). rvr, wh cmpsig th dstiati trs, w shuld try t us up all th wavlgth capacitis, as th uutilizd capacity cat b shard by thr dstiati trs. Basd th abv bsrvatis, w prpsd th fllig grdy algrithm t cmput th miimum dstiati tr st, as shw i igur 3 Our Tr-Cstructi algrithm first grups th traffic dmads accrdig t thir dstiati. This ca b d by

4 4 TrCstructi(G, R) { rsults = {}; Grup th traffic dmads accrdig t thir dstiatis. fr (ach dstiati grup DG(D i )) { Iitializ a w dstiati tr T j (D i ). T j (D i ).capcity = wavlgth capacity. curtr = T j (D i ). fr+ (ach traffic dmads r i DG(D i )) { rg th light path frm r.s t r.d it T j (D i ). if(r.bw < curt r.capacity) curt r.capacity -= r.bw. ls { if(r.bw > curt r.capacity) Isrt a w dmad (s, d, r.bw capacity) it DG(D i ). Add curt r it rsults. Iitializ a w dstiati tr T j+ (D i ). curtr = T j+ (D i ). } } } rtur rsults; } ig. 3. Th grdy algrithm fr Tr-Cstructi simply scaig th dmad st c. r ach grup, th crrspdig dstiati trs ar cstructd grdily. If addig th currt light path t th currt dstiati tr wuld xcd its wavlgth capacity, w split th currt rqust it tw sub-rqusts. Th first part jis th currt tr ad uss all its rmaiig capacity. Th scd part starts a w dstiati tr it which w attmpt t mrg th rmaiig paths i th currt grup. Th ptimality f this grdy algrithm is bvius as th umbr f rsult trs is miimizd fr ach dstiati ds. Th tim cmplxity f this tr cstructi algrithm is O( V R ), whr R is th siz f th traffic dmad st ad V is th umbr f ds i th twrk, which buds lgth f all pssibl light paths. V. TREE-WAVELEGTH ASSIGET Wh a data burst is rady t b st ut, th surc d d t kw which wavlgth it will us t trasmit th burst fr its itdd dstiati. I TWI twrk, this is dcidd by th tr-wavlgth assigmt prcss. I traditial ptical twrks, wavlgths ar assigd t spcific light paths. Hwvr, i TWI twrks ach dstiati tr is assigd a wavlgth. I this scti, w discuss diffrt stratgis f assigig wavlgths t dstiati trs. Th dstiati trs ar cstructd i th prvius tr cstructig phas. Our gal is t miimiz th umbr f wavlgths that w us t accmmdat all th trs. I Scti V-A, w itrduc th gric frm f th tr-wavlgth assigmt prblm ad prv that cmputig th ptimal tr-wavlgth assigmt is P-Hard. I Scti V-B, fur grdy huristics ar prpsd t apprximatly slv th prblm i rasabl tim. A. Gric rm f th Tr-Wavlgth Assigmt Prblm W t that i TWI twrks, tw dstiati trs that shar sm liks cat b assigd t th sam wavlgth, as th TWI switchs will t b abl t distiguish thir traffic. S, trs that hav cmm liks ar csidrd i cflict fr wavlgth assigmt. O th thr had, trs that d t t shar ay lik ca b assigd th sam wavlgth withut itrfrc. Such trs ar said t b cmpatibl. Athr bsrvati fr tr-wavlgth assigmt is that a dstiati tr may b assigd mr tha wavlgth. That is, sm surc-dstiati paths may us wavlgth whil th thr paths us a diffrt wavlgths. I particular, w ca divid a dstiati tr it a cmpatibl part ad a cflict part with rspct t a currt wavlgths that has alrady b assigd t sm thr trs, ad assig th cmpatibl part t th currt wavlgth. t that th split always starts frm th surc ds (laf ds), ad ds at th dstiati (rt). Sic th dstiati ds is abl t rciv data flw frm multipl wavlgths simultausly, splittig dstiati tr as dscribd ds t affct th crrctss f th data trasmissi. Hwvr, it prvids mr flxibility wh w rslv th cflicts amg dstiati trs. Basd th abv bsrvatis, th gric frm f th tr-wavlgth assigmt prblm is as fllw: Giv a st f dstiati trs DT = (t,, t i ) a TWI twrk G < V, E >, miimiz th ttal umbr f wavlgths that ar dd t accmmdat all th trs i DT, withut vilatig th fllwig cstraits: ). Dstiati trs that shar a wavlgth shuld b cmpatibl with ach thr. 2). Dstiati tr btaid frm th tr cstructi phas is ithr assigd a sigl wavlgth, r split it svral parts with ach part big assigd t diffrt wavlgths. Thrm : Th abv tr-wavlgth assigmt prblm is P-Hard. Prf: W prv this by rducig th Graph-Clrig prblm t th tr assigmt prblm. Graph-Clrig is a wll-kw P-Cmplt prblm. Giv a graph G < V, E >, w wat t clr all th vrtics with a miimum umbr f clrs such that tw adjact vrtics hav th sam clr. W first cstruct a crrspdig TWI-WA istac basd a Graph-Clrig istac G < V, E >. r ach d v i i G, w iitializ a crrspdig tr t i, which ly ctais its rt d r i. r ach lik (v i, v j ) i G, w isrt a w dg ( ij, ij 2 ) t bth trs t i ad t j. W appd this w dg t th last isrtd d i th tr, s th tr has a chai-lik structur. igur 4 givs a simpl xampl. d v ad v 2 ar adjact i G. S w hav dg ( 2, 2 2 ) appdd t ds r ad r 2 fr trs t ad t 2 rspctivly. r th sam ras, dg ( 3, 3 2 ) is appdd t d 2 2 i t ad d r 3 i t 3. Aftr w fiish th abv stps fr all liks i E, w hav a dstiati tr st DT = (t, t 2,, t ).

5 G G T 2 5 W cstruct a TWI twrk G t frm DT by mrgig th tplgy f all th trs i DT. I th xampl, w btai a 7-d graph G t by mrgig trs t, t ad t 3 i DT. v v 2 v 3 2 r 2 T T 2 T r T r S t r r 2 r r 3 2 ig. 4. Rducti frm Graph-Clrig prblm t tr-wavlgth assigmt prblm. rm th cstructi f DT ad G t, w ca s that if tw vrtics v i ad v j ar adjact i G, tr t i ad t j must hav a cmm lik ij ad ij 2, which mas t i ad t j ar i cflict i th tr-wavlgth assigmt prcss fr twrk G t. O th thr had, if tw trs t i ad t j ar i cflict fr wavlgth assigmt, thy must shar th dg frm ij t ij 2 ad that dg is th ly lik that is cmm t bth trs. rm th cstructi, thr must b a lik btw vrtics v i ad v j i G. awhil, if th trs ar all i th shap f a chai, as i ur cstructi, splittig a tr brigs bfit t th wavlgth assigmt prcss. Thrfr, i th ptimal assigmt fr DT G t, vry tr i DT is assigd t a sigl wavlgth. w, lt k b th miimum umbr f clrs w d t clr G ad m b th miimum umbr f wavlgths w d t accmmdat all th trs i DT. Basd th abv bsrvati, w claim that k = m. irst, w shw that k wavlgths is sufficit, if w ca clr G usig at mst k clrs. Our wavlgth assigmt schm is t assig tr t i th wavlgth W j, j k if th crrspdig vrtx v i i G is clrd usig clr C j. Sic t i will t b split, ad all th vrtics i G that ar clrd with C j cat b adjact t ach thr, w ca guarat that t i will b cmpatibl t ay thr trs that ar assigd th wavlgth W j. w, w shw that G als ca b clrd withut cflict usig at mst m diffrt clrs, whvr m wavlgths ar sufficit fr th cstructd tr st DT. r ach d v i i G, if its crrspdig tr t i is assigd t wavlgth W j, it will b clrd with C j. Sic thr is cflict i W j, ds with clr C j will t b adjact t ach thr i G. Thrfr th clr f v i is valid. rm th abv statmts, Graph-Clrig ca b rducd t th tr-wavlgth assigmt prblm i plymial stps. S tr-wavlgth assigmt is a P-Hard prblm. B. Grdy Huristics I this scti, w prps a st f grdy huristics t cmput a apprximatly ptimal assigmt i rasabl 3 tim. Ths huristics hav a similar mai prcss wh cmputig th wavlgth assigmt. Hwvr, thy diffr frm ach thr i th rdr th iput dstiati trs ad th xistig wavlgths ar assigd. Th mai ida f ur grdy huristics is as fllws. Th dstiati trs i DT ar chckd by accrdig t th tr srtig rdr. A dstiati tr is matchd agaist alrady assigd wavlgths accrdig t th wavlgth srtig rdr. r tr t i ad wavlgth W j, if part f t i ca fit it wavlgth W j, t i is dividd ad a part f it is assigd th W j. Th rst f t i is th matchd agaist th wavlgths W j+ ad s. If all th i-us wavlgths tgthr cat accmmdat t i, a w wavlgth is pd fr th uassigd part f t i. W prps 2 diffrt apprachs t srt th dstiati trs. ) st Cflicts Tr irst (C): Th trs ar srtd i dcrasig rdr f t th umbr thr trs i DT with which thy hav a cflict, dtd as C i. This is srtig critri is basd th ida that if w assig trs with mr cflicts first, w may rach th miimum umbr f rquird wavlgths vry quickly. Th, fr ths trs with lss cflicts, thr is a highr chac that thy will fit it th xistig wavlgths. 2) st Prcssd Tr irst (P): Lt P i b th umbr f cflictd trs f t i that hav alrady b assigd wavlgths. Istad f chsig trs with largr C i valus, w pick up trs that has highr P i valus. Each tim aftr a tr is assigd, th P i valus f all th uassigd trs ar updatd ad th with th largst P i valu is chs as th xt tr t b assigd wavlgths. Wh multipl trs hav th sam P i, th ti brakr will b th valu f thir C i valu. Th thught bhid this rdrig is similar t th C rdrig. rvr, P rdr is hpd t imprv th C rdr by kpig th priritis sychrizd with th rsult f th xistig assigmts. W als prps tw srtig rdrs fr wavlgths. ) Bst-it Wavlgth irst (B): Th i-us wavlgths ar srtd i th dcrasig rdr f th umbr f liks i th twrk that d t us this wavlgth. This rdr is updatd vry tim a tr-wavlgth assigmt is cmpltd. 2) st-it Wavlgth irst (): Evry tim bfr a dstiati tr is big assigd, th xistig wavlgths ar srtd by th siz f th subtr thy ca accmmdat fr th currt tr. W masur th subtr siz by cutig th umbr f surc ds that ca b ctaid i th currt wavlgth. If wavlgth ca hld a largr umbr f th surc ds ad thir crrspdig light paths, it will hav highr pririty durig th matchig. Th wavlgths r-rdrig is triggrd at rutim whvr th currt tr is chagd. Eithr a split th currt tr, r a w tr is tak ut frm DT fr assigmt. W als t that thr is d t cmpltly srt all th wavlgths durig th updats. Th ly wavlgth w ar itrstd i is th that ca accmmdat th largst subtr. Thrfr

6 6 w ly d t fid th st-it wavlgths, rathr tha srt all wavlgths. Cmbiig th diffrt tr srtig ad wavlgth srtig mthds tgthr, w btai 4 diffrt huristics fr trwavlgth assigmt: C-B, C-, P-B ad P-. Th cmplxity f ach f ur huristics is as fllws: ) C-B: Lt V b th umbr f vrtics i th TWI twrk ad T b th umbr f dstiati trs i DT. Wh w dtrmi th cflicts btw ach pair f trs, it taks O( V ) tim as ach tr may ctai at mst V dgs. Sic vry pair f trs i DT is chckd, cutig th cflicts fr th whl DT st taks O( V T 2 ) tim. Th srtig taks athr O( T lg( T )) tim. Thrfr cmputig th C rdr taks O( V T 2 ) tim. Durig th assigmt prcss, th maximum umbr f wavlgth dd is T. S th umbr f matchs fr ach dstiati tr is O( T ). r ach tr wavlgth pair, it taks O( V ) tim t match thm. S th prcssig tim fr sigl dstiati tr is budd by O( V T ). T maitai th B rdr, w d t updat th wavlgth capacitis ad srt thm. It taks athr O( T lg( T )) tim. S th vrall prcssig tim fr dstiati tr is O( V T + T lg( T )). Th ttal cmplxity fr C- B algrithm is O( V T 2 +( V T + T lg( T ) T ) = O( V T 2 ). 2) C-: T fid th st-it wavlgths t th currt tr, w d t match th tr agaist all th wavlgth, This taks O( V T ) tim. A dstiati tr will split at mst V tims durig th assigmt, s O( V 2 T ) tim is tak t prcss dstiati tr. Th vrall cmplxity fr C- is O( V T 2 + V 2 T 2 ) = O( V 2 T 2 )), whr O( V T 2 ) is th C srtig tim ad O( V 2 T 2 ) is th trwavlgth matchig tim. 3) P-B: If th tr rdr is updatd dyamically, xtra O(T) pratis ar addd t th prcssig f ach dstiati tr. Hwvr, ths xtra pratis d t chagd th asympttic cmplxity fr th trwavlgth matchig prcss. Th vrall P-B cmplxity is th sam as fr C-: O( V T 2 ). 4) P-: Similar t P-B, th xtra pratis rquird t maitai th P rdr is dmiatd by th thr tr-wavlgth assigmt pratis. Thus, ths xtra pratis d t affct th asympttic cmplxity f P-, which is still O( V 2 T 2 ). A. Exprimtal ramwrk VI. EVALUATIO I this scti, w masur th prfrmac f th wavlgth assigmt huristics dscribd i Scti V ad valuat hw diffrt srtig schms affct th prfrmacs i varius scaris. Bsids cmparis th ptimality f th thir assigmts, w als masur th xcuti tim f ach huristic ad study hw xcuti tim varis with twrk siz ad wrklads. W implmtd a -srt vrsi f th grdy huristics that ds t d th srtig stps fr ithr th dstiati trs r th wavlgths. By cmparig th -srt ig. 5. (a) CI (b) sh twrk Tplgis. huristic with th s w prpsd i Scti V-B, w ca ivstigat th impact f th srtig stps. r vry tst cas, w als prvid a lwr-bud fr th ptimal sluti (LB). Th lwr-bud is cmputd by cutig th ccurrcs f ach twrk liks i all dstiati trs. Th maximum cut amg all th liks is th lwr bud fr th miimum umbr f wavlgths w d. With this bud, w ca stimat hw wll ur huristics ca d i th xprimts. T simulat a ptical twrk, w us a 25-d msh-trus tplgy, a ral wrld 9-d CI twrk (igur 5) ad svral radmly gratd tplgis. r radmly gratd tplgis, w st th ut-dgr f ach d t b a radm itgrs btw 5 ad 7. T sur twrk cctivity, th radm twrk has bidirctial liks btw ds i ad i + fr vry i <, whr is th umbr f ds. Sic th tst rsults frm CI ad sh tplgy ar vry similar t ach thr, i this papr w ly prst th rsults frm CI ad Radm tplgis. Th traffic dmads ar als sythtically gratd. Each rqust is dscribd by a 3-tupl (s, d, BW). W first idtify th sts f surc ds ad dstiati ds frm th all graph vrtics V. I th xprimts, w mark 4% f th vrtics i V as surc ds ad athr 2% ds as dstiati ds. Th rmaiig 4% ds ar srvd as cmmuicati ds i th twrk. Th prcss f markig ds is ttally radm. Th surc s ad dstiati d ar th slctd usig a uifrm radm umbr gratr frm th rspctiv sts s that th wrklad is distributd uifrmly amg diffrt d pairs. Th rquird flw siz BW is gratd usig a chppd rmal Distributi (., ). Usig this distributi, abut 96% f th flws sizs ar i th itrval (,.2). Gratd flw siz ar discardd if its valu is utsid th rag (, ). As th xpctati f traffic dmads is ly., mst admissibl dstiati trs gratd will cmpris multipl light paths. r ach tst cas, th maximum umbr f traffic dmads is budd by th umbr f surc-dstiati pairs. This umbr is dtd as axlad. r xampl, i a -d

7 s gt l v a f b r m h s gt l v W a f b r m h s gt l v W a f b r m 5 7 radm twrk, if w mark 4% f th ds as surc ds ad 2% ds as dstiati ds, w will hav at mst 8 diffrt surc-dstiati pairs. That wuld b th maximum umbr f light paths that w d t hadl i th tst cas. Durig th xprimts, ur wrklads ar varid frm 2% f axlad t % f axlad. B. Evaluati Rsults h s W a v u L B B C " C " B P " P " " S r t T r a f f i c L a d ( % f t h a x L a d ) ig. 6. Th prfrmacs f wavlgth assigmt huristics udr diffrt umbr f rqusts i CI twrk. u L B C Z B C Z P Z B P Z Z S r t T r a f f i c L a d ( % f t h a x L a d ) ig. 7. Th prfrmacs f wavlgth assigmt huristics udr diffrt umbr f rqusts i -d radm tplgis. igurs 6 ad 7 prst th valuati rsults fr ur wavlgth assigmt huristics udr varius traffic lads i CI ad radm twrks. I th xprimts, w prduc traffic lads that ar 2%, 4%, 6%, 8% ad % f th axlad. rm th xprimtal rsults, w mak th fllwig bsrvatis: ) All 4 huristics that w prps i Scti V-B grat bttr assigmts tha th -srt huristic i all tst scaris. Ths huristics utprfrm th -srt huristic with mr bvius margis i th light traffic lads (lss tha 6%) tha i havy traffic lads. This shws that th srtig th trs ad wavlgths prvids mr hlp t th wavlgth assigmt wh th twrk is lss ccupid. Wh th twrks liks ar saturatd, rarrag th rdr f match will t b abl t imprv th schdulig much. 2) Amg th fur grd huristics, P- huristic givs th bst prfrmac i all tst cass. Rgardig th srtig mthds fr th dstiati trs, th P huristics prvids bttr assigmts tha th C huristics. This mas adjustig th tr rdr dyamically prvids mr rasabl matchig rdrs durig th tr-wavlgth assigmt. O th thr had, th huristics utprfrm th B huristics wh th wrklad is high (mr tha 8%). Hwvr, wh th wrklad is lss tha 4%, th prfrmacs f huristics ad B huristics ar cmparabl. This shws that wh th traffic lad is high, a mr carful chic th wavlgth, lik, is cssary t prvid a bttr assigmt. Wh thr ar plty f rsurcs availabl, a rlativly crud srtig, lik B, is sufficit. 3) Wh th traffic lad is light, th srtd huristics prvid a rsults cls t th lwr bud, i.. a vry gd apprximati th ptimal sluti. Wh traffic lad is high, th assigmts frm th huristics ar rlativly far away frm th lwr bud. Hwvr, this ds t cssarily ma that th huristics cat apprximat th ptimal slutis udr high wrklads, as th lwr buds may t tightly bud th ptimal slutis wh traffic lad is high. 4) Th umbr f wavlgths dd icrasd with th traffic lad. I small twrks lik CI ad sh, th d fr xtra wavlgths icrass fastr tha i larg radm twrks. Th ras is that it is lss likly t fid disjit light paths fr diffrt surc-dstiati pairs. Wh traffic lad icrass, cflicts ar mr frqut i small twrks tha i larg twrks. u t w r k S i z ( u m b r f d s ) L B C B C P B P S r t ig. 8. Th prfrmacs f wavlgth assigmt huristics i radm twrks with varius sizs. igur 8 prsts th prfrmac f th huristics radm tplgis f varius sizs wh th umbr f traffic dmads is 8. W ca s that with th icras f th twrk capacity, fwr wavlgths ar rquird t accmmdat th rqust st. Hwvr, wh th twrk siz is mr tha 4 ds, th imprvmts ar almst gligibl. Rcall that durig th tr cstructi phas, multipl dstiati trs ar built if th ttal traffic siz xcds th wavlgth capacity. S wh th twrk tplgy is larg ugh t rslv mst cflicts i th tr tplgis, th miimum umbr f wavlgths dd i such twrks is havily iflucd by th maximum umbr f admissibl trs that shar th sam dstiati, i.. th capacity f th wavlgth agai bcms th mai cstrait. igur 9 prsts th ruig tim f ur wavlgth assigmt huristics udr diffrt wrklads ad igur

8 ) c g i ' g r Al ) c m i T g i u ms h r g l A P P ) C É B d s 3 C É ( s c P É B 2 5 P É r t Ti m 2 É S R u h m s i t T r a f f i c L a d ( % f t h a x L a d ) ig. 9. Th algrithm ruig tim f wavlgth assigmt huristics udr diffrt umbr f rqusts i d twrks. 2 5 B ) C d s C 2 B ( s c S r t ' R i t t w r k S i z ( u m b r f d s ) ig.. Th algrithm ruig tim f wavlgth assigmt huristics i radm twrks with varius sizs. givs th ruig tim as a fucti f th twrk siz. W s that th -srt huristic is always th fastst algrithms. Th diffrc i th ruig tim icras as th twrk siz grws, as wll as th traffic lads icras. Huristics usig th sam tr srtig algrithms grally hav th sam ruig tim, which mas th vrhad brught by th tw wavlgth srtig mthds ar similar t ach thr. r huristics usig diffrt tr srtig schm, w t that th C srtig is fastr tha th P srtig. Hwvr, thir prfrmac gap is much smallr tha th gap with th -srt huristic. Althugh th srtd huristics ar rlativly slw cmpard t th -srt huristics, thir vrall cmputatial csts ar still accptabl. I bst cass, th avrag schdulig tim fr rqust is lss tha 5 scds. I th wrst cas, th avrag schdulig tim is lss tha 3 scds fr th slwst huristic. As a summary, th srtig schms prvid csidrabl bfits t th TWI wavlgth assigmts. Th imprvmt is mr with rlativly lw wrklad. Hwvr, th srt huristics ruig tim is affctd by th xtra vrhad brught by th srtig prcss. vrthlss, th ruig tims ar still rasabl v i th wrst cas. Th -srt huristic is cmptitiv wh th twrks ar larg ad traffic dmads ar havy. It prvids much fastr schdulig spd whil yildig littl i th assigmt ptimality. assigmt that us th miimum umbr f wavlgths. W shw that this wavlgth assigmt prblm is P-Cmplt by rducig th Graph Clrig prblm t it. ur grdy huristics ar prstd t cmput a apprximatd sluti withi rasabl tim. Extsiv xprimts ar prfrmd t valuat th ptimality ad th ruig tim f th huristics. Th rsults shw that srtig th dstiati trs by thir dgr f cflicts ad srtig th wavlgths accrdig t thir availabl rsurcs ar ffctiv apprachs t imprv th huristics prfrmac. Hwvr, crtai vrhads th Huristics ruig tims ar itrducd by th srtig prcss. W als tic that i sm xtrm scaris, th -srt huristic ca prvid cmptitiv rsults usig much smallr ruig tim. REERECES [] I. Widjaja, I. Sai, R. Gils, ad D. itra, Light cr ad itlligt dg fr a flxibl, thi-layrd, ad cst-ffctiv ptical trasprt twrk, 23, vl. 4,. 5, pp [2] C. uzma ad I. Widjaja, Tim-dmai wavlgth itrlavd twrkig with wavlgth rus, i IOCO, 26. [3] K. Rss,. Bambs, K. Kumara, I. Sai, ad I. Widjaja, Schdulig bursts i tim-dmai wavlgth itrlavd twrks, 23, vl. 2,. 9, pp [4] J. S. Turr, Trabit burst switchig, 999, vl. 8,., pp [5] K. SHIOURA ad Y. KAWAKITA, Wavlgth slctiv switch usig arrayd wavguids with liarly varyig rfractiv idx distributi, i Phtics Basd Wavlgth Itgrati ad aipulati, 25, pp [6]. Kaur, Trabit burst switchig, 22, p [7] H. Ishi, J. iwa, ad K. su, Rviw ad status f wavlgthdivisi-multiplxig tchlgy ad its applicati, 984, vl. 2, pp [8] R. S. Tuckr, Optical packt switchig: A rality chck, Optical Switchig ad twrkig, vl. 5,., pp. 2 9, 28. [9] Y. Ch, C. Qia, ad X. Yu, Optical burst switchig: a w ara i ptical twrkig rsarch, IEEE twrk, vl. 8,. 3, pp. 6 23, 24. [] I. Sai, I. Widjaja, ad J. rris, Prfrmac f a distributd schdulig prtcl fr twi, SIGETRICS Prfrmac Evaluati Rviw, vl. 32,. 2, pp. 38 4, 24. [] D. Xu, Y. Qi, ad C. K. Siw, Prfrmac aalysis f a vl traffic schdulig algrithm i slttd ptical twrks, Cmputr Cmmuicatis, vl. 3,. 8, pp , 27. [2] Bipackig prblm, [3] Graph clrig prblm, [4] E. Jug, Y. Li, S. Raka, ad S. Sahi, Prfrmac valuati f rutig ad wavlgth assigmt algrithms fr ptical twrks, i 3th IEEE Sympsium Cmputrs ad Cmmuicatis, 28. [5] Q. a ad P. Stkist, O path slcti fr traffic with badwidth guarats, i 5th Itl. Cf. twrk Prtcls (ICP), 997, pp [6] R. Guri ad A. Orda, twrks with advac rsrvatis: Th rutig prspctiv, i Prcdigs f th 9th Aual Jit Cfrc f th IEEE Cmputr ad Cmmuicatis Scitis IOCO, 2, pp [7] Y. Li, S. Raka, ad S. Sahi, I-advac first-slt schdulig with wavlgth cvrsi fr -scic applicatis, i I Prcdigs f Th IEEE Sympsium Sigal Prcssig ad Ifrmati Tchlgy, 2. [8] A. Gadkar ad S. Subramaiam, Wavlgth-rus i ptical timslttd twrks, Optical Switchig ad twrkig, vl. 7,. 4, pp , 2. VII. COCLUSIO I this papr, w discuss th wavlgth assigmt fr TWI twrks. W prps a 2-stp prcss t cmput th wavlgth assigmt fr a giv st f th traffic dmads. Th gal f ur schdulig algrithm is t fid th

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