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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004 1613

Topological Design Optimization of aYottabit-Per-Second Lattice Network

Jules R. Dégila, Student Member, IEEE, and Brunilde Sansò, Member, IEEE

Abstract—This paper deals with the topological design of a yotta-bit-per-second (1 yotta = 1024) multidimensional network. TheYottaWeb is a recently proposed architecture based upon agile op-tical cores that provides fully meshed connectivity with direct op-tical paths between edge nodes that are electronically controlled. Inorder to arrange the edge nodes around the agile cores (ACs) into asuitable and efficient YottaWeb, one proposal is to create a multidi-mensional lattice structure of ACs. The problem of designing sucha structure is highly combinatorial. In this paper, we present theproblem, that we call nodal arrangement problem, and we proposea meta-search procedure based on Tabu and VNS to solve it. Theperformance of the algorithm is gauged using a set of randomlygenerated networks with different distribution of traffic.

Index Terms—Agile optical core, lattice structure, meta-search,next-generation Internet, nodal arrangement problem (NAP), op-tical internet, PetaWeb, tabu search, variable neighborhood de-scent, YottaWeb.

I. INTRODUCTION

TO KEEP UP with the progression of the current In-ternet, new proposals led by the next-generation Internet

(NGI) initiative [1], supported by Defense Advanced ResearchProjects Agency (DARPA), have been considered. They con-sist of the development of protocols, standards and testbednetworks. One such development is an architecture called thePetaWeb [2]–[5]. The PetaWeb scales to a total capacity ofseveral petabits per second (Pb/s), three orders of magnitudehigher than the external capacity of the current global Internet.The concept of the PetaWeb is based on the development of anagile optical core (using the wavelength-division multiplexed(WDM) fibers and optical cross-connectors (OXCs) [5]) thatcan provide a high-capacity interconnexion between a transportnetwork edge nodes. It also allows to overcome the problemswith the current Internet by providing a direct high-capacityinterconnection between the edge nodes (see Fig. 1). ThePetaWeb architecture is intended to accommodate thousands ofsuch high-capacity edge nodes distributed nationwide.

Based on the motivation that a global high-capacity network,such as the Internet, could contain millions of high-capacityedge nodes. Beshai et al. proposed an architecture that couldreach an external capacity in the order of yottabit-per-sec-onds, called the YottaWeb [6]. The main idea of the YottaWeb

Manuscript received February 26, 2003; revised January 26, 2004. This workwas supported in part by a Collaborative Research and Development (CRD)Grant between Nortel Networks and the National Sciences and Engineering Re-search Council of Canada.

The authors are with the GERAD and the Department of Electrical Engi-neering, École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/JSAC.2004.829642

Fig. 1. Fundamental concept of the AC.

Fig. 2. General optimization model of the YottaWeb.

sketched in [6] is to use the agile cores (ACs) from the PetaWebas building blocks for an expanding network. Hence, theinvolved network design problem implies the need for theefficient use of the AC’s enabling technology. One importantparameter to reach the required efficiency, is the choice of agood topology which could allow tremendous growth of theglobal capacity and a low number of hops between edge nodes.

Fig. 2 illustrates the optimization challenge involved in thedesign of the YottaWeb: it consists in deciding which edgesnodes (ENs) should be connected to which AC. Such a solu-tion can be embedded into a virtual conceptual frame.

One such virtual frame is a lattice structure, proposed byBeshai et al. [6] for the YottaWeb. The structure is built onthe basis of the number of ACs attached to the same edge nodesand on the number of edge nodes that are connected to thesame AC. Thus, an important and highly combinatorial problemthat gives rise in this context is the nodal arrangement problem(NAP), which defines which edge nodes should be connectedtogether in the conceptual lattice structure.

0733-8716/04$20.00 © 2004 IEEE

1614 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004

Fig. 3. Structure of PetaWeb.

The object of this paper is to present a powerful metaheuristicfor the solution of the NAP. The algorithm was developed afterformally exploring the properties of the lattice structure for theYottaWeb, which is presented in this paper for the first time.This paper is organized as follows. In Section II, the basic ele-ments of the YottaWeb Architecture and their relationship withthe PetaWeb, the lattice structure, and the NAP are described.Section III is devoted to a literature review on problems that aresimilar or are related to the NAP. In Section IV, the resolution al-gorithm, that we have called Lattice Arrangement Meta-SearchProcedure (LAMP) for is presented in detail. Numerical resultsfollow in Section V and the conclusions and recommendationsfor further work are finally presented in Section VI.

II. YOTTAWEB TOPOLOGY AND THE

NODAL ARRANGEMENT PROBLEM

The enabling technology behind the YottaWeb proposal givenin [6], is the PetaWeb architecture and in particular, its high-capacity, distributed, edge controlled, optical core. Indeed, theconfiguration of the AC allows to consider one PetaWeb as asubnet of a greater network.

In what follows, we present the basic configuration of thePetaWeb, and we show how it gives rise to the YottaWeb ar-chitecture and the virtual lattice structure. The section is con-cluded with an explanation of the NAP, its definition, and someprevious algorithms that attempted at a solution.

A. PetaWeb

Fig. 3 (modified form [5]) sketches the structure of thePetaWeb, which is a composite-star network. The PetaWebuses a channel-switching core and a distributed control systemthat dynamically modifies the routing of individual channels asthe need arises. The access nodes to the core could be electronicswitches in packet switching or adaptative circuit mode, withhigh capacity of multiple Terabits 10 per second. Theoptical core nodes, which should be an array of parallel spaceswitches, provide fully meshed connectivity with direct opticalpaths between edge nodes. The connection of the ENs to theoptical core through the parallel space switches is depictedin Fig. 4 (similar to the one proposed in [5]). For the sake

Fig. 4. Parallel-planes optical core node in the PetaWeb.

of clarity, only two ENs are considered. As explained in [5],an edge node is connected to a core node with one or morefiber links, each link having several channels. Fig. 4 showsthe parallel space switches composing a core node. Incomingoptical signals are demultiplexed and associated to differentchannels. Outgoing channels associated with a given edge nodeare multiplexed into a fiber link going back to the edge node.

The concept allows information delivery at the optical rate,within the network, since the optical cores are bufferless withoutconnections between them. A time-locking mechanism coordi-nates the edge nodes, enabling the PetaWeb core to resembleone geographically distributed switch. From now on, to explainthe YottaWeb in Section II-B, we will use the term “agile core”(AC) to signify the set of high-capacity optical core nodes of aPetaWeb. This set of optical core nodes forming a PetaWeb isgrouped within the dotted circle of Fig. 3 and is symbolized bythe star at the right which represents an AC.

B. Lattice Topology of the YottaWeb

The YottaWeb structure defines a way of efficiently con-necting the edge nodes to the ACs. A schematic view isportrayed in Fig. 2. In the figure, the edge nodes can be con-nected to several ACs. The fundamental question of the designis to determine which are the best links for the connection; inother words, to determine which edge node should be connectedto which AC. Of course, the term “best” is used in reference tothe optimization metric chosen by the designer.

DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1615

Fig. 5. YottaWeb arrangement into lattice structure.

In the case of the YottaWeb, one of the objectives of the de-sign is that the connections can be carried out as directly as pos-sible from the origin to the destination node. Thus, a suitableoptimization measure relates to the number of hops between theorigin and the destination. Let be the setof edge nodes of the Network, and let bethe set of ACs. Let us also assume that there is traffic demandbetween the pairs of edge nodes. Then, the problem addressed isto find the minimum hop path between every origin–destination.Such a path is constituted only by links from the edge nodes tothe ACs, as no direct link exist between nodes or between ACs.The minimum hop should also be weighted by the demand inorder to favor the use of direct paths for pairs of edge nodes ex-changing high-traffic volumes. This will be further classified inSection II-C.

Bearing in mind that the addressing and the routing must be asquick as possible even in an expanding network, one proposal isto use the lattice structure to represent the topology. That struc-ture is pointed up by the notion of dimension of the network,which is defined as the number of ACs at which an edge nodeis connected to at the same time. Fig. 5(a) illustrates a unit-di-mension YottaWeb, where each of the four edge nodes is con-nected to only one AC, represented by a single line in the latticestructure.

The difference between the representation by the latticetopology and the traditional one is the use of a “tunnel” (line) tolink all the edges within a subnet, instead of arcs linking them.Assuming that two lines (ACs) can only intersect at one pointand defining them to be perpendicular to a regular lattice struc-ture (each AC is connected to exactly edge nodes, beingthe AC’s capacity), one could easily increase the dimension ofthe YottaWeb to 2 as in Fig. 5(b) and to more than 2 as needed.

A more complex example is portrayed in Fig. 6, where thereare two types of lines connecting the edge nodes: the light onesand the shadowed ones. Each type represents a different AC to

which the edge nodes in the line are connected. From this defi-nition of the lattice topology, we deduce some of its propertiesin Section II-C.

C. Some Structural Properties of the Lattice Topology

The lattice structure, as defined above, corresponds to a reg-ular form. In that form, each pair of lines representing an AC areparallel or perpendicular. In addition, every AC links the samenumber of nodes. Those assumptions could be different in anirregular form, which is not studied in this article. In addition,let us classify the ACs following the dimensions. An exampleis given in Fig. 6, where the AC’s 1, 2, 3, and 4 are the ACs ofthe 2nd dimension as they are parallel to the axis , and theAC’s 5, 6, 7, and 8 are named the ACs of the first dimension. Ingeneral, we define by AC of the th dimension, an AC parallelto the axis . The next proposition gives the conditions of theexistence of the regular lattice structure.

Proposition 1: Let , , and be respectively thenumber of nodes, the number of ACs, the dimension of the net-work and the capacity of each AC. For a regular lattice structurerepresentation, the following two equalities are satisfied.

1) .2) .

Proof: Both these two equalities come from the necessityof proportionality between the components (nodes, ACs, andtheir capacity) of a regular lattice topology as defined above.Indeed

1) Since there are edge nodes per ACs of each dimension,there are on the whole edge nodes in a latticestructure of dimension two and edge nodesin a regular lattice structure of dimension three. Then, indimension with , there on the whole

edge nodes.2) Let us consider , the subsets of the

ACs of the same dimension, as defined above. The readercould easily see that each class of ACs of dimension

covers the whole set of edge nodes, since each edgenode is connected to one and only one AC of each dimen-sion. Then, with nodes per AC, there are on the whole

divided by ACs per dimension. Finally, there areACs in the whole

regular YottaWeb.The structure of the lattice topology is comparable to the

topology of the generalized hypercube [7] defined for a multi-processor interconnection architecture. From that point of view,the lattice structure holds many interesting properties from thehypercube, while its advantage remains its simplicity. Indeed,the “tunnel” line linking the nodes simplifies the representation,and the manipulation of subnets defined by ACs.

In terms of global capacity, the range of the yotta-bits-per-second can be reached by following a simple calculation. Let usconsider that the nodes are grouped into a lattice structure andthat the access capacity per node could be equal to 1 terabit/s10 bit/s . Let us suppose 1000 edge nodes per subnet. In a

two dimension YottaWeb, the total external capacity within thenetwork is computed as 1000 1000 10 bits/s 10

exa bits/s. This total external capacity reaches 10


Fig. 6. 16-Node YottaWeb arrangement into two dimensions lattice.

zetta bits/s in dimension three and 10 yotta bits/sin dimension four.

D. Nodal Arrangement Problem (NAP)

The problem of arranging the nodes into the regular latticestructure, keeping the number of hops as low as possible, ishighly combinatorial. In fact, the size of possibilities increasesnonpolynomially for every node added to the network.

Formally, the nodal arrangement problem, which we callNAP could be stated as follows.

Given:

• set of edge nodes;• set of ACs;• capacity of each AC (that is equivalent to the maximum

number of nodes that can communicate at the same timethrough an AC);

• the dimension of the lattice structure;• , the traffic matrix between

the edge nodes.Find: the arrangement of the edge nodes into the lattice struc-

ture to optimize a certain performance metric.Performance Metric: There are several possible metrics that

can be used to optimize the design of the system. For instance, inthe works related to the well-known problem of logical topologydesign in WDM, several authors have used different types ofmetrics such as the mean hop distance, the maximal congestion,the queueing delay or the maximum offered load (see [7]–[10]).However, in this paper, we have chosen to concentrate on onemetric that links the directness of the routing with the impor-tance of the traffic carried.

The metric, that we consider is the traffic weighted mean hopdistance between the origin and the destination edge nodes.This is an interesting metric since minimizing it can lead to otherimportant benefits for the network and imply the optimization ofothers metrics. For instance, as it is mentioned in [9] and [10], inminimizing the traffic weighted mean hop, one maximizes theone-hop traffic metric.

In what follows, we describe previous algorithms designedspecifically for the NAP.

TABLE ISTEPS OF THE “GREEDY” HEURISTIC

TABLE IISTEPS OF THE TETRIS HEURISTIC

E. Previous Solution Approaches for the NAP

Two heuristics have been proposed up to now by the authorsof the new architectures [3], [5] to solve the NAP. These arethe “Greedy” and the “Tetris” heuristics based on the notion of“Friendship” between nodes which represents the “interaction”or the amount of the traffic between these nodes. Those heuris-tics are sketched in Tables I and II, where the objective of thearrangement is also to minimize the mean hop distance weightedby traffic demand.

These two algorithms start by a step by step construction ofthe lattice structure. The decision of selecting the next node, isbased only on “friendship” comparisons between the untreatednodes and a few number of the nodes that have been alreadyplaced (only one node in the case of the Greedy algorithm and atmost nodes in the case of the Tetris algorithm, where is thedimension of the lattice structure). The bubble sort at the thirdpoint of the Greedy algorithm (Table I) could be time consumingand not efficient. The authors have concluded that the Tetrisalgorithm is more efficient than the Greedy heuristic, as eachnode is placed only once.

In Section III, we explore the optical networking literature toplace the problem of nodal arrangement NAP addressed here,within its context.

III. LITERATURE REVIEW

The problem most similar to our NAP is what in the litera-ture has been coined as the optimal node assignment problem


TABLE IIISTEPS OF THE VND

(ONAP) [10]. In what follows, we emphasize the similaritiesand the differences between the NAP and the ONAP.

In the ONAP, the physical topology can be defined as theset of nodes and the set of physical (most of the time, bidi-rectional) links. The logical topology, also referred as virtualtopology or as lightpath topology, or as lightpath network [11]is the higher level in a WDM transport network. The logicaltopology consists of the set of nodes and the set of lightpaths.Each link in the logical topology is a directed lightpath. Fromthese definitions, the ONAP is generally applied to the designof the lightpath network, given a physical topology defined by:the number of nodes, the number of fiber links between pairs ofnodes and their length, the number of wavelength per fiber, andthe number of transmitters at each node. Given the demand ma-trix, a routing algorithm and a regular existing topology ( MSN,hypercube, shufflenet, multidimensional torus,…) and a perfor-mance metric, the goal of the ONAP is to find different routedlightpaths to accommodate the demand matrix. As it is shown in[10], the problem is known to be NP-hard and some heuristicshave been proposed to tackle it.

We now state the difference between the NAP and the ONAP.

• The ONAP addresses the design of lightpaths betweenedge nodes, while the NAP considers aggregate communi-cation “tunnels” (ACs) between ENs. The number of com-munication links to seek is drastically reduced in the NAP.This is extremely important for a very large high-capaci-tated network such as the future Internet.

• Another interesting feature of the NAP is the fact that it isdefined within a lattice topology. This allows the plannerto use the elegance of such a topology to address simplerrouting schemes. The lattice topology is in fact well usedfor such properties in different applications [12].

In Section IV, we explained in detail the proposedmeta-search procedure for the NAP.

IV. META-SEARCH PROCEDURE FOR THE

NODAL ARRANGEMENT PROBLEM

As previously stated, the difficulty of the YottaWeb latticeNAP comes from its huge combinatorial nature. An efficientway to improve the quality of the solution is to use a meta-heuristic, that is, a procedure consisting of a master strategythat guides and modifies others heuristics to produce solutionsbeyond those that are normally generated in the quest for localoptimality. In this work, we use a combination of two meta-

heuristics: Tabu Search (TS) from [13] and the variable neigh-borhood descent (VND) from [14]. They both refer to the pro-cedure for searching a solution among the set of feasible so-lutions , following certain rules and using the notion of theneighborhood structure. The goodness of the solution is evalu-ated by an objective function with being an element ofthe search space .

In Sections IV-A–E, we first review some basic propertiesof the TS and the VND. Then, we detail the tools of the Meta-Search Algorithm, called lattice arrangement metasearch proce-dure (LAMP) that we propose for the NAP.

A. Tabu Search (TS)

Initially proposed by Glover [13], TS is a procedure that ex-plores the solution space beyond local optimality, by avoidingcycling. TS uses a tabu list which is a set of the solutions de-termined by historical information from the last iterations ofthe procedure, where is fixed or is a variable that depends onsome properties of the problem. At each iteration, given the cur-rent solution and its corresponding neighborhood , theprocedure moves to the solution in the neighborhood thatimproves the objective function, while the moves that lead to so-lutions on the tabu list are forbidden. Then, the tabu list avoidsreturning to the local optimum from which the procedure hasrecently escaped. An additional basic element of the TS is theaspiration criterion, which determines when a move is admis-sible despite being on the tabu list. The termination criterionfor the TS is often based on a number of iterations without im-provement. The reader is referred to [13] for additional expla-nations of the TS and its efficient application to several classicalcombinatorial optimization problems. We now explain the VNDconcept.

B. Variable Neighborhood Descent (VND)

Proposed by Hansen and Mladenovic [14], the VND meta-heuristic refers to a local improvement with the use of severalneighborhood structures, instead of just one as is usually thecase. The extended neighborhood allows the method to escapelocal optima with respect to a smaller neighborhood.

Defining by the neighborhood struc-tures such that is the set of the solutions of the thneighborhood of . Table III presents the steps of the VND pro-cedure as it is designed by its author.

The loop step (2) of Table III is repeated at most times,starting at each iteration with a random solution . Then, a localsearch algorithm is performed to find the best neighbor of


Fig. 7. Illustration of d-moves.

, using neighborhood . If this local optimum is betterthan the incumbent, then set and restart the search with

else, the search proceeds with .Given that VND provides a way to efficiently explore mul-

tiple neighborhoods, it is particularly well suited for a meta-heuristic procedure for the NAP.

We describe in Sections IV-C–E, the neighborhood struc-tures, the evaluation function, and the fitting of TS and VNDprocedures to design our proposed procedure for the NAproblem.

C. Neighborhoods for the Lattice Structure

From now, let us consider as the search space representedby all the possible arrangements (by permutations) of the edgenodes into the lattice structure. Before explaining the neighbor-hood used in the search, and using the classification of the ACsgiven in Section II-C, we define three kind of moves: -moves,

-moves, and -moves. Fig. 7 illustrates the -movewhich performs node swapping within an AC of the th dimen-sion. Thus, edge node and are swapped within the ACnumber 1 by a 1-move. Also, a 2-move swaps the edge nodes

and within the AC number 5.In the same way, a -move represents a swapping of

nodes belonging to an AC of the th dimension with nodeswhich belong to an AC of the th dimension, respecting theorder of the nodes. This move is illustrated in Fig. 8 where,for the sake of clarity, only the concerned ACs are completelyportrayed. The other ACs are represented by dashed lines.

In the example, the AC1 and the AC7 are swapped. The op-eration is called a -move, since AC7 is an AC of thefirst dimension and AC1 is an AC of the second dimension. Theswapping of these two ACs, as it is sketched in the figure, isachieved by performing successively the moves , , ,and . In addition, we define an ac-move, depicted in Fig. 8,as the swapping of two ACs. This move leaves every node at itsplace and swaps only the ACs.

In the example of Fig. 8, agile core AC1 is permuted withagile core AC2, leaving the nodes , , , , , and atthe place they had before the operation. It is important to noticethat the ac-move does not influence the number of hops betweeneach pair of nodes, since the nodes remain in the position. How-ever, the ac-move is intended to vary a measure due to the lo-cation of the ACs, with respect to the edge nodes and could beused in a procedure assessing such measure.

Fig. 8. Illustration of d � d -moves.

With these notions, the following paragraphs define three cat-egories of neighborhoods for the LAMP.

Neighborhoods Definition: Let be one possible ar-rangement for the lattice structure and let us define asthe th neighborhood of . is a set constituted of ar-rangements that can be obtained by some -move(s) from ar-rangement .

Let us further define as a neighborhoodof arrangement . An arrangement is said to be an elementof the set , when can be obtained by - movesfrom arrangement .

Finally, let us define by , the AC’s neighborhood ofarrangement . An arrangement is said to be an element ofthe set , when can be obtained by few numbers ofac-moves from arrangement .

D. Objective Function

As stated before, the objective function that we consider in theoptimization process of the NAP is the mean hop value weightedby the demand .

We explain in this paragraph the evaluation of this function.To formally formulate that function, let

be the given traffic matrix, and the matrix ofthe distances (in hops) between the different pairs of location

of the arrangement . Using the same notionas in [10], we also define the binary-valued decision variables

, , . if node is assigned to locationin arrangement ; otherwise, . Assuming that only

the shortest path distance is used between nodes of the latticestructure, the distance matrix is symmetric, i.e, . Thus,the mean hop value weighted by the traffic is given by

(1)

which can be rearranged as

(2)

where represents the total amount oftraffic entering the network. The second expression (2) takesinto account the symmetry of the distance matrix. Such an equa-tion is useful in practice to differentiate the value of the trafficweighted mean hop.


TABLE IVSTEPS OF THE LAMP

E. LAMP Algorithm

Table IV shows the different steps of the proposed Meta-Search algorithm. This algorithm is based on the VND previ-ously reviewed, in which a TS is incorporated. Globally, TS isperformed using a first subset of neighborhoods , while an-other subset is used to escape local optimum and to diver-sify the search. All of the neighborhood structures defined inSection IV-C are considered. On the whole, there are neigh-borhoods structures of type , of type and

of type .The initialization, step 1, is generally performed by setting the

order in which the neighborhood structures will be examined.The ordering distinguishes two subsets of neighborhood types.The first subset contains the neighborhood of types , with

. The neighborhoods of type and thoseof type are both in the second subset. The ordering of thefirst set is random. However, the ordering of the neighborhoodstructures of types and depends on the objectivefunction and any preference between ACs and ENs.

The neighborhoods of the second class are used to diversifythe search, but most often they are used to perturb, in order toescape the local optima. The subsequent paragraph details theother main points of the algorithm.

Initial Solution: Different initial solutions could be chosenat step 1. In this paper, we consider a random arrangement as agood candidate because of its efficiency in time. We also foundit interesting to start the LAMP with a result given by anotherarrangement algorithm as the Tetris heuristic, for instance.

Tabu List: Given that the TS is done considering a neigh-borhood with which, the moves are done within an AC, thelength of the tabu list does not exceed the capacity of an AC. Thetabu list allows to direct the moves to avoid cycling. Hence, fora specific dimension , each edge node is moved at mosttimes.

Function Evaluation: The mean hop value weighted by thetraffic request is computed by considering the minimum dis-tance between each pair of nodes. This consideration does notrestrict the scope of the results as the lattice structure offersmany alternate minimum distance paths between every pair ofnodes which are not connected to the same AC.

The evaluation of the function is done once at the initial-ization, step 1. Thereafter, only the differentiate of the objec-tive function is computed. This differentiation is done, by usingonly the ENs and, or the ACs, which are involved in the moves.

TABLE VTEST PROBLEMS CHARACTERISTICS

Moves are accepted when the derivatives of the objective func-tion are under a certain very small threshold.

V. NUMERICAL RESULTS

In this section, we present numerical results to assess the per-formance of the proposed LAMP algorithm. Several scenarioshave been constructed for different test network problems in dif-ferent dimensions. Comparative node arrangement results havebeen computed using the Random, the Tetris, and the LAMP al-gorithms. In each case, the computed optimization metric wasthe mean hop traffic.

In Sections V-A–D, we detail each of the elements involvedin the numerical tests such as the test network problems and thedifferent traffic matrix generators.

Next, the test results are presented, drawn, and analyzed.

A. Test Problems

Table V presents the characteristics of the network problemschosen for the tests. We used seven problem tests (from to


TABLE VITWO TYPES OF TRAFFIC PATTERNS FOR AN EIGHT-NODES TEST NETWORK (CENTRALIZED TRAFFICS ARE HIGHLIGHTED)

TABLE VIICOMPARATIVE MEAN HOP VALUES AND PERFORMANCE IMPROVEMENTS WITH RVTG TRAFFIC

) of dimension two, seven (from to ) of dimensionthree, and six problems of dimension four.

For each problem, Table V gives the numbers of nodes ,the number of ACs , the dimension of the network, and thecapacity of the ACs .

B. Traffic Matrix Generators

To simulate the behavior of the YottaWeb, which is supposedto be as versatile as the Internet, two types of traffic generatorshave been used that are described as follows.

• Random volatile traffic generator (RVTG): With this,an exponential distribution traffic pattern is generated ran-domly between nodes. Afterwards, an index of volatilityis evaluated, to control how different traffics tend to varyvery often within the network. If such index is higher than

1, the generated traffic is volatile, otherwise, a new trafficmatrix is generated. Let be the arithmetic mean of thetraffic , , . Then

An illustrative example of a matrix pattern generated inthis fashion is given in Table VI(a), where the traffic isgiven in petabits per second. The outgoing and incomingports of each edge node have, respectively, 2 Pb/s as max-imal capacity. The total traffic within the network is ap-proximately 16 Pb/s.

• Random centralized traffic generator (RCTG): Here,about 1% of the edge nodes of the network are consideredto be server edge nodes with high intensity and the rest


TABLE VIIIREPARTITION OF THE TRAFFIC PER THE NUMBER OF HOPS WITH RANDOM TRAFFIC (RVTG)

present low intensity. In Table VI, an example of a central-ized pattern is given where the highlighted node numbersix holds for the server node.

For each of these traffic generators, a high-pass filterreduces the noises (in significant traffic) in the simulatednetwork.

C. Results

1) Experimental Conditions: The tests were performed ona 2-GHz Pentium 4 computer. The results computed for eachalgorithm and each test problem are the averages obtained byapplying the algorithm hundreds of times to that problem. Eachtime, a RVTG traffic matrix and a RCTG traffic matrix weregenerated. Then, Random, Tetris, LAMP (with random arrange-ment as initial solution), and LAMP (with Tetris arrangement asinitial solution) algorithms, were applied to each instance of testnetwork , , using each traffic matrix.

To further assess the performance of the given algorithms, aperformance improvement (PI) measure has been used for Tetrisand LAMP algorithms. PI represents the percentage of improve-ment of a given algorithm over the random solution. It is com-puted as

(3)

where represents the objective value of the solution givenby the specific algorithm (“algo”), and the objective valueof the solution given by a random arrangement.

2) Tables Description: Tables VII–IX show detailed resultsof these four arrangement algorithms applied to each instanceof the problem tested.

Table VII is organized in two parts. The computed mean hopvalues for a random volatile traffic type are presented in thefirst part, while the second part contains the same measure forcentralized traffic patterns. For each problem and each type oftraffic, we present the computed mean hop value, evaluated innumber of hops, the consumed CPU time, evaluated in seconds,and the computed , evaluated in percentage. The best com-puted mean hop value is highlighted for each test problem. Wewould mention that the maximum variance of the computedmean hop values is 1/1000.

D. Discussion

At this point, some major remarks can be made.

• In the case of random traffic, the lowest computed meanhop values occur mostly with the LAMP algorithm with aninitial Tetris arrangement. On the other hand, for the caseof centralized traffic, the lowest values are computed mostfrequently by the LAMP with a random arrangement asinitial solution. This remark gives some insights of the be-havior of the different algorithms with respect to the trafficpattern that is next explained when comparing graphicallythe algorithms performance.

• Given that a straightforward lower bound for the problemis one mean hop, one can say that, for the centralizedtraffic, and in particular for dimension two, optimal or nearoptimal solutions are attained.

• Also, since the variances of the computed mean hopvalues are insignificant, the use of the average meanvalues to assess the performance of the different algo-rithms is justified.


TABLE IXREPARTITION OF THE TRAFFIC PER THE NUMBER OF HOPS WITH CENTRALIZED TRAFFIC (RCTG)

Tables VIII and IX show the percentage of the traffic that isdelivered in a given number of hops using the four algorithms.One could see the repartition of the traffic given by the LAMPalgorithms compared with the repartition given by the Randomand the Tetris. Clearly, the use of the LAMP produces a hugeimprovement over the Random and Tetris solutions. The im-provement is significant for the one-hop traffic. For instance,for problem , 82% of the traffic found by the random solu-tion was three-hop and only 2% was one-hop. With the use of theLAMP, we were able to reach a solution where 46% of the trafficis one-hop and just 6% is three-hop, thus drastically reversingthe inefficiencies of the previous solution. We also underlinethat, once again, the beneficial effect of the LAMP is even morestriking for centralized traffic. A graphical view of the abovementioned results is summarized in Fig. 13. Comparing the evo-lution of the repartition of the traffic in each case, one could seethat the LAMP algorithms convert higher multiple hops trafficgiven by the Random and Tetris algorithms, to one-hop traffic.Thus, another performance assessment of LAMP algorithm overthe Tetris and the Random is given by the measure of the per-centage of one-hop traffic.

Figs. 10–12 that portray the mean hop values and the ,respectively, in dimension two, three and four, show that theLAMP algorithm is more efficient in any case, than the Tetris.We now point out some additional observations.

• If we evaluate the difference between the of the LAMPand the of the Tetris we see that, in dimension two, it isabout 15% [see Fig. 10(b)]. On the other hand, in dimen-sion four, it can reach more than 45% [see Fig. 12(d)].

Fig. 9. Illustration of ac-moves.

Thus, there is a great variability of the performance im-provement depending on the dimension.

• When the RVTG traffic is used, both the LAMP algo-rithm with the Tetris arrangement as an initial solutionand the LAMP algorithm with the initial random arrange-ment have a similar performance [see Figs. 10(b), 11(b),and 12(b)]. However, with RCTG traffic, the difference,in dimension two, between the performance of the LAMP(with Tetris) and the LAMP (with Random) becomes sig-nificant [see Fig. 10(d)]. Indeed, LAMP, with a randomarrangement as initial solution, outperforms the LAMPcombined with Tetris. This could be explained by the factthat, with a centralized traffic, the Tetris algorithm encoun-ters specific local optima from which the LAMP algorithmhas difficulty escaping thereafter.

• As previously mentioned,the LAMP algorithm couldreach a quasioptimal solution (with a gap of less than


Fig. 10. Comparative mean hops and performance improvements results for problems of dimension 2.


10 ), while the Tetris algorithm produces results that arealways higher than a 1.5 mean hop value.

Summarizing, we have evaluated the performance of theLAMP over the Tetris and Random using several measures.



Fig. 13. Evolution of the repartition of the traffic per the number of hops for random and centralized traffic pattern.


• performance improvement with respect to the randomgrouping;

• mean hop value given over 100 instances of the same testproblem;

• percentage of one-hop traffic for a given test problem;• CPU time taken by each algorithm.

In all the cases, except for the CPU time, the LAMP pro-vides significant improvement with respect to the Tetris and theRandom performance.

VI. CONCLUSION

The great contribution of the YottaWeb architecture is its ex-pandability and reduction of intermediate nodes between edgenodes. The proposed lattice structure for a certain dimensionnetwork guarantees that the number of hops for any O/D pairwould be at most . However, the aim of the planner shouldbe to reduce such number of hops to the minimum. This workhas shown how to reduce the mean hop values, weighted by thetraffic, for communications within the network, and that for net-works of more than 4000 nodes in dimension four. In dimen-sions two, three, and four, the LAMP has a superior perfor-mance than the Tetris or the Random grouping. Moreover, wefound that the performance improvement of the LAMP is evengreater when the traffic is centralized. This is particularly impor-tant given that, as the network increases in size, concentrationof traffic to some major origin-destinations is quite likely. Asfurther work, we are currently exploring lower bounding proce-dures for the method and studying update and expansion issuesrelated to the NAP.

ACKNOWLEDGMENT

The authors are grateful to M. Beshai and F. Blouin of NortelNetworks for their fruitful comments and for lending us theirtraffic generator, to M. Labonté who helped with the implemen-tation of the Tetris algorithm, and to the anonymous referees thathelped improve the final version of the paper.

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Jules R. Dégila (S’01) received the B.S. degree inapplied mathematics from the University of Dakar,Dakar, Senegal, in 1996 and the M.S. degree in op-erations research from the University of Sherbrooke,Sherbrooke, QC, Canada. He is currently workingtoward the Ph.D. degree in electrical engineeringat the École Polytechnique de Montréal, Montréal,ON, Canada. His Ph.D. dissertation was conductedin collaboration with Nortel Networks.

His research interests include modeling, next-gen-eration network optimization, optical networking,

large scale system, and performance analysis.

Brunilde Sansò (M’92) was born in Rome, Italy, in1960. She received the E.E. degree from the Univer-sidad Simon Bolivar, Caracas, Venezuela, in 1981,the M.S. degree in reliability and the Ph.D. degreein operations research from École Polytechnique deMontréal, Montreal, QC, Canada, in 1985 and 1988,respectively.

After Postdoctoral studies at the CRT, Universityof Montreal, and a Research Fellowship at theGERAD, she joined the faculty of École Polytech-nique de Montréal in 1992, where she has been a

Full Professor since 1997. She is currently with the Department of ElectricalEngineering, where she is the Director of the LORLAB, a research laboratorydevoted to the performance, reliability, design and optimization of operationalplanning of broadband networks. She is Co-Editor of the book Telecommuni-cations Network Planning (Norwell, MA: Kluwer, 1998) and the forthcomingbook Performance and Planning Methods for the Next Generation Internet(Norwell, MA: Kluwer).

Dr. Sansò is a recipient or co-recipient of several awards and honors, amongthem, the 2003 DRCN Best Paper Award, the Second Prize in the 2003 CORSPractice Competition, the 1995 IEEE/ASME JRC Best Paper Award, the 1992NSERC Women Faculty Award, and the 1992 FCAR Young Researcher Award.She is an Associate Editor of Telecommunication Systems and has been a refereeand technical committee member for major journals and scientific conferences,reviewer for government agencies, and industry consultant. She was the Pro-gram Co-Chair of the Fifth INFORMS Telecommunications Conference.

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