Multicasting for all-optical multifiber networks

Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 219

Multicasting for all-optical multifibernetworks

Fatih Köksal and Cem Ersoy

NETLAB, Department of Computer Engineering, Bogaziçi University, 34342 Bebek,Istanbul, Turkey

[email protected]; [email protected]

Received August 8, 2006; revised November 20, 2006;accepted November 30, 2006; published January 29, 2007 �Doc. ID 73883�

All-optical wavelength-routed WDM WANs can support the high bandwidthand the long session duration requirements of the application scenarios suchas interactive distance learning or on-line diagnosis of patients simulta-neously in different hospitals. However, multifiber and limited sparse lightsplitting and wavelength conversion capabilities of switches result in a diffi-cult optimization problem. We attack this problem using a layered graphmodel. The problem is defined as a k-edge-disjoint degree-constrained Steinertree problem for routing and fiber and wavelength assignment of k multicasts.A mixed integer linear programming formulation for the problem is given, anda solution using CPLEX is provided. However, the complexity of the problemgrows quickly with respect to the number of edges in the layered graph, whichdepends on the number of nodes, fibers, wavelengths, and multicast sessions.Hence, we propose two heuristics layered all-optical multicast algorithm[(LAMA) and conservative fiber and wavelength assignment (C-FWA)] to com-pare with CPLEX, existing work, and unicasting. Extensive computational ex-periments show that LAMA’s performance is very close to CPLEX, and it issignificantly better than existing work and C-FWA for nearly all metrics, sinceLAMA jointly optimizes routing and fiber-wavelength assignment phases com-pared with the other candidates, which attack the problem by decomposingtwo phases. Experiments also show that important metrics (e.g., session andgroup blocking probability, transmitter wavelength, and fiber conversion re-sources) are adversely affected by the separation of two phases. Finally, thefiber-wavelength assignment strategy of C-FWA (Ex-Fit) uses wavelength andfiber conversion resources more effectively than the First Fit. © 2007 OpticalSociety of America

OCIS codes: 060.4250, 060.4510.

1. IntroductionIn all-optical networks, the network nodes or elements are connected with opticaland/or photonic switches, and data stay on the optical domain using WDM. An all-optical channel (lightpath) is created between the source and the destination for uni-cast communication. A point-to-multipoint extension of the lightpath concept (lighttree) is proposed to better support the multicast and the broadcast traffic [1]. Alterna-tive forms of WDM networks range from small LAN in which single-hop strategy isemployed, to WAN where the fibers may span the whole country such as in the NSF-Net backbone or even the whole world. There are two current strategies in WDMWAN networks: Wavelength-routed ones such as in [1–3] and optical burst switching(OBS) based networks such as in [4,5]. In the former case, the multicast data will becarried over a logical (or virtual) topology that we built by using lightpaths or lighttrees [1], and in the latter case, the data are carried via OBS in a more bursty fash-ion that is suitable for relatively short duration of dense transportation.

The main issues related with the characteristic of the multicasting are the size ofthe network, the duration of the session, and the amount of information exchanged. Inour application scenarios, the connection should supply large bandwidth to accommo-date the multicasting of sessions. Moreover, it may be done in a very broad region,and its duration is expected to last many minutes or hours, not seconds. Remote diag-nosis and operations in hospitals would be an appropriate example. Thus, our sce-nario includes a relatively long duration of multicast transportation that requireslarge bandwidth, and wavelength-routed WDM WAN is the appropriate environment.

We solve the multicasting problem on the logical topology, which is determined bylightpaths and light trees in WAN [6]. An important parameter is related with the

1536-5379/07/020219-20/$15.00 © 2007 Optical Society of America


constraints about the capabilities of switching elements. A switch may have no, lim-ited, or full wavelength conversion and light splitting capabilities. In limited wave-length conversion, all nodes can convert an incoming wavelength to a subset of avail-able output wavelengths. Similarly, a node with limited light splitting capability cancopy incoming data to a subset of output links, if the number of output links is lessthan the limited splitting capability of that node. Sparse light splitting and wave-length conversion means that not all but some nodes in the network have full wave-length conversion and light splitting capabilities.

New solutions in multicasting domain would usually come from the ideas in uni-casting domain. Similarly, we propose to use a layered graph approach, which hasbeen previously proposed for unicasting, to have a more general, realistic, and flexiblemodel of an all-optical multifiber network for multicasting. This new presentationenables us to state the problem of static multicasting for all-optical multifiber net-works with sparse light splitting and wavelength conversion restrictions so that it isformulated as original mixed integer linear programming (MILP) as described in Sub-section 4.A. The MILP formulation is solved using CPLEX, which is a state of the artoptimization tool [7]. CPLEX finds the optimal solution within a given precision, andit also gives a lower bound by relaxing the integrality constraints. However, it is onlypossible to solve MILP problems to optimality for small networks and a number ofsessions, since the problem is NP hard (Section 3). Therefore, we also propose two effi-cient heuristics layered all-optical multicast algorithm [(LAMA) and conservativefiber and wavelength assignment (C-FWA)] for larger problems and dynamic multi-casting requests. Extensive computational experiments demonstrate that LAMA per-forms at most 20% worse than the optimal with 99% statistical confidence. LAMA alsoperforms 2.5 times better than its competitor member-only (M-ONLY) [8] in terms ofcost. However, the main contribution is to demonstrate that LAMA works better thanits alternatives, since we jointly optimize routing and fiber-wavelength assignmentphases compared with the other candidates that attack the problem by decomposingtwo phases. Experiments show that important metrics (e.g., session and group block-ing probability, transmitter wavelength, and fiber conversion resources) are adverselyaffected by the separation of routing and fiber-wavelength assignment phases in mul-ticasting. Finally, we also propose a new fiber-wavelength assignment strategy (Ex-Fitin C-FWA), which uses wavelength and fiber conversion resources more effective thanthe First Fit, if we have to separate two phases.

The rest of the paper is organized as follows: Section 2 covers the related work inthe literature. Section 3 describes the problem definition, mathematical formulation,and the underlying assumptions. Section 4 is devoted to solution techniques, and com-putational experiments are given in Section 5. Finally, Section 6 summarizes the con-clusions and has a discussion about future works.

2. Multicasting for All-Optical NetworksA comprehensive survey on optical multicasting over wavelength-routed WDM net-works is given in [9]. Therefore, we only summarize works that are close to our formu-lation in the literature. Benefits of all-optical multicasting are studied in a simulationenvironment by Malli et al. and it is concluded that with a reasonably large multicast-ing group size, multicasting can reduce the bandwidth consumed [3]. However, thiswork does not consider multifiber, or nodes that have the sparse wavelength conver-sion and/or sparse splitting capabilities but are distributed nonuniformly.

The static virtual topology design (VTD) problem is examined in terms of multicast-ing in [10–12]. However, wavelength conversion issues are not examined in [12].Power loss in WDM networks is an important issue, and [13,14] examine multicastingfrom this perspective. Another static VTD formulation is given in [1], but the authorsdid not consider the sparse splitting cases. If we assume full light splitting, the prob-lem just reduces to finding minimum Steiner trees (MST) in a layered graph describedin Section 3. Moreover, the objective function is different in [1]. In [15], wavelengthassignment of one tree problem is shown to be solvable by a polynomial algorithm.Similarly, the special case of a general multicasting that includes only one source inthe formulation with static traffic is given in [16]. Their scenario is very similar toours in terms of setting, but they have only one source that is the video server store,and all the other nodes are groups of users. Multicast routing under the delay con-


straint problem (MRDCP) is defined as a minimal Steiner tree (MST) problem withdifferent light splitting and delay constraints in [17] for one multicast session. In [18],the problem of minimizing the number of wavelengths is considered for one multicastsession and for multihop WDM optical networks. Another approach is given in [19] forefficient use of wavelengths by adding wavelength conversion constraints. Yang andLiao formulated their problem as a static VTD [6]. They allow multihop traversing,i.e., electro-optical conversion, which is not allowed in our formulation, since our net-work is all optical. In [20], authors evaluate the trade-off between capacity and wave-length continuity by developing analytical models and concluding that even a smallamount of wavelength conversion capability helps shifting the advantage to the light-tree approach. In [21], the problem of wavelength assignment is studied in order tomaximize the network capacity for one multicast in the case of no wavelength conver-sion. Full wavelength conversion is assumed without mentioning sparse splitting in[22], but delay constraints are also added, and a distributed algorithm is proposed forthe solution. Similar assumptions are also considered in [23] with many fibers, and adistributed reinforcement algorithm is proposed.

In [24], the sparse splitting case is examined only for one fiber and one multicast(tree) session by minimizing the number of wavelengths used. The closest work to ourformulation in the literature proposes four different heuristics [8], and we choose toimplement the M-ONLY heuristic to compare CPLEX, LAMA, and C-FWA. Moreover,all nodes have some limited splitting capability without wavelength conversion in[25]. The problem is formulated to optimize mostly quality-of-service (QoS)-based met-rics (e.g., maximizing the number of destinations and/or minimizing the wavelengthcost), since wavelength conversion and light splitting resources are very scarce in thissetting.

3. Problem Definition for All-Optical MulticastingImportant issues for the formulation of the problem are how to handle multifiberinstallations among nodes, representation of different wavelengths in a fiber, and howto represent nodes with a sparse wavelength conversion capability. We propose to usethe layered graph for modeling the WDM network as described in [26–28] for all-optical multicasting. The layered graph in a multifiber WDM network, which has Nnodes, F fibers for each pair of nodes in the network, and W wavelengths in each fiber,is constructed by replicating the original directed network FW times (both nodes andlinks among them) in addition to adding only one node (main node) for each node ofthe original network (no link is added). In each replication, the original bidirectionallinks of the network are preserved, but they represent a specific wavelength channelin the given fiber and wavelength. Thus, we have �FW+1�N nodes in the final directedlayered graph, and a main node is represented by one node in each FW layer. If weuse a link from a source main node to any one of the corresponding nodes (subnodes)for the routing of a multicasting request, then a transmitter is used to create a treeoriginating from the source node. Consequently, a main node is connected to its repre-sentative nodes (subnodes) with bidirectional links. For example, the usage of a linkfrom a main node to its subnode in the layer for F=2 and W=2 represents the usageof a transmitter for the second wavelength in the second fiber on the correspondingnode of the original network. Similarly, the usage of a link from a destination subnodeto the corresponding main node indicates that this destination is reached in the mul-ticasting tree. Therefore, main nodes are used to represent the root and leaves of themulticasting tree, and corresponding subnodes are for interior nodes. Finally, the sub-nodes of a main node in different fiber-wavelength layers are connected with respectto the wavelength conversion and switching capabilities of the main node of these sub-nodes. For example, corresponding subnodes in different fibers are connected to repre-sent wavelength conversion, if the main node of these subnodes has full wavelengthconversion property. Similarly, corresponding subnodes in different wavelengths areconnected to represent fiber conversion, if the main node of these subnodes has aswitching property.

There may be a different number of fibers between some nodes, and we present anexample to explain the difference from the homogeneous case. A layered graphexample is given in Fig. 1 for a simple WDM network having three nodes, four bidi-rectional fiber links, two wavelengths on each link, and two heterogeneous fiber lay-


ers, which indicate a differing number of fibers between nodes rather than differentfiber types. The WDM network is replicated as different layers corresponding to differ-ent fibers and wavelengths. However, the structure of each layer of fibers may be dif-ferent, because connections among nodes may contain a different number of fibers.For example, we have two bidirectional fibers between nodes 1 and 3, but there is onlyone bidirectional fiber between nodes 1 and 2, and nodes 2 and 3 in Fig. 1(a). Nodes 1and 2 have only switching capability, but node 3 has both switching and wavelengthconversion capabilities in Fig. 1. Therefore, the subnodes of main node 3 in the samefiber layer are connected to each other to allow conversion among different wave-lengths. However, this is not valid for main nodes 1 and 2, since they can only doswitching between fibers.

After establishing the layered graph of a network topology, finding a multicastingtree in the original network turns out to be finding a MST in the layered graph. Thereare multiple types of Steiner tree problems. We prefer a generic MILP solver such asCPLEX instead of using a specialized Steiner solver, since our problem differs fromthe basic MST problem in several ways. First, there is more than one MST �k�, andthe total cost of all MSTs is minimized. Second, each MST should be edge disjoint inthe layered graph, since each wavelength in a fiber can only be used by one multicast-ing session. Third, the representation of the sparse splitting capability of nodes alsoforces us to add degree constraints to the formulation. Thus, our problem can be calleda k-edge-disjoint (nonoverlapping) degree-constrained Steiner problem in graphs forrouting and wavelength assignment of k multicasts in WDM networks.

The Steiner tree problem, its variants and heuristic based approaches are mostlyexamined in the literature, and [29,30] summarize the results. Although it is evenpossible to solve very big Steiner tree problems to optimality [31], the problem is NPhard [32]. Thus, its generalized version that includes k-edge-disjoint MSTs with addi-tional degree constraints in our case is even harder to solve.

Fig. 1. (a) Simple WDM network having three nodes, four bidirectional fiber links, twowavelengths on each link, and two heterogeneous fiber layers. (b) Notation. (c) Layeredgraph model of (a).


3.A. Assumptions• A wavelength on a given fiber between two nodes carries only the data of one

multicast session, since the route for one multicast constitutes a tree, and aggregationof more than one multicast session on one wavelength is not allowed, because multi-hopping is not allowed in light trees, that is, all transportation occurs in the opticaldomain without optic-to-electronic domain conversion.

• The capacity of one wavelength channel is enough to carry one multicast session,since one wavelength channel offers a very large bandwidth with current technologies,and it is quite difficult today to imagine a multicast application requiring a fulllambda channel for each destination. Moreover, we can divide a multicast stream tomore than one tree, if the required bandwidth is greater than the capacity of onewavelength channel. Therefore, capacities of the fibers and the required bandwidthfor each multicast are not input parameters for the formulation, and each sessionoccupies one wavelength channel.

3.B. Notation Used in the FormulationG�V ,E�: The graph G representing the WDM network with vertex set V and the

edge set E. It contains N nodes and L bidirectional links.F: Number of fibers.W: Number of wavelengths.SS: The set of nodes that have a sparse splitting property.SC: The set of nodes that have a sparse conversion property.Nls: The number of nodes that have a full light splitting property.Nwc: The number of nodes that have a full wavelength conversion property.

The parameters of the network are used to construct the layered graph, and thecost between two nodes is cij in the layered graph. We minimize the objective of thesum of all cijs for the routing and wavelength assignment of k multicast sessions [Eq.(1) in Subsection 3.C]. These cij are either given and dictated by the problem; oradjusted to favor different metrics, which we demonstrate in Subsection 5.B for theblocking probability. Thus, this flexible cost assignment enables us to have a generalframework in which different objectives can be simultaneously achieved or balanced.

A demonstrative example is given in Fig. 2 for the routing and wavelength assign-ment of the multicast session �2�→ �1,3� to clarify what sort of costs are involved. Thecost of the link 2→5 represents using a transmitter on node 2. The link 5→6 denotes

Fig. 2. Routing and wavelength assignment of a multicast session �2�→ �1,3�, which isshown with bold arrows for the network, that is given in Fig. 1.


the cost of using the first wavelength channel on fiber 1 between nodes 2 and 3. Simi-larly, the link 13→12 is for the cost of using the second wavelength channel on fiber2 between nodes 3 and 1. The link 6→11 denotes the fiber conversion cost, and thelink 11→13 denotes the wavelength conversion cost on node 3. The links 13→3 and12→1 are used to ensure that all destinations are reached, and they do not add to theobjective function.

We can adjust cijs to reflect the underlying relative cost of these operations so thatall four different cost terms can be minimized simultaneously. For example, if thewavelength conversion or the fiber conversion is not costly for us in our routers, thenwe can make the corresponding costs zero. Similarly, we can discourage the creation ofmore than one tree from the source by setting the cost of using a transmitter to a highvalue if we also want to minimize the number of transmitters that are used for multi-casting. We use the following variables for static traffic generation:

S: Number of sessions.Rmn: Fraction of multicast nodes.M: Set of multicast session sets, M= �Mk �k=1,2, . . . ,S�.uk: The first node (source) in the kth multicast set Mk.n: Number of nodes in a multicast group determined by Rmn.Mk�: Kth multicast set excluding the source, Mk�=Mk− �uk�.

We also have two decision variables:

f klij : The flow of multicast k to be sent to node l over the link �i , j�, f kl

ij �R+� �0�:xij

k : The binary decision variable that determines whether link �i , j� is used by mul-ticast k, xij

k � �0,1�.

3.C. Problem FormulationThe k-edge-disjoint (nonoverlapping) degree-constrained Steiner problem �kEDSP�can be formulated as follows:

Minimize the total cost of served sessions,

Total cost: z = min �k

��i,j��E�

cijxijk , �1�

where E� contains all types of links (fiber conversion, wavelength conversion, links forwavelength channels, and links for using a transmitter on the source node) and E�contains all types of links except the ones with zero costs:

• The cost for the links from destination subnodes to the main nodes are excludedfrom the objective function. For example, such links 13→3 and 12→1 in Fig. 2 areused to ensure that all destinations are reached. A subtle point should not be missedthat if we would assign a cost for this sort of link, then the links that are alreadyincluded as a cost could be used for the routing of other multicasting sessions to mini-mize the objective function, since a path from a source to a destination can be com-pleted in any layer (any fiber and any wavelength). Another multicasting session withnode 3 as a destination would have a tendency to use node 13 to reach the destinationnode 3 instead of using nodes 6, 9, and 11 in Fig. 2, since this cost is already includedby the multicast session, which is routed previously.

• The cost for the links from the source subnodes to the main nodes and from themain destination nodes to the subnodes are also excluded, since they are irrelevantand normally not used for the routing and wavelength assignments of multicast ses-sions.

xijk � fkl

ij , ∀ �i,j� � E�, ∀ k, ∀ l � Mk� , �2�

��j,i��E

fklji − �

�i,j��E

fklij = �1, ∀l � Mk�, ∀ k, and i = l

0, ∀l � Mk�, ∀ k, and i � l� , �3�
� �


The multicast data in trees should be nonbifurcated, that is, multicast data in asession cannot be split into two streams in any node of an all-optical network. Con-straints (2) and (3) ensure that the solution is in the form of a set of trees.

�k

xijk � 1, ∀ �i,j� � E�, �4�

where E� contains only the wavelength channel links among the same layer and itexcludes the links from main nodes to subnodes and vice versa and fiber and wave-length conversion links, since these links can be used more than once. Consequently,this wavelength continuity constraint (4) makes sure that a link representing a wave-length, and a fiber can be used once, and all degree-constrained trees should be edgedisjoint.

�j�V�, �i,j��E�

− xijk + �

i�V�, �j,i��E�

xjik � 0,

∀i � SS � V�, ∀ k,i is not a source for multicast k, �5�

where V� contains all the nodes (it includes the main and the subnodes) in the layeredgraph and V� contains only the subnodes.

The degree constraint (5) (in/out degree) forces routes (or specifically trees) toaccommodate the sparse splitting restrictions. It should be noted that if a sparse split-ting node is a source node, then it should not have this restriction, since a source sub-node represents the main source node, and it can send as many packets as it wishes,since it is not copying packets, instead it is creating these packets without copying.Moreover, one subtle distinction should not be missed, that a main source node shoulduse the link that is coming from the main source node to a subnode to create a newtree from the same source, instead of using the fiber or wavelength conversion linksamong the source subnodes. Thus, Restriction (6) is added to the formulation:

xijk = 0, if i and j represent the same main source node for multicast k. �6�

The decision variables xijk determine routing and wavelength assignment of the net-

work, that is, a degree-constrained Steiner tree for each multicast session and wave-length assignment in each of these trees. Moreover, the number of decision variablesalso determines the complexity of the problem. The number of edges in the layeredgraph is �2WFL�+ �F�F−1�N�+ �W�W−1�Nwc�+ �2WFN�. Thus, we multiply that num-ber by S to find the number of binary decision variables. If we take F=1, W=1, andS=1 and remove degree constraints due to sparse splitting nodes, the special case(Steiner tree problem) would still be NP hard [32]. Therefore, the problem is solvedusing CPLEX [7] for smaller networks, and we propose a heuristic algorithm, LAMA,for larger problems.

4. Proposed Solutions: LAMA and C-FWA4.A. MILP Solution: CPLEXWe solve the MILP formulation, which is given in Subsection 3.C with CPLEX [7] anduse both the solution of CPLEX and the lower bound, which is produced by relaxingthe integrality constraints. However, it takes a very long time for CPLEX to find theoptimal results within a given precision for problems, which contain many binaryinteger variables. Therefore, we propose the LAMA for larger problems.

4.B. LAMAA layered graph is created from the graph representing the WDM network asexplained in Section 3 and LAMA works on this graph. LAMA is a directed degree-constrained minimum spanning tree (MST) heuristic, which is based on finding short-est paths from the current spanning tree to the remaining nodes in the multicast ses-sion, and it uses this method for all multicast sessions consecutively. The Bellman–Ford all-pairs shortest path algorithm is run for each multicast destination in eachsession dynamically [33]. Thus, the complexity of the algorithm is O�Sn��FW+1�N�3�,since there are �FW+1�N nodes in the layered graph. In LAMA, the routing paths ofnode pairs are not precomputed and fixed. Therefore, LAMA can dynamically change


the cost assignment, while joining new multicast members to the group. Thepseudocode for LAMA is given in Algorithm 1.

The ability to change different cost terms in the layered graph is an importantadvantage for the LAMA heuristic to support the minimization of different metricssimultaneously. If we have an idea of the relative costs of these operations, then wecan use LAMA to minimize the ultimate objective function. However, LAMA could alsobe used for totally different aims. We might not know or care about relative costs, butwe might want to minimize a specific metric such as: average bandwidth, delay, high-est wavelength index, or blocking probability (Subsection 5.B). Thus, the LAMA heu-ristic could be trained in a batch mode to favor and optimize some metrics for a givennetwork and given workload. In our experiments, we follow the second use of theLAMA heuristic to favor and optimize specific metrics. However, we use different traf-fic loads, which are created randomly, for batch mode training and on-line tests to befair for all algorithms.

The total cost that has to be minimized consists of four different terms: the cost ofusing wavelength channels, wavelength conversion, fiber conversion, and using

Algorithm 1. LAMA Algorithm1: Z=the set of nodes in a multicast session;2: RemainingConnectionNumber for degree constrained nodes;3: Multicast Tree=the links and nodes in a multicast session (a tree);4: AllMulticast Tree=set of MulticastTree;5: for each multicast session do6: set the cost of links from source main node to its subnodes with

TransmitterUsageCost;7: initialize RemainingConnectionNumber with degree constraints;8: initialize MulticastTree as empty and add the source node;9: initialize the multicast set Z with multicast nodes except the

source;10: while (Z is not empty) and (current path finding is successful)

do11: calculate shortest paths from nodes in MulticastTree to nodes

in Z12: for all pairs of nodes from nodes in MulticastTree to nodes in

Z do13: find the shortest path that does not violate degree con-

straints on the layered graph;14: end for15: if current path finding is successful then16: remove the links corresponding to used wavelengths from

the layered graph;17: add this path to the MulticastTree;18: update Z, RemainingConnectionNumber;19: else20: current path finding is not successful;21: end if22: end while23: if the session is successfully created (all path determinations are

successful)then

24: load all the calculated metrics to an output file;25: add MulticastTree to AllMulticastTree;26: else27: undo changes on the layered graph for that session;28: end if29: end for

transmitters. The cost of using a wavelength channel can be assigned to minimize two


different metrics. If we assign equal cost (or simply 1) for wavelength resources, onepart of the optimization becomes minimizing the average bandwidth (AB). Alterna-tively, this part of the optimization becomes minimizing average delay (AD), if weassign the time duration of the communication in these wavelength resources as costs.We can either assign costs with respect to bandwidth or delay, and this decisionaffects all heuristics, which are using this information. Therefore, LAMA can optimizethe preferred metric by changing the type of the cost assignment. Moreover, we alsorealize that we can balance these two metrics, if we assign the cost of a wavelengthresource as the average of the corresponding delay value in this wavelength resourceand the mean of all delays in the network. We can also use a weighted averageinstead of an arithmetic average to favor one metric over another. CPLEX, LAMA,C-FWA, and M-ONLY should use the same type of cost assignment to be able to fairlycompare all competitors. Therefore, we omit the work related with the effect of thecost assignment on two different metrics, since all heuristics are evaluated fairly, if weuse one type of cost assignment for all. Then, we interpret the cost of using a wave-length channel as the delay in this wavelength channel, and we define all other typesof costs with respect to the average delay in the network to normalize different com-ponents of the total cost. The relative importance of these costs to the average delay inthe network is represented by three ratios, respectively Rwcc (wavelength conversioncost/average delay), Rfcc (fiber conversion cost/average delay), and Rtuc (transmitterusage cost/average delay). However, we can heterogeneously assign different valuesfor different costs of a particular cost term.

The LAMA heuristic does not separate routing and wavelength assignment stepsand considers both of them jointly: its adjustable parameters on the layered graphmake it very flexible. LAMA can be applied to dynamic multicasting without adjust-ment, since it routes sessions consecutively in a dynamic fashion. LAMA can also pro-duce partial solutions, even if there is no feasible solution to the problem. Forexample, LAMA can route 19 out of 20 sessions when CPLEX finds the infeasibility ofrouting all sessions. Thus, CPLEX cannot give partial solutions and is not applicableto dynamic mutlicasting. Moreover, the running time of LAMA linearly increases withrespect to the number of sessions, contrary to CPLEX.

A demonstrative example is created to clarify how LAMA runs on the layeredgraph. Network size, number of wavelengths and fibers are kept small to make thedemonstration understandable (N=5, F=1, W=2) and the sparse splitting and conver-sion node sets are given as follows: SS= �1,4� and SC= �4,5�. The original network andthe layered graph of that original network are given in Fig. 3. There are two copies ofthe original network for two wavelengths with subnodes 1�, 2�, 3�, 4�, 5� and 1�, 2�, 3�,4�, 5�, and 1, 2, 3, 4, 5 represent the main nodes. The arrows with dots represent thelinks from main nodes to subnodes and vice versa. The links that are shown by smallarrows represent the wavelength conversion, and nodes 4 and 5 do not have this capa-bility. We assume that the values of Rwcc and Rtuc are set as 1. Thus, the costs of these

Fig. 3. Layered graph model of a network having five nodes and one fiber carrying twowavelengths (only delay costs are shown for simplicity).


links equal the average delay in the network, since the cost assignment of wavelengthresources is done to optimize average delay. We show only delay costs in the demon-strative example for simplicity. Additionally, nodes 1 and 4 do not have light splittingcapability and cannot multiplex the incoming data to transfer to more than one node.

The first multicast session is �1�→ �3,4,5�. The shortest path, which does not vio-late degree constraints, from 1 to the multicast destinations is 1→1�→2�→4�→4.There is another path on the other wavelength’s network, and one of them is chosenarbitrarily in this setting. However, the transmitter usage costs and the wavelengthconversion costs can be differentiated to let the LAMA heuristic minimize other met-rics such as average highest wavelength index (AHWI). The next shortest path to thecurrent tree is 4�→5�→5, and the last one seems 4�→3�→3, but it violates thedegree constraint on the sparse splitting node 4. Thus, the next shortest path 2�→3�→3 is chosen, and the first session is routed on the layered graph. The resultingroutes are shown in Fig. 4 with bold arrows. The cost assignment of wavelengthresources is done with respect to the delay, but it adversely affects the wavelengthresources consumed, since the variance of the delay is high in the example. However,we could also minimize average bandwidth by assigning the cost of wavelengthresources consumed equally. In this case, LAMA produces 1-4-5-3, which consumes 17units of delay (compared with 15 units of delay of the previous solution), but it alsoconsumes one wavelength resource less than the previous solution.

After routing the first session, we need to remove the links that are used for wave-length channels and come up with a new layered graph in which the links represent-ing wavelength and fiber conversion and transmitter usage are not removed. (Becausewe assume that we have enough transmitters, and fiber-wavelength conversionresources are not restricted, if the node has that capability.) Similarly, LAMA routesmulticast requests in the order that they are requested until blocking occurs.

4.C. Member-Only HeuristicFor comparative evaluation, we have also implemented the M-ONLY heuristic thatwas originally proposed in [8] and further modified it to handle multifiber cases withthe First-Fit strategy. It separately considers routing and fiber and wavelengthassignments. The first step is to build a forest for routing by adding multicast mem-bers consecutively to the current tree. The closest member to the current tree is addedfirst and it goes on until all multicast members are included. However, nodes with asparse light splitting property in the current tree are not used to connect a new mem-ber to the current tree. Additionally, a new tree from the source node is created, if itis not possible to add any remaining multicast members to the current tree. Then the

Fig. 4. Routing on the layered graph for the first multicast session ��1�→ �3,4,5��,which is shown with bold arrows.


First-Fit algorithm is preferred to assign fibers and wavelengths for the final forest. Ifthere is more than one fiber, we do not have to find a wavelength assignment in whichonly one fiber is involved. Thus, we try to find the first available fiber and wavelengthfor a given segment, which are a part of the tree that should use the same wavelengthfor routing. The other details can be found in [8].

The complexity of the algorithm is the sum of the complexities of routing and wave-length assignment steps. We use the Bellman–Ford all-pairs shortest path algorithmfor finding the shortest paths. It should be noted that we do not have to run this algo-rithm for all sessions and multicast members separately, unless all wavelength chan-nels in all fibers are occupied between any two nodes, and this link is removed fromthe original network. As a result, this algorithm should be run only when a changeoccurs in the original network, but it can be run for each session, and each multicastmember at the worst case and the complexity of routing becomes O�SnN3�. Similarly,we need to do fiber and wavelength assignment for each link in multicast forests (oneforest for each session) with complexity O�FW�. At the worst case, we can have n sepa-rate trees each containing at most N−1 edges, since a tree with maximum N nodescan contain at most N−1 links. Thus, the total worst-case complexity would beO�SnNFW+SnN3�. However, the shortest path calculations seem to dominate interms of running time in the experiments.

We especially prefer this algorithm to compare with LAMA, since it is easy to showthe effect of separating routing and wavelength assignment steps, and it also runsvery well in practice [8], i.e., it is a strong competitor.

4.D. Conservative Fiber and Wavelength Assignment HeuristicThe First-Fit fiber and wavelength assignment strategy in the M-ONLY heuristic isnot the only option. Moreover, we also realize that the number of wavelength and fiberconversions can be decreased if we also try to minimize fiber and wavelength conver-sions among segments. Therefore, we have also modified the wavelength assignmentstrategy of M-ONLY [8] and created our own alternative heuristic (C-FWA). The maindifference between C-FWA and M-ONLY is that it assigns fibers and wavelengthsafter a path is added to the current multicast tree in a way in which it tries to use thesame fiber and wavelength of the link that connects this path to the current tree. If itis not possible to complete this assignment, then it uses the First-Fit algorithm as inM-ONLY. We call this new fiber and wavelength assignment strategy Ex-Fit.

The worst-case complexity of the C-FWA heuristic is the same as M-ONLY, since thecomplexity of checking the availability of the old fiber and wavelength for all seg-ments is much less than the current complexity of the fiber and wavelength assign-ment step �O�SnNFW��. In practice, this algorithm is expected to run in shorter timeduration than M-ONLY, since it is expected to do less computation in the fiber andwavelength assignment step. Finally, we call this algorithm C-FWA so that it reflectsthe fiber and wavelength assignment characteristics, because it first tries to use theold fiber and wavelength in a conservative way. In this respect, we would name theM-ONLY heuristic as Greedy-FWA, since it always tries to use the first available fiberand wavelength for assignment.

4.E. UnicastingIt is possible to route a multicast session by separately routing each request in a uni-cast manner. However, it wastes resources and using a multicasting solution canreduce the bandwidth that is consumed [3]. During the comparisons, we use CPLEXresults as the lower bound. Similarly, we also include unicasting only in the firstgroup of experiments to fully assess the benefit of using multicasting algorithms.

5. Computational Experiments5.A. Experiment Design

5.A.1. Network Model and WorkloadWe experimented with different size random networks with different properties, andall experiments were performed on a Pentium IV 3.2 GHz computer with 1 Gbyte ofRAM. The following characteristics are considered for the creation of random net-works that are used in the experiments:


N: Number of nodes.D: Average nodal degree.E: Number of edges= �ND� /2.PC: Physical connectivity=E / �N�N−1� /2�=D / �N−1�.Dmin: Minimum nodal degree.Dmax: Maximum nodal degree.R: Network diameter (max shortest path).H: Average internodal distance (average shortest path).

We try different values for an adjustable parameter (Alpha) to have artificial ran-dom networks that resemble real networks. Table 1 contains the important character-istics of some real life networks and some random networks that are used for theexperiments.

The source and destination nodes of a multicast session are created randomly byusing the following parameters:

S: Number of sessions.M: Number of multicast nodes.Rmn: M /N.

We use the following factors in addition to N and D for the experiment design:

F: Number of fibers.W: Number of wavelengths.Rwc: Number of full wavelength conversion capable nodes/N.Rls: Number of full light splitting capable nodes/N.Rtuc: Transmitter usage cost/AD.Rwcc: Wavelength conversion cost/AD.Rfcc: Fiber conversion cost/AD.

The total cost consists of four different cost terms: delay, transmitter usage, wave-length conversion, and fiber conversion. The parameters Rtuc, Rwcc, and Rfcc representthe relative weight of these terms with respect to the average delay.

5.A.2. Evaluation MetricsAll evaluation metrics are strictly dependent on the traffic load (number of sessions,S). Therefore, we prefer to normalize all metrics with respect to the traffic load viadividing them by the number of sessions, but the number of successfully routed ses-sions can be different for different algorithms because of blocking. We use the percentgap with respect to the lower bound in terms of different parameter values and algo-rithms compared: Percent gap= �metric value−lower bound� / lower bound. The firstthree metrics are very similar to the ones used in [8]. We especially include them forcomparative evaluation of LAMA, M-ONLY, and C-FWA:

1. AB: Total bandwidth/S.2. AD: Total delay/S.

Table 1. Different Parameters of Real and Random Networks that AreCreated with Alpha=0.20

Network N D E Dmin Dmax PC H R

ARPANet 20 3.10 31 2 4 0.16 2.81 6UKNet 21 3.71 39 2 7 0.19 2.51 5EON 20 3.9 39 2 7 0.20 2.36 5

NSFNet 14 3.0 21 2 4 0.23 2.14 3Network2 10 3.0 15 1 6 0.33 2.00 4Network7 15 4.0 30 2 6 0.29 1.99 4Network10 20 3.0 30 1 6 0.16 2.65 6


3. AHWI: Sum of the highest wavelength index for each fiber/S. This metric is alsodependent on the number of fibers �F� and wavelengths �W�, but we do not want tonormalize this metric by dividing it with the number of fibers, since we are fixing theavailable wavelength channels for a given experiment design. For example, we usethe following combinations for an experiment for F /W: 1 /8−2/4−4/2.

4. Average wavelength conversion (AWC): Number of wavelength conversions/S.5. Average fiber conversion (AFC): Number of fiber conversions/S.6. Average number of tree (AT): Number of trees in forests/S. It should also be

noted that a multicast session may consist of more than one tree routed from thesame source. This number exactly equals the number of transmitters used per session(forest).

7. Total cost: It is given in Eq. (1). CPLEX is used to minimize the total cost, andwe use the percent gap of this metric with respect to the lower bound, which is alsofound by CPLEX. It consists of four different terms, which are explained in Subsection3.B. The parameters Rtuc, Rwcc, and Rfcc are used to normalize different cost terms.

8. Average running time: Total running time of the algorithm/S.9. Group blocking probability (percent): In the static multicasting problem, a group

consists of S sessions. If an algorithm fails to route any of these sessions in a group,then it is considered to fail to route this group. Thus, this metric measures the qual-ity of service in terms of the overall group performance. CPLEX determines the feasi-bility of a routing of a group, since if the routing is possible, it finds the solution what-ever the cost is.

10. Session blocking probability (percent): The whole group performance is notenough to measure the QoS experienced for each session in a group. Then, we alsomeasure separately the number of sessions that are blocked and divide it by the num-ber of sessions that can feasibly be routed. Similar to the group blocking probability,CPLEX determines the optimal number of successfully routed sessions for feasibleexperiments, and other heuristics are measured with respect to how many of thesesessions are successfully routed.

5.B. Minimization of Blocking Probability and Total CostIn dynamic multicasting, session requests are done consecutively, and any dynamicalgorithm should establish sessions one by one in the order that they are requested.However, all session requests are done at once in a batch mode in static multicasting,and the establishment order for sessions may affect the metrics. Before mentioningthe main results of the experiments, we want to clarify this effect for LAMA, C-FWA,and M-ONLY, which are iterative algorithms. We performed all tests many times byonly changing the order of destinations at each replication. The results indicate thatall metrics almost do not change at all with respect to the order of destinations for allheuristics. Moreover, we changed both the order of destinations and the sessions, andwe repeated these tests with many different random combinations. The results arenot almost the same for this time, but there is no statistically significant difference forany metric and any heuristic. However, we also investigated the percent and absolutedifferences by spotting the pairs, which have the maximum separation in terms ofeach metric. The performance of LAMA for the average highest wavelength index,wavelength and fiber conversion, and number of tree metrics changes at most lessthan 0.01 in terms of absolute difference, the average bandwidth and delay change atmost 0.16 and 0.06, respectively. In terms of percent change, it is at most less than2.5% for all metrics. Group and session blocking probabilities change at most 0.2%.

We decided to find the best values for Rfcc, Rwcc, and Rtuc to minimize the QoSrelated metrics that are the group and session blocking probabilities for LAMA. Thus,the number of sessions that are successfully routed by LAMA before blocking is cho-sen as a metric to decide the values of these parameters. If there is a feasible solutionfor the problem, then CPLEX routes all the sessions in a group. However, it cannotgive the maximum number of sessions that can be routed before blocking. CPLEXuses the same parameter values for Rfcc, Rwcc, and Rtuc with LAMA. We designed thefollowing experiment to be able to find a good combination of these parameters (ninefactors):


1. D: {3,4} (2).2. N: {30} with two different networks (2).3. F /W: �2/4,2/8,4/4� (3).4. Rwc=Rls: {0, 0.5, 1} with two different SS and SC set for 0.5 (4).5. Rmn: {0.25, 0.50} with two different multicast sets (4).6. Rfcc /Rwcc /Rtuc:�0.01/0.01/0.01,1/1/1,100/100/100,0.01/1/1,100/1/1,1/0.01/1,1/100/1,1/1/0.01,1/1/100� (9).

Number of experiments: �2.2.3.4.4�.9=192.9=1728.We determine the baseline success for each level with combinations

�0.01/0.01/0.01,1/1/1,100/100/100� and the other six combinations are intended tomake each parameter on and off with respect to the baseline �1/1/1�. For each 192combinations, we apply nine different parameter sets and keep track of the maximumnumber of successfully routed sessions. Then we calculate the 99% confidence intervalfor the number of successfully routed sessions for each 192 combinations. Finally, werecord the number of cases in which a particular parameter set (e.g., �0.01/0.01/0.01�)is outside the given confidence limits.

Table 2 denotes that the most intuitive parameter combination �1/1/1� gives thebest results in terms of general performance, and all four terms of the total cost(delay, transmitter usage, wavelength conversion, and fiber conversion) are equallyimportant. The results of 20 experiments were beyond the upper confidence limits,and the results of ten experiments were below the lower confidence limits. Therefore,the success ratio is two. Moreover, we also examine the effect of Rmn, D, Rwc�=Rls�, F,and W on the performance, and this parameter set almost works best for different val-ues of these factors as well.

After determining the right parameter set, we designed final tests so that the work-load is distributed evenly from very light traffic conditions to very heavy ones that cancause blocking, and the blocking rate is kept at approximately 20%. The objective isminimizing the total cost, which includes delay, wavelength, and fiber conversioncosts, and the transmitter usage cost. Heuristics try to route most of the requests, andthey are also expected to use fewer wavelength and fiber conversions, fewer transmit-ters, and less delay.

The design aims to cover a very broad spectrum of combinations of factors (D, N, F,W, Rwc, Rls, S, and Rmn), since the percent gap metric is fair to compare all algorithmsfor all combinations, if the number of successfully routed sessions are the same. Wealso examine the different terms of the total cost separately by examining the averagewavelength and fiber conversions, the average number of transmitters (trees), and theaverage delay. We fix the parameters Rfcc, Rwcc, and Rtuc to one, and there are eightdifferent factors for the experiment design:

Table 2. How Many Times a Parameter Set Is Outside the Upper and Lower99% Confidence Limits

Parameter Sets

Upper Lower Upper/LowerRfcc Rwcc Rtuc

0.01 0.01 0.01 12 10 1.20.01 1 1 15 19 0.8

1 0.01 1 15 8 1.91 1 0.01 15 11 1.41 1 1 20 10 2.01 1 100 12 10 1.21 100 1 8 24 0.3

100 1 1 17 37 0.5100 100 100 30 38 0.8


Network:(1) D: {3,4} (2).(2) N: {10, 15, 20, 25, 30} with two two different networks (10).(3) F /W: �1/4,2/2� (2).

Thus, we have 20 different network instances and two fiber and wavelength combina-tions for each of them.Capabilities of network nodes:

(4) Rwc=Rls: {0, 0.5, 1} (3).

There are three different restrictions about the capabilities of nodes, and we have fivedifferent N {10, 15, 20, 25, 30} values. Thus, we have 15 different sparse splitting andsparse conversion node sets.Workload:

(5) S: {4, 6, 8, 10} (4).(6) Rmn: {0.25, 0.50} (2).

There are eight different combinations, and we have five different N values. Thus,we have 40 different multicasting sessions.

There are a total of 960 �=2�10�2�3�1�2� groups of sessions in the experi-ment, and 204 (22.4%) of them are found to be infeasible by CPLEX. The group block-ing probability measures the percentage of 756 �960−204� multicast groups, whichconsist of a different number of sessions {4, 6, 8, 10}, that are blocked. Similarly, weuse these 756 groups to calculate the session blocking probability. All multicast groupsare used for the calculation of running time, and all remaining metrics are calculatedonly for 652 groups, since LAMA, M-ONLY, and C-FWA heuristics successfully routeall the sessions in these 652 groups. Apart from the other heuristics, unicasting couldonly route 273 multicast groups, and only these are used for the calculation of non-QoS metrics (average bandwidth, delay, highest wavelength index, fiber and wave-length conversion, and number of trees). The CPLEX optimizer produces a solutionand a gap, and a lower bound (LB) on the problem can be calculated by subtractingthe gap from the cost of the current solution. Other metrics are also calculated for thesolution found by CPLEX.

The parameters of the LAMA heuristic are optimized to reduce the session blockingprobability. Figures 5 and 6 clearly demonstrate that LAMA is almost as good asCPLEX in terms of both the group and the session blocking probabilities. C-FWA isslightly better than M-ONLY, but they perform very poorly against LAMA in terms ofboth metrics. The average wavelength and fiber conversion performance of UNICASTseem to be better than the other algorithms. The reason is that UNICAST could onlyroute 273 groups, and these are the easier ones. Therefore, it consumes less resourcesfor easier problems. The poor performance of unicasting for other metrics shows howmuch we gain by multicasting.

The mean values, the upper and the lower 95% confidence limits of the mean for allother metrics are given in Table 3. LAMA gives the best results in terms of the aver-

Fig. 5. Group blocking probability metric for a different number of sessions �S� forLAMA, M-ONLY, C-FWA, and UNICAST (Percent).


age highest wavelength index, but CPLEX is superior to all others for the remainingperformance metrics. LAMA is statistically better than M-ONLY and C-FWA for theaverage highest wavelength index, wavelength and fiber conversions, number of trees,and the total cost percent gap. It uses 26.9% less wavelength, 7.7 times less wave-length conversion, 3.9 times less fiber conversion, 13.5% less number of trees (trans-mitters), and 2.5 times less total cost percent gap than M-ONLY. Moreover, LAMA is

Table 3. Average Bandwidth, Delay, Highest Wavelength Index, WavelengthConversion, Fiber Conversion, Number of Trees, Running Time in Seconds,the Total Cost Percent Gap Metrics’s Mean Values and, the Lower and the

Upper 95% Confidence Limits of the Mean for CPLEX, LAMA, M-ONLY,C-FWA, and UNICAST.

Metrics CPLEX LAMA M-ONLY C-FWA UNICAST

Averagebandwidth

Lower 9.56 9.98 9.96 9.94 10.67Meana 9.91 10.34 10.33 10.31 11.36Upper 10.26 10.70 10.70 10.68 12.04

Averagedelay


Average highestwavelength

index


Averagewavelengthconversion


Averagefiber

conversion


Averagenumber of

trees


Averagerunning

time


Total costpercent gap


aThe mean values are represented using boldface type.

Fig. 6. Session blocking probability metric for a different number of sessions �S� forLAMA, M-ONLY, C-FWA, and UNICAST (Percent).


only 19% worse than the optimal with 95% statistical confidence. We repeat the analy-sis with 99% statistical confidence, and it is also less than 20%. Similarly, it consumes27.9% less wavelength, 2.2 times less wavelength conversion, 3.3 times less fiber con-version, 12.6% less number of transmitters, 1.8 times less total cost percent gap thanC-FWA. The difference between M-ONLY and C-FWA is that C-FWA uses nearly 3.5times less wavelength conversion and 20.7% less fiber conversion, and it has slightlybetter results for the group and session blocking probabilities.

There is no statistically significant difference among LAMA, M-ONLY, and C-FWAin terms of the average bandwidth and delay, but CPLEX is sometimes superior toothers in terms of the average delay. Confidence limits seem to be wide for the averagebandwidth and delay, but we cover a very broad spectrum of parameters, and thisincreases the standard deviation. Moreover, we also examined the result for eachparameter separately, but we did not spot any significant difference.

The CPLEX running time is determined mainly by the number of nonzero integervariables, which are affected by D, N, S, and Rmn, and also Rwc�=Rls�. CPLEX totalrunning times are 18 min on average, and it takes only 2 s for the LAMA heuristic tofind high-quality solutions for all multicast sessions. M-ONLY and C-FWA use a verysmall fraction of a second to run. The running time of CPLEX is more seriouslyaffected by the number of sessions, contrary to LAMA, M-ONLY, and C-FWA, and it isalso unpredictably affected by the complexity of the problem, since there is a greatgap between the maximum and minimum running times for CPLEX.

Table 4 shows the total cost percent gap with respect to the number of fibers andwavelengths, the average nodal degree, and the fraction of multicast nodes. LAMAperforms better while increasing the number of fibers and decreasing the fraction ofmulticast nodes, since we are increasing resources and decreasing the traffic load (interms of the number of multicast members), respectively, for these two cases. How-ever, an increase in connectivity of the network (average nodal degree) slightly affects

Table 4. Total Cost Percent gap as a Function of the Number of Fibers „F…

and Wavelengths „W…, the Average Nodal Degree „D…, and the Fraction ofMulticast Nodes „Rmn… for LAMA, M-ONLY, and C-FWA

Parameters Values LAMA M-ONLY C-FWA

Fiber/Wavelength 1/4 22.24 45.87 33.59�F /W� 2/2 15.20 47.49 32.32

Avg. nodal degree 3 17.54 45.09 31.33�D� 4 19.29 47.82 34.02

Fraction of multicast nodes 0.25 15.29 40.38 28.85�Rmn� 0.5 23.35 55.91 38.84

Fig. 7. Total cost percent gap as a function of the number of nodes in the network �N�for LAMA, M-ONLY, and C-FWA.


the total cost of LAMA. C-FWA always performs better than M-ONLY with 41.9% lesstotal cost on average, and it behaves similar to LAMA with respect to the parameterchanges.

LAMA performs consistently better and its total cost percent gap changes slightlyin terms of the number of nodes as shown in Fig. 7. However, M-ONLY’s performancedeteriorates when we increase the number of nodes. C-FWA performs closer to LAMA,and there is not an increasing trend in terms of the total cost percent gap as a func-tion of the number of nodes in the network.

The performance of LAMA, C-FWA, and M-ONLY with respect to the number ofsessions and the percentage of nodes with wavelength conversion and light splittingcapability are given in Figs. 8 and 9, respectively. It is important to have consistencyin terms of the broad range of traffic load from very light conditions to conditions witha high blocking rate. All three algorithms seem consistent for a different number ofsessions. C-FWA performs better when we increase the wavelength conversion andlight splitting capabilities. Similarly, for that case, LAMA behaves slightly better, butM-ONLY’s performance deteriorates.

Fig. 8. Total cost percent gap as a function of the number of sessions in a multicastgroup �S� for LAMA, M-ONLY, and C-FWA.

Fig. 9. Total cost percent gap as a function of percentage of nodes with wavelengthconversion and light splitting capability for LAMA, M-ONLY, and C-FWA.


6. Conclusions and Future WorkThe future application scenarios will require a long duration of multicast sessionsthat needs large bandwidth, and all-optical wavelength-routed WDM WANs will bethe appropriate infrastructures. In this setting, we proposed a MILP formulation thatutilizes a layered graph approach for the static multicasting in all-optical wavelength-routed WDM WAN with multifibers, sparse wavelength conversion, and light splittingcapable routers. The formulation combines four different cost terms: delay, wave-length, fiber conversion costs, and the transmitter usage cost, and we can play withthe relative importance of these cost terms by changing their weights. Although theseparameters can be adjusted to be able to reflect the underlying relative cost of theseoperations so that all four different cost terms can be minimized simultaneously,LAMA can use them to optimize specific metrics (e.g., the blocking probability).

The adjustable parameters of LAMA make it very flexible to balance different objec-tives. Although LAMA, C-FWA, and M-ONLY have the same worst-case complexities,which may be important for the dynamic multicasting problem, LAMA runs slowerthan its competitors. However, it can be applied to moderate size dynamic multicast-ing problems without adjustment and produce partial solutions. Moreover, the run-ning time of LAMA linearly increases with respect to the number of sessions, contraryto CPLEX. In extensive computational experiments, we show that LAMA performsvery close to CPLEX and significantly better than M-ONLY and C-FWA in terms ofnearly all metrics, since the LAMA heuristic does not separate routing and wave-length assignment steps compared with the other candidates. Experiments also showthat important metrics (e.g., the session and group blocking probability, transmitterusage, wavelength, and fiber conversion resources) are adversely affected by the sepa-ration of routing and fiber-wavelength assignment phases in multicasting. Finally, wealso propose a new fiber-wavelength assignment strategy (Ex-Fit in C-FWA), whichuses wavelength and fiber conversion resources more effectively than the First Fit.

The performance of the LAMA heuristic is quite satisfactory in terms of quality, butthe processing time may be a problem for networks having many fibers and wave-lengths. It takes 1 min for LAMA to route one session when the number of nodes �N�is 30, and the number of fibers and wavelengths �F /W� are �1/32,2/16,4/8� (32 lay-ers). A time-optimized implementation of LAMA currently uses approximately 1 minfor the routing of a session when the number of nodes �N� is 30, and there are 64 lay-ers. Additionally, we also evaluated another version of LAMA which is the Fast-layered all-optical multicast algorithm. It reduces shortest path calculations signifi-cantly, but it also performs poorly compared with LAMA. Therefore, we currentlysolve the problem to optimality using CPLEX for small problems. We propose LAMAfor medium size problems and C-FWA for large problems. We have also completed thetesting of a modified version of LAMA so that its complexity is independent of thenumber of fibers and wavelengths. We are currently running experiments on this newversion. The layered graph approach is very flexible, and we can optimize differentmetrics by changing the cost assignment. This work includes the optimization ofblocking probability. We have recently finished another group of tests with a highernumber of fibers and wavelengths for the optimization of three different metricssimultaneously (the average bandwidth, delay, highest wavelength index). Similarly,we will design an experiment in which we find the minimum number of transmittersand minimum number of fibers used without a performance loss in other metrics.

References and Links1. L. H. Sahasrabuddhe and B. Mukherjee, “Light-trees: optical multicasting for improved

performance in wavelength-routed networks,” IEEE Commun. Mag. 36, 67–73 (1999).2. G. Sahin and M. Azizoglu, “Multicast routing and wavelength assignment in wide area

networks,” Proc. SPIE 3531, 196–208 (1998).3. R. Malli, X. Zhang, and C. Qiao, “Benefits of multicasting in all-optical networks,” Proc.

SPIE 3531, 209–220 (1998).4. M. Jeong, C. Qiao, and Y. Xiong, “Reliable WDM multicast in optical burst-switched

networks,” in Opticomm’00 (SPIE, 2000).5. C. Qiao, M. Jeong, A. Guha, X. Zhang, and J. Wei, “WDM multicasting in IP over WDM

Networks.” in Proceedings of International Conference on Network Protocols (ICNP) (IEEE,1999), pp. 89–96.

6. D. Yang and W. Liao, “Design of light-tree based logical topologies for multicast streams in


wavelength routed optical networks,” in Proceedings of IEEE INFOCOM (IEEE, 2003), Vol.1, pp. 32–41.

7. ILOG CPLEX: high-performance software for mathematical programming andoptimization, http://www.ilog.com/products/cplex/.

8. X. Zhang, J. Y. Wei, and C. Qiao, “Constrained multicast routing in WDM networks withsparse light splitting,” J. Lightwave Technol. 18, 1917–1927 (2000).

9. Y. Zhou and G. S. Poo, “Optical multicast over wavelength-routed WDM networks: asurvey,” Opt. Switching Netw. 2, 176–197 (2005).

10. M. Ali and J. S. Deogun, “Power-efficient design of multicast wavelength-routed networks,”IEEE J. Sel. Areas Commun. 18, 1852–1862 (2000).

11. M. Ali and J. Deogun, “Allocation of multicast nodes in wavelength-routed networks,” inIEEE International Conference on Communications (ICC) (IEEE, 2001), Vol. 2, pp. 615–618.

12. M. Ali and J. Deogun, “Cost-effective implementation of multicasting in wavelength-routednetworks,” J. Lightwave Technol. 18, 1628–1638 (2000).

13. X. D. Hu, T. P. Shuai, X. Jia, and M. Zhang, “Multicast routing and wavelength assignmentin WDM networks with limited drop-offs,” in Proceedings of IEEE INFOCOM (IEEE, 2004),Vol. 1, pp. 494–500.

14. Y. Xin and G. N. Rouskas, “Multicast routing under optical layer constraints,” inProceedings of IEEE INFOCOM (IEEE, 2004), Vol. 4, pp. 2731–2742.

15. B. Chen and J. Wang, “Efficient routing and wavelength assignment for multicast in WDMnetworks,” IEEE J. Sel. Areas Commun. 20, 97–109 (2002).

16. J. He, S. H. G. Chan, and D. H. K. Tsang, “Routing and wavelength assignment for WDMmulticast networks,” in Proceedings of IEEE GLOBECOM’01 (IEEE, 2001), pp. 1536–1540.

17. M. T. Chen and S. S. Tseng, “A genetic algorithm for multicast routing under delayconstraint in WDM network with different light splitting,” J. Inf. Sci. Eng. 21, 85–108(2005).

18. Y. Wan and W. Liang, “On the minimum number of wavelengths in multicast trees in WDMnetworks,” Networks 45, 42–48 (2005).

19. H. Honda, H. Tode, and K. Murakami, “Optical WDM multicasting design underwavelength conversion constraints,” IEICE Trans. Commun. E88-B, 1890–1897 (2005).

20. S. Sankaranarayanan and S. Subramaniam, “Comprehensive performance modeling andanalysis of multicasting in optical networks,” IEEE J. Sel. Areas Commun. 21, 1399–1413(2003).

21. J. Wang, B. Chen, and R. N. Uma, “Dynamic wavelength assignment for multicast inall-optical WDM networks to maximize the network capacity,” IEEE J. Sel. Areas Commun.21, 1274–1284 (2003).

22. C. Huang, X. Jia, and Y. Zhang, “A distributed routing and wavelength assignmentalgorithm for real-time multicast in WDM networks,” Comput. Commun. 25, 1527–1535(2002).

23. P. Garcia, A. Zsigri, and A. Guitton, “A multicast reinforcement learning algorithm forWDM optical networks,” in 7th International Conference on Telecommunications-ConTEL,Zagreb, Croatia (2003), Vol. 2, pp. 419–426.

24. C. Hsieh and W. Liao, “All optical multicast routing in sparse-splitting optical networks,” inProceedings of the 28th Annual IEEE International Conference on Local Computer Networks(LCN’03) (IEEE, 2003), p. 162.

25. J. Wang, X. Qi, and B. Chen, “Wavelength assignment for multicast in all-optical WDMnetworks with splitting constraints,” IEEE/ACM Trans. Netw. 14, 169–182 (2006).

26. S. Xu, L. Li, and S. Wang, “Dynamic routing and assignment of wavelength algorithms inmultifiber wavelength division multiplexing networks,” IEEE J. Sel. Areas Commun. 18,2130–2137 (2000).

27. T. Cinkler, D. Marx, C. P. Larsen, and D. Fogaras, “Heuristic algorithms for jointconfiguration of the optical and electrical layer in multi-hop wavelength routing networks,”in Proceedings of IEEE INFOCOM (IEEE, 2000), Vol. 2, pp. 1000–1009.

28. G. Maier, A. Pattavina, L. Roberti, and T. Chich, “Static-lightpath design by heuristicmethods in multifiber WDM networks,” in Proceedings of OPTICOMM (2000), Vol. 4233, No.1, pp. 64–75.

29. P. Winter, “Steiner problems in networks: a survey,” Networks 17, 129–167 (1987).30. F. K. Hwang, D. S. Richards, and P. Winter, The Steiner Tree Problem (Annals of Discrete

Mathematics) (North-Holland, 1992).31. T. Koch and A. Martin, “Solving Steiner tree problems in graphs to optimality,” Networks

32, 207–232 (1998).32. R. M. Karp, “Reducibility among combinatorial problems,” In Complexity of Computer

Computations, R. E. Miller and J. W. Thatcher, (eds.) (Plenum, 1972), pp. 85–103.33. E. Horowitz and S. Sahni, Fundamentals of Data Structures in Pascal (Freeman, 1988).

Multicasting for all-optical multifiber networks

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