a survey of topologies and performance measures for large-scale networks

IEEE Communications Surveys & Tutorials • Fourth Quarter 200418

ollowing the progression of the Internet, communica-tion networks are growing and developing rapidly. Newservices are being required and developed. This evolu-

tion involves the improvement of the performance of theexisting network technologies. Also important is the need tofind efficient scalable networking methods at different levels:topological layout, access rules to the communication medi-um, traffic routing, transport procedures, applications han-dling, etc. An important point for achieving good results is tosuitably choose the network topology at the beginning. Thetopological layout defines the schematic description of thearrangement of a network, including its nodes and connectinglines. For an in depth introduction to topological networkdesign problems, readers are referred to the book by Sharma[5].

The topological layout is an important factor for reducingmany bottlenecks that happen when dealing with large-scalenetworks. Those bottlenecks could be congestion at nodes oron links, when a network evolves to a larger size. Thus, thetopological design of the envisaged versatile, scalable, andhigh-capacity networks requires the use of meaningful andreliable performance measures that help the designer in pre-dicting and tuning the behavior of the networks. The perfor-mance measures become important decision tools to decidethe best topology, and must therefore be carefully chosen.

This article aims to survey and discuss both performance

measures and topological proposals for expanding high-capac-ity networks. In addition, we discuss optimization tools fordesigning topologies that meet the requirements of the chosenperformance measures.

The content of this article could be summarized in foursections. The first section is devoted to the definition of con-cepts and notations of topological design. The second sectionconcerns the definition and the classification of fundamentalperformance measures. The most important metrics are sur-veyed and grouped into two sets, structural and functional,and they will be defined later. The third section is about thechoice of topologies in both physical and higher levels. Exist-ing topologies could be classified into two large groups: arbi-trary and regular topologies. Moreover, in light of the abovecited topological studies, we evaluate the new proposed regu-lar YottaWeb lattice-based topology [1–3]. The lattice topolo-gy is constructed by aggregation of connection arcs betweenedge nodes into “tunnels” between those edges nodes. Theconcept of the YottaWeb is enabled by recent advances inagile optical core technology, and it is well suited to large-scale optical networks. In the fourth section, the optimizationtools used in topological design are surveyed. In this sectionwe describe the application of the measures to the choice ofthe topologies. The practical use of the performance measuresin designing the topologies is presented.

This article is organized as follows. Concepts and illustra-

F

S U R V E Y SI E E EC O M M U N I C A T I O N S

T h e E l e c t r o n i c M a g a z i n e o fO r i g i n a l P e e r - R e v i e w e d S u r v e y A r t i c l e s

JULES R. DÉGILA AND BRUNILDE SANSO, GERAD AND ECOLE POLYTECHNIQUE DE MONTRÉAL

ABSTRACTThis article surveys important parameters for the design of large-scale network

topology such as the YottaWeb topology [1–3]. First, a wide range of performance measures to evaluate the behavior of envisaged topologies are

presented, discussed and classified according to their meaning and their effectivenesson large-scale networks. Second, different types of topologies, from simple to morecomplex, are identified, and the features of k-ary n-cube topologies [4], such as ring,torus, and hypercube topologies, are surveyed and discussed. Full details and advan-tages of the recently introduced YottaWeb topology are pointed out, in the light of

the predefined concepts. Finally, the application of the performance measuresto the design of the topologies is surveyed.

A SURVEY OF TOPOLOGIES ANDPERFORMANCE MEASURES FOR

LARGE-SCALE NETWORKS

FOURTH QUARTER 2004, VOLUME 6, NO. 4

www.comsoc.org/pubs/surveys

IEEE Communications Surveys & Tutorials • Fourth Quarter 2004 19

tions of topological design are given. We deal with the surveyof fundamental performance measures. We devote a sectionto the classification of some usual topologies compared to thenew proposed lattice topology. The optimization strategies arepresented, and results are discussed. General conclusions arepresented in the final section.

TOPOLOGICAL DESIGN:CONCEPTS AND NOTATIONS

Before surveying the fundamental performance measures, letus define some basic concepts, notations, and assumptions.

PHYSICAL AND LOGICAL TOPOLOGY

In telecommunication network design, the topological designintervenes at two levels: the physical level and the logical level.The physical topology can be defined as the set of physicalnodes and the set of physical links that connect the nodes.Links can be counted, for instance, as distinct optical fibers ordistinct physical channels that connect two adjacent nodes.

The logical topology [6, 7], also referred to as virtual topol-ogy [8, 9], lightpath topology [10], lightpath network [8], IPtopology, and optical topology [11], is a higher level. As anillustration, in WDM networks signals are routed and switchedbased on wavelengths composing the fibers. The current tech-nologies allow more than 150 distinct wavelength channelswithin the same fiber. The logical topology consists of the setof logical switching nodes and the set of lightpaths, where alightpath refers to an optical path between two adjacent nodes,formed by a sequence of physical fibers and associated wave-length(s). Each link in the logical topology is a directed light-path. An illustration of the physical level and its embeddedlogical topologies is shown in Elmirghani and Mouftah [12], towhich the reader is referred for a complete description ofWDM network technologies and architectures for scalablenetworks. In Fig. 1, which we retrieved from [12], an exampleof a five-node network is given, with three wavelengths per bi-directional link (solid lines) between the nodes. Figure 1ashows that there is no direct physical link between nodes Cand E. However, based on the wavelength assignment to thelightpaths (dotted lines), a direct logical path between nodesC and E is established in the resulting logical topology of Fig.1b.

Most of the time in the literature addressing the particularcase of optical transport topological design, the physical topol-ogy already exists or is chosen, and the challenge consists ofdesigning the logical topology to be embedded onto the physi-cal one. The independence of the logical topology from thephysical topology has been underlined in Mukherjee et al.[10], Banerjee et al. [6], and most recently in Mellia et al. [7].

TOPOLOGY REPRESENTATION

The topology is generally represented by a graph. Figure 2gives an example of the configurations of physical and logicaltopologies of a typical transport network. Using the notationintroduced by Plante and Sansò [13], the network layer l isrepresented by the graph G l(V l, E l) where V l is the set ofnodes and E l is the set of links between nodes. The pair ofnodes (x, y) in G l +1 is connected by a set of routes R in G l

consisting of a sequence of adjacent links e in E l, with R ={e1, …, ei}.

For the remainder of this article, we consider the two net-work sets using the preceding notation as:

G0 : the physical layerG1 : the logical layerWe also use the notation of graph G(V, E) (without any

subscript) to refer to any of the predefined physical or logicallayers. Let |V| = n be the number of nodes and |E| = m thenumber of links within the network. Unless otherwise stated,physical edge nodes here could independently represent all-optical network access nodes, optical core routers, or switches.The difference between physical and logical nodes generallyarises from the way logical connections are chosen.

For example, when in the physical network the switch at anode is already tuned in order to enable one-hop communica-tion between two adjacent nodes to the given switch, the node

n Figure 1. Wavelength routing networks: a) physical topology and wavelength assignment; b)virtual topology. (Retrieved from [12].

(a)

λ1

λ1

λ1

λ2 λ2

λ2λ3

λ3

ED

C

B

A

(b)

λ1

λ1

λ1

λ2

λ2

λ2

λ3

λ3E

D

C

B

A

n Figure 2. The configurations of physical topology and logicaltopology of an optical network.

Physical optical node Optical switching node

Logicallevel

Physicallevel


related to the switch could be ignored in the logical topology.An illustration is given in Fig. 2 with the number of nodes nreduced from seven to four, from the physical to the logicalnetwork.

On the other hand, we consider in this article that arcsbetween two edge nodes are bi-directional at both the physicaland logical levels. This assumption is well admitted in the lit-erature, specially for physical optical topology, for which theassumption matches the reality of optical fibers that are gen-erally deployed per pair, for communication in the two direc-tions.

In what follows, most of the performance measures definedare related to the logical topology. Indeed, most of the time,any logical layout can be put on top of a physical layer; theperformance of the whole network depends most intimatelyon the logical layout. However particular attention must bepaid to the physical constraints.

PERFORMANCE MEASURES

The goals of the performance metrics are to provide tools forthe classification, comparison, and evaluation of the featuresof the studied topologies. Numerous metrics exist in litera-ture, but only those that are fundamental for the high-capacityoptical topologies in this study will be presented and dis-cussed.

Based on the design concepts pointed out later, the perfor-mance measures involved in a topological design problemcould be classified into two categories: structural metrics andfunctional metrics. Structural metrics are designed to charac-terize the topology without any knowledge of traffic engineer-ing (flows on the links, routing schemes, etc.). For example,the considered number of nodes within a topology is a struc-tural metric. On the other hand, functional metrics take intoaccount the traffic engineering within the network, and theexpression of a functional metric is a function of parameterssuch as flows on the links, routing schemes, node functionali-ties, etc. Thus, the total amount of traffic within a network isconsidered as a functional metric.

Since many of these measures are interrelated, we willgroup them around the most fundamental and explain therelationship between them.

In following sections we present for each metric or groupof metrics a formal definition, its utilization, and its impact ingauging and comparing topologies.

THE STRUCTURAL METRICS

Some fundamental structural metrics encountered in litera-ture are the following.

The Number of Nodes n: This metric is also known as theorder of the network. This is a basic metric that is required tobe flexible, especially in a large network, in order to facilitatenetwork expansion. A “good” topology must easily enable thesimple operations of addition or withdrawal of a single node.Some authors [6] refer to the required flexibility for the orderas the extensibility, which allows a topology to easily expandwhile preserving its desired features.

In terms of large networks, we consider in this work that n ≥ 1000.

The Diameter D and the Degree ∆: The diameter andthe degree are, respectively, the longest distance (generallyexpressed in number of hops [14] or in fiber length [15])between a pair of nodes and the maximum number of linksattached to a node within the topology. When the diameter isexpressed in fiber length [15] it can be seen as an indicator of

the network cost, and then must be as low as possible. On theother hand, the degree is directly linked to the number ofports at a node, and then represents not only an indicator ofthe amount of functionalities of the node, but also an indicatorof the cost of the node. The degree is related to the capacity ofnodes and is an index of the node complexity.

Thus, for an appropriate design a trade-off is necessarybetween the needed increasing capacity and the cost or thedegree of a topology. Also, lowering the diameter and thedegree at the same time is contradictory, since the two mea-sures evolve in opposite directions. Thus, there is a need for ametric that combines the previous parameters. For instance,there is the cost effectiveness defined in [16] as C = ∆ × D.Other authors [17] define the node degree distribution, which isthe average node degree computed as 2m/n.

The relationship between the order, the degree, and thediameter could vary widely. However, there is the well knownMoore’s bound [11] nMoore(∆,D), which represents an upperbound for the order of a topology with given degree anddiameter. In fact, the fundamental goal of the design is to findthe densest possible topology (graph) for which, given thedegree and the diameter, the order is equal or close to:

(1)

Important progress has been made in constructing suchtopologies, specially in the field of multicomputers [18], butresults are still far away from Moore’s bound. For now, mostof the proposed constructed topologies that are close toMoore’s Bound, for n > 1000, are impractical because of theircomplexities [18].

The Average Inter-nodal Distance D—: Rather than thediameter, the average inter-nodal distance D

–is often considered

[14]. It is calculated as

(2)

where δ(x,y) is the distance (in hop or fiber length) betweenthe pair of nodes (x,y) . Such a metric has been used by Sen etal. [19], Banerjee et al. [6, 20], and Li and Ganz [21]. Thisstructural metric leads to a functional metric when the trafficwithin the network is taken into account. The derived extend-ed metric will be presented later as applied to functional met-rics.

The Number of Links m: This metric corresponds to thenumber of arcs in the graph and is equal to 1/2 Σi=1

n ∆i, where∆i is the degree of node i. The number of links is a cost factorand must be reasonably low. It is equal to n∆/2 for regulartopologies. The number of links is also an indicator of therequired number of wavelengths, and then the number ofrequired optical fibers in the case of optical networks.

The Fault Tolerance: Fault tolerance is counted as thenumber of alternate paths (with disjoint nodes and disjointarcs) existing between each pair of source and sink nodes. Ahigh fault tolerance is desirable for high-capacity topologies.This number can be expressed as the maximum number ofnodes and/or links that can be removed (or broken) from thenetwork without disconnection [6]. In brief, a good designaims at a logical connected topology where the traffic couldbe easily re-routed in case of faults in the network. More pre-cisely, Oh and Chen [22] have defined the metric of strongfault tolerance to characterize the property of parallel routing.A network G with degree ∆ is said to be strong fault tolerantwhen with at most ∆ – 2 faulty nodes within the network, eachremaining pair of nodes is still connected by min{∆ f (x), ∆ f (y)}

D =∈

∑ δ ( , ),

,

x y

nx y V2

n i

iMoore( , ) ( )∆ ∆ ∆D

D= + −

=

−∑1 1

0

1


disjoint paths, where ∆ f (x) and ∆ f (y) are the numbers ofneighbors of the nodes x and y, respectively, that are notfaulty. Due to the importance of satisfying this metric, manymethods have been studied to guarantee a high fault toler-ance.

The Regularity of the Topology: Regularity is also anattractive feature, and is defined as a symmetry in the distribu-tion of the nodes, one compared to others. Many regulartopologies, such as hypercubes, torus, etc., have been pro-posed. In a later section we will present their advantages anddrawbacks while surveying particular regular topologies.

FUNCTIONAL METRICS

We list in this section the functional metrics used to charac-terize the topologies.

The Traffic Weighted Metrics: Instead of using thenotion of distance presented above, [21] proposes a metric ofprobable average distance or weighted average distance calculat-ed as:

(3)

where Φ(δ) holds for the probability that an arbitrary message(packet) transmitted from a source node to a sink node isdelivered in a distance of δ hops. Li and Ganz [21] remindthat the probability Φ(δ) takes into account not only thetopology of the network, but also the routing schemes to beused. The probability could also be replaced, in a more pre-cise manner, by the fraction of the real traffic, when this frac-tion is known. The authors deduce from their analysis that asmall average inter-nodal distance D

–is not a guarantee of a

small communication delay. Indeed, packet delivery delayscould be long, despite a small D

–, according to the loading and

the processing of traffic at the nodes. Then, more sophisticat-ed measures are needed to catch the network complexity.

The Total External Traffic: The configuration of thetopology determines the total external traffic Tt that could beaccommodated by the network. This measure can be on theorder of several peta (1015), and even yotta (1024) bits per sec-ond [1] in high-capacity networks. The total external traffic issometimes defined by incorporating in it a certain quality ofservice within the network. Then, for the study of the PetaWebthat gives rise to the YottaWeb, Blouin et al. [15] define Tt asthe highest traffic load that a network could support, whileproviding a specific level of service such as no more than fivepercent of blocked channels requests occur for any pair of ori-gin-destination, and no more than 1 percent of blockingoccurs throughout the entire network.

Node Processing-Related Metrics: In [21] other function-al measures, mostly related to hardware performance, havebeen defined. One could name the average processed traffic pernode γ, the density of traffic per node µ, and the processing effi-ciency ρ of a node. These measures impact the performanceof the entire network, and play an important role in networkoptimization. They are also dependent on the previously citednode capacities.

Flow Number: With the effort to construct measures thatmore accurately guide the behavior of optical networks, Sen etal. [19] proposed the flow number, fn, which is a measurerequired to be small for a desirable network, as it is an indica-tor of the total number of wavelengths needed to reach fullconnectivity. Formally, let us define fn for a graph G(V,E),where R1, …, Rk are the set of all possible paths destined toroute traffic from any source to any destination. The trafficflow is set as dx,y = 1 ∀x, y, x ≠ y. Let T be the matrix of

dimension m × k where each entry T [i, j] indicates the num-ber of flows traversing link ei under the routing procedure Rj.Then the flow number of the graph G = (V, E) is computedas:

(4)

The authors specify that the flow number defined above isclose to the load metric defined in [23], but with the differ-ence that the load metric is a function of three parameters:the optical path requests (represented by pairs of origin-desti-nations), the routing procedure used to establish the opticalpaths, and the topology of the network. Indeed, the flow num-ber would depend only on the topology of the network.

To conclude this list of significant performance measuresused in topological design, let us say that multiple combinedforms of measures are also presented in the literature. Thosecombined forms are used to evaluate certain aspects of thenetworks that only one metric could not evaluate in a preciseway [24].

USUAL AND YOTTAWEB TOPOLOGIES

The optical network topology proposals encountered in theliterature could be considered from different points of view.As a first consideration, flat versus hierarchical topologies aredesigned. The important obstacles in designing such topologystructure, especially for large networks, is the very high num-bers of variables involved. The modeling of networks such asthe Internet could lead to a problem size of at least thousandsof nodes, requests, and paths. Most of the procedures to opti-mize the design are focused on bottlenecks caused by size.Hierarchization has been opposed to the flatness of the graph,with the goal to divide and conquer the size difficulties. It is aquestion of classifying nodes following different functionali-ties.

Indeed, in the flat topology, depicted in Fig. 3, each nodecould communicate in a peer-to-peer fashion, where thenodes are equipped with the same functionalities. The mainadvantages of the flat topology is to enable more obviousaddressing and routing procedures, even when the size of thenetwork increases.

On the other hand, the hierarchical topology comprises atleast two levels, as depicted in Fig. 4. Indeed, at the lowestlevel there are peer-to-peer exchanges between nodes, butsome of the nodes hold for gateways for a network of higherlevel. In this case, the functionalities at the nodes are differ-ent. The problem with the hierarchical topology is that thelimitation of the hierarchical structure is quickly reached atthe level of the intermediate nodes, when the size of the net-work increases.

On the other hand, the usual topology proposals, flat orhierarchical, could be classified into two different types: arbi-trary topologies and regular topologies.

The arbitrary topologies are represented by graphs withouta known structure, while the regular topologies are well struc-tured and present a certain symmetry between its compo-nents. Figures 5a and 5b provide examples of four-nodearbitrary and regular topologies. The regular topologies arepopular in the literature, since they possess interesting, pre-dictable properties such as easy addressing and routing. How-ever, they present some limitations, especially in the case oflarge-scale networks where some performance measures arevery sensitive, or can be affected by the size of the networks.

Using the notation above defined for n and m, the totalamount of possible non-oriented graphs is

fn G T i jj k i m

( ) min [ max [ , ]].=≤ ≤ ≤ ≤1 1

DD

prob ==

∑ δ δδ

Φ( ),1


which corresponds to the m non-null entries in the triangularconnection matrix of size n × n. In these possible topologies,there is an important number that does not have any knownor apparently obvious recognizable structure: these are the socalled arbitrary topologies. In the following sections we pre-sent the outcomes of each topology group.

ARBITRARY TOPOLOGIES

Some authors have studied the possibility of producing ran-dom arbitrary topologies [25] in designing optical networks.As it is noted in Rose [26], the use of full random graphs witha given number of links randomly distributed presents in prac-tice some difficulties. For example, the connectivity, which isan important characteristic, cannot be insured by a randomgraph. Thus, many authors assume additional constraints thatlead to a “semi-random” or “semi-arbitrary” graph.

Although they often produce excellent performance in termsof the mean hop value, arbitrary topologies require more com-plex routing procedures. Moreover, some authors consider thatthe total external capacity of arbitrary topologies is low [26].Furthermore, sufficient fault tolerance is not guaranteed. Rose[26], under certain conditions, has recommended the use of“semi”-random topologies, for their low mean hop value. Other-wise, network designers prefer well structured regular topolo-gies, which are defined and discussed in the next paragraph.

REGULAR TOPOLOGIES

Among the well structured, regular topologies, one can distin-guish two classes: simple structures and composed structures.The first class consists of simple topologies such as the ring,the bus, and the stars. The composed structures are moreelaborate structures, which are the repetition of simple struc-tures in a regular fashion. In the following paragraphs, we firstdefine the different topologies, the we describe their struc-tural behavior for large-scale networks.

For this survey, we have chosen a progressive approach tointroduce the topologies. The topologies are presented start-ing with simple structures and moving to more complex struc-tures, with the complex topologies derived from thecomposition of the simple topoloogies. This approach reveals

the method to progressively improve the topologies by addingnodes or links.

The Simple TopologiesThe Bus: This configuration is given in Fig. 6b, where the

five nodes are connected one after one, with a bi-directionallink.

Structural behavior: As in the previous topology, the busoptical topology also has the great advantage of being simple.Moreover, it requires a few number of links (n – 1) to beoperational and to be extended, with low cost. However, theoptical signal attenuation problem due to the length of thefibers could quickly become significant. In addition, a networkbased on a bus topology could quickly be the victim of conges-tion, since all the traffic passes through the same link. More-over, the problem at one node is easily transmitted to theothers, or could isolate a whole part of the network.

Thus, the bus topology is not appropriate for networkexpansion. The degree is 2 while the diameter is equal to thenumber n – 1 of links. In comparison to the ring, the meanhop distance in a bi-directional bus topology is of n/3 nodescompared to n/4 nodes for the ring [6].

The Ring: Depicted in Fig. 6a, the nodes are connected ina loop configuration.

Structural behavior: Because of its regularity and its sim-plicity, the ring allows all nodes equal access to the network,even when the number of nodes increases. However, the lackof reliability is an important drawback, as ring topologies donot provide enough alternative paths between their nodes.Also, the simple operation of adding and removing a nodefrom such a topology involves the interruption of the commu-nication within the network. Therefore, ring topology is notextensible, and is not appropriate for large networks. Howev-er, its simple interface, and its unidirectional version is mini-mal, in the sense that it requires a minimum number of linksto reach full inter-connectivity. Degree in the simple ring isconstant and equal to 2, while its diameter is about n/2 (forlarge n). As an illustration, the mean hop distance in a bi-directional ring topology is n/4 nodes.

The star: In this simple topology, illustrated in Fig. 6c, thenodes are connected through the same central node.

Tn

rand

m

=

2

2

n Figure 4. Sketch of a hierarchical topology.

n Figure 3. Sketch of a flat topology.

n Figure 5. Arbitrary and regular topologies.

(a) Arbitrary topology (b) Regular topology


Structural behavior: When compared with the ring topolo-gies, the standard star-based topology presents the drawbackthat it requires more links, and thus more optical fibers. Also, afaulty central node implies the rupture of communication withinthe network. The degree of the central node is n – 1, while it is1 for the other nodes. However, the star has the advantage ofbeing simple and it is easy to add new components, or to modifyexisting devices. The monitoring and management of the starcould be centralized. In addition, the rupture of a link does notaffect the whole network. Moreover, the central node could beswitch-controlled from the edge node as it is in the “composite-stars” of the PetaWeb [15] discussed earlier.

In the subsequent sections we give descriptions of interest-ing composed regular topologies, and we discuss their perfor-mance according to several metrics.

The Composed Topologies: The composed structure iscalled regular since, for each node, it presents the same viewof the topology. The reasons to recommend the choice of aregular topology are multiple [8, 21]. Before presenting exam-ples of such composed regular topologies, let us enumeratetheir important common desirable characteristics:• They enable simple routing procedures, so that in most

cases there is no need to have a routing table at thenodes. Node addresses are sufficient for each traffic flowto self-route to the destination. Network engineering isthen simplified.

• They provide a small diameter and a rather low meanhop value.

• Their fault tolerance is good, given that many alternativepaths between nodes are provided.

• They facilitate network reconfiguration of the networkwhen the traffic varies.On the other hand, the drawbacks of the composed regular

topologies could be listed as follow:• The resource consumption is sometimes too gluttonous

[21], resulting from the use of several wavelengths tobuild the regular structure.

• The composed regular structure is in practice moreappropriate to uniform traffic [8]. Hence, its perfor-mance decreases when the traffic pattern is not uniform.

• The order of the network n is rigid, since it varies propor-tionally with other resources, thus preventing a softexpansion of the network.Despite all these disadvantages, the properties of regular

composed topologies remain very attractive for optical net-works. A wide range of proposals exist in the literature,including the hypercube [27, 28], the metacube [29], the gen-eralized mesh [19], the d-neighbors regular topology [14], themulti-dimensional torus [20], and the Shufflenet [6, 30–32].Also, in order to circumvent the weak points of these topolo-gies, “semi”-regular topologies have been derived from theprevious topologies.

In the following sections we present some of the most usedregular topologies: the mesh, the multi-dimensional torus, andthe hypercube. Those topologies finally lead to the definitionof the uniquely designed regular YottaWeb topology.

The Mesh: A D-dimensional regular mesh topology (with krow) is defined as the interconnection of k1 × k2 × k3 × … × kDnodes, where ki is the number of nodes in the ith dimensionof the radix D. Each node is identifiable by its position in eachdimension and is represented by the vector (x1, x2, x3, …, xD),with 0 ≤ xi < ki. An illustration of the mesh topology is givenin Fig. 7, where an example of 2 × 2 × 3 mesh is sketched. Ascan be seen in the figure, two nodes (x1, x2, x3, …, xD) and (y1,y2, y3, …, yD), in general notation, are neighbors if and only ifit exists a digit i such that xi = yi ± 1, and xj = yj for all i ≠ j.

Structural behavior: Routing in the regular mesh is self-performed with knowledge of the origin and destinationaddresses. The expansion of such a topology is performed byincreasing the number of nodes for a specific dimension, whilenodes in other dimensions remain at their original position.

Various types of meshes were also proposed in the litera-ture to improve the structure. There are, for example, the tori(see the paragraph below), which are derived from meshes byadding toroidal bonds for a general three-dimensional torus.The authors in [19] introduced the generalized multimesh(GM) for lightwave networks. The GM is a semi-regular struc-ture comparable to the torus, but with less diameter.

The multi-dimensional torus: A multi-dimensional torustopology structure, also called D-cube and k-row, is defined asthe interconnection of k nodes in each of the D dimensions.Nodes are addressed in the same fashion as in meshes, andtwo nodes (x1, x2, x3, …, xD) and (y1, y2, y3, …, yD) are neigh-bors (interconnected) if and only if it exists a digit i such thatxi = (yi ± 1)mod k, and xj = yj for all i ≠ j. Thus, there arelooping bonds in the D-cube with k-row, which is not the casein the meshes. Figure 8 gives an illustration of those loopingbonds.

Structural behavior: When k = 2, each node has D neigh-

n Figure 6. The simple topologies.

(a) Ring

(b) Bus

(c) Star


bors, and with k ≥ 2, the number of neighbors for each nodescales to 2D. The number of neighbors corresponds to thedegree of each node, and in contrast with the meshes wherethe node degree varies with its position, every node has exact-ly the same degree in the torus. Such toroidal topology is reg-ular and symmetric. For D = 2, directed bi-dimensional toriare named MSNs (for Manhattan Street Networks) through-out the literature.

To generalize, a D-dimensional torus is a D-cube with loop-ing bonds, where each node is addressed as a D-tuple (x1, x2,x3, …, xD), with 0 ≤ xi < ki and ki is the number of nodes inthe ith dimension. The number of nodes in each dimensioncould vary as long as the product of the number of nodes perdimension is equal to the total amount of nodes in the net-work. Routing is facilitated within such topology, since it onlyneeds the destination node address in order to have the rout-ing path. The expansion principle remains the same as in thecase of meshes, except that re-connections are needed for thelooping bonds.

The hypercube: One of the most popular topologies foundthroughout the literature on multi-processor interconnection isthe hypercube. Proposed by Bhuyan et al. [28], its extended gen-eral properties have been derived in Saad et al. [33]. The general-ized structure of the hypercube presented in [28] consists of ar-dimension structure with ki nodes in the ith dimension, where anode in a particular axis is connected to every other node in thesame axis. A number n of nodes (with n not prime) in a simplemixed radix representation system leads to a variety of hypercubestructures, where the integers k1, k2, …, kr, which are the numberof nodes per dimension, are chosen such that:

(5)

An example of a generalized hypercube is depicted in Fig. 9,where only viewed components are drawn and each node isaddressed in a radix system by a r-tuple (x1, x2, …, xr–1, xr)with 0 ≤ xi ≤ (ki – 1) for i = 1, 2, …, r. The popular case of abinary hypercube corresponds to k1 = k2 = … = kr = 2 and isintensively studied in the literature.

Structural behavior: The properties of the hypercubestructure [33] could be summarized as follow:

• The degree of each node is ∆ = Σi=1r

(ki– 1); this number increases with the sizeof the network. This is a main weaknessof the hypercube topology.

• The distance δ(x,y) between two nodes xand y, in terms of number of hops, isequal to the Hamming distance. Indeed,the Hamming distance between any twonodes is the sum of the number of coor-dinates in which the addresses of the twonodes are different. This distance is upperbounded by D. Thus, the diameter of thehypercube is D = r.

• The total amount of links needed for ageneralized hypercube is Σi=1

D(ki – 1).

• The total amount of nodes at a distanceof d hops from a given node is calculatedas: Nd = Σ(ki – 1) (kj – 1)… where i, j, k∈ {1, 2, …, D} and i ≠ j ≠ k and the sumincludes (d

D) terms. As a particular case,let ki = k for i = 1, 2, …, D; we have Nd = (d

D)(k – 1)d

and the average inter-nodal mean dis-tance within the network becomes:

D–

= D.(k – 1).kD–1/(n – 1), (6)

with Saad et al. [33] has shown that the hypercubestructure of diameter r is optimized when the total amount ofneeded link is minimized. Formally, this happens with

• There are d node disjoint shortest paths and d! alternatepaths between every pair of nodes separated by the dHamming distance. Several schemes of fault tolerancehave been presented in the literature [22].

• The hypercube structure allows simple routing proce-dures based on a binary representation addressing systemthat is equivalent to the basic radix system. In fact, theith digit in the radix system address is at most (ki – 1),and could be expressed as log2ki binary bits, with krepresenting the least digit greater or equal to k. Then,every node in the network could be addressed withΣi=1

D log2ki bits. A part of the message unit header con-tains the destination address. Then, for the routing, each

k k k nrr

1 = = = =� .

k n= D .

k nii

==∏ .

1

D

n Figure 8. 2 × 2 × 3 Torus in dimension 3.

(0,0,0)(0,1,0)

(1,1,0)

(1,1,1)

(1,1,2)

(0,1,2)

(0,1,1)(0,0,1)

(1,0,1)

(1,0,0)

(0,0,2)

(1,0,2)

n Figure 7. 2 × 2 × 3 mesh in dimension 3.

(0,0,0)

(0,0,1)

(1,0,0)

(0,0,2)(1,0,1)(1,1,1)

(1,1,0)

(0,1,0)

(0,1,1)

(0,1,2)

(1,0,2)(1,1,2)


node compares its own address with the destinationaddress in a bit-per-bit fashion, and routes the messageto the direction of the first different bit.Important works have been devoted to the study of the

hypercube and its derived topologies. The authors in [21] and[34] have studied the generalization of the hypercube struc-ture. On the other hand, the authors in [27, 35] and in [36]have introduced and studied the notions of incomplete andmore incremental hypercube structures.

From Usual Topologies to the Lattice Topology: So far wehave presented regular topologies suited to optical topologydesign. The list is not exhaustive since there are also regulartopologies such as de Bruijn and Kautz graphs [6, 11]. Howev-er, only directed versions of de Bruijn and Kautz graphs areregular and do not comply with the hypothesis of bi-direction-ality we have admitted in this article. On the other hand, theunidirectional Shufflenet [6] has been mostly proposed as aninterconnection topology for communication networks com-posed of n = k p k nodes. A Shufflenet is associated with acouple (k, p), and its nodes are arranged into k columns of pk

nodes each. For the unidirectional version, each node has pincoming and k outgoing links, as shown in Fig. 10. Gerla etal. [30] have studied the behaviors of the bi-directional versionof the shufflenet.

In addition, de Bruijn and Kautz graphs are not well suitedfor expansion. However, composed topologies based on deBruijn and Kautz graphs, such as the Gem Net [6] that is acomposition of the Shufflenet with a de Bruijn graph, havebeen proposed to support large optical networks.

Our previous survey has shown that both arbitrary and reg-ular topologies have been proposed for next-generation opti-cal networks. The properties of regular topologies seem to besuited for a high-performance large topology design. Howev-er, even with a flat or hierarchical structure, drawbacks stillpersist. Naturally, resource aggregation could enhance thedesign task by reducing the size of the variables. A fundamen-tal question is how to proceed with the aggregation. The Yot-taWeb topology that we discuss in the next section is ananswer to that question. In the YottaWeb, a multi-dimension-al lattice is defined, where nodes in each dimension belong tothe same sub-network and could communicate through a“tunnel.” The final structure that resembles the hypercubepresents additional advantages that will be explained later.

The Regular YottaWeb Topology: The YottaWeb topologyis derived from the PetaWeb architecture proposed in [1] and[15] and is designed for a high-capacity, distributed, edge con-trolled, optical core network. The PetaWeb topology is a“composite” star, as shown in Fig. 11a. Therefore, the config-uration of the agile core (AC) makes it possible to considerone PetaWeb as a subnet of a greater network. We will usethe term “agile core” to signify the set of high-capacity opticalcore nodes of a PetaWeb. This set of optical core nodes form-

ing a PetaWeb is grouped within the dotted circle of Fig. 11and is symbolized by the big star at the right, which representsan agile core. This constitutes the physical level G0 of theYottaWeb composed of a plurality of edge nodes and agilecores as it is represented by the left graph of Fig. 12.

The YottaWeb lattice structure defines a method of effi-ciently connecting the edge nodes to the agile cores. A pathbetween each origin-destination is constituted only by linksfrom the edge nodes to the ACs, as no direct link existsbetween nodes or between ACs. The YottaWeb regular latticestructure is pointed up by the notion of dimension D of thenetwork, which is defined as the number of ACs to which anedge node is connected at the same time. The lattice structurerepresents the logical level. Figure 12a illustrates a unit-dimension YottaWeb, where each of the four edge nodes isconnected to only one AC, represented by a single line in thelattice structure. The difference between the representation bythe lattice topology and the traditional topolgy is the use of a“tunnel” (line) to link all the edges within a subnet, instead ofarcs linking them. Assuming that two lines (ACs) can onlyintersect at one point and defining them to be perpendicularto a regular lattice structure (each AC is connected to exactlyK edge nodes, K being the AC’s capacity), one could easilyincrease the dimension of the YottaWeb to 2 as in Fig. 12band to more than 2 as needed.

A more complex example is portrayed in Fig. 13, wherethere are two types of lines connecting the edge nodes: thelight lines and the darker lines. Each line represents a differ-ent AC to which the edge nodes in the line are connected. Itis called “Yotta”Web topology since in terms of global capaci-ty, the range of the yotta bits/s can be reached by the follow-ing simple calculation. Let us consider that the nodes aregrouped into a lattice structure and that the access capacityper node could be equal to one terabit per second (1012

bit/sec). Let us suppose 1000 edge nodes per subnet. In a two-dimensional YottaWeb the total external capacity within thenetwork is computed as 1000 × 1000 × 1012 bits/sec = 1018 =

n Figure 9. A generalized 4 × 3 × 2 hypercube.

021 121 221

321

320311211111011

001

101 201 301310

300200100

000

n Figure 10. A Shufflenet with k = 2 and p = 3. The first andthird columns represent the same nodes.

0000

0101

0202

1010

1111

1212

2020

2121

2222

0 1 2


1 exa bits/sec. This total external capacity reaches = 1021 = 1zetta bits/sec in dimension three and = 1024 = 1 yotta bits/secin dimension four. From this definition of the regular latticetopology, every AC connects the same number K of nodes.

Structural behavior: In Dégila and Sansò [2, 3] some prop-erties of the lattice structure have been derived. The structureof the lattice topology is comparable to the topology of thegeneralized hypercube [28] defined for a multi-processorinterconnection architecture. From that point of view, the lat-tice structure holds many interesting properties from thehypercube, while maintaining the advantage of being simple.Indeed, the “tunnel” line linking the nodes simplifies the rep-resentation and the manipulation of subnets defined by ACs.

Fundamental properties of the lattice topology are enu-merated as follows.

On the degree: From the lattice perspective, the notion ofnode degree is slightly different. Here the degree representsthe number of ACs to which a node is connected. Thus, the“degree” of each node is ∆ = D the dimension of the Yot-taWeb. It is an important fact that the considered “degree” ateach node does not depend on the size of the network, but isonly related to its dimension. After the degree is chosen, itremains fixed.

On the inter-nodal distance: The distance δ(x,y) betweentwo nodes x and y, in terms of the number of hops, is equal tothe Hamming distance, as in the case of the hypercube, andthus, the distance is upper bounded by D and the diameter ofthe lattice is equal to the dimension. The total amount ofnodes at a distance of d hops from a given node is calculatedas: nd = (d

D)(K – 1)d, and the average inter-nodal mean dis-tance within the network is:

D–

= D.(K – 1).KD–1/(n – 1), (7)

where K = n1/D. Then, substituting the value of K in Eq. 7, wehave, after simplifications:

D–

= D.(n – n1–1/D)/(n – 1). (8)

On the number of links: To compare with the total amountof links needed for a generalized hypercube, which is Σi=1

r (ki – 1), there is the total number of needed ACs that iscomputed as m = D × KD–1 = D × n1–1/D.

On the fault tolerance: There are d node disjoint shortestpaths and d! alternate paths between every pair of nodes sep-arated by a d Hamming distance. Thus, the lattice structure is

highly-fault tolerant, as the hypercube [22] and the number ofalternate paths remains constant.

On the addressing and routing: The addressing is the sameas in the hypercube and is simply based on a mixed radix sys-tem where the ith digit in the radix system address is at most(K – 1), and could be expressed with log2K binary bits, withK representing the least digit greater than or equal to K.Thus, every node in the lattice network could be addressedwith Σi=1

D log2K = D × log2K bits. Also, the routing proce-dure is lightened, since there is no need to conserve (memoryconsumption) and to frequently update (CPU and time con-sumption) a routing table at each node.

DESIGN OF TOPOLOGIES WITHPERFORMANCE MEASURES

In this section we explain how the performance measures areused in practice for the design of topologies for large-scaleoptical networks. It is well known [8] that the problem of opti-cal topology design, which is a sub-case of the general net-work design problem, leads to difficult optimization problems.Hence, to simplify the problem two approaches are generallyused in a complementary manner as successive sub-problems:a logical topology structure is first chosen according to itsstructural properties [6]; then the nodes are arranged into thechosen structure by solving an optimization problem of thetype proposed in [7, 10], where the traffic flows within thetopology is given by the traffic matrix.

This section has two parts. In the first part we survey thedifferent methods by which the designer analyzes the struc-tural properties of the topologies with the different structuralmetrics. The second part is devoted to the topological designby means of mathematical programming.

TOPOLOGICAL DESIGN BY STRUCTURAL ANALYSES

Here the topological design considers the structural behaviorof the graph representing the networks. One compares quali-tative and quantitative aspects of the considered graphs. Ingeneral, the surveyed structural metrics from earlier are usedand their values for each topology are computed. Then,depending on the obtained values, which are compared torequired values, a topology is chosen by the designer.

n Figure 11. Structure of PetaWeb.

(B) Agile core

AC

(A) Composite-star structure of the PetaWeb

Fiber link

Edge node Parallel-plane optical core


In the next section we offer an overview of the literatureon optical topology design by the means of structural metrics.Thereafter, an explicit illustration of the application of thestructural measures to the surveyed topologies is given. A sen-sitivity scale is given for each measure in order to allow anoth-er level of comparison between topologies.

Key Literature Review: In the papers written on the sub-jects, new network topologies are generally introduced andcompared to previous topologies, or additional properties ofexisting topologies are exhibited. Based only on structuralproperties, the numerical comparison results for optical net-works dealt with the size on the order of one thousand nodes[6]. Then, their results for large-scale networks were present-ed.

Early in the 1990s Li and Ganz [21] studied regular virtualtopologies for optical networks. The following regular topolo-gies are investigated: Shufflenet, Binary Hypercube, General-ized Hypercube, and 2-Dimensional Torus. Uniform traffic isconsidered within the network, i.e., it is assumed that there isthe same amount of traffic between the pair of nodes. Their

results showed various hardware/performance compromisesfor high-speed networks.

Later, in order to allow a variable number of nodes in theoptical networks, Tan and Du [36] introduced the embeddingof Incomplete Hypercubes [27, 35]. The performance of theproposed scheme of design is comparable to those of bothunidirectional and bi-directional hypercube.

In 94, Banerjee et al. [20] extended the studied torus formultihop lightwave network from dimension two to a higherdimension. In fact, until this work only two-dimension torushad been applied to the design of optical networks. Here, theauthors analyze the properties, especially the average hop dis-tance, of three-dimension torus and hypothesized approximateresults for higher dimensions.

Recently, Banerjee et al. [6] provided a comparative intrin-sic presentation of regular topologies widely proposed foroptical networks, considering different performance metrics,such as the structural properties, broadcasting issues, and thescalability of these topologies. The following topologies havebeen considered: bus, ring, star, multi-connected ring, shuf-flenet, de Bruijn graph, hypercube, HCRNet, Cayley graphconnected cycles, and TreeNet. Further reference and proper-ties of those topologies are provided.

Currently, new topologies and metrics are being proposed,using other known topologies and avoiding some of theirshortcomes.

Comparison of Topologies and Sensitivity Scale: In Table 1 we present an illustration of the performance of thetopologies presented in this article, compared to the new Yot-taWeb topologies, according to the following fundamentalstructural performance measures: number of nodes, numberof links, diameter, degree, and average internodal degree. Tofacilitate the comparison, we assume that all the topologieshave the same number n = k2 of nodes.

In addition to the value M of each performance measure,we propose in this article to also discuss the evolution of the

n Figure 12. YottaWeb arrangement into lattice structure: a) One dimension lattice; b) two-dimension lattice.

Agile core = communication channel

2

3

41

(b) Two-dimension lattice

Agile core Edge node

(a) One dimension lattice

Agile core = communication channel

A

A

1

A

D

B

C

D

B

D

C

A C

B D

2 3

4

n Figure 13. 16-Nodes YottaWeb arrangement into two dimen-sions lattice.

Sub network: 1 agile core + 4 edge nodes

Agile cores Edge node


measures. For this purpose we define its sensitivity scale SSM asthe minimum variation of its value in the course of the time.For example, the number of nodes for ring, star, and bustopologies have 1 as their sensibility scale, while it is 2K + 1for the other regular topologies. This information is importantfor the designer when faced with the task of upgrading thenetwork. Also, the networks are generally in place, and theirevolution toward large-scale is the primary concern. From thatperspective two types of structural measures should be distin-guished. The first type groups the measures of the number ofnodes and the number of links, which we call “incrementalstructural measures.” The sensitivity scale of these incremen-tal measures should ideally be equal to 1. The second groupof measures are the “consequential structural measures” sincetheir evolution is inducted by the evolution of the first groupof measures. The consequential measures are, thus, the diam-eter, the degree, and the average inter-nodal distance. Thesensitivity scale of the inducted measures should ideally beequal to zero, since they are generally preferred to be con-stant.

The second column of Table 1 shows the notation of eachstructural measure M followed by its sensitivity scale SSM(I), withI the ideal value of the sensitivity scale. The values of themeasures are computed for both the simple topologies andthe two-dimensional type of regular topologies. Based uponthe sensitivity scale, one could notice that only the simpletopologies met the “incremental structural measures” require-ments. This confirms of one of the drawbacks of the regularstructure. On the other hand, considering the consequentialsensibility scale, only the YottaWeb topology met all therequired properties for large values of the number k. Indeed,for the measures of the YottaWeb, only one sensitivity scale,

is not equal to zero. However, it that sensitivity scale tends tonull for large n.

To conclude this illustration, the sensitivity scale provides afast method to classify and compare topologies according totheir structural properties when dealing with the enlargement

of those networks. On the other hand, on the same basis wehave shown the structural performance of the YottaWeb overthe other topologies for large-scale networks.

TOPOLOGICAL DESIGN BY FUNCTIONAL ANALYSES

Mathematical programming tools have been intensively usedfor designing networks, using the functional measures. Onegenerally formulates the design problem as a mathematicalprogramming problem. Then solving tools are generallyapplied to find the topology that optimize the functional mea-sures. In this section we illustrate how the optical networkdesign problems have been tackled. In the next section we citein detail three papers that surveyed and widely addressed thesubjects. In the next section we illustrate the formulation.

Key Relevant Literature: Mukherjee et al. [10] were one ofthe first to explore widely the principles for designing large-scale optical network topologies. The authors formulated thelogical topology design as an optimization problem with theobjectives to minimize the network’s average packet delay (apacket switching network is considered) or to maximize thescale factor by which the traffic matrix can be scaled up (toprovide the maximum capacity upgrade for future trafficdemands). The considered objectives related to the delay area non-linear function, leading to non-linear difficult optimiza-tion problems. A survey of previous heuristics methods fol-lowed by simulated annealing combined with flowdeviation-based methods were presented. Numerical resultsare finally discussed, as are open problems.

Later in 2000, Leonardi et al. [7] addressed the linear pro-gramming formulation of the problem by considering the max-imum congestion level in the network. In this article theauthors classified the heuristics approaches for the sub opti-mal solution for the logical topology design problem into fourclasses: heuristics solutions of the MILP problem; maximiza-tions of the single-hop traffic flows; heuristics maximizationsof the single-hop and multi-hop traffic flows; and finally, algo-rithms based on the adoption of a pre-established regular logi-cal topology and on the optimization of the node placement

2

1 2( )( ),

k k+ +

nnnn Table 1. Measurements of the structural behavior of the k2-nodes topologies.

Structural measures Not.

Number of nodes n k2 k2 k2 k2 k2 k2 k2

SSn(1) 1 1 1 2k + 1 2k + 1 2k + 1 2k + 1

Number of links m k2 k2 – 1 k2– 1 2k(k – 1) 2k2 2k2(k – 1) 2k2

SSm(1) 1 1 1 4k 4k + 2 6k2 + 2k 4k + 2

Diameter D k2/2 2 k2 – 1 2(k – 1) k 2 2SSD(0) 1/2 0 1 2 1 0 0

Degree ∆ 2 k2 – 1 2 4 4 2(k– 1) 2SS∆(0) 0 1 0 0 0 2 0

Average inter-nodal distance

D—

k2/4 2 k2/3 k – 1 k/2

(approx.) SSD—(0) 1/4 0 1/3 1 1/2 2

1 2( )( )k k+ +2

1 2( )( )k k+ +

2

1

k

k +2

1

k

k +

Simple topologies for 2 – D Regular topologiessmall networks

Generalized YottaWebRing Star Bus Mesh Torus Hypercube Topology


according to the traffic pattern.Numerical results are also dis-cussed for networks with atmost 120 lightpaths. The resultsshowed heuristics that considerboth single-hop and multi-hoptraffic, starting from a fully-connected logical topology andremoving lightpaths. Anotherimportant result is that optimalrouting algorithms should becoupled with the logical designalgorithms in order to havegood solutions.

In the same year, Dutta andRouskas [9] presented a widesurvey of the topics related tothe virtual topologies design forwavelength-routed optical net-works. Those topics are relatedto the context and scope of thecomponents of the wavelength-routed optical architecture.Additional covered topics arethe survey of network perfor-mance optimization tools, fol-lowed by the reconfigurabilityconsiderations. The last pointrefers to the problem of recon-figuring a network from onevirtual topology. This problemis interesting to save the cost of having to reconfigure thewhole network, and it helps achieve a global strategy to opti-mize the initial problem of designing a logical topology.

Using the Functional Measures: In this section we first pro-vide the general framework for the formulation of the designof virtual optical networks as a mathematical programmingproblem. Table 2 presents a different part of the frameworkwith key references for further reading. Our classification isbased on the survey of Dutta and Rouskas [9]. When addi-tional elements are added, their references are provided.

This table shows the position of the functional measuresthat we have surveyed in this article, in the general logicaloptical topologies design. The functional measures are used astargets of the objectives 4–5, when finding the points 13–14,given 1–3. In fact, the routing scheme could be neglected [10],thus considered as an input of the problem of finding virtualtopologies. Otherwise, routing is given by the lightpath-fiber-wavelength indicator variable (h).

On the other hand, our functional measures are subdividedinto two groups. The first group is associated with the maxi-mization problem (4); the second group is associated with theminimization problem (5). Different formulations of thosemeasures could be found in the literature. The formulationsincluding the delay lead generally to non-linear functions, andthus to non-linear problems [10]. Otherwise, linear formula-tions of the problem are often proposed when the delay func-tions are neglected, or simply linearized.

Solving the resulting problems is not easy. Numericalexamples in the literature are obtained for networks with asize less than one hundred nodes. The exact methods for solv-ing mathematical optimization problems are generally ineffi-cient. Most of the time the global problem is subdivided intodifferent successive sub-problems. Successive approximativesolutions are found by heuristics [9] and could lead to a worsefinal solution.

However, the methods that consist of first choosing a virtu-al regular topology, such as the YottaWeb, and then tacklingthe problem of nodal arrangement into the chosen structure,are still often preferred. These methods lead to combinatorialoptimization problems, for which heuristic methods couldyield good solutions [3].

DISCUSSIONS

In this section, we summarize the results of this article andfinally extrapolate on their usage in real-world networks.

In Tables 3 and 4 the performance measures surveyed inthis article are grouped around their fundamental metrics. Inthe first column of each table, fundamental structural or func-tional performance metrics are given, followed in the secondcolumn by related measures found in the literature.

At a glance these tables show that the proposed classifica-tion around the fundamental metrics facilitates the compari-son between the numerous performance measures proposedin the literature. Further application of these measures to thestudied topologies has shown the highest performance of theYottaWeb for large-scale networks, assuming that we aredealing with regular networks, with its known drawbacks.

Concerning the topologies, our presentation follows a pro-gressive construction toward large-scale networks. In fact, onecould transform an optical bus topology to an optical ringtopology simply by adding an additional link between the firstand the last node. Thus benefits are gained with the diminu-tion of the diameter and the unweighted mean average hops,while fault tolerance is also improved. These simple topolo-gies are not suited for large-scale networks. However, mostexisting deployed optical networks are long-haul and do nothave many nodes, so they are based on such simple topolo-gies.

Furthermore, several optical buses or rings lead to mesh or

nnnn Table 2. General functional measures optimization framework for the topological design foroptical networks.

Given 1. Physical topologies/hardwares2. Traffic matrix3. (Routing scheme)

Objectives 4. Maximization a. Total external traffic [10]b. One-hop traffic [7]

5. Minimization c. Maximal congestion [7]d. Average weighted internodal distance [7]e. Mean delay [10]

Variables 6. Flow f. Portions of traffic over links [7, 10]

7. Link g. Lightpath-fiber indicators [7, 10]h. Lightpath-fiber-wavelength indicators [10]

Constraints 8. Virtual structure i. Virtual node degreej. Lightpath hop or length limitation

9. Flow k. Flow conservationl. Flow delay

10. Coupling constraints m. Flow-lightpathn. Lightpath — wavelengtho. Wavelength — fiber

11. Wavelength constraints

12. Variables range m. Binary variablesp. Positive real variables

Find 13. Virtual topologies14. (Routing scheme)


torus, respectively. A mesh also could be transformed easily inTorus, simply by adding looping bonds. Continuing, a torusgives rise to the hypercube by linking directly the nodes of thesame line. Thus, the evolution of the network could be seenfollowing such a perspective of converting one topology toanother, simply by adding additional links. On the other hand,YottaWeb proposal drastically changes the topology construc-tion philosophy. Using the optical technologies, composed-stars networks have been aggregated in a conceptual mesh-likenetwork with highest structural performances. These networksare costly to be stabilized, since they require additional opticaldevices (fibers, optical parallel planes, etc.) [15]. However,they make it possible to reach enormous overall networkcapacities with fewer drawbacks.

The other challenge in the design of optical topologies isrelated to functional analyses, which we have shown to beimportant. Our survey shows that, despite all the mathemati-cal formulation of the problems, only heuristics methodsyielded acceptable results. We have shown the general frame-work of such analyses. In the reviewed literature, only simpli-fied versions of the problem have been investigated. On theother hand, results depend intimately on the considered traf-fic pattern. However, the traffic of a next-generation networkis difficult to predict. Hence, it is encouraging that differenttypes of heuristics are designed for different types of traffic.Further research should investigate the integra-tion of multiple heuristics in order to efficientlyhandle different variation of the traffic pattern inthe course of time.

CONCLUSIONS

Parameters of optical topology design have beensurveyed. Performance measures as well astopologies have been classified, compared, andreferenced. Admitting that any simple topologiessuch as the bus, the ring, and the star cannot sup-port large-scale networks, the favored array-based regular topologies have been graduallydescribed and discussed, from the basic mesh tothe more popular hypercube. The characteristicsand the limitations of each type of topology havebeen explained, in particular when its size increas-

es, and the results have been analyzed. The per-formance of the hypercube topologies over theprevious topologies have been gradually shown.The lattice-structure topology proposal has beendescribed from the physical to the logical repre-sentation. The lattice structure appeared as asolution for many shortcomings of the hypercube.Further work on the optimization and updatingof such a lattice structure could be found inDégila and Sansó [24]. A sensitivity scale mea-sure has been introduced, in order to group thestructural measures under the same analysisbasis. The functional optimization of large-scaleoptical topologies have also been surveyed. Themain challenges in this case come from the vari-able traffic patterns. The reader is referred toDégila and Sansó [3] for an example of designingan efficient heuristic for logical optical topologydesign that works for different types of traffic.

Until recently [6] the regular topologies wereonly used in testbed optical networks such as Ter-anet at Columbia University. However, there is aneed to ensure that the next-generation networks

will not be a patchwork [15] of current networks. One theother hand, it is more difficult to study the evolution of cur-rent topologies to arbitrary large-scale topologies. Thus, ourfeeling is that the regular networks will play an important rolein future networks. The easy accessibility of optical deviceswill accelerate the process.

ACKNOWLEDGMENTS

This work was supported by CRD grant numberCRDPJ248035-01 between Nortel Networks and the NationalSciences and Engineering Research Council of Canada. Theauthors wish to thank Maged Beshai and Francois Blouinfrom Nortel for fruitful discussions. We also thank the Editor-in-Chief and the anonymous referees for their helpful sugges-tion.

REFERENCES[1] M. Beshai and A. Yu, "YottaWeb topology,"

Disruptive Network Solutions, Nortel Networks, Ottawa, IEEEPers. Commun, vol. 7, Aug. 2000.

[2] J. Dégila and B. Sansó, "Design Optimization of a Next-Genera-tion Yottabit-per-second Network," Communications, 2004 IEEEInt'l. Conf., vol. 2, 20–24 June 2004, pp. 1227–31.

nnnn Table 3. Classification of some fundamental structural performance mea-sures.

Number of nodes n –Extensibility [6]

Number of links m –Fibers length [15, 37]

Diameter D –Hop-depth distribution [14, 17]–Length-depth distribution [15, 17]–Maximal distance [14]

Degree ∆ –Node degree distribution [18]

Average inter-nodal distance D—

–Normalized average inter-nodal distance [18]

Symmetry S –Node-symmetry [18]–Edge-symmetry [18]–Homogeneity [18]–Combinatorial properties [34]

Fault tolerance R –Reliability [6]–Strong fault tolerance [22]–Connectivity [19]–Number of biconnected components [38]

Performance measures M Related performance measures

nnnn Table 4. Classification of some fundamental functional performance mea-sures.

Total external traffic –Throughput Tt [15, 26]

Node processing –Average processing traffic per node γ [21]–Density of traffic per node µ [21]–Node processing efficiency ρ [21]

Traffic weighted inter-nodal –Probable average inter-nodal distance [21]metrics –Mean hop traffic weighted by the traffic [2, 3]

–Maximal congestion [7, 9, 10]–Mean delay (latency) [7, 8, 10]–One-hop traffic [7, 9, 10]–Multiple-hops traffic [7, 9, 10]

Flow number fn –Load [20]

Performance measures Related performance measures


[3] "Topological Design Optimization of a Yottabit-per-second Lat-ticeNetwork," IEEE JSAC , vol. 22, no. 9, Nov. 2004, pp.1613–25.

[4] W. Mao and D. Nicol, "On k-ary n-Cubes: Theory and Applications,"ICASEReport 94-88, Tech. Rep., 1994.

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BIOGRAPHIESJULES R. DÉGILA ([email protected]) received the Ph.D.degree in electrical engineering from Ecole Polytechnique de Mon-tréal, Canada in 2004. His Ph.D. thesis was conducted in collabo-ration with Nortel Networks, discussing the design optimizationof a next-generation multidimensional agile photonic network. Hisresearch interests include network planning (wireless or wired),next-generation Internet, voice over IP, converged networks,large-scale network/systems optimization, and performance analy-sis.

BRUNILDE SANSO ([email protected]) earned a degree inelectrical engineering from the Universidad Simon Bolivar (Cara-cas, Venezuela) in 1981, an M.S. degree in sciences in the areaof reliability in 1985, and a Ph.D. in operations research fromEcole Polytechnique de Montréal in 1988. After postdoctoralstudies at the CRT of the University of Montreal, and a researchfellowship at the GERAD, she joined the faculty of the EcolePolytechnique de Montréal in 1992, where she has been a fullprofessor since 1997. She is currently with the department ofelectrical engineering, where she is the director of the LORLAB,a research lab devoted to the performance, reliability, design,and optimization of operational planning of broadband net-works. She is an associate editor of telecommunication systemsand has been a referee and technical committee member formajor journals and scientific conferences, a reviewer for govern-ment agencies, and an industry consultant. She was the pro-gram co-chair of the 5th INFORMS TelecommunicationsConference. She is a co-editor of the book “TelecommunicationsNetwork Planning” and the forthcoming book “Performance andPlanning Methods for the Next Generation Internet” (KluwerAcademics). She is a recipient or co-recipient of several awardsand honors, including the 2003 DRCN best paper award, 2ndprize in the 2003 CORS Practice Competition, 1995 IEEE/ASMEJRC best paper award, 1992 NSERC Women Faculty Award, andthe 1992 FCAR young researcher award.

a survey of topologies and performance measures for large-scale networks

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