Maximum Weight Matching Dispatching Scheme in Buffered Clos-Network Packet Switches

Transcription

1 Maxmum Weght Matchng Dspatchng Scheme n Buffered Clos-Network Packet Swtches Roberto Roas-Cessa, Member, IEEE, E Ok, Member, IEEE, and H. Jonathan Chao, Fellow, IEEE Abstract The scalablty of Clos-network swtches make them an alternatve to sngle-stages swtches for mplementng largesze packet swtches. Ths paper ntroduces a cell dspatchng scheme, called maxmum weght matchng dspatchng (MWMD) scheme, for buffered Clos-network swtches. The MWMD scheme s based on a maxmum weght matchng algorthm for nputbuffered swtches. Ths paper shows that, wth request queues n the buffered Clos-network archtecture, the MWMD scheme s able to acheve a 00% throughput for ndependent admssble traffc, wthout allocatng any buffers n the second stage and wthout expandng the nternal bandwdth. As a practcal scheme, a maxmal oldest-cell-frst matchng dspatchng (MOMD) scheme s also ntroduced. MOMD shows that usng a fnte number of teratons n the dspatchng scheme, the throughout under unbalanced traffc pattern can be hgh. Index Terms Packet swtch, clos-network, dspatchng, throughput, maxmum-weght matchng I. INTRODUCTION It s well known that sngle-stage swtches have lmted scalablty, n terms of the number of ports. Ths lmted scalablty s the result of the lmted number of connecton pns that a swtch chp can allocate. The three-stage Clos-network [], [2] swtch can provde a hgher degree of scalablty than snglestage swtches, because of the smaller number of swtch chps needed for a large sze swtch. We can categorze the Clos-network swtch archtecture nto two types: bufferless and buffered. A bufferless Clos-network swtch has no memory n any stage. To avod contenton n any stage, schedulng needs to be performed at the nput ports pror to sendng the packets through the swtch. Ths approach has the advantage of smplfyng the desgn of the swtch modules. However, the matchng process may be complex and requre a long resoluton tme. Schedulng of bufferless Clos-network swtches can also be used n optcal swtches where swtch reconfguraton may not have to change as fast as per tme slot. There are several studes on schedulng for bufferless Clos-network swtches [7], [8], [9]. They, however, are out of the scope of ths paper. Here, we assume that varable-length packets are segmented nto several fxed-szed packets, or cells, when they arrve. Cells are swtched through the swtch fabrc, and reassembled nto packets before they depart. The tme to transmt a cell through the swtch s called tme slot. Roberto Roas-Cessa s wth the Department of Electrcal and Computer Engneerng, New Jersey Insttute of Technology, Unversty Heghts, Newark NJ, Emal: rroas@nt.edu E Ok s wth NTT Network Innovaton Laboratores, NTT Corporaton, 3-9- Mdor-cho Musashno-sh, Tokyo Japan. E-mal: ok.e@lab.ntt.co.p H. Jonathan Chao s wth the Department of Electrcal and Computer Engneerng, Polytechnc Unversty, Brooklyn, NY, 20. Emal: chao@poly.edu One way to ease the complexty of schedulng n Closnetwork swtches s by allocatng memory n the frst and thrd stages. In ths way, f contenton for an nternal lnk occurs, loser cells are stored n the buffers n the frst stage modules. These swtches can be referred to as buffered Clos-network swtches. As the memory technology evolves, the memory amount that can be embedded nto a chp s no longer a strct lmtaton. Wthn buffered Clos-network swtches, we can consder two groups: wth and wthout buffers n the second-stage modules. A ggabt ATM (asynchronous transfer mode) swtch usng buffers n the second-stage was presented n [3]. In ths archtecture, every cell s randomly dstrbuted from the frststage to the second-stage modules to balance the traffc load n the second-stage. Implementng buffers n the second-stage modules resolves contenton among cells from dfferent frststage modules [5]. However, t requres a re-sequencng functon at the thrd-stage modules, because the buffers n the second-stage modules cause an out-of-sequence problem. A three-stage swtch wth buffers n the frst and thrd stages and bufferless second stage s called a buffered Closnetwork swtch. In [4], an ATM swtch was developed. Ths approach does not suffer from the out-of-sequence problem. Snce there are no buffers n the second-stage modules to resolve contenton, dspatchng cells from the frst stage to the second stage becomes an mportant ssue. A random dspatchng (RD) scheme s used for cell dspatchng from the frst stage to the second stage [4], as adopted n the case of the buffered second-stage modules n [3]. However, RD s not able to acheve a hgh throughput unless the nternal bandwdth s expanded, because the contenton at the second stage cannot be avoded. To acheve 00% throughput for unform traffc by usng RD, the nternal expanson rato s set to about.6 when the swtch sze s large [2], [4]. Ths expanson makes a hgh-speed swtch dffcult to mplement n a cost-effectve manner. It has been shown that t s possble to provde 00% throughput under unform traffc wthout expandng the nternal bandwdth on a buffered Clos-network swtch wth a round-robn-based dspatchng scheme [2]. Moreover, dverse dspatchng schemes have been proposed to reduce the average cell delay [5], [] under unform traffc. However, real traffc patterns are not only unform, but a wde varety of admssble traffc patterns, ncludng those wth nonunform dstrbutons. One queston arses: Is t possble to acheve a 00% throughput under ndependent admssble traffc, wthout allocatng any buffers n the second stage to avod the out-ofsequence problem and wthout expandng the nternal bandwdth? Ths paper proposes a cell dspatchng scheme, called IEEE Communcatons Socety 075

2 maxmum weght matchng dspatchng (MWMD) scheme, for buffered Clos-network swtches. The MWMD scheme s based on the maxmum weght matchng algorthm for nput-buffered swtches [6]. It s known that for ndependent admssble traffc, a maxmum throughput of 00% n an nput-buffered swtch s achevable by usng a maxmum weght matchng algorthm [6]. We show that by the consderng request queues n a buffered Clos-network swtch and the MWMD scheme t s possble to acheve a 00% throughput for ndependent admssble traffc, wthout allocatng any buffers n the swtch modules at the second stage and wthout expandng the nternal bandwdth. Furthermore, we ntroduce an teratve dspatchng scheme, maxmal oldest cell frst matchng dspatchng (MOMD), based on the oldest-cell-frst (OCF) scheme [3] for sngle-stage nput-buffered swtches, and show that t can provde 00% throughput under our nonunform traffc model, called unbalanced. Ths paper s organzed as follows. Secton II descrbes our swtch model. Secton III ntroduces the MWMD scheme. Secton IV ntroduces the MOMD scheme. Secton V shows the performance study of MOMD. Secton VI presents the conclusons. II. BUFFERED CLOS-NETWORK SWITCH MODEL Fgure shows a buffered Clos-network swtch. The frst stage conssts of k nput modules (IMs), each of whch has an n m dmenson. The second stage conssts of m buffer-less central modules (CMs) each of whch has a k k dmenson. The thrd stage conssts of k output modules (OMs), each of whch has an m n dmenson. The termnology used n ths paper s as follows: : ( +)th nput module, where 0 k. CM(r): (r +)th central module, where 0 r m. OM(): ( +)th output module, where 0 k. n: number of nput/output ports n each IM/OM, respectvely. k: number of IMs/OMs. m: number of CMs. IP(, h): (h +)th nput port (IP) at, where 0 h n. OP(, l): (l +)th output port (OP) at OM(), where 0 l n. VOMQ(, ): vrtual output-module queue at that stores cells destned for OM(). RQ(,, r): Request queue (RQ) at that stores requests of cells destned for OM() through CM(r). RQ(,, r) also keeps the watng tme W (,, r), whch s the number of slots a head-of-lne (HOL) request has been watng. L I (, r): output lnk of that s connected to CM(r). L C (r, ): output lnk at CM(r) that s connected to OM(). λ(, h,, l): arrval rate at IP(, h) for OP(, l). λ(, ): arrval rate at for OM() = h l λ(, h,, l). An has k vrtual output-module queues (VOMQs) to elmnate HOL blockng. A VOMQ s smlar to a vrtual output queue (VOQ), whch s used n nput-buffered swtches [3], [6]. When a cell enters VOMQ(, ), the cell request s randomly dstrbuted and stored n RQ(,, r) among m request queues. A request n RQ(,, r) s related to VOMQ(, ), but s not related to a specfc cell n VOMQ(, ). Every tme a request n RQ(,, r) s granted for transmsson, one cell s dequeued from VOMQ(, ) n a FIFO manner. A VOMQ can receve at most n cells from n nput ports and can send at most m cells to m CMs n one tme slot. Each has m output lnks. Each output lnk L I (, r) s connected to each CM(r). IP(, h) IP(0, 0) IP(0, n-) VOMQ(0, k-) IP(, 0) IP(, n-) IP(k-, 0) VOMQ(0, 0) VOMQ(, 0) VOMQ(, k-) VOMQ(k-, 0) IP(k-, n-) VOMQ(k-, k-) RQ(0, k-, m-) RQ(, 0, 0) RQ(, k-, 0) RQ(, 0, m-) RQ(, k-, m-) RQ(k-, 0, 0) RQ(k-, k-, 0) RQ(k-, k-, m-) CM(0) CM(r) CM(m-) OM() OP(0, 0) OP(, l) OP(0, n-) OP(, 0) OP(, n-) OP(k-, 0) OP(k-, n-) Fg.. Clos-network swtch wth vrtual output-module queues (VOMQs) n the nput modules A CM(r) has k output lnks, L C (r, ), connected to OM(). AnOM() has n output ports, each of whch s denoted as OP(, l) and has an output buffer. Each output buffer receves at most m cells n one cell tme slot, and each output port at the OM forwards one cell n a frst-n-frst-out (FIFO) manner to the output lne. Scalablty, n terms of the port speeds, has also been of research nterest. A sngle-stage parallel packet swtch resembles the swtch fabrc of a Clos-network swtch (.e., the set of IMs, CMs, and OMs) because of the way the nternal lnks are connected n the latter. Parallel swtches consder usng swtch planes runnng at lower speed than the external connecton lnks to provde hgh-speed ports [4]. We consder that Clos-network swtches use nternal lnks transferrng data at the same speed as the external lnks. Moreover, followng the prncple of the parallel swtch, Clos-networks swtches can also use parallel swtchng planes to provde hgh port speeds. However, the dscusson about ncreasng the port speed n a Clos-network swtch s out of the scope of ths paper. III. MAXIMUM WEIGHT MATCHING DISPATCHING (MWMD) SCHEME The MWMD scheduler conssts of m subschedulers, each of whch s denoted as S(r), as shown n Fgure 2. Subscheduler S(r) selects up to k requests from k 2 RQs, where correspondng cells to the selected RQs are transmtted through CM(r) at the next tme slot. In S(r), k RQs: RQ(, 0,r),..., RQ(, k,r), are the requests from to all OMs An L-bt cell must be wrtten to and read from a VOMQ memory n a tme less than L Cn and L, respectvely, where C s a lne speed. For example, Cm L when L =64 8 bts, C=0 Gbt/s, and n = m =8, Cn = L s 6.4 Cm ns. Ths s feasble when we consder 2-port memores wth current avalable CMOS technologes. IEEE Communcatons Socety 076

3 Fg. 2. Subscheduler S(0) RQ(, 0, 0) RQ(, k-, 0) RQ(k-, 0, 0) RQ(k-, k-, 0) Subcheduler S(m-) RQ(0, k-, m-) RQ(, 0, m-) RQ(, k-, m-) RQ(k-, k-, m-) OM() OM() Maxmum Weght Matchng Dspatchng (MWMD) Scheduler through CM(r), and k RQs: RQ(0,,r),..., RQ(k,,r), are the requests from all IMs to OM() through CM(r). S(r) selects one request from each and one request to each OM(). S(r) fnds an approprate match accordng to ts schedulng algorthm as a scheduler of an nput-buffered swtch does. In the request selecton n S(r) at each tme slot, a maxmum weght matchng algorthm, the OCF algorthm [6], s employed. RQ(,, r) keeps the watng tme W (,, r), whch s the number of slots a HOL request has been watng. S(r) fnds a match M(r) at each tme slot, so that (,) M(r) W (,, r) s maxmzed. Note that each S(r) behaves ndependently and concurrently. S(r) uses only k 2 W (,, r) to fnd M(r). S(r) and S(r ), where r < r, do not exchange any nformaton to fnd M(r) and M(r ), respectvely. When RQ(,, r) s granted by S(r), the HOL request n RQ(,, r) and a cell n VOMQ(, ) are dequeued. The dequeued cell s one of the HOL cells n VOMQ(, ). Here, the number of dequeued HOL cells s equal to the number of granted requests by all S(r)s. In order to send cells n sequence to an output buffer at OP(, l), we adopt a smple rule as follows. Consder that Y RQs for the same (, ) par, whch are RQ(,, r ), RQ(,, r 2 ),..., RQ(,, r Y ), where r < r 2 < r Y, are granted by more than one subscheduler. Y cells from the HOL n VOMQ(, ) are transmtted to OM() through CM(r ), CM(r 2 ),..., CM(r Y ) at the same tme slot. The yth cell from the HOL n VOMQ(, ) goes through CM(r y ).Note that each request n RQ(,, r) s related to VOMQ(, ), but s not related to a specfc cell n VOMQ(, ), as descrbed n Secton II. When more than one cell goes to an output buffer at OP(, l), a cell from CM(r) enters the output buffer earler than the cell from CM(r ), where r<r. As a result, cells are sent n sequence. A. 00% throughput by MWMD We prove that MWMD acheves 00% throughput for all admssble ndependent arrval processes wthout nternal bandwdth expanson,.e., wth n = m. Theorem : MWMD acheves 00% throughput for all admssble ndependent arrval processes wthout expanson of the nternal bandwdth. Proof: Wth ths archtecture, we use the rate matrx of the arrval process as: Λ [, h,, l], where the assocate rate vector s λ (λ 0,0,0,0,...,λ 0,0,0,n,...,λ 0,0,k,0,..., λ 0,n,0,0,...,λ k,0,0,0,...,λ k,n,k,n ). We assume that our admssble nput traffc condtons, λ(, h,, l) < and λ(, h,, l) <, () h are satsfed. The arrval process A (t), whch s the aggregated cell arrvals of n nput ports at that are destned to VOMQ(, ), s statonary and ergodc. The arrval matrx representng the sequence of arrvals A(t) [A, (t)], where, at tme t: g, g n f arrval(s) occurs at VOMQ, A, (t) = (2) and the assocated rate vector s A(t) (A 0,0 (t),...,a 0,k (t),...,a k,k (t)) T. The servce matrx, Γ(t) [Γ, (t)] ndcates whch VOMQs are servced at tme t: p, p n f VOMQ, gets servce at t Γ, (t) = (3) and the assocate vector Γ(t) s smlarly defned as A(t). However, Γ(t) may not be used as a permutaton matrx. To overcome ths, consder usng request queues, RQ(,, r), as descrbed n Secton III. As the MWMD scheme, performed by the rth subscheduler consders the RQs related the rth CM for dspatchng cells through CM(r), the set of RQs can be consdered ndependent of the others. They can be separated as a local set of RQs, denoted as LRQ(, ), for each CM(r). Each LRQ has a correspondng arrval request matrx (t) [, (t)]. The request arrval matrx s such that f an arrval to LRQ, occurs at tme t, (t) = (4) as RQs accept n average one request from nput to output each tme slot, and the RQ dstrbuton s ndependently and nformngly dstrbuted. The assocated vector, (t), s smlarly defned. The servce receved by LRQs s determned by the servce matrx Θ(t) Θ, (t), where: f LRQ, s served at tme t Θ, (t) = (5) and the assocated vector, Θ(t), s smlarly defned. Therefore, the LQRs and CMs are equvalent to the VOQs of an nputbuffered swtch, consderng (t) and Θ(t) as the arrval l IEEE Communcatons Socety 077

4 and servce processes. The servce receved by LRQs s the result of MWMD, where only the HOL requests at LRQs are matched. From (), we obtan that λ(, ) = λ(, h,, l) <n, and h l λ(, ) = λ(, h,, l) <n. (6) h l Here, = =n s used. By dvdng (6) by m, λ(, ) < n m m and λ(, ) < n m m. (7) By usng n = m, we obtan λ(, ) < and m m λ(, ) <. (8) Therefore, the total servce receved by VOMQ(, ) s r Θ,(t). If the receved servce by Θ, (t) for each CM and ts LRQ set s such that the RQs are stable, then MWMD s stable. Note that mλ(, ) s the request arrval rate for RQ(,, r). In addton, snce the cell arrval process at VOMQ(, ) s ndependent, a request arrval process to RQ(,, r) s also ndependent. Ths s because, when a cell enters VOMQ(, ), ts request s randomly dstrbuted and stored among m request queues. As MWMD s based on the maxmum-weght matchng scheme, Lemma 7 n [6] can be appled. Therefore, the subscheduler model n MWMD s equvalent to the scheduler model n the maxmum weght matchng (MWM) of the nput-buffered swtch model defned n [6]. Usng Lemmas 8-0 and Theorem 4 n [6], where the OCF-based MWM algorthm has the queue occupancy stable, achevng 00% throughput n an nput-buffered swtch for all admssble and ndependent arrval processes, a subscheduler n MWMD provdes 00% throughput. As all subschedulers have equvalent performance, MWMD provdes 00% throughput for all admssble and ndependent arrval process wth n = m. IV. MAXIMAL OLDEST CELL FIRST MATCHING DISPATCHING (MOMD) In ths secton, we ntroduce the maxmal oldest-cell frst matchng dspatchng (MOMD) scheme wth multple memory access (.e., queues can dspatch up to m cells) as a maxmal dspatchng scheme wth lower complexty than MWMD. MOMD uses the swtch model presented n Secton II. There are k VOMQs at each IM, each denoted as VOMQ(, ), and m request queues, RQs, each assocated wth a VOMQ, and denoted as RQ(,, r). MOMD has dstrbuted arbters n IM and CM. In, there are m output-lnk arbters. In CM(r), there are k arbters, each of whch corresponds to OM(). The VOMQs and dstrbuted lnk arbters are shown n Fgure 3. When a cell enters a VOMQ(, ), the cell gets a tme stamp assgned, whch s entered along wth a request to RQ(,, r), where r s randomly selected. We consder lnk arbters for the output lnks at IMs and CMs. To determne the matchng between VOMQ(, ) and the output lnk L I (, r), IP(, h) IP(0, 0) IP(0, n-) VOMQ(0, k-) IP(, 0) IP(, n-) IP(k-, 0) IP(k-, n-) VOMQ(k-, k-) Fg. 3. VOMQ(0, 0) VOMQ(, 0) VOMQ(, k-) VOMQ(k-, 0) RQ(0, k-,m-) RQ(, 0, 0) RQ(, k-, 0) RQ(, 0, m-) RQ(, k-, m-) RQ(k-, 0, 0) RQ(k-, k-, 0) RQ(k-, k-, m-) IM lnk arbter CM(0) CM(r) CM(m-) CM lnk arbter OM() Dstrbuted arbters n a buffered Clos-network wth MOMD OP(0, 0) OP(, l) OP(0, n-) OP(, 0) OP(, n-) OP(k-, 0) OP(k-, n-) the output lnks arbters select a VOMQ (through ther RQs) based on tme stamps. The scheme s descrbed as follows: Phase : Matchng wthn IM Step : A non-empty RQ(,, r) sends a request to the unmatched output lnk arbter assocated to L I (, r). The request ncludes a weght, whch s the tme stamp of the HOL request. Step 2: Each output-lnk arbter L(, ) chooses one RQ request by selectng the oldest tme stamp. If a te occurs, the output lnk arbter selects the RQ(,, r) by selectng the VOMQ wth the largest ndex. The output lnk arbter sends the grant to the selected RQ and VOMQ. Phase 2: Matchng between IM and CM Step : After phase s completed, L I (, r) sends the request to CM(r) belongng to the selected VOMQ. At CM(r), each arbter assocated wth OM() chooses one request by selectng the oldest tme stamp. Tes are broken by selectng, among the older requests, the largest ndex. The arbter at CM(r) sends the grant to L I (, r) of. Step 2: If an IM receves a grant from a CM, the IM sends a (HOL) cell from that VOMQ at the next tme slot. Note that up to m cells, at the head of lne, may be dspatched from a VOMQ. MOMD dspatches cells as n MWMD to get cells arrve the output buffer at OP(, l) n sequence. The descrbed scheme refers to a sngle teraton between IM and CM. To perform more teratons, the unmatched VOMQs, lnks, and CM arbters are consdered, and the reected requests are nhbted n the remanng teratons, performed between IM and CM, n the tme slot. V. PERFORMANCE EVALUATION OF MOMD We show the delay and throughput performance of swtches, where n = m = k =8, usng the RD [4], CRRD [2], and MOMD schemes. Note that MOMD does not use teratons n IM, denoted as IM, as n the CRRD scheme. MOMD uses request queues that smplfy the process, such that one lnk arbter selects one out of m request queues. However, we consder that a number of teratons between IM and CM n all schemes, denoted as IM-CM, can be IEEE Communcatons Socety 078

5 Average cell delay (cell slot) CRRD, IM-CM= RD MOMD, IM-CM= MOMD, IM-CM= Input load Fg. 4. MOMD under bernoull unform traffc n a (n = m = k =8) swtch used. 2 We note that MOMD cannot acheve a 00% throughput under unform traffc wth a sngle IM-CM teraton. However, MOMD gets hgher throughput as the number of IM- CM teratons ncreases, as the fgure shows. In the swtch under smulaton, the number of teratons to provde 00% throughput s four (IM-CM = 4). The smulaton shows that the round-robn based scheme, CRRD, s more effectve under unform traffc than MOMD, as CRRD acheves hgh throughput wth one teraton, IM-CM =. Throughput Unbalanced probablty, w RD MOMD, IM-CM= MOMD, IM-CM=2 MOMD, IM-CM=4 MOMD, IM-CM=8 CRRD, IM-CM = CRRD, IM-CM =8 Fg. 5. MOMD under bernoull unbalanced traffc and n = m = k =8 We also consdered a nonunform traffc pattern, called unbalanced [5]. The unbalanced traffc pattern has one fracton of traffc wth unform dstrbuton and the other fracton w of traffc destned to the output wth the same ndex number as the nput. When w =0, the traffc s unform; when w = the traffc s totally drectonal. We evaluate RD, CRRD, and MOMD under ths traffc pattern. To obtan the best performance, CRRD adopts IM = m (.e., the number of teratons performed n IM to match the VOQs used n CRRD and output lnks [5]) for any number IM-CM. Fgure 5 shows that RD, wth one teraton, and CRRD, wth multple IM-CM teratons, cannot offer 00% throughput under unbalanced traffc. Although CRRD wth IM-CM = 8 ncreases the throughput, ths s not close to 00%. The reason for that s that CRRD, based on round-robn selecton, tres to allocate the same servce for all queues, ndependently of the HOL cell age or queue load n each tme slot. MOMD, as t uses weghtbased selecton, provdes hgher throughput as the number of IM-CM teratons ncreases. MOMD wth IM-CM = offers low throughput. However, ts throughput approaches to 00% when IM-CM =8. Ths presents the challenge of performng a large number of teratons, therefore, we would need to consder the adopton of a ppelnng technque [0] for mplementaton of MOMD. These results also show that wth a fnte number of IM-CM teratons n ths swtch can provde hgh throughput under ths traffc pattern. VI. CONCLUSIONS Ths paper proposed the MWMD scheme for buffered Closnetwork swtches, whch s based on a maxmum weght matchng algorthm for nput-buffered swtches. Ths paper showed that the combnaton of MWMD and request queues n a buffered Clos-network swtch provdes 00% throughput under ndependent admssble traffc, wthout allocatng any buffers n the second stage and wthout expandng the nternal bandwdth. In addton, we ntroduced a maxmalweght matchng dspatchng scheme, MOMD, based on the OCF scheme for sngle stage swtches, whch can provde hgh throughput under a nonunform traffc pattern, called unbalanced. REFERENCES [] C. Clos, A Study of Non-Blockng Swtchng Networks, Bell Sys. Tech. Jour., pp , March 953. [2] E. Ok, Z. Jng, R. Roas-Cessa, and H. J. Chao, Concurrent Round- Robn Dspatchng Scheme n a Clos-Network Swtch, IEEE ICC 200, pp. 07-, June 200. [3] T. Chaney, J. A. Fngerhut, M. Flucke, and J. S. Turner, Desgn of a Ggabt ATM swtch, Proc. IEEE INFOCOM 97, pp. 2-, Apr [4] F. M. Chuss, J. G. Kneuer, and V. P. Kumar, Low-Cost Scalable Swtchng Solutons for Broadband Networkng: The ATLANTA Archtecture and Chpset, IEEE Commun. Mag., pp , Dec [5] E. Ok, Z. Jng, R. Roas-Cessa, and H. J. Chao, Concurrent Round-Robn-Based Dspatchng Schemes for Clos-Network Swtches, IEEE/ACM Trans. Networkng, vol. 0, no. 6, pp , Dec [6] N. McKeown, A. Mekkttkul, V. Anantharam, J. Walrand, Achevng 00% Throughput n an Input-queued Swtch, IEEE Trans. Commun., vol. 47, no. 8, pp , Aug [7] C. Y. Lee and A. Y. Qruc, A Fast Parallel Algorthm for Routng Uncast Assgnments n Benes Networks, IEEE Trans. Parallel Dstrbuted Sys., vol. 6, no. 3, pp , Mar [8] T. T. Lee and S-Y Lew, Parallel Routng Algorthm n Benes-Clos Networks, Proc. IEEE INFOCOM 96, pp , 996. [9] K. Pun, M. Hamd, Dstro: A Dstrbuted Statc Round-Robn Schedulng Algorthm for Bufferless Clos-Network Swtches, IEEE Globecom 2002, vol. 3, pp , [0] E. Ok, R. Roas-Cessa, H. Jonathan Chao, PCRRD: A Ppelne- Based Concurrent Round-Robn Dspatchng Scheme for Clos-Network Swtches, IEEE ICC 2002, vol.4, pp:22-225, [] P. Konghong, M. Hamd, Statc Round-robn Dspatchng Schemes for Clos-network Swtches, IEEE HPSR 2002, pp , May [2] T. Anderson, S. Owck, J. Saxe, and C. Thacker, Hgh speed swtch schedulng for local area networks, ACM Trans. Comput. Syst., vol., no. 4, pp , Nov [3] N. McKeown, Ph. D. Thess, Unversty of Calforna at Berkeley, May 995. [4] S. Iyer, N. McKeown, Analyss of the Parallel Packet Swtch Archtecture, IEEE/ACM Trans. Networkng, vol., No. 2, pp , Apr [5] J. Turner and N. Yamanaka, Archtectural Choces n Large Scale ATM Swtches, IEICE Trans. Commun. vol., E8-B, no. 2, pp , Feb Ths s an extended verson of CRRD, presented n [5]. IEEE Communcatons Socety 079