sible number of wavelengths. The wave~~ngt~ ~ ~ ~ c ~ n b~dwidth is set low eno~gh to interfax One of the most im

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1 sible umber of wavelegths. The wave~~gt~ ~ ~ ~ c ~ b~dwidth is set low eo~gh to iterfax vices. Oe of the most im ed trasmitters ad ysis much more CO "The author is also f Cumputer sciece Departmet, Uiversity of address: w~~cg.-.~u the geeral hardware cofiguratio. Two kid of $igle-hop lightwave etworks will be studied. Oe cotais self-loop /96 $ IEEE 486

2 at each ode. The other does ot cotais ay self-loop. The rest of this paper proceeds as follows. Sectio 2 describes a protocol for embeddig a regular graph ito the optical passive star whe each ode has multiple trasceivers. I sectio 3, we study the performace of the TWDM siglehop lightwave etworks with self-loops. I sectio 4, we study the performace of the "DM sigle-hop lightwave etworks without self-loops. Fially, sectio 5 cocludes this paper. 2. Cosecutive Partitio Assigmet Protocol The cosecutive partitio assigmet (CPA) protocol is proposed to embed regular digraphs ito the optical passive star coupler uder the hardware cofiguratio that each statio has multiple trasmitters ad multiple receivers. A regular digraph is a digraph where every ode has the same out-degree ad i-degree ad hece referred to as degree oly. Suppose that the regular digraph to be used as the virtual topology has degree d ad at each ode the outgoig liks ad icomig liks are sorted properly. Each ode a areassumedtohavettrasmitters{(a,t) lost IT-1) ad receivers ((a, r) I 0 5 r I - l}, where both T ad are factors of d. a ad t are called as the ode idex ad local idtrr respectfully of the trasmitter (a, t). a ad r are called as the ode idex ad local idex respectfully of the receiver (a, r). The the embeddig is performed as follows. First at each ode a, its outgoig liks are cosecutively partitioed ito T groups evely, wd all the liks i group t are assiged to the trasmitter (a, t) where 0 5 t 5 T - 1. Similarly, at each ode a, its icomig liks are also cosecutively partitioed ito groups evely, ad all the liks i group r are assiged to the trasmitter (a, T) where 0 5 r 5-1. The each lik i the virtual topology is implemeted by tuig the trasmitter ad receiver associated with the lik to the same wavelegth. The above embeddig ca be formulated by a trasmissio graph G(T, ). The trasmissio graph G(T, ) is a bipartite digraph. The vertex set of G(T, ) is the uio of the trasmitter set {(a,t) I ais avertexofthevirtudtopology,o 5 t 5 T - 1) ad the receiver set ((b,r) I bisavertexofthevirtualtopology,o _< r _< - 1). Each edge of G(T, ) is from a vertex(or trasmitter) i the trasmitter set to a vertex (or receiver) i the receiver set. Each trasmitter has # outgoig liks, ad each receiver has $ icomig liks. There is a oe-to-oe correspodece betwee the liks i the regular virtual topology ad the liks i the trasmissio graph. For ay lik a4b i the regular virtual topology, if this lik is the a'-th outgoig lik of a ad j-th icomig lik of b, the the correspodig lik i the trasmissio graph is I the trasmissio graph G(T, ), a set of trasmitters ad receivers form a compoet if there is a path betwee ay two of them assumig the edges i this bipartite graph are bidirectioal. I other words, forgettig the uidirectioal ature of the virtual^ lik betwee a trasmitter ad a receiver, a compoet is ib coected compoet i graph theoretic termiology. I each compoet, a wavelegth ca be assiged startig at ay trasmitter (receiver). The all receivers (trasmitters) coected to this trasmitter (re ceiver) are forced to receive (trasmit) at this wavelegth. Cotiuig i this maer, we ed up with all trasmitters ad receivers withi ai compoet assiged to the same wavelegth. Thus we have the followig lemma. Lemma 1 All trasmitters ad receivers costitutig a compoet i the trasmissio graph are assiged to the same wavelegth. The w'mum umber of wavelegths that ca be employed, Wmax, is equal to the umber of compoets i the trasmissio graph. The above lemma provides a approach to fid Wmax, the maximum umber of wavelegths that ca be employed. It should be poited out that if the umber of wavelegths actually available, W, is k~ tha Wma,, we may allow SVeral compoets to share a wavelegth to reduce the umber of wavelegth required. Fbr the trasmissio schedule with ay give umber of wavelegths, the reader is referred to m. Whe max(t, ) = d, the umber of compoets oly depeds o the etwork size ad mi(t, ) rather tha the topology. To be specific, we have the followig lemma. Theorem 1 Suppose thai'max(t, ) = d, the Wmax = N mi(t, ) where N is the umber qf odes i the etwork. Proof We cosider the followig three possible cases. Case 1. T = = d. I this case, each trasmitter coects to oly oe receiver, ad each receiver coects from oly oe trasmitter. Therefore, each compoet cotais oly oe trasmitter ad cly oe receiver. So Wmax = Nd. Case 2. < T = d. I this case, each trasmitter coects to oly oe receiver, ad each receiver coects from $ trasmitters Therefore, each compoet cotais oly oe receiver ad 4 itrasmitters. So W, = N. Case 2. T < = - 1. I this case, each receiver coects from oly oe xeiver, ad each receiver coects 487

3 t So w~eev~ the CPA (T, ) < d. We first briefly revisit ode 0 5 Q 5-1, its i-th outgoig Lik is d its i-th icomig lik is a --+ (a+ 1 +i) mod, (Q i) mod 12 -b Q, where0 5 i < - 1. se that each ode has T trasmitters ad re- Q + i, ceivers where both T ad a~ factors o. The i the trasmissio graph uder the CPA embeddig protocol, the d its i-th icomig lik is set of receivers a trasmitter (a, t) coects to is i -+ Q, i a-1-1 {((a+l+i) mod, L x J ) I t- I i 5 (t+l)--l}, - T T ad the set of trasmitters a receiver (b, r) coects from is the set of trasmitters a receiver (b, r) coects from is We deote by r the least commo multiple of %j$ ad - Let 1c For ay 0 5 t < T ad 0 5 r <, let At,, = {(r- + i,t) IO 5 i < -}, Bt,r = {(t5; + j, r) I Q 5 j < T}* The the mai result for the sigle-hop lightwave et- fore, Wmax = T. Proof It's easy to veri y &at ad0 < r <, the receivers i the set Bt,r mkio graph. There-. m The -_--- T TI I m ' The mai result for the sigle-hop lightwave etworks without self-loops ca be stated i the followig theorem. 3 Zfmax(T, ) = - 1, the Zfma~(T, ) < - 1, the Wmax = mi(t, ). ects to i the tras h receiver i Bt,, co- Whe max(t, ) = - 1, the theorem ca followed h m Theorem 1. So i the remaiig of this sectio, we assume that T, < - 1. It'seasy to show for ay 0 5 t < T ad 0 5 i < 9, h. This implies that Wmax = T. 488

4 ad for ay 0 5 r 5-1 ad0 5 i 59-1, Therefore, for ay compoet there exists a uique iteger 05 k.e ~suchthatforaytrasmitter(a,t)adreceiver (b, r) i this compoet, 1-J t = 1-J TI For ay 0 5 k , let ' r = k. t Ak = {(u,t) 10 5 a 5-1,L-J T' Bk = {(b,r) r IO 5 b 5-1, L.?z;] = k), = k}, Wewillprovethatforayo 5 k < %,thetrasmittersi Ak ad the receivers i Bk are i the same compoet. To prove this, we first give the followig lemma, Lemma2 Foray0 5 a 5-1, 1. if t mod T' = 0, the the WO trasmitter (a, t) ad ((U + 1) mod, t) are i the same compoet; 2. if r mod ' = 0, the the WO receivers (6, r) ad ((b- 1) mod, r) are i the same compoet. Ptoof. (1). Cosider the followig two liks. ad IftmodT'=O,the(t%j$)modq =O. As% > 1, = 1%~. This meas that the two trasmitters 7I- T (a, t) ad ((U + 1) mod, t) coects to the same receiver ad therefore are i the same compoet. (2). The proof is similar to (1). 0 From the above lemma, we ca immediately have the followig corolloary. Corollary 1 If t mod T' = 0, the all trasmitters with local idices t are i the same compoet. If r mod ' = 0, the all receivers with local idices r are i the same compoet. Lemma3 ForayO<a<-1, I. if t mod T' > 0, the the MO t"itters (a, t) ad ((a + 1) mod, t -. 1) am i the same compoet; 2. if r mod ' > 0, the the WO receivers (b, r) ad ((b + 1) mod, r -- 1) are i the same compoet. Proof (1) The lemma is mivial whe T' = 1. So we assume that T' > 1. Cosider the followig two liks. ad -1 t+-1 ((a+l) mod, t-1) - ((a+l+tt) mod, L-;-?-J).- Ift mod T' > 0, the (t q?) mod > 0, which implies that~s~ = L? ~ J. 7t- 71- Thismeasthatthetwoaasmit- ters (a, t) ad ((Q + 1) mod, t - 1) coects to the same receiver ad therefore are i the same compoet. (2). The proof is similar to (1). 0 Now we ca prove the followig lemma characterizig the copoets i the trasmissio graph. the trm'tters i A& ad the receivers i Bk am i the same compoet. Lemma 4 For ay 0 _< k c e, Proof From Corolary 1, all trasmitters with local idices kt' are i the same compoet. Furthermore, by Lemma 3, all trasmitters i Ak are i the same compoet. By similar argumet, all receivers i Bk are also i the same argumet. As we metioed at the begiig of this subsectio, the set of receivers ay trasmitter i Ak coects to are i Bk, ad the set of trasmitters ay receiver i BE coects from are i Ak. So the trasmitters i Ak ad the receivers i Bk are i the same compoet. 0 The above lemma impl~ies that w, = 9. Theorem 3 are true whe max(t, ) < Coclusio Therefore, I this paper, two kidls of TWDM sigle-hop lightwave etworks are studied. Oe is with self-loop. The other is without self-loop. For both kids of etworks, the maximal cocurrecy that car1 be achieved whe multiple fixed trasceivers are used at each statio. The embeddig protocol used i this paper is the CPA embeddig protocol. Oe iterestig problem wheither or ot there is ay other embeddig scheme such that the TWDM sigle-hop lightwave etworks ca have higher maximal cocurrecy. Fially, the author wish to thak Prof. Dig-Zhu Du for his kid ad isightful surggestios. 489

5 . Du. A media-access protocol isio mu~tiplex~ passive star etworks. T ~ ~ c a Computer l ~ Sciece ~ Dept., ~ [8]. Like. Frequecy divisio mu~tiple~~ optical et-. ~ ~ a s W ~. works based o de hij graphs. 1011,1990. versity of M ~ e s1996. o ~ [ 111 K. W~~ ad D. Du. Time ad waveleg~d~isio~ multitectures for optical passive 92-44, Computer Sciece 490