Design of Light-Tree Based Logical Topologies for Multicast Streams in Wavelength Routed Optical Networks

Transcription

1 Desig of Light-Tree Based Logical Topologies for ulticast Streams i Wavelegth Routed Optical Networs De-Nia Yag ad Wajiu Liao Departmet of Electrical Egieerig Natioal Taiwa Uiversity Taipei, Taiwa wjliao@cc.ee.tu.edu.tw Abstract this paper, we formulate a optimizatio problem for the desig of light-tree based logical topology i Wavelegth Divisio ultiplexig (WD) etwors. The problem is comprised of two parts: (1) multicast routig ad wavelegth assigmet of light-trees, ad (2) the desig of light-tree based logical topology for multicast streams. the first part, we use ixed teger Liear Programmig (LP) to solve the optimal routig ad wavelegth assigmet problem of light-trees with a ed-to-ed delay boud, ad obtai the optimal placemet of power splitters ad wavelegth coverters. The umerical results show that etwors with just a few power splitters ad wavelegth coverters ca efficietly carry multicast data. the secod part, we exted the above formulatio to desig the logical topology based o light-trees for multicast streams. our approach, a light-tree ca carry data of multiple multicast streams, ad data of a multicast stream may traverse multiple light-trees to reach a receiver. The umerical results show that our approach use etwor resources more efficietly, as compared to the approach with a separate light-tree for a multicast stream ad to the approach of trasportig multicast streams over lightpath based logical etwors. Keywords- ixed teger liear programmig (LP); WD; multicas; light tree; logical topology. NTRODUCTON Optical etwors based o Wavelegth Divisio ultiplexig (WD) are the most promisig cadidates for the ext geeratio bacboe etwors. a wavelegth-routed etwor, data are trasported i all-optical WD chaels referred to as lightpaths. The set of lightpaths forms the logical topology of a WD etwor. Data are processed electroically oly at each ode of a logical topology (i.e., at the edpoits of the lightpath), ad switched optically at itermediate odes of the uderlyig physical etwor. Desig of logical topologies has bee studied i recet years [1-4], ad the goal is to miimize the cogestio (i.e., the load o each chael), or the resources used i physical etwors. ulticast is a efficiet way to trasport the data of oe-tomay ad may-to-may applicatios. Supportig multicast i logical etwors is less efficiet tha physical etwors sice the latter provides higher coectivity. [-]. To support multicast i the WD layer, Sahasrabuddhe ad uherjee [6] itroduce the cocept of the light-tree, which is a poit-tomultipoit extesio of a lightpath. Brachig odes of a lighttree are equipped with power splitters. t is widely believed ad desired that oly a portio of the etwor odes are required to be equipped with power splitters due to their expesive cost. This id of etwors is called a sparse-splittig etwor. compariso with multicastig i logical etwors, trasportig multicast streams o light-trees uses less umber of wavelegth chaels 1, optical trasmitters, ad receivers [6]. However, trasportig each multicast stream o a differet light-tree is also iefficiet if the data rate of a stream is much lower tha the data rate provided by a wavelegth chael. this case, it is more efficiet to multiplex multiple multicast streams o a light-tree. Sice the seders ad receivers of differet streams may be differet, a multicast stream may eed to be trasmitted o multiple light-trees to reach all the receivers. other words, data of a multicast stream may eed to traverse multiple lighttrees to reach each receiver. To provide such service efficietly, the followig three issues eed to be addressed: (1) to determie the destiatios of each light-tree, (2) to determie the routig ad wavelegth assigmet of each light tree, ad (3) to determie the set of light-trees used by a multicast stream, provided that the set of multicast streams to be trasmitted is give. terestigly, this problem is just equivalet to desigig a light-tree based logical topology for multicast streams. The light-tree based logical topology is a hypergraph [], i which each li of the logical topology is a hyperedge. Here a hyperedge is a li by which more tha two odes ca be coected. Each hyperedge represets a light-tree i WD etwors. ixed teger Liear Programmig (LP) is a popular techique used to solve optimizatio problems. Previous wor based o this techique for WD etwors maily focus o the followig problems: Routig of light-trees. Optimal routig for multicast streams i pacet etwors has bee discussed i [11]. However, this formulatio caot be applied directly to WD etwors because it assumes that every ode is 1 A wavelegth chael meas a edge o a light-tree.

2 capable of multicastig. The formulatio i [12,13] obtais the optimal routig for light-trees i sparsesplittig optical etwors. However, the umber of variables or costraits grows expoetially with the etwor size, i.e., the umber of odes ad lis i a etwor. As a result, their formulatio is ot practical for large etwors. The formulatio i [14] uses less umber of variables ad costraits for the same problem. Note that [12-14] do ot cosider wavelegth assigmet i the etwor. Oly the routig problem is discussed. Desig of logical topologies. Formulatio to desig the light-path based logical topologies for uicast streams are provided i [1-4]. ellia et al. [1] proposes a formulatio to desig the logical topologies for multicast streams. Both approaches use lightpaths, istead of light-trees, to form the logical topologies. oreover, they assume that either all odes are capable of wavelegth coversio, or o ode is capable of wavelegth coversio. Optimal placemet of power splitters ad wavelegth coverters. Both power splitters ad wavelegth coverters are cosidered scarce resource, ad should be allocated carefully [13,16-18]. Ali ad Deogu [13] formulate a optimizatio problem to allocate power splitters i WD etwors. However, this formulatio is ot scalable because the umber of variables ad costraits agai grows expoetially with the etwor size. Subramaiam et al. [16,18] propose a aalytical method to aalysis the blocig probability of uicast traffic with differet umber of wavelegth coverters i physical etwors. A dyamic programmig method is proposed to fid the optimal placemet of wavelegth coverters. Xiao et al. [17] icorporates Liear Programmig (LP) formulatio ito their algorithm to determie where to place wavelegth coverters i physical etwors for uicast traffic. Light-tree based logical topologies have bee studied i [19,2] usig heuristics. A special id of hypergraph, Kautz hypergraph, is used to desig the logical topologies. O routig ad wavelegth assigmet of a hyperedge, the authors assume that each physical ode is capable of both wavelegth coversio ad power splittig. this paper, we study the desig of optimal light-trees based logical etwors for multicast streams i WD etwors. The problem is comprised of two parts: (1) multicast routig ad wavelegth assigmet of light-trees ad (2) the desig of light-tree based logical topology for multicast streams. the first part, we use LP to solve the multicast routig ad wavelegth assigmet of light-trees. The umber of odes capable of power splittig or wavelegth coversio is give as a parameter (i.e., ow a priori). Our formulatio has the followig characteristics: Ed-to-ed delay of a light-tree is bouded. Optimal placemet of wavelegth coverters ad power splitters is obtaied. the secod part, we exted the above formulatio to desig the logical topology based o light-trees for multicast streams. The light-tree based logical topology is a hypergraph. Our formulatio has the followig characteristics: Data carried o the logical topology are multicast streams. The ed-to-ed delay of each multicast stream is bouded. The routig of each multicast stream i the logical topology is obtaied. A commo problem of LP formulatio is the difficulty i requirig some costraits to hold oly uder a certai coditio. For example, the flow coservatio rule states that the amout of flows icomig to a ode is the same as the amout of flows outgoig from the ode. This rule holds for a ode oly if the ode is ot capable of power splittig. t is hard to mae the costrait liear if the capability of a ode is to be decided (e.g., whether a ode should be equipped with power splitters or wavelegth coverters). To solve this problem, we icorporate a large umber, say, ito the formulatio. our approach, for the ode which does ot follow the flow coservatio rule, will force the costrait to always hold. This techique eables LP to formulate more complicated scearios for WD etwors, while the umber of variables ad costraits oly icreases slightly. We will demostrate how to use this techique i the ext sectio. The rest of the paper is orgaized as follows. Sectio, our formulatio is preseted ad explaied i detail. The formulatio to solve the routig ad wavelegth assigmet of light-trees is preseted first. The, we exted the formulatio to solve the logical topology desig problem. Sectio, umerical results obtaied by our formulatio for a example etwor are show. Sectio V, we discuss the complexity of our formulatio. Fially, we coclude this paper i Sectio V.. PROBLE FORULATON The problem to be studied is comprised of two parts: routig ad wavelegth assigmet of light-trees, ad logical topology desig. We assume the physical topology, the umber of power splitters ad wavelegth coverters i the etwor, ad the umber of optical trasmitters ad receivers at each ode are give as parameters. We assume that the power loss of power splittig ca be eglected because optical amplifiers are used. We also assume that each wavelegth coverter is a full rage wavelegth coverter [16], i.e., each wavelegth o a chael ca be coverted to ay other wavelegth. The otatio is defied as follows. V the set of all odes i a etwor; E the set of all lis i a etwor; Γ the set of light-trees to be created; H the set of multicast streams; Λ the set of wavelegths supported i a fiber; () the set of lis icidet to ode, i.e., is the si of the lis, V ;

3 Out() the set of lis icidet from ode, i.e., is the source of the lis, V ; T the set of light-trees rooted at ode, V ; Up(e) the trasmittig ode of li e, e E ; D(e) the receivig ode of li e, e E ; a wavelegth i a fiber, Λ ; t a light tree, t Γ. ultiple fibers o a li are supported i our formulatio. We assume that the set of wavelegths that a fiber ca support is the same for all fibers. There is oe source ad multiple destiatios o a light-tree. We defie a icomig chael of a ode as a wavelegth chael of a fiber o a li which belogs to (). Similarly, a outgoig chael of is a wavelegth chael of a fiber o a li belogig to Out(). We also defie the outgoig wavelegths of used by lighttree t as the wavelegths assiged to outgoig chaels of which are used by t. For a light-tree, we call m a upstream ode of if there is a path from m to o the light-tree. the above case, we ca also call a dowstream ode of m. A. Routig ad Wavelegth Assigmet of Light-trees We use LP to fid (1) the optimal routig ad wavelegth assigmet of light-trees with a ed-to-ed delay boud, ad (2) the optimal placemet of power splitters ad wavelegth coverters i a WD etwor. The parameters which are give as a part of problems are deoted as follows. N S the umber of power splitters i a etwor; N C the umber of wavelegth coverters i a etwor; Rx the umber of optical receivers at ode ; Tx the umber of optical trasmitters at ode ; F e the umber of fibers o li e; P the propagatio delay of li e; we assume e P e > for e E ; PT t the ed-to-ed delay boud of light-tree t; D if ode is a destiatio of light-tree t, D = 1; otherwise, D = ; a costat used to eforce coditioal rules to the costraits; the value of is set as follows. = max Λ PTt, E Fe, Pe (1) t Γ e E We ote that the umber of light-trees i which a ode participates must be equal to or less tha the umber of optical receivers at. The variables to be decided are defied as follows. the umber of fibers of li e i which wavelegth e, is used by light-tree t. must be a positive e, iteger or ; J if ay wavelegth of a fiber o li e is used by e light-tree t, J = 1 ; otherwise, J = ; e e Q the ed-to-ed delay from the source of light-tree t to ode, Q ; Φ if ode is capable of power splittig, Φ = 1 ; otherwise, Φ = ; Θ if ode is capable of wavelegth coversio, Θ = 1 ; otherwise, Θ = ; G if light-tree t uses wavelegth of ay fiber icidet from the source of t, G = 1 ; otherwise, G = The objective fuctio is formulated as follows. e, t Γ e E Λ miimize (2) Similar to [14], this objective fuctio is to miimize the resource, i.e. the umber of wavelegth chaels, used by all light-trees i a WD etwors. The goal of most existig wor [-9] is to maximize the umber of light-trees which ca be created i the etwor. Note that our objective fuctio is highly correlated with this goal because more light-trees ca be created as the resource used by all light-trees decreases. oreover, some other objective fuctios proposed i existig wors ca be used as the objective fuctio i our formulatio. For example, with some mior modificatio to our formulatio, the followig two objective fuctios ca be achieved: (1) to miimize the umber of optical trasmitters ad receivers i a etwor, i.e., the objective fuctio i [6] ad (2) to miimize the overall profit of creatig light-trees, i.e., the objective fuctio i [12,13]. The set of costraits is show as follows.. 1) Routig ad wavelegth assigmet costraits: D + t Γ e, Fe, e E, Λ (3) 1 e, J e, e E, Γ (4) Λ ) Λ e, e ( ) Λ e,,, Γ, t T the covetioal multi-commodity flow assigmet problem, a flow coservatio costrait is required to esure that the amout of outgoig flows of a ode must be equal to the amout of icomig flows if the ode is either the source or a destiatio. optical multicast etwors, this costrait does ot hold because the flow, i.e., optical sigal, ca be duplicated. We use eqs. (3)-() to mae sure that there is a path from the source to each destiatio o the light-tree. Eq. (3) esures that each wavelegth chael of a li is used by at most F e light-trees. Eq. (4) guaratees that J e equals oe if at least oe wavelegth chael of a fiber o li e is used by light-tree t. f ode is ot the source of t ad at least oe of the followig coditios holds, ()

4 is a destiatio of light-tree t at least oe outgoig chael of is used by t the eq. () guaratees that t uses at least oe icomig chael of. Sice the icomig chael of must be a outgoig chael of aother ode, say, m, eq. () also esures that at least oe icomig chael of m is used by t if m is ot the source of t. Therefore, there must exist a path from the source of t to, or ad m are i a loop, i.e., is also a upstream ode of m. The delay costraits specified later will guaratee that there is o loop i ay feasible solutio. 2) Node costraits: ) Λ e ( ) e, e, ) + D ( Φ ) +, Γ, t T ( Φ + Θ ), Γ, t T e, ) + e ( ) Λ e, e ( ), Λ, e, e, Θ,, T, t T, Λ Φ Θ N N S C, (6) (7) (8) (9) () Eqs. (6)-() esure that the result will ot cotradict with the capability of a ode. We explai i detail the above costraits i the followig four cases: Φ = ad Θ = : the wavelegth assiged to a icomig chael ca be either coverted to ay other wavelegth for a outgoig chael or split ito multiple outgoig chaels. For all icomig/outgoig chaels of ode usig wavelegth, eq. (7) guaratees that for ode, the umber of icomig chaels used by light-tree t must be equal to or larger tha the umber of outgoig chaels used by t if is ot the source of t. Eq. (6) states that ode eeds oe more icomig chael if is a destiatio of t. Eq. (8) holds if eq. (7) is satisfied. Φ = ad Θ = 1 : the wavelegth assiged to a icomig chael ca be coverted to ay other wavelegth but caot be split to multiple chaels. Because ode is icapable of power splittig, eq. (6) esures that the umber of icomig chaels of used by t must be equal to or larger tha the umber of outgoig chaels of used by t. Oe more icomig chael is eeded if is a destiatio of t. Sice ode is capable of wavelegth coversio, eqs. (7) ad (8) are uecessary. Here maes eqs. (7) ad (8) always hold for ay assigmet of values to the variables. Φ = 1 ad Θ = : the wavelegth assiged to a icomig chael ca be split but caot be coverted. For all icomig/outgoig chaels of ode usig wavelegth, eq. (8) guaratees that at least oe icomig chael is used by light-tree t if ay outgoig chael is used by t. Sice ode is capable of power splittig, eqs. (6) ad (7) are uecessary. Here maes eqs. (6) ad (7) always hold for ay assigmet of values to the variables. Φ = 1 ad Θ = 1 : maes eqs. (6)-(8) always hold for ay assigmet of values to the variables. Eqs. (9) ad () esures that the umber of odes equipped with power splitters or wavelegth coverters must be less tha or equal to N S or N C, both of which ca be give accordig to the budget of a etwor provider. 3) Delay costraits: Q =,, T (11) Q + P Q Up( e) e D( e) 1- J e, e E, Γ (12) Q PTt D + 1,, Γ (13) Eqs. (11)-(13) esure that the delay of the path from the source of a light-tree to each destiatio must be equal to or less tha the delay boud. Eq. (11) states that the delay from the source of a light-tree t to a ode equals zero if is the source of t. Eq. (12) esures that if li e from ode m to is used by t, the the delay from the source of t to is equal to or larger tha the delay from the source of t to m plus the propagatio delay of e. f e is ot used by t, i.e., J e =, maes eq. (12) always hold. Eq. (13) guaratees that the delay from the source of t to must be equal to or less tha the delay boud if is a destiatio of t. Eq. (12) also esures that there is o loop i ay feasible solutio. f there is a loop ad odes m ad are both i the loop, the delays from m to ad to m must both be zero, which cotradict our assumptio that the propagatio delays of all lis are larger tha zero. With the delay costrait ad the routig ad wavelegth costraits specified above, there must exist a path from the source of a light-tree to each destiatio. With the objective fuctio, redudat paths ca be prued. Therefore the optimal solutio must be a tree. 4) Optical trasceiver costraits: ) t T Λ e, ) G e, Tx + Φ,, T,, Λ (14) (1)

5 t T Λ G t Γ Tx D + Θ, Rx, (16) (17) Eqs. (14)-(17) esure that the umber of optical trasmitters ad receivers used by a light-tree must comply with the capability of the source ad destiatios, i.e., whether the source ad destiatios is capable of power splittig ad wavelegth coversio. We explai i detail the above costraits i the followig four cases: Φ = ad Θ = : i this case, the umber of optical trasmitters used by all light-trees rooted at is equal to the umber of outgoig chaels of used by all light-trees. Eq. (14) guaratees that this umber must be equal to or less tha the umber of optical trasmitters at. Eqs. (1) ad (16) also hold if eq. (14) is satisfied. Φ = ad Θ = 1 : the umber of optical trasmitters eeded by a source is equal to that i the first case. Φ = 1 ad Θ = : Sice ode is capable of power splittig, each outgoig wavelegth requires oly oe optical trasmitter. maes eq. (14) always hold for ay assigmet of values to the variables. Eqs. (1) ad (16) esure that the umber of outgoig wavelegths of the source must be equal to or less tha the umber of optical trasmitters i the source. Φ = 1 ad Θ = 1 : maes eqs. (14)-(16) always hold. The umber of light-trees rooted at ode must be equal to or less tha the umber of optical trasmitters at ode. We assume that the give parameters always satisfy this costrait. Eq. (17) states that the umber of light-trees i which a ode participates must be equal to or less tha the umber of optical receivers at. Sice both D ad Rx are parameters, here we assume this costrait always holds. Later i the ext sectio, we will eed this costrait because D becomes a variable. B. Desig of Light-Tree Based Logical Topologies this sectio, we exted the formulatio described above to solve the logical topology desig problem. A logical topology is a hypergraph, i which each ode represets a switch capable of electroic processig. Each hyperedge represets a light-tree i the physical etwor. A ode icapable of electroic processig ca t be the root of a lighttree. We use the formulatio of light-trees described i the last sectio to fid the hyperedges for the logical topology. The traffic over the logical topology is composed of multiple multicast streams to which a ed-to-ed delay boud is guarateed. Each multicast stream has a seder ad several receivers. We assume that the propagatio delay domiates the ed-to-ed delay because the li ca be maitaied relatively lightly utilized by settig λ max properly, leadig to egligible queueig delays. The goal of this sectio is to fid the optimal desig of logical topologies ad the optimal routig of multicast streams usig LP. Some ew parameters are defied as follows. j a multicast stream, j H ; λ j the data rate of multicast stream j; λ the data rate supported o a wavelegth chael; max R if is a receiver of stream j, R = 1; otherwise, R = ; S if is the seder of stream j, S = 1; otherwise, S = ; PS j the ed-to-ed delay boud of stream j; Some ew variables are defied as follows. j ) B if the data of multicast stream j are carried o light- tree t, B = 1; otherwise, B = ; ( t, Z if ode is a destiatio of light-tree t, ad the data of multicast stream j are carried o light-tree t, Z = 1 ; otherwise, Z = ; Y the ed-to-ed delay from the seder of stream j to ode, Y ; D, as a parameter i the last sectio, becomes a variable i this sectio. oreover, variable PT t ad the costrait i eq. (12) are of o use i this sectio. Here the value of should be set as follows. = max Λ Γ PS j, E H Fe, Pe (18) j H e E The objective fuctio may either be to miimize the cogestio [1,3] or to miimize the average pacet hop distace [2]. For the former case, λ max is set as a variable. For the latter case, additioal variables represetig the data rate of each stream carried o light-tree t trasmitted o each li are required. this paper, we use the same objective fuctio as i the previous sectio (i.e., eq. (2)) i order to compare the followig three desig scearios fairly. Each multicast stream is carried o a differet lighttree. The problem is just the routig ad wavelegth assigmet of light-trees as discussed i [7-9,12-14]. Optimal solutio ca be obtaied usig formulatio i previous sectio. ulticastig is oly available o logical etwors. Each ode of a logical etwor is capable of multicastig. Each edge of a logical etwor is a lightpath. t is the problem as discussed i [1]. ulticastig is available both o physical etwors ad logical etwors. A light-tree ca carry data from multiple multicast streams. t is the focus of this sectio.

6 The set of costraits is listed as follows. 1) ulticast routig costraits: R Z t t Γ t T :, R,, j H (19) B + D 2 Z,, Γ, t T, j H (2) 1 ( 1 S ) B ( 1 S ) Z,, j H ( (21) t T t: t Γ, t T Eqs. (19) to (21) esure that data of each multicast stream will pass through oe or several light-trees to each receiver. Eqs. (19) to (2) guaratee that each receiver of stream j must be a destiatio of a light-tree which carries the data of j. Eq. (21) esures that the source of the light-tree which carries the data of stream j must be a destiatio of aother light-tree which also carries the data of stream j, or the source must be the seder of stream j. Similar to the routig ad wavelegth assigmet costraits i the last sectio, there must exist a path from the seder of j to each receiver, or the receiver must be i a loop. The delay costraits specified i the ext paragraph will esure that the latter case will ot exist.. 2) Delay costraits: Y Y R S m Y + Q =,, j H ( 1 Z ), m,, m, T, j H ( PS Y ),, j H j m (22) (23) (24) Eqs. (22) to (24) esure that the delay of the path from the seder of a multicast stream to each receiver must be equal to or less tha the ed-to-ed delay boud of the stream. Eq. (22) states that the delay from the seder of stream j to a ode equals zero if is the seder of stream j. Eq. (23) esures that if ode is a destiatio of light-tree t rooted at m, ad t carries the data of stream j, the the delay from the seder of stream j to must be equal to or larger tha the delay to ode m plus the delay of the path from m to. Eq. (24) guaratees that the delay from the seder of stream j to each receiver must satisfy the ed-to-ed delay boud. 3) Rate costraits: j H λ B λ Γ (2) j max, Eq. (2) esures that the data rate carried o each light-tree must be equal to or less tha the maximum data rate which a wavelegth chael ca support. NUERCAL EXAPLES our experimets, the CPLEX optimizatio pacage [21] is used to obtai optimal solutios. We use a 14-ode NSFNET bacboe etwor as the topology of our umerical examples (as show i Fig. 1). The umber beside each ode idicates the D of the ode. The umber beside each li shows the propagatio delay (i millisecod) of the li. We add some auxiliary costraits to improve the speed of fidig optimal solutios. our experimets, there is oly oe fiber o each li. The goal is to fid the miimal umber of wavelegth chaels i each experimet. oreover, the miimal umber of trasmitters plus receivers is obtaied by slightly modificatio of eq. (2) 2. The first part of our experimets is to fid the optimal routig ad wavelegth assigmet of light-trees uder differet etwor parameters. The etwor parameters are listed as follows: the umber of power splitters; the umber of wavelegth coverters; the umber of wavelegths supported i a fiber; the umber of light-trees to be created; the umber of destiatios i a light-tree. Fig. 2 shows the results of differet umber of power splitters whe the umber of destiatios i a light-tree is chaged. There are two light-trees rooted at odes 2 ad 11, respectively. The ed-to-ed delay boud of each light-tree is ms. The destiatios are chose radomly. Each fiber supports five wavelegths. The etwor supports oe wavelegth coverter. The results of more tha oe wavelegth coverters are similar. The results of light-trees with more destiatios use more etwor resources. Besides, a small umber of power splitters (e.g., splitters) is eough to reduce the amout of resources used by the two light-trees. With more power splitters, the umber of wavelegth chaels, optical trasmitters, ad receivers reduced by addig oe more power splitter decreases. We ote that a light-tree with more destiatios beefits from more power splitters because more etwor odes ca be used as splittig odes of the light-tree. With less power splitters, a separate lightpath has to be created from a further splittig ode or the source to a destiatio. Table shows the placemet of power splitters ad wavelegth coverters i optimal solutios whe each lighttree has 13 destiatios. Table focuses o the amout of resources used by the two light-trees with 11 ad 13 destiatios, respectively. Some experimets produce o feasible solutios. With more destiatios, the etwor requires more power splitters, wavelegth coverters, or more umber of wavelegths supported i a fiber to fid feasible solutios. With more power splitters ad wavelegth coverters, it is easier to obtai optimal solutios with less umber of wavelegths supported i a fiber. The reaso is that a light-tree i a etwor with more power splitters uses less umber of wavelegth chaels. With more wavelegth coverters, more cadidate wavelegths i a fiber ca be selected by a light-tree. 2 The total umber of trasmitters ad receivers is added to eq. (2). However, its coefficiet is much smaller tha oe to esure that the optimal umber of wavelegth chaels is ot chaged.

7 Figure 1. A 14-ode NSFNET etwor TABLE OPTAL PLACEENT OF POWER SPLTTERS AND WAVELENGTH CONVERTERS N S Splitter Coverter , ,8, ,,6,9 6 2,4,,9, ,,6,8,9, 4 7 2,4,8,9,,11,13 4 Fig. 3 shows the results of differet umber of power splitters whe the umber of light-trees is chaged. The source ad destiatios of each light-tree are chose radomly. average, there are six destiatios i a light-tree. Each fiber supports four wavelegths. The etwor supports oe wavelegth coverter. The results of more tha oe wavelegth coverters are similar. The results of more light-trees use more resources. The proportio of etwor resources reduced by power splitters is roughly the same, idepedet of the umber of light-trees i a etwor. t is because the average umbers of destiatios i all light-trees are idetical. The secod part of our experimets is to compare the followig strategies of trasmittig two multicast streams i WD etwors. Two multicast streams are trasmitted o two separate light-trees, without multiplexig together. Two multicast streams are multiplexed i logical etwors. We assume the data rate of a wavelegth chael is larger tha the sum of data rate required by the two streams. There are five odes i the logical etwor, icludig the seders of the two streams. The seders, receivers, ad the odes i logical etwors are chose radomly. The ed-to-ed delay boud of each multicast stream is ms. average, there are six receivers i a multicast stream. There are three wavelegths i a fiber. The lis of logical etwors, i.e. RWA of lightpaths ad light-trees, the placemet of power splitters ad wavelegth coverters, ad the routig of the two multicast streams are to be decided i our experimets. TABLE COPARSONS OF RESOURCES USED BY TWO LGHT-TREES WTH DFFERENT NETWORK PARAETERS D Λ NC Number of wavelegth chaels N S X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X D Λ NC Total umber of trasmitters ad receivers N S X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X D: the umber of destiatios i each light-tree Λ : the umber of wavelegths i a fiber N C : the umber of wavelegth coverters i the etwor N S : the umber of power splitters i the etwor X: o feasible solutio Fig. 4 shows the results of differet umber of power splitters with differet trasmissio strategies. The results are averaged over samples. The results of multiplexig two streams i lightpath based logical etwors are just the results of multiplexig i logical etwors without power splitters, i.e., N S =, the leftest poit of the lower curve. Compared with lightpath based logical etwors, multiplexig two

8 Number of wavelegth chaels Total umber of trasmitters ad receivers Figure 2. Comparisos of the resources used by two light-trees with differet umbers of destiatios i each light-tree. Number of wavelegth chaels Total umber of trasmitters ad receivers Figure 3. Comparisos of the resources used by differet umber of light-trees. Number of wavelegth chaels With multiplexig Without multiplexig Total umber of trasmitters ad receivers With multiplexig Without multiplexig Figure 4. Comparisos of the resources used by two multicast streams with differet trasmissio strategies. streams i light-tree based logical etwor,i.e., N S 1 uses less umber of wavelegth chaels, optical trasmitters, ad receivers, with the help of wavelegth coverters ad power splitters. compariso with trasmissio o two separate light-trees, multiplexig i light-tree based logical etwors uses less umber of wavelegth chaels sice both streams ca be carried o a sigle wavelegth chael. creasig the umber of power splitters reduces less umber of wavelegth chaels whe two streams are multiplexed i logical etwors. The reaso is that each ode capable of electroic processig, i.e. the ode i logical etwors, ca act as a splittig ode of multicast tree i logical etwors. The total umbers of trasmitters ad receivers required by trasmissio with ad without multiplexig are close. The reasos are described as follows. For trasmissio o two separate lighttrees, the source of a light-tree uses more trasmitters. For multiplexig o light-tree based logical etwors, eve though the seder of a multicast stream ca use fewer trasmitters, odes capable of electroic processig i a multicast tree eed

9 some trasmitters ad receivers to covert data betwee optical ad electroic domais. V. DSCUSSON this paper, we solve the routig, wavelegth assigmet of light-trees, placemet of power splitters ad wavelegth coverters, ad the logical topology desig by usig LP with a limited umber of variables ad costraits. The umber of variables ad costraits used i our formulatio are 2 O ( Γ ( E Λ + V H )) ad O ( V Γ ( Λ + H )), respectively. Cosiderig the routig ad wavelegth assigmet light-trees or placemet of power splitters, the umber of variables ad costraits i previous wor [12,13] grows expoetially with etwor sizes i order to iclude all feasible light-trees i the formulatio. The umber of all possible light-trees i a etwor is the expoetial with the etwor size. O the other had, the umber of variables ad costraits i our formulatio are 2 oly O ( E Λ Γ ) ad O ( V Γ Λ ), respectively. Note that we do ot compare with [14] because it does ot cosider the problem of wavelegth assigmet. Cosiderig the desig of lightpath based logical topology for uicast traffic, the umber of variables ad costraits i 2 [3] are both O( q V ( E Λ Fe + H ' )), where q is the maximal umber of logical lis which ca be created for a source-destiatio pair, ad H is the umber of uicast streams. For a fair compariso with our formulatio, if q is the maximal umber of light-trees which ca be created for a source-destiatio pair, the umber of light-trees i a etwor for our formulatio should satisfy the followig equatio: Γ = q V (26) Accordig to eq. (26), our formulatio eeds much less umber of variables. Note that we do ot compare with [1] because it does ot cosider the problem of wavelegth assigmet. Toratore et al. [4] propose a ovel formulatio, amed source formulatio, which ca reduce the umber of variables ad costraits. The idea is that for each li the set of variables represetig the amout of flows from a source to each destiatio ca be substituted by a sigle variable represetig the total amout of flows from the source to all destiatios. The flow coservatio rule for the source formulatio is that the amout of flows origiatig from a source icomig to a ode is equal to the amout of flows outgoig from the ode. Ufortuately, the source formulatio caot be applied here because the flow coservatio rule does ot hold for light-trees. Therefore, we caot aggregate the set of variables correspodig to the light-trees with the same source ito a sigle variable. V. CONCLUSON this paper, we have preseted a ixed teger Liear Programmig (LP) formulatio to solve the optimal routig ad wavelegth assigmet problem of light-trees with a edto-ed delay boud, ad obtai the optimal placemet of power splitters ad wavelegth coverters i the etwor. The results show that etwors with just a few power splitters ad wavelegth coverters ca efficietly carry multicast data. We have also exteded the above formulatio to desig a light-tree based topology for multicast streams with a ed-to-ed delay boud, ad obtai the optimal routig of multicast streams. We have demostrated that this approach ca use the etwor resources more efficietly, as compared to the approach i which each multicast stream is trasmitted o a differet lighttree, ad the approach i which all streams are multicast i lightpath based logical etwors. ACKNOWLEDGENT This wor was supported i part by OE program for Promotig Academic Excellece of Uiversities uder grat umber 89E-FA , ad i part by the Natioal Sciece Coucil, Taiwa, uder grat umber NSC E REFERENCES [1] R. Ramaswami ad K. N. Sivaraja, Desig of Logical Topologies for Wavelegth-Routed Optical Networs, EEE J. Select. Areas Commu., vol. 14, o., Ju. 1996, pp [2] D. 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