DBG MANs and their routing performance 0.W.W.Yang and ZFeng Abstract: Tlic authors proposc liic de Brubn graph (dBG) LIS ii topology for multi-hop lightwave networks and thcir hierarchical structures. Three routing algorithms that would progressively improve on the mean path length ti lid miwork throughputdclay performance arc studied under various topological variations and locality factors. Thc study shows tirat thc pcrfnrmance of tlic bidircctionul dBC iiclworks and their liicrarchical derivalivcs are desirnblc nnd arc coinpatable to othcr types of tnulti-hop systems. Thcrcforc, they are candidates for metropolitan a i w networks. 1 Introduction Oplical fibre over thc ycats has proved itself to be an advaticed transrnissioii technology Tor next-generAon nct- works such as thc metropolitan arca nctworks (MANs) that would cover a large area. In gcncrd, optical architcc- tiircs for MANS ftlll into two main categories; single-hop itnd multi-hop [l]. Pcrrorniance evaluations of multi-hop nctworks often includc meastires such :is nieaii path length, average dclays, etc. Thcsc nicasiirm vmy in tiifl’crcnt topol- ogics and depend 011 nctwork characteristics such as the degrcc of a network. Conscquently, filldillg a suitable nel- work topology has ken ti prim iritcicst in coinpulcr’ corn- muniaition iictwork designs. Various andid;ite topologies liaw been proposed including llic perfect shuffle (c.g [I]), the Loroid (Manhattan stixt network) [Z], h e hypeicuhc (c.g. [3]), Ktuk graph (e.g. [4]) and tlic de Bruijn graph (dBG) (e.g. [5, 61). The history of unidirectioiial dl3G (UdBG) nctworks can be round in [7], and Ihcir properties in [S, XI. The bidirectional de Bruijn graph (BdBG) has been studied with respect to its fault-tolerance charactcris- tics and conncctivity [X-IO]. RLIL to the best of our knowl- edge, no computation has becii done on the mcm path length of the BdBG networks bccausc 110 shortest-path routing method has been found so far. By clustering small networks and interconnecting them via chosen bridge nodes, a nctwork can take on a hierarchi- cal structure that can often imp-ovc nctwork performancc when traffic locality exists. Work has been done on liicrar- chical hypcrcuhe networks [3], Manhattan strccl nctworks [Z], etc. As Far as wc know, 110 work liiis bocn done on the hierarchical de Bruin graph. This paper proposcs three differen1 routing mnettiods for tlie BdBG network and conipares tliern to other routing mclhods in terms of tlicir inem path length, delay and throughput. In ttie liicrarcliical UdBG and Bd BG net- works, we study in addiiion the traffic locality and the cri- teria to chodsc a node as thc appropriate bridgc for interconncction at the uppcr Icvel. 2 Routing algorithms in UdBG and BdBG networks Shortcst pat11 routing is vcry simple in R UdBG nclwork, as there is only one such path belwccn the source and dcstina- tiori riodcs [7]. The patti can bc obtaiiicd by overlapping the postfix portion of 51 source address [hat is in common with the prefix portion of the destination address. For cxample, in ti (2, 4) UdBG network intlicalcd by the solid iines of Fig. 1, nodc 0010 wi~nts to scnd a packet to iiodc 1011. As shown in Fig. 2, the path-hits are ‘001011’, and therdorc nodcs ’OOIO’, ‘0101’ and ‘IO1 I’ are on thc path. 1011 1101 . .......................... ~ \ - ‘I..., , Unlike tlic UdBG network, a shortesl-palh algorittim for a gcncral BdBG network has nol bccn found. In this Scc- tion we propose tliree routing algorihms that would improve on thcir capahility in seeking out tlic shortest path. They HIT rcslmtivcly cdled ttie rcrrcshing (RFK), neigh- hour searching (NSC3 arid pattern matching (PMC) algo- rithms. They will be compared with thc shortest consistent IEE PT~IC..~.’~IIIHIIIII., Vul. 147, Are, 1. Fdirrrrrry ZDM 32
Transcript
DBG MANs and their routing performance
0.W.W.Yang and ZFeng
Abstract: Tlic authors proposc liic de Brubn graph (dBG) LIS ii topology for multi-hop lightwave networks and thcir hierarchical structures. Three routing algorithms that would progressively improve on the mean path length ti lid miwork throughputdclay performance arc studied under various topological variations and locality factors. Thc study shows tirat thc pcrfnrmance of tlic bidircctionul dBC iiclworks and their liicrarchical derivalivcs are desirnblc nnd arc coinpatable to othcr types of tnulti-hop systems. Thcrcforc, they are candidates for metropolitan a i w networks.
1 Introduction
Oplical fibre over thc ycats has proved itself to be an advaticed transrnissioii technology Tor next-generAon nct- works such as thc metropolitan arca nctworks (MANs) that would cover a large area. In gcncrd, optical architcc- tiircs for MANS ftlll into two main categories; single-hop itnd multi-hop [l]. Pcrrorniance evaluations of multi-hop nctworks often includc meastires such :is nieaii path length, average dclays, etc. Thcsc nicasiirm vmy in tiifl’crcnt topol- ogics and depend 011 nctwork characteristics such as the degrcc of a network. Conscquently, filldillg a suitable nel- work topology has k e n ti p r i m iritcicst in coinpulcr’ corn- muniaition iictwork designs. Various andid;ite topologies liaw been proposed including llic perfect shuffle (c.g [I]), the Loroid (Manhattan s t ix t network) [Z], h e hypeicuhc (c.g. [3]), Ktuk graph (e.g. [4]) and tlic de Bruijn graph (dBG) (e.g. [5, 61). The history of unidirectioiial dl3G (UdBG) nctworks can be round in [7], and Ihcir properties in [S, XI. The bidirectional de Bruijn graph (BdBG) has been studied with respect to its fault-tolerance charactcris- tics and conncctivity [X-IO]. RLIL to the best of our knowl- edge, no computation has becii done on the mcm path length of the BdBG networks bccausc 110 shortest-path routing method has been found so far. By clustering small networks and interconnecting them
via chosen bridge nodes, a nctwork can take on a hierarchi- cal structure that can often imp-ovc nctwork performancc when traffic locality exists. Work has been done on liicrar- chical hypcrcuhe networks [3], Manhattan strccl nctworks [Z], etc. As Far a s wc know, 110 work liiis bocn done on the hierarchical de Bruin graph.
This paper proposcs three differen1 routing mnettiods for tlie BdBG network and conipares tliern to other routing mclhods in terms of tlicir inem path length, delay and throughput. In ttie liicrarcliical UdBG and Bd BG net- works, we study in addiiion the traffic locality and the cri- teria to chodsc a node a s thc appropriate bridgc for interconncction at the uppcr Icvel.
2 Routing algorithms in UdBG and BdBG networks
Shortcst pat11 routing is vcry simple in R UdBG nclwork, as there is only one such path belwccn the source and dcstina- tiori riodcs [7]. The patti can bc obtaiiicd by overlapping the postfix portion of 51 source address [hat i s in common with the prefix portion of the destination address. For cxample, in ti (2, 4) UdBG network intlicalcd by the solid iines of Fig. 1, nodc 0010 wi~nts to scnd a packet to iiodc 1011. As shown in Fig. 2, the path-hits are ‘001011’, and therdorc nodcs ’OOIO’, ‘0101’ and ‘ IO1 I’ are on thc path.
1011 1101
. .......................... ~ \ - ‘I..., ,
Unlike tlic UdBG network, a shortesl-palh algorittim for a gcncral BdBG network has nol bccn found. In this Scc- tion we propose tliree routing algorihms that would improve on thcir capahility in seeking out tlic shortest path. They HIT rcslmtivcly cdled ttie rcrrcshing (RFK), neigh- hour searching (NSC3 arid pattern matching (PMC) algo- rithms. They will be compared with thc shortest consistent
path (SCP) :algorithm, which i s cffctively the routing algo- rithm used in I h c UdBG network [IO].
2.1 Addressing in BdBG networks A bidircclioiial de Btaijn graph (BdBG) caii he constructed froin ii UdRG by cliangiiig :ill unidiiwtional links to bidi- rectional. Mathematically, node A = (AD, A D _ , , ..., A , ) is coiineckd to nodc B = (& B,.,, ..., 8,) i T oiic of the fol- lowing two conditions is salisficd:
(i) Node U is EL lcft-ncighbour, i.e. Ui = A;-1
where U,, Ai E {U, 1 , 2 , . . . ~A ~ 1) and 2 5 ,i 5 (ii) Nntlc D is a right-nciglhour, i .n. l3i = Ai.1.I \vhcre Bi, Ai E {0,1.,2,, . , ,A-l} MI^ 1 5 i 5 ,JJ-l
(1) Therefore, a BdBG ekl ivc ly corisists of two supcrirn- poscd UdRGs; the original UdBC iictwork and another onc with nodc addresses that are revei-sc in bits order. For cxample, in the (A, 13) = (2, 4) nctwork in Fig. 1, nodcs 0001 and 001 1, which are connected by lhc solid unidirec- tional link (representing thc original UdBG network), take on the addresses 1000 and 1100, respectively, when con- riet;ted by ii dolkd unidirectional link. In olhcr words, R
nodc address can take on two bit-rcpresentations. This can sonletinlev be uliliscd in the routing algorithins oi“ a BdBC to identify thc neighbours or to wmputc disrancc in either direction.
Since a node ciin send ii packcl to either its left neigh- bours or its right miglibours in a BdBG, wc say a packet follows i t forward palh (W) when it always chooses B left ncighbour from its current nodc. For example, fhc solid lines in Fig. 1 arc Ihc Fl’s. However, a packct follows A
backward path (BP) when it always chooscs R right neigh- bour t i l l it arrives at the riestiiiation nodc. Also, matcli-- fwcl(S, D) is defimd as a n operation which rclurns the numbcr of hops requii-ed to reach tlic dcstiiialion along an FP, where S and n arc, respectively, tlie sourcc address and the destination address. Similarly, matcllbwd(S, D) returns Ihc iiuniber of hops along a BP. The four I-outiiig algo- rithms menlioncd bcforc caii now be descrihd as l‘ollows.
2.2 Shortest consistent path (SCP) routing algorithm I n this algorithm, R source node determines which one (using FP or RP) or the two UdBG networks ofl‘crs a shortcr path length. Once the piith (FP 01’ SI’) is chosen, iI packet simply follows thc sanic path a s in thc original UdBG algorithm, and will not c h m g ils coum at any iiitermedbte nodes. Thc path length perforinancc is improved becausc a source node lias now two pith choices. begill
if (D == Pr) rcccivc the packet; iT (S ;r Pi.) following the path fixed by llic source; clsc begin
FP lmgth i = matcli-fwd(S, U); RP Iciigth , j = mimli-bwd(.Y, D);
iT (i <j) path = PI’; else if (i == .i, path = FP or BP which caii be chosen 1-mdomly with qua l probability;
else path = BP; end
cnd
IEE Pmr!.-C~orrrr~tirrr., I ’d 147, MI, I . Fi-lirrrrrw 20M
2.3 Refreshing (RFRI routing algorithm The RFR algorithm finds n shortcr path by taking advan- Lage of the bidirectional links in the BdBG wliicb may offer a shorter path if a picket is allowcd to changc its direction (hoii i FP to U P or vice i ~ c r . ~ ) at an intcrnicdiatc node Pr. begin
if (D == Pr) rcccivc the packet; clsc begin
FP length i = maich-rwd(I+, D); RP lciiglh j = match-bwd(P’r, U); if (i < J] FP is chosen to scnd packet to the left neighbour; clsc il’ (i == .i, FP or B1’ is chosen randoinly with qual pi-obability;
else BP is choscn to send packet to the right wiglibour;
end end For example in Fig. I with S = (0001) and D = (lOOl), one caii see ttiwt the packcl can go along the route 0001-00LCc 1001 which requires thc packcl lo switch from the FP to I3P a t node 0010. This route has a length of two hops com- pared with three hops achicvcd by h e SCP algotitlim.
2.4 Neighbour searching (NSCI routing a1go rithm As ai1 improvement 011 thc RFR algorithm, thc NSC dgo- rithin allows a node to send a packct to a neighbour thal would take the least iiurnbcr or hops to reach the destina- tion. Thus a node can makc a siiiartcr decision on the nexl iwdc l o mid by cxploitiiig the bidirdonul properly of the BdRC. Nolc that neighbour[k] in the algorithm bclow iudi- caks the nddi~css or rhc lcth neighbour of tlie prccscnt node PI.. begin
if (U == P r ) rcccivc the packet; else begin
k = 1 whilc (k 5 number of neighbours) do 1 I e g i I I
i = match-fwd(Pr; neighbour[k]); , j miitch-bwd(Pr, neighbour[k]); P i.cquircd path length at tieigltbourcounlcr */ ptIi[k] = niinimum(i,j]; IC = I C + I ; I,,
eiid next nodc = ncighbour x whose pxth[x] is the smaHIest; send piicket to thc ncxt nodc;
cnd cnd For example, Ibr iiodc I001 to send to node 1000 in Fig. I, the RFR algorithm would choose node 0010 as Ihc next node first (this is dccidcd by obtaining FP(lOO1, 1000) = 3 and BP(1001, l0W) = 41, litid the total path lcnglli to the destination is thrcc hops (the remaining path may bc 0010.. 0001-1000 by iqxcating tlie RFR algorithm). However, inspection shows that nodc I [ )U l should have sent to node 0100 using BP, and tlicn to node I000 for a total path length of only two hops. This c m k t i ch ied by thc NSC dgorilhm.
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2.5 Pattern marching (PMCl routing algorithm Thc idca bchiiid pattern inatching is to find out the criteria that cui orient the packcl. towards the destination by tak- ing advantage of thc atldrcss bit pattern between the present and the dcsliiiation nodes. Two of these critriteria ai'c the following: (i) Width of the cominoii palkrn (WCP): this is the number of digits in thc largcst common pattern between two addresses. The lurgcr thc WCP, the closer the two nodes. For exempie, nodc 001 1 is closcr to node 01 10 than node 0001 becniisc Lhc WCP of (001 I, 0110) is three whilc the WCP of (0009,~lO) i s IWO. Olwiously, WCP is largest when thc prcscnt node equals the destination nodc. (ii) Distance of the coinmon patterns (DCP): this is the 1x1- ative ofiset of the coininon pallcm huiid in two addresses. Using 1hc addrcssiiig property in the tlc Rriiijn graph, one can vcriry that for a given commoil paucrn, thc largcr the DCP is, thc closer the soul-ce and destination nodcs arc to each otlicr. For example, although the WCPs or both (0011, 0110) and (0111, 0110) tire thixe, their DCPs arc I and 0, rcspcc~ivcly. One then deduces (and vcrilics) that node 01 10 is closer to nodc 001 1 than to node 01 11.
Thc hollowing PMC routing algorithm bctwccn the prcscnt iiadc (Pr) and the destination node (D) is wrilten based on lhe two criteria above. When hvo nciglibnur nodes have the same WCP and DCP, either one can be chosen randomly. bcgin
if (D == Pr) i:cccivc the packet; else hgin
WCP-current = WCP of (Pr, D); DC1'-current = DCP of (Pr., D); while {there is iiii iinchcckcd neighbour) do kgin
WCP-temp = WCP of (this neighbour, U); IXP-temp = DCP of (this neighbour, D); i T (WCP-tcmp) > WCP-current) hgin
ir (DCP-temp > UCP-current) bcgin ncxt node = this neighboul.; DCl'-c~i rrent = D CP-temp;
end a id
cnd scnd packet to the next node;
end cnd
3 Performance analysis of the UdBG and BdBG networks
In ii high-gccd lighlwave system it is very desirable to pro- vide a small mcao path length because it routing protocol that diverts traffic along R longer path will gencratc inorc internal traffic. This would inore likcly m a t e traffic bottle- necks because of thc optoclcclronic signal processing requircd at each node. We shall use the mean pdth length, average packet delay nnd network throughput to sludy and compare both the routing dgorithms and vaiious dUG topologies.
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3.1 Mean path length {average number ofhopsl Define mean path lcnglh as the avcragc number of hops among all sourcedestination pairs in the network. An emulation program coded in thc C programming language has been inipletnented to inject external tl.aKtc into a source and extract it from a d c s t i n a h nodc according to a given roiiting algorilhm. Due to this implenientation, we find i t convenient to use enumeration to dctciminc f l via the rela- Lionship B = (total internal trsflic)/(totaI external w a k ) [l I]. This is simply done by rcmrding thc rcsultant internal traffic along each link for all N(N - 1 ) sourdest inat ion pairs in the network.
Table 1: Number of nodes and mean path length as a func- tion of A and D
Table 1 shows 1hc dcrivcd mcan path lengths of the UdBG using the three proposed routing scherncs. Thc pcr- fonnance of SCP is for comparison purposcs. For small networks, thcrc is no difference in the mean path length among all routing algorithms since SCP has ;ilready achieved the shortest path. For larger networks, thcrc is improvetnent in RFK mid further irnproveinent in NSC. Furthcimorc, PMC produces shorter paths in most low dcgree cases, while in some higher degree u s e s the incan path lciiggth of PMC may be larger.
Table 2: Comparison among networks of the same size and same degree
Mean oath lengths I
No. of
nodesin TWO Bilayered BdBG net nemork superimposed ShufflaNets using NSC
Tablc 2 and Fig. 3 compare the dBG networks with the bilayei-ed ShulfleNcls. To makc Ihc coniparison more
fE/7 Pror,-<.'otn!iiuii., V d 147, hro. 1, I'ehriwry 2000
mncaninghl, we lime only chosen several exatnplcs in which the BdBG networks and the bilnycrcd ShuffleNets have the same degree and numbcr of nodes. Unlike other studies [ 121, the UclBG networks do iiot always have better per- formance than il ShuMcNct. Flowcvcr, our r d t s stlaw that BdBG networks Iiavc bctter mean path length per- formance and delay perforinancc. Natc also that the mean pi th lenglhs arc oblaincd by the NSC routing algorithm (not the best algorithm), wliilc those of bilayered Shuf- flcNcts arc obtaiiied by the shortest roilling algorithm alrcady.
7 P 50 -
- 2 40- %
90- 3 E 5 c 20-
0 0.05 0.10 0.1 5
3.2 Edge loading, delay and throughput performance The end-to-end delay h n i a soul-ce to a destination nodc consists mainly of the queueing dehys atid the propagation delays dong the path. In oui analysis, the propagation delay can be assuicd IO bc x r o because it is usutilly a con- stant. The nodc qucucing dclay, however, is intimately rclalcd IO the internal trafic of a network which in turn IS related to the routing algorithms. Let N bc h e number of nodes in the network, J the numbei of links, C Ihe capacity of a link in bit& and ]/,U the mean packet Icngth in bits, then using thc siinplXyying assumptioiis of an M/M/I model, the mcaii qiictiing dclay B for a picket in ii ncl- work, whcii iinrmalised with respect to ii pslckcl lranmis- sion time (pq-', is givcn [6, 1 1 J as
In eqti. 2, Li is thc cdgc load at link i, which is ihc amount of intcrnal traffic on link i in resgoiisc io cxtemal trafic
injected into the network. Note ttiat even if the oKcrd traffic to the BdBG nclwork is M y syinmctric (i.e. one unit of flow pci. sourcc-dcslination pair.) as assumed in eqn. 2, the edge loading is no1 iicccssarily cqual at each link in a dBG network [I].
A link i is stablc so loiig as the normalised traffic load lJpC < l/Lj and llic system stability limit in cqii. 2 c m also be identified from the rnaxiinuni cdgc load L, among all links in the network. Sinze network throughput is relaid to the maximuin throughput among thc links, it i s reasonable to dclinc the 'maximum normalised network throughput' as the reciprocal of the maximum edge load in ti network.
Table 3 gives thc maximum cdgc load and thc niaxiinum normalised network throughput of various (A, D) dBG nct- works. Tlic throughput improvement factors, defined as the ratio or Ilic throughput of B d W networks to UdBG net- works, show ha1 11ic BdHG nctworks liavc more than dou- bled the throughput even though the number of links is al most doubled. This tradeoff p i t i denionstrates ttic em- ciency of the BdBG networks. It is also observed that for nclworks with thc same degire A, the improvement factor increases with respxl to iiicrcasing iictwork diamcter D. Howcvcr, by holding nclwork diarnctcr IJ constant, increasing the nclwork dcgroc A docs 1101 ncccssarily pro- duce a better improvement. For cxainplc, lhc (2, 6) HdBG network hws an itnprovement fiictor of' 2.447 wliilc the (2, 5) BdBG iiclwork has only 2.250; however, the (4, 5 ) BdBG iictwork has a 2.210 improvemcnt factor which is worsc than ha1 or h c (2, 5 ) nclwork.
k ! !
ic k
I
I
.U 2 ,.....,....~..... . . .
ado2 o.im o i o e o.ios o.i io o.dia normalised affered load
Fig. 4 further camparcs RdBC nctworks will1 similar UdBG networks in lciins of Ihcir. iionnaliscd delays B,uC 21s it function or thcir normaliscd ofkrcd load #@U. Three
Table 3: Maximum edge load and throughput performance of UdBG and BdBG networks
groups of networks with (A, D) = (2, 6), (3, 5) and (4, 4), respectively, are presented. SCP routing is wxumed in BdBG nctwot.ks. . Along with llic observations from Tables 1 and 2, we established that the BdBG can fare bet- ter than their UdRG counterparts.
Fig. 5 iiscs ihc SCP as B reference to wniparc h c dclay performance of the thrcc proposed routing schemes in a (2, IO) BdBG network; it has I O 2 4 nodes which is about Ihc size of a metropolitan area nclwork. In the low offered- load region, the norinaliscd dclay achieved by euch routing algorithm is nearly the saiiic as tlic magnitude of the mean path length because it is tlic avcragc number of hops that pickets go through. As tlic load increases, it is intcresling to see that the RFR and NSC algorithms are more prornis- ing beuiuse they a n achieve relatively smaller delay biii higher throughput in the high offered-load rcgion. The inaximum throughput or thc RFR is better than the NSC, but its delay is aclually slightly larger than the NSC for most of the ofkrcd-load ranges. However, the PMC has a lower dehy than that or Ihc SCI’, but it offers the lowcst throughput aipabilily.
4 Hierarchical dBG networks
Since the dBG networks can support il large number of nodes with considerably short iiican path lengths, they are certainly alternatives as M A N structures. As the network grows in size, tlic dRG iictwork is also mpable of supporl- ing a hierarchical structure which is appe;iling to such con- sideriitions as traffic lwdity, implementation coinplcxity and cost.
local
00 100
clus
Y Y
To build a hierarchical d3G network, we cm first divide ii largc iictwork into smell clusters, each being a complete (AL, D3 dBG (UdBG or BdBG) network by itsclt‘(the sub-
36
script L indicates local cluslcrs). A particular node ‘z’, called the ‘bridgc nodc’, is chosen and interconnecled with the bridge nodes of otlicr clusters to form a (Au, Du) dBG network(the subscript U indimks an upper-level network). We shall iisc Ihc notation (A!,, DL)/(Al,, U,) to indicale a gciicral two-level network, ancl ‘xLJ for an address where its clnslcr addrcss is x, and its low1 node nddrcss is y . Fig. 6 is an example of a (2, 3142, 2) RdDG network with local addrwes, 100 chosen a s bridgc nodes. During rout- ing, ii node a n identify l0c;d t r d k by jus1 checking the first pail ‘x’ of the destindtion addrcss. If‘ ‘x’ does not malch [hat of the local cluster, the picket will lx routed to the bridge node ‘z ’ ; otlierwisc, thc packct will be directed to the local nodc ‘y’.
Dclinc the n o ( ~ ~ - ~ E e a a - p a l i i - l c ~ ~ ~ ~ f R bode 11 as thc avcr- age niinibcr or hops in opposite diimtions betwecn nodc i and all other nodes. Unlike othcr symnictric networks such as ShuffleNets, the paih lcngths in opposite djrcxtions arc not necessarily equal! and liencc H (node /9 is ti function of the node i location. It woiild bc dcsirablc to locate R bridge node with a bridge inem pal11 length of
/I = ipin (T(no$e i))
TO capture traffic locality that chi i f i ickks a large number of nclwork applications, we deliiic thc locsliiy factor CI to bc thc probability that both source desliiialion nodes of a mcssage are& the h ime cluster. Let m bc thc izumber of clusters, snd H I ;ind Z& be, rcspcctivcly, h c inean path lengths of the local cluslm arid thc second-level network. In the Appendix the incan path length of a two-level hierar- chical iie~ivork is shown to he
- a€ I\‘
- $1 = aF -I- 2 ( 1 - a)B + (I - &)% (3)
As shown in the Appendix, undcr unilbimily distributed traaffic
4.1 Hierarchical UCBG networks The determination of B in cqii. 4 is an interesting and challenging ask bccausc we have not been able to obt;iin a closcd-form cxprcssinn Tar the computation of the nodc- mean-pa,ath-lcngth, and therefore its minimum. Consc- qucntly, we resorted to II computer proogrm lo scarch each node to obtain its nodc-incaii-path-length.
Fig. 70 shows the distribution of tlic nuinher of nodes with respect to the notie-mcan-patklcngllis for a (2, 6) UdUG nctwork. In a total of 64 nodes, it is obscrvcd that H(011110) = H(10000I) = 4.141, which arc the siiiallest vdues in the network. Eilhcr oiic of thcsc two nodes m n be considered as the b d g c nodc. Fig, 711 shows the distribu- tion of node-nican-path-lcngtli in a (3, 4) UdBG network. Six nodes ’0112’, ‘0221’, ‘1002’, ’1220’, ‘2001’ and ‘2110’, have the Same ininimuin nicm path Icngths H . It is observed h a t in all UdUG nctworks with A 2 3, nodes with an a d d m pattcrn ‘xyy .., yz’ always h w e the miniinuni node-incan-path-lengths, where x, y , z are integers belong- ing to thc set (0, I, ..., A - I) and x + y # z. Similarly, il is found that in a11 binary (A = 2) UdBG networks, the nodes with address of all zeros or oms, cxccpl tlic first and the last digits, alwiiys have Ihc shortcst nodc-mean-path- lengths. This gives LIS a sirnplc guidcliric to choose an opti-
IEE P~OE~-COIIII I I I I I I . , vol. 147. XU. 1. i . i h , ~ , r ~ ~ 20DO
mum bridge node whcn constracting the hier:urchical UdBG networks.
nodemean path lsngth a
Wc ncxt consider ti 1024-notlc binary network srid vary the cluster sizc: NL = /Vim from 4 to 256 nodcs. Under the uniforim traffic condition, tlic locality factor cy ninges from 0.00293 to 0.249. l'able 4 shows Ihcir mean patti length pcrforiiisnce. The onelevel UtlBG nciwork has a shorter mean pith length than any other UdBG hiciwchical net- work, and this suggcsts that a single level is adcquate for UdBG nclworks that have uniform triillic pattcms. Fig. 8 shows explicitly the incan path length perhmancc as a function of thc locality Factor a. The relationship is lineur :IS expectd. Thc onc-lcvel (2, 10) network i s uscd as corn- parison. The locality and tlic pcrrormarice of the four nck works in the c ; ~ ol' unifoiin traffic (indicakd by an asterisk *) a1.e also included h r reference. Our analysis results show 1ha1 thc hierarchinil structures will cvcntually pdorni better thiiii a one-level iictwork if the ri exceeds :t
ccrtain value, e.g. a > 0.13 jii tlic (2, 4)/(2, 6) network. Fig. 9 shows the delay ;incl throughput performance or lhc (2, 4)/(2, G) UdBG network with diflcrcnt locality factors. Wc scc h a t when a increases, the nccwork performance is getting better, both in tcrins of dclay and throughput
Table 4: Mean path length in two-level hierarchical UdBG and BdBG networks with 1024 nodes in a uniform traffic case
First level/ No. of nodes No. of path length
second level per cluster clusters UdsG BdsG
9.383
8.397 8.665 9.188
9.892 10.680 1 1.405
11.793 8.377
7.565
7.097
7.148
7.598
8.438
9.479 10.478 11.138 7.547
0 ' I I I I 2.5 5 ~ 0 7 L i 0.0
5 normalised offered load (xlo )
Fig .9 Com/rcr,.i?on ~ 1 0 r g r l f l k n r /o~iiK}J$lctals in (1 (2, 4j/Q G j UdM; IwIlbTirk ~ c1 = 0.015 (iiliikirni 1r:ific)
( I = 0.8 . . . Ei = 0,4 _ _ _ _
4.2 Hierarchical BdBG networks Unlike the UdBG networks, our investigations have shown that there is no spccilic rulc to pick the desirable bridge localions in a hicraichical BdBG network becausc the best bridge location depends on the routing algotithims as well as tlic bit pattern. Our experience docs suggest that, if the nchvork i s laid out in B plane! the nodes closcr Lo Ihc centre are morc likely to have smaller node mean path lcngtli. We also use tlic RFR routing algorithm in our evaluation bccausc of its good throughput and dclay pcrroimance.
Fig. 10h shows R 729-node network witti a nodc degree of three atid with cluster s ix varying rrom 9-81, Just like the UdRG nctworks, as c1 inclrases the ineati path Icngths or Ilic liicrsrchicd BdBG networks decitasc lincat-ly and cvcntually becoine lower than for thc oii~lcvcl network. However, Fig. 1011 shows an intcrcshg ohscrvatioii for a 1024-node network, in thal Ihc entire mean patti Ici~glh performance line of thc (2, 2)/(2, 8) network lies under thc onc-lcvcl (2, 10) network. This Ineatis ttrat cvcn when there is no Imality at all, i.e. U = 0, the (2, 2)/(2, 8) network call still ncliicvc a shorter ineaii pi th length than thc onc-level countcrpart. This clearly demonstratcs thc advantage of
37
12.5 r
2
0' I I a
o.., .. . .o
I , I
I
0 2.5 5.0 7.5 10.D CI b
1-.
l2 I
wlieeti the number of clusters equals 256 (clustcr size = 4), tlic network has the optimum meail path iength pcrlbimi- ance, and this minimum is appirently indepeiidciil 01' a (similar obscrvations are iiiade for. the UdBG nehvorks be€ore).
Fig. 12 compares the delay and throughput perlbrmaiicc aiiiong the thi-ix routing algorithms. Since NSC has vcry similar performance to that of RFR, it is ornitled here for clarity. The (2, 2)/(2, 8) networks arc u s d and two locality factors are considered; CI = 0.00293 which corresponds to the uniform tratfic, and a = 0.8. Notc that the delq- throughput pcerformance has improved with respoct to locality. In both cases, not only the RFR routing algorithm can achieve the lower delay of lhc nctwork but also the higlier throughput perhrmancc. AIthough thc PMC has lower dclay in the low offered-load region, it also has H rel- ativcly lower throughput. By comparison, thc dclay- throughput improvement of RFR is higher for a largcr a.
m4r,.-..-.7-..:--..;i
0~0002 o.0004 0.0006 0 0
ii4 n
Experimcnls have also bccn pcrfonned by having differ- ent routing algorithms at diIfcrciil levels (c.g. RFR in the upper- and SCP in thc lowcr-level). In every combination, rcsirlh show tlial Lhc p~foimatlce is veiy close to that when the lower-level uscy thc sainc routing incthod as the upper-level (e.g. whai boll] lcvcls use RFR). However, as llic localily ractor becomes larger, the routing in the lower- lcvcl inay become more and more effective, Conscquenlly, one sliodd use methods that would achieve highcr throughput in thc uppcr lcvcl wliilc kccping the routing mctliod in the lower level as simple BY possible, unless tlic lower lcvel becomes more congested than the upper level. For cxample, since the PMC method has a worse through- puL perkmance, it should not be used in the second levcl where the traffic intensity may be tnudi higher than in tlic fii-st level,
5 Discussion
From the observiitions above, the SCP algorilhm is Lhc siinplcsi. Thc naturc or this algorithm effectively treats a BdBCi network as a 'UdBG network' with the wmc degrcc and same size by supxposing two UdBGs together. It can easily provide source routing capability by etnLxdding in the packet header the assigned direction (FP 01' BPI, and no further routing computation is required at intcrmcdiatc nodes later. In comparison, thc olhcr threc routing algo- rithms nrc self-routed and simple in the sense that a node can decide which neighbour it should send to by just look- ing at the destination uddress stored in the header of Ihc
IKR ~ r ~ r , - ~ . ~ ~ i i / } i ~ / ~ i , , Vol. 147, :Yo, 1. Fohrttwy 2iJOO
packet. The imphnentations can also be realised easily. Although hvo criteria air: provided in the PMC algorithm, any other criteria that cm correctly orient the packet mn dso be used.
We note here that the maxiinm dcgrcc of a RdBG neb work i s 2A which is doubb thal ol‘ a UdBG wliilc tlic min- itnum degree is 2A - 2 [8], Both types ol‘nclworks havc the same number {A”) of nodes. The diameter (the tnaximum hops between any node pairs) is also the same because the distance between two extremity nodes does not change with rcspccl to their unidirectional counterptirts. The number of links required is actually less than double bacause sotne nodcs in the UtlBG already have links in both directions hetween them.
When coiishwcling x gcncral iictwork, onc m y decide first on 1hc routing algorithm that gives an acceptable dclay-throughput perlbrmance (e.g. RPR in the binaiy nctwork}. Then one would use the derived rule of thumb to pick a bridge lordtion. When designing a metropolittln area network (MAN) with 1024 nodes, our performance evalua- tion on mean path Icngh and dclay-througlipul siiggcst that the (2, 2)/(2, 8) hierarchy may be the best choice.
It mdy be worthwhile to point out that, from a network design point of view, the de Btiiijn graph (dBG) networks m n be used U either a logical or a physical topology, and our analysis results are valid in both cases. I t is interesting to note that when used as 11 physicd topology, WDM is not uccessary because all connections are strictly point-to- point. This may be RII advantage when implementing ii large MAN becwse logical topology inay require a large number of wavelengths, and present technology may put a cap on the nunibcr of wavclcngths supportcd in R sharcd medium. Howcwr, the real physical dimension and the scalability 01‘ a rcgular network become ;i concern.
6 Concluding remarks
Multihop lightwave networks based on UdBG, BdBG and their hierarchical structures have been proposed and stud- icd in this paper. Thi-ee routing algorithms, RFR, NCS and PMC, have bcen proposed and studicd. For small ncl- works, all routing algflrithmr; can achievc thc shorlcst path. Therclbrc, onc inay simply use the SCP routing algorithm for its low complcxity and SOLITCC routing capability. For larger networks, lower mean path length rind edge load are desirable in obtaining lower delay. PMC has the shortest paths when network diameter i s small. However, in net- works with R longer dianietcr, the RFR and NSC arc rcc- ommendd For a vcry large m a covcragc, Iiicrarchical UdBG and BdBG networks with simple self-routing char- acteristics can bc uscd. This call dccrcasc tlic mcan pall1 length especially where locality may exist. Our previous experience has produccd guidclims Lo choose the bridge node for a local cluster. Our rcsults also show that nct- works with degree o f two appear to be the best choice. Not only has it the simpler hardware implementation, but also no locality is required to take x lmnhge of the hierarchical nctwork.
Network performances have been studied in terms of mcan path length, maximuin network throughput, delay, locality and the hierarchical capability. For BdBG net- works, RFR slid NSC appear to have a better throughput perI‘oimance than the PMC. In general, the larger the local- ity factor the better the performance of a clustering topol- ogy. Based on our pcrlormancc study, along wilh rcsulls comparing UdBG and bilaycrcd ShuMcNcts, we danon- strate that the BdBG networks can be alternatives to the MAN topology; they are plausible MAN candidates.
1!X Proc..Coinmirii.. Vol, !47, No, I , Febninry 2000
7 Acknowledgments
This work hiu been supported by a TRIO Research Assisl- antship and an NSERC Opcraling Grant under contract #OGP0042878.
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References
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Appendix: Derivation of the mean path length of two-fevel hierarchical networks
Under the uniform tr.afic assumption, each node sends a packct to cveiy other node in the network so [hat Lhe total offered load (external traffic) to the nctwork is N ( N - I) Tor a network of N nodcs. Although the real offered load of the nctwork may not be N(N - I), the corresponding change in thc intcrnal traffic will iesult in the uime mean path length sincc Ihc dihcrcnce will be cancelled out in the ratio. Lct A! = total number of nodes in thc nctwork VI = number of clusters
= mean path length of the local clustcr = mean path length of the uppcr lcvcl network
= mcan path length of the network = mean pdth length of the bridge nodc
(bridgcs)
B a = locality factor. Assuming all c luskr~ are identiwl, the mean path length of two-lcwl hicrarchiml networks can be derived as
inlcrnal tra.liic E external traffic
- H -
,mCinhrnnl traffic in one cluster + intemal traffic in the second level
( 5 ) - - N(N - 1)
39
For the second temi in the numerutor, we have
internal traffic in the sccorld level
=
= H a 4 (I - a ) . AT ' (A- - I) extemaJ traffic onerecl to t ~ i c second level
(6) -
For the first teiin in the numerator, wc have
irilernal trafic in m e clustcr
local traffic -t- remote traffic
: H~ - oi - C e x t s r n a ~ t r d i c to one cluster -
1 4-2 [B.(I - .)-E fixt,ernd traffic to one cluster
- N N m m = 11~ . a I - - ( N - 1) -+ 2 + B - ( 1 - a) . - ( N - I j
(7) The rcason for the factor two is that a incssage would go through i ls own cluster and then the cluster or die destina- tion node; thereforc, two bl-idg~mea~~-patli-lengths have to bc considered. Substituting eqns. 6 and 7 in eqn. 5, we gct
77 = *z + 2 ( l - @)E + (I - a ) Z (8) Eqn. 8 is i i s d to calculate the mean path lcngth of the hierarchical networks throughout this paper. Tf the traffic of lhc network is assuincd LO be uniformly distributed, i.e.
Nfm - 1 c y = N - 1
eqn. 8 becomes
I1 = (9) - (AT - . m ) K + N(n.t - 1) (273- -t H2 1
m(N - 1) Since wc can always chooseAbridge such that is less than 7&. When 3 is equal to H I , we can simplify eqn. 9 to
'(E+.) (10) - N ( m - I)
nz( AT - I) H=K+ II is obsei-ved that i fm >> I and N >> 1, cqn. 10 can fur- ther be simplificd to - 2% + s. Thc rcsult is obvious because on avcixgc, each message wodd have to go through its ow11 clustcr, the upper-level nctwork and the other cluster. Therefom, the total mean path length of the nelwork is simply the sun1 of the mean path Icngths of these three components.
