Advanced Information Technology of Slot-Switching Network Schemes for on All-Optical Variable-Length Packet

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1 Joural of Computer Scece 6 (): -, 200 ISS Scece Publcatos Advaced Iformato Techology of Slot-Swtchg etwork Schemes for o All-Optcal Varable-Legth Packet Soug Yue Lew, 2 Edward Sek Kh Wog ad 2 Choog Kwa Fatt Departmet of Iformato Techology Uversty Tuku Abdul Rahma, 46200, Petalg Jaya, Malaysa 2 Departmet of Iformato Systems, Faculty of Busess ad Accoutacy, Uversty of Malaya, 50603, Kuala Lumpur, Malaysa Abstract: Problem statemet: The purpose of ths paper was to vestgate all optcal packet swtchg, because t was the key to the success of the future Iteret. It ca meet the strget badwdth requremet of future Iteret applcatos, such as real-tme vdeo streamg. Due to the lack of optcal Radom Access Memory (RAM), however, the all-optcal schemes studed the lterature were ether ot fleble eough to accommodate Iteret packets, whch were varable-legth ature, or fal to schedule packets at swtches to acheve low loss rate. Approach: The am of ths paper was thus to tackle the fleblty ad utlzato ssues all-optcal packet swtches, eve at the absece of optcal RAM. We approached ths paper by frst studed a ew slotted model for all-optcal varable-legth packet swtchg, whch was called Varable-legth-Packet Fed-Legth Slot (VPFS) swtchg. Results: We proved by mathematcal aalyss the theoretcal mamum utlzatos that ca be acheved by the model two varat schemes. The we proposed a ew schedulg algorthm for shared-fber-delay-le swtches order to acheve low loss rate whe the utlzato approaches the mamum. We justfed our desg by smulato. I our fdg, through mathematcal aalyss ad computer smulato, our proposed swtchg model ad schedulg algorthm ca be coupled well to acheve good performace for all-optcal packet swtches. We also foud that the selecto of the slot sze the etwork was very crtcal as t determed the trasmsso overhead ad hece the utlzato of the all-optcal etwork. Our research lmtato depeded o slot sze. Although a small slot sze was crtcal for hgh utlzato wth our model, t was ot always preferable. It was because a small slot sze creased the swtchg ad schedulg complety at the swtch. Thus the selecto of a optmum slot sze for the etwork was a compromse betwee utlzato ad complety. Cocluso/Recommedatos: A fast schedulg algorthm has bee studed order to reduce the schedulg complety so as to crease the utlzato wthout much pealty. I regard to the practcal mplcatos, the VPFS was a promsg model to fully utlze the huge capacty of all-optcal etworks ad to accommodate varable-legth packets for future Iteret traffc. Wth VPFS, the selecto of the slot sze was crtcal, ad t was a compromse betwee the etwork utlzato ad schedulg complety. Key words: All-optcal etwork, varable-legth packet, slot swtchg, fber delay le, schedulg algorthm ITRODUCTIO I the past decades, research has bee actvely coducted to make all-optcal packet swtches ad subsequetly all-optcal packet etworks, a realty. A major dffculty remas-packet coteto. Two prmary coteto-resoluto methods have bee tesvely studed wth the lterature to attempt to resolve ths problem. The frst s the wavelegth coteto-resoluto scheme, where a optcal packet a coteto stuato ca be coverted to aother avalable wavelegth order to avod coflct (Jaso, 2000). A major problem of ths scheme s cost (Ramamrtham ad Turer, 2003). The secod method s the tme coteto-resoluto scheme, where Fber Delay Les (FDLs) are requred to delay or buffer optcal packets, ths process acts as a form of optcal memory, whe a coteto stuato arses. As FDLs Correspodg Author: Edward Sek Kh Wog, Faculty of Busess ad Accoutacy, Uversty of Malaya, 50603, Kuala Lumpur, Malaysa

2 J. Computer Sc., 6 (): -, 200 are bulky, oly a lmted amout of such memory s usually recommeded whe buldg these optcal swtches. To effcetly make use of the FDL resource has become a mportat tmg ssue wth the coteto-resoluto stuato. umerous swtch archtectures ad schedulg algorthms have bee proposed for the effcet tmg, swtchg ad schedulg of optcal packets a alloptcal swtch that uses FDLs. Some eamples are, Tme-Slce Optcal Burst Swtchg (Ramamrtham ad Turer, 2003), Karol s Algorthm (Karol, 993), Optcal Packet Swtchg (Cha et al., 200; Chao ad Choa, 999; Karol, 993; Huter et al. 998; Gullemot et al., 998), Optcal cell swtchg (Chao ad Lew, 2003; Masett et al., 993) ad the Sequetal FDL Assgmet (SEFA) Algorthm ad ts varat (Lew et al., 2005; Sh Jag et al., 2005). These algorthms however have placed costrats upo the amout of traffc processed, from fg the packet legths, to requrg packets to be alged before swtchg, makg these swtches mpractcal a realworld evromet where throughput s crtcal for ther ecoomc justfcato ad use. Algorthms for routg varable-legth, ualged packets are eeded order to overcome these problems. As FDLs are used to buffer optcal packets, a fed graularty of the swtchg or schedulg ut, that s, the slot sze, eeds to be mposed depedg o the buffer maagemet scheme used (Hroak ad Masayuk, 2006; Hroak et al., 2006). A Varable-legth-Packet Fed-legth-Slot (VPFS) swtchg model s oe that accommodates these varable-legth packets by usg the fed-legth slots wth curret optcal swtches. I a VPFS etwork, all swtches adopt the same slot sze but slot boudares are ot sychrozed from oe swtch to the et. Wth referece to Fg., whe a packet of ay legth arrves at a VPFS swtch, a slot or a cha of cosecutve or cotguous slots must be assged to carry that packet from the put ad f bufferg s ecessary, passg through the same FDL path, to the dested output. ote, t s ot possble for a cha of slots that are carryg a gve data packet to be broke up ad routed separately wth the swtch, as ths s due to the physcal propertes of a optcal data packet. Sce a slot s the mmum swtchg or schedulg ut ad caot be further splt, ay slot carres data for a gve packet oly. Fgure 2 shows head-tal clashg. Ths occurs f the tal of a packet ad the head of ts followg packet fall to the same slot wth a VPFS swtch that s wthout a algg capablty. The followg packet wll the be dropped ad ot swtched to ts output. 2 Packets Fg. : A VPFS swtch Slots Swtch Fg. 2: Packet 2 wll have to be dropped due to head-tal clashg Head-tal clashg deterorates the performace of a VPFS ad solutos to ths problem have ot bee wdely studed wth the avalable lterature. To allevate or eve elmate ths pheomeo, three solutos are possble. The frst s to select a small slot sze for the VPFS etwork, as ths reduces the chace of head-tal clashg. The advatages of a small slot sze s ot oly lmted to the reducto of the occurrece of ths problem, but also, to the tmg effects of mmzg processg overheads whle mamzg the lk utlzato of VPFS. The VPFS model ca be further geeralzed to a geerc varable-legth packet swtchg as the selected slot sze s allowed to approach zero, aga a stuato where o head-tal clashg occurs, but lttle swtch processg as well. The apparet ad real challege of swtchg ad schedulg s the selecto of the sze of the slot a VPFS. The secod soluto s to force all sources ad swtches a VPFS etwork to mpose a gap betwee ay two packets departg from the same output lk, so that whe the packets arrve at the et swtch, there wll be o head-tal clashg. I ths case, the sze of the gap must be at least the slot sze. Ths approach s termed costraed VPFS scheme wthout algmet. Although ths scheme ca elmate head-tal clashg, the prce to be pad s large overheads ad low lk utlzato, or swtch throughput. The thrd soluto s to employ algers (some papers may refer t to as sychrozers (Chao et al., 2000) at each swtch to alg each packet s head to a slot

3 J. Computer Sc., 6 (): -, 200 boudary. As log as the etwork traffc satsfes a gve ad certa codto, we show that the VPFS scheme wth algmet ca elmate the head-tal clashg problem throughout the etwork wth mmum overheads. Here, t s regarded as a best-case bechmark for other VPFS schemes. There s a prce to be pad for ths approach; the optcal sgals may eed to be swtched a umber of tmes wth the algers before they eter the swtch fabrc (Chao et al., 2000) ad ths ca result uacceptable levels of sgal atteuato. As we study the performaces of () geerc VPFS wthout algmet, () costraed VPFS wthout algmet ad () VPFS wth algmet, we propose a varable-legth Packet FDL Assgmet (VAPFA) algorthm for shared-fdl all-optcal packet swtches. Wth referece to Fg. 3, a shared-fdl all-optcal swtch cotas a umber of feedback FDLs that are shared amog all put ports ad each FDL ca delay packets by a fed umber of slot tmes. Slot tme s a cocept computer etworkg. It s twce the tme t takes for a electroc pulse (OSI Layer -Physcal) to travel the legth of the mamum theoretcal dstace betwee two odes. Assume that there are Z feedback FDLs, Y put ports ad Y output ports. The outputs (puts) of FDLs ad the puts (outputs) of the swtch are collectvely called the lets (outlets) of the swtch fabrc, yeldg Y+Z lets ad Y+Z outlets. The shared-fdl optcal swtch has bee studed etesvely the lterature (Hroak et al., 2006; Karol, 993; Sh, 2005) ad (Hroak et al., 2006) t has bee proposed to hadle varable-legth packets. However, the algorthm proposed (Hroak et al., 2006) s a o-reservato algorthm whch does ot provde departure schedulg ad t caot guaratee the packets beg buffered are able to access the desred output ports after comg out of the FDLs. Sce the departure tme s ot scheduled advace, the delay boud of the algorthm s udefed ad t may requre a packet to be swtched ad re-crculated may tmes. Fg. 3: A sgle-stage shared-fdl optcal swtch 3 I ths study, we focus o the reservato schedulg algorthms for varable-legth packets the sgle-stage shared-fdl swtch, whch s kow as the VAPFA algorthm. The VAPFA algorthm s eteded from the SEFA algorthm proposed (Lew et al., 2005). I cotrast to the o-reservato schedulg algorthms, the VAPFA algorthm performs the FDL assgmet for the etre jourey of a delayed packet so that t ca be scheduled to match wth the desred output port a future tme. As optcal packets caot be fragmeted, t should be oted that a vald FDL assgmet, all slots that belog to the same packet must always pass though the same FDL path from the put to the output. If a packet that eeds to be delayed fals to be assged a FDL path to some future tme for the desred output port owg to FDL ad/or output-port coflcts, t s dscarded wthout eterg the swtch so that t does ot occupy ay resources. Frstly, we aalyze the arrval overhead ad mamum lk utlzato of the geerc VPFS wthout algmet. Uder methodology, we dscuss the VAPFA schedulg algorthm for shared-fdl swtches. The, we provde the smulato model ad results for the geerc VPFS. Uder dscusso, we propose, aalyze ad evaluate the costraed VPFS wthout algmet, whch s desged for elmatg the head-tal clashg throughout the VPFS etwork. As a best-case bechmark, the VPFS wth algmet scheme s also beg studed uder cocluso. MATERIALS AD METHODS Geerc VPFS wthout algmet: Here we cosder a geerc VPFS wthout algmet ad part of the cotet show the followg appears (Lew et al., 2007). Cosder a packet wth legth arrvg at a VPFS swtch, where a fed slot sze of s s adopted. Let m deote the umber of cosecutve slots requred to carry the packet, where m ca ether be /s or /s +. For eample, f s ad 3.5, the value of m ca ether be 4 or 5, depedg o the arrval tme of the packet. The arrval overhead of the packet s defed as δ ms-, whch cossts of the head overhead ( Head ) ad tal overhead ( Tal ), as show Fg. 4. The value of δ ca ether be /s s- or /s s+s-. Calculatg the average overhead s mportat gaugg the effcecy of the swtch that we wat to study. Ths s because t shows how much data a slot actually carres o average ad how much space s wasted due to the slot space allocato the swtch desg. Through our vestgatos, we have come to the cocluso that for varable-legth, ualged packets, the average overhead added upo each arrvg packet s eactly equal to the sze of a sgle slot, rregardless of the packet sze, as prove below.

4 J. Computer Sc., 6 (): -, 200 upper boud of U the VPFS wthout algmet scheme s derved as follows. Theorem : Gve that the legth of a packet s a radom umber,, wth a average value of, the effectve lk utlzato of the VPFS swtchg scheme wthout algmet s bouded by + s. Fg. 4: Arrval overhead, δ Proof: Cosder observg a VPFS swtch over durato of T, whch vald packets have arrved ad bee cosdered for swtchg. Let be the legth ad δ the arrval overhead of the th packet, where. ote that: T ( + δ ) We have: Fg. 5: α, the data sze that may cause overhead Lemma : For ay packet, the epected arrval overhead at a VPFS swtch wthout algmet s equal to slot sze s. Proof: Wth referece to Fg. 5, let α +s- /s s be the o-zero data sze from the packet that may cause arrval overhead, where 0<α s. By the defto of α, the arrval overhead ca ether be (s-α) or (2s-α), wth probabltes: Prob{δ s-α} (s-α)/s Prob{δ 2s-α} α/s by: The epected arrval overhead ca the be gve s α α (s 2s α + α ) + 2sα α E[ δ ] (s α ) + (2s α ) s s s 2 s s s As a result of Lemma, a smaller slot sze reduces wastage of badwdth. More mportatly, t creases the boud of the effectve utlzato of VPFS. Let U deote the effectve lk utlzato, where dropped packets due to head-tal clashg are dscouted. The 4 U lm lm lm T + E[ δ ] ( + δ) δ + + s From Theorem, to acheve a better utlzato, aga, a smaller slot sze for the VPFS swtchg scheme s preferred. However, there are several costrats that restrct a etwork desger s ablty to choose a very small slot sze. () Compared wth the electroc swtches, all-optcal swtches requre loger cofgurato tme to accommodate packets from dfferet puts to dfferet outputs; therefore the slot sze caot be too small. (2) For a smaller slot sze, more cosecutve ad cotguous slots are requred to cota a data packet ad all these slots must be scheduled together from the put to the output, passg through the same FDLs wheever bufferg s ecessary, ths creases the swtchg ad schedulg complety of the all-optcal swtch. RESULTS Varable-legth packet FDL assgmet algorthms for shared-fdl swtch: VAPFA s prcpally modfed from SEFA for accommodatg varablelegth packets ad ts program logc s smlar.

5 J. Computer Sc., 6 (): -, 200 (a) (b) Fg. 6: Shared-FDL swtch ad cofgurato table (a) 4 4 shared FDL swtch (b) cofgurato table Wth each VAPFA swtch, a cofgurato table s mataed. The cofgurato table s used for storg the coectvty schedule of the swtch fabrc. A eample s gve Fg. 6, where Fg. 6b shows the cofgurato table of the 4 4 swtch gve Fg. 6a. ote that the rows of the table represet the outlets of the swtch fabrc ad the colums the tmeslots. I ths eample, the frst 4 rows represet FDLs wth delay values,, 2 ad 4 slots, respectvely, followed by 4 rows represetg 4 output ports. Whe a route s reserved for a packet, the table stores the let, from whch the packet s comg, to the correspodg etres of the table. I ths eample, two packets, both dested for output port 4, are arrvg at put ports ad 3, requrg 4 slots ad 3 slots to carry, respectvely. Assumg that there s o other traffc, the packet from put ca be mmedately arraged to be trasferred to output 4 from tmeslot t (curret tmeslot) to tmeslot t+3, as show the cofgurato table. However, as output 4 s o loger avalable from t to t+3, the secod packet has o choce but to travel through FDL 4, whch has a delay value of 4 tmeslots ad oly the t ca be scheduled to be trasferred to output 4 from t+4 to t+6. Ths FDL route s reflected the cofgurato table by placg put 3 to the row of FDL 4 from t to t+3 (as ths packet requres 3 slots to carry) ad the placg FDL 4 to the row of output 4 from t+4 to t+6. If a packet arrves ad t has o reserved route, the packet s dropped ad cosdered lost. Ths route reservato s performed usg the followg steps: 5. Repeat for each tmeslot 2. Repeat for each put port 3. If there s a ew packet 4. Check cofgurato table for a mmedate soluto 5. If there s a mmedate soluto 6. Update output port cofgurato table to reserve the route 7. If there s o mmedate soluto 8. Search for a possble combato of delays usg FDLs that wll delay the packet tll future tme where output port s avalable a umber of cosecutve slots, the total durato of whch s log eough to trasmt the packet 9. If there are more tha oe soluto 0. Select the best soluto ad update the cofgurato table to reserve the route.. If there s o soluto, packet s dropped For greater uderstadg, we break dow each step show above to greater detals. For step, we loop the remader of the algorthm for every tmeslot. Ths s because the VAPFA swtchg scheme does ot route the optcal packets as s, but the cha of slots that each packet s placed to. Although packets are comg to the swtch a cotuous fasho, by placg these packets to slots we ca vsualze the routg of these packets as a dscrete stepped process. Ths s because oce the route of a packet has bee set, the algorthm oly has to wat for the et tmeslot before t checks aga for ew packets ad process them, whle the optcal packet tself s routed through cofguratos the hardware based o the cofgurato table. If we assume the tme sze of the

6 J. Computer Sc., 6 (): -, 200 slots to be /00 of a sec, the we ca assume that the algorthm wll loop step for 00 tmes every sec. I step 2, we loop through the remader of the algorthm for every put port that s attached to the swtch. Ths meas, for every tmeslot, all the put ports are looped through for processg. If there are 32 put ports, the step 2 wll be repeated 32 tmes for every tmeslot. The put ports are cycled through sequetally, from smallest to largest borrowg from the sequetal ature of SEFA. Step 3 checks f there s a sgal for a ew packet at that curret tmeslot. Ths s mportat because ot every tmeslot wll have a ew packet arrvg for each respectve put port, because we assume that packets are varable legth ad some packets may cosume more tha slot, requrg a cha of slots. For stace, a packet arrvg at tme t, 3 slots log, for a partcular put port would mea that correspodg put port wll have o ew packets arrvg at least tll tme t+3. If there s a ew packet arrvg at a partcular put port at the curret tmeslot of the swtch, we mmedately check to see f there are eough vacat slots the output port to cota the comg packet wthout usg ay FDLs for delay (step 4). If the output port s deed avalable, the cofgurato table of the swtch s updated (step 5) to reserve the output port for the umber of slots the comg packet cosumes. It s also mportat to ote at ths pot that due to the physcal propertes of a optcal data packet, t s ot possble for a cha of slots that are carryg a certa packet to be broke up ad routed separately. Therefore, wheever the cofgurato table s look-up to fd f there are eough empty slots to cota the comg packet, these vacat slots must be cotguous order to cota the complete cha of slots for the packet. Whe there s o mmedate soluto (step 7), we search through all the FDLs usg Breadth Frst Search (BFS) to fd a combato of delays that wll delay the slot or slots log eough tll there s a vacat cha of slots the output port whch s suffcet to cota the comg packet (step 8). For ths porto, we represet each FDL as a ode a search tree ad the depth of the search tree represets how may FDLs (recrculato) a packet has to traverse before reachg a soluto. (I graph theory, Breadth-Frst Search (BFS) s a graph search algorthm that begs at the root ode ad eplores all the eghborg odes. The for each of those earest odes, t eplores ther ueplored eghbor odes ad so o, utl t fds the goal). Fally, f the algorthm maages to retur a set of solutos (step 9) we select the soluto wth the lowest recrculato (least FDLs used). Ths s because each 6 recrculato wll cause sgal atteuato. I the evet that there are a umber of solutos that use the same umber of crculatos, we pck the soluto that causes the least delay to a packet. For eample, f soluto A uses two FDLs wth delay values 2 ad 6 tmeslots ad soluto B uses two FDLs wth delay values 4 ad 4 tmeslots, we choose soluto B because the total delay tme (4+4 8) s less tha that of A (2+6 8). Assumg there ests a soluto C that uses three FDLs, each wth the same delay value of 2 tmeslots, we stll choose soluto B because t uses fewer crculatos, although the total delay curred soluto C s less tha that soluto B. I fact, soluto C would ot eve be cosdered as a breadth frst search, oce a soluto s foud the search tree at depth 2, the algorthm would ot proceed to search for a soluto at depth 3. However, a ormal breadth frst search wll stop whe oe soluto s foud, but we epad the algorthm to fsh searchg the etre chld odes that have the same depth where a soluto ca be foud. Ths allows the algorthm to pck the best soluto wth the same depth stead of pckg the frst soluto foud, whch may ot be optmal. Fally, step 0 we update the cofgurato f a best soluto ca be foud. If there are o solutos, we cosder the packet lost ad dscard t whe t arrves because there are o possble routes reserved for t the cofgurato table. Fgure 7 shows the updatg of the cofgurato table wth the oly possble soluto for a packet from put port 4, wth slot legth 3, arrvg at tme t, dested for output port 3. It s delayed for 2 tmeslots usg FDL3, the delayed aga for tmeslot usg FDL2 ad ets at tme t+3 at output port 3. FDL () FDL 2 () FDL 3 (2) FDL 4 (4) Output Output 2 Output 3 Output 4 t t+ t+2 t+3 t+4 t+5 t+6 t+7 Iput 3 Iput Iput 4 Iput 2 FDL 2 FDL 3 FDL 2 FDL FDL 4 Fg. 7: A packet legth 3 arrves for output port 3 ad cofgurato table s updated

7 J. Computer Sc., 6 (): -, 200 The tme requred to assg a FDL route for a packet s proportoal to m, where m s the umber of slots that the packet eed to load. Ths s because we eed to access a ode o the search tree m tmes to eame whether the FDL/output s avalable for m cosecutve tmeslots. Sce m s versely proportoal to the slot sze, the tme complety of the VAPFA algorthm s versely proportoal to the slot sze as well. I other words, the smaller the slot sze, to a certa packet, the more tme t takes to assg a FDL route. Smulato of VAPFA algorthm geerc VPFS: To focus o the swtch performace whe smulatg a geerc VAPFA swtch the geerc VPFS, we assume that comg traffc s regulated such a way that o head-tal clashg wll occur ad thus o packet wll be dropped due to such a pheomeo. It should be oted that such a assumpto s vald at those swtches mmedately et to the sources because sources ca choose to sed packets uder ay traffc regulatos. For other termedate swtches, however, oly whe the codto descrbed the costraed VPFS apples, the head-tal clashg ca the be elmated, assumg o packet algmet. The costraed VPFS wll be dscussed the followgs. Gve the value of the effectve lk utlzato, U ad the packet legth probablty desty fucto (pdf), f(), two radom umbers eed to be geerated order to determe () the arrval of a packet ad () the umber of cosecutve slots that the packet eed to load. To desg the radom umber geerator whch determes the arrval of a packet, let P deote the probablty that a packet s arrvg a udetermed slot. Corollary : I the VPFS scheme wthout algmet, the probablty that a packet s arrvg a udetermed slot s gve by: su P ( U) Proof: Let M be the umber of empty slots ad the umber of packets the durato of T. ote that M+ s the total umber of evets observed T, where: P lm ad, T ( + δ ) + M s + M By the defto: 7 U lm T lm Thus: ( + δ ) + M s lm lm ( + s) + M s ( + δ) + M s lm + M M ( + s) + s + M + M P P P( + s) + ( P)s P + s su P ( U) From corollary, oe ca traslate a gve lk utlzato to the probablty that s requred the packet arrval geerator for smulato. After the arrval of a packet s geerated, the et step s to geerate the packet legth ad determe the umber of loaded slots based o the gve pdf f(). To smplfy the aalyss, we assume that there s a mamum sze for the packets, the value of whch s ormalzed to. I other words, the packet legth,, falls the rage of (0, ]. Sce ut s the mamum packet legth, we further assume that the slot sze of the VPFS swtch to be s, because havg slots larger tha the mamum packet sze would cause wastage due to overhead ad vod spaces. Assume that s /, where s a postve teger. I ths case, m +, where m s the umber of slots that a packet may occupy. A radom umber geerator s therefore eeded to geerate m wth respect to f(). The probablty dstrbuto of m s thus gve by: Prob{m y} ( )f ()d, for y ; 0 y { (y 2)}f ()d y 2 + (y )f ()d for 2 y ; y y + { ( )}f ()d, for y

8 J. Computer Sc., 6 (): -, 200 (a) Fg. 8: Packet loss Vs Lk utlzato a geerc VPFS swtch For stace, f f() from 0-, the Prob{m } Prob{m +} 2 ad Prob{m y} for 2 y. Wth the dstrbuto, oe ca use aother radom umber geerator to geerate packet legths for packets. I the followg, we show the smulato model ad results of the VAPFA algorthm wthout algmet. We study a shared-fdl swtch wth 32 FDLs ad assume that the ormalzed packet legth s a radom umber uformly dstrbuted from 0-, that s, f() ad 0<. I order to evaluate how the selecto of slot sze affects the performace of the swtch, we further assume the slot sze s /, where s a postve teger. I ths case, the largest packet the smulato wll have a equvalet legth of slots. Aother factor that may affect the swtch performace s the legths of the 32 FDLs. I our model, we assume a total FDL legth of aroud 500 (ormalzed value) or 500 slots ad each FDL ca have a delay value from slot, 2 (6 log2 ) slots, 4 slots ad so o up to 2 + slots. Wth referece to Fg. 8, we plot the packet loss rate agast the utlzato for, 2, 4, 8 ad 6. From the observato, the larger the value (whch also meas the smaller the slot sze), the better the performace s terms of packet loss rate. For eample, whe, packet loss begs whe U 0.28 ad the loss rate reaches 0. whe U However, whe 6, packet loss oly begs whe U 0.6 ad the loss rate reaches 0.28 whe U ote that U 0.89 s the mamum effectve utlzato that the system ca acheve for 6, whch s a result from Theorem. Whe jectg more traffc tha gve by ths value to the VPFS swtch that s wthout algmet, the head-tal clashg ca o loger be avoded. 8 (b) Fg. 9: Head-tal clashg may occur f the output traffc s ot regulated. (a) packet ad 2 are from dfferet puts, but depart from the same output at the jth swtch; (b) due to asychroous a lot boudary, packets ad 2 may ecouter head-tal clashg at the (j+) st swtch Costraed VPFS wthout algmet: So far, we have ot placed ay costrat to the output packet stream whe schedulg packets at a VPFS swtch. However, f the output packet stream from a swtch s ot regulated, the head-tal clashg may occur the et swtch due to asychroous slot boudares betwee the two swtches, as llustrated Fg. 9. Ths wll further troduce packet loss for the etre etwork. To totally elmate the chace of head-tal clashg, we regulate the output packet stream departed from ay swtch. A smple soluto s to further mpose a gap of at least s to the tal of ay outgog packet from a swtch. I other words, whe the slot cha of a packet departs from a swtch, t s always followed by a empty slot. Ths scheme s referred to as the costraed VPFS wthout algmet. I ths case, the departure overhead of a packet at a termedate swtch s equal to ts arrvg overhead plus a slot gap of s. ote that such a empty slot gap followg each packet may also be ecessary for the optcal swtch to cofgure ts swtchg state order to accommodate the et packet. Characterstcs of the costraed VPFS: Lemma 2: For ay packet, the epected departure overhead at a termedate swtch the costraed VPFS scheme s 2 sec. Proof: The epected departure overhead the VPFS scheme s equal to E[δ]+s, where E[δ] s the epected arrval overhead. From the result of Lemma, E[δ] s. Thus, the epected departure overhead at a termedate swtch the costraed VPFS scheme s:

9 J. Computer Sc., 6 (): -, 200 s+s 2s Theorem 2: Gve that the legth of a packet s a radom umber,, wth a average value of, the effectve output lk utlzato of the costraed VPFS scheme wthout algmet s bouded by. + 2s Thus: su P ( U) su DISCUSSIO Proof: Cosder observg a output of a costraed VPFS swtch over durato of T, whch packets have departed from the swtch. Let be the legth ad δ the arrval overhead of the th departg packet, where : U lm T lm lm + δ + δ + 2s + + s ( + δ + s) E[ ] s As the output packet stream from a swtch s the put packet stream to the et swtch, we ca derve the probablty that a packet s arrvg a udetermed slot at a termedate swtch as follows. Corollary 2: I the costraed VPFS scheme wthout algmet, the probablty that a packet s arrvg a udetermed slot at a termedate swtch s gve by: Smulato for costraed VAPFA wthout algmet: We smulate ad study the costraed VAPFA algorthm wthout algmet usg t he same swtch model as that has bee dscussed. Fgure 0 shows the smulato result. As ca be see, the costraed verso of the VAPFA algorthm has degraded terms of performace compared wth the geerc ucostraed VAPFA. However, t s worth otg that the costraed VPFS ca be cosdered as the worst-case scearo for the VPFS schemes. Sce o packet loss ca be observed at U 0.52 the smulato (ad P loss at U 0.54) for 6, t ca be cocluded that the VPFS scheme, together wth the VAPFA algorthm, ca work qute smoothly throughout the all-optcal etwork whe the lk utlzato s aroud 0.5. VPFS wth algmet: I VPFS wth algmet, we assume a alger s employed at each put such a way that the head of a arrvg packet ca always be alged perfectly wth the frot boudary of a slot, as depcted Fg.. Such a perfect alger may ot be practcal, but to study such a scheme helps us uderstad the boud of the best-case performace of a VPFS swtch. su P ( U) su Proof: U lm T lm ( + δ + s) + M s P lm ( + 2s) + M s P( + 2s) + ( P)s P P + Ps + s 9 Fg. 0: Packet loss Vs lk utlzato for a costraed VAPFA a VPFS swtch

10 J. Computer Sc., 6 (): -, 200 Packet Alger (a) Alger Packet Swtch (b) Fg. : Swtch wth packet algmet. (a) the packet loads 4 slots before the alger; (b) the alger ca alg the packet to the slot boudary ad mmze the slots loaded before swtchg Fg. 3: Comparso betwee the three schemes for 4 ad 6 Fg. 2: + Tal s always a costat wth algmet throughout the etwork Wth a alger, the arrval overhead of a packet ca oly occur at the tal of the packet. The value of such a overhead totally depeds o the packet legth pdf f(). For eample, f the packet legth s a dscrete radom umber whch s always a teger multple of the slot sze, the the arrval overhead ca be totally elmated by the algers ad packets ca come oe mmedately after aother packet throughout the etwork wthout worryg the head-tal clashg problem. The models dscussed (Ramamrtham ad Turer, 2003; Karol, 993; Cha et al., 200; Chao ad Choa, 999; Huter et al., 998; Gullemot et al., 998; Chao ad Lew, 2003; Masett et al., 993; Sh, 2005) all fall ths specal category where the packet legth s fed, beg eactly equal to the slot sze. Although the above eample may ot be vald for other packet legth pdf, t mples a very mportat characterstc of the VPFS scheme wth algmet, as descrbed below. Let be the packet legth ad Tal be the tal overhead of a packet after algmet. Wth Fg. 2, + Tal s always a costat (whch s also a teger multple of the slot sze s) throughout the VPFS etwork wth algmet. I other words, as log as the subsequet packet always keeps a dstace of Tal wth the prevous packet, the head-ta l clashg ca ever 0 occur at the et swtch. Let s refer the above codto to as the Tal -codto. It should be oted that the etre packet stream departg from a output of a swtch must always satsfy the Tal -codto ature. Therefore, we ca coclude that the head-tal clashg ca be totally elmated the VPFS etwork wth algmet, as log as the traffc from all sources satsfy the Tal - codto. Fgure 3 provdes the comparso betwee the three VPFS schemes, cosderg the same model as dscussed earler. It s obvous that the performace of the swtch wth algmet mproves dramatcally whe compared to the costraed VAPFA model ad also a marked mprovemet over the geerc ucostraed VAPFA. For eample, whe 6, the packet loss begs oly at U Aother terestg observato s, whe becomes larger, the performaces of the three VPFS schemes get closer. COCLUSIO I ths study we have proposed a varable-legth packet FDL assgmet algorthm for shared-fdl swtches ad used a model to study ad smulate three VPFS swtchg schemes, amely the geerc VPFS wthout algmet, the costraed VPFS wthout algmet ad the VPFS wth algmet (Wog, 2006; 2007). All of these VPFS schemes are desged for swtchg varable-legth packets a slot swtchg evromet, but they dffer from each other by the maer of treatmet of head-tal clashg. Through our aalyss ad smulato, we foud that the selecto of slot sze greatly affects the performace of the three

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