Physical topology design for all-optical networks

Optical Switching and Networking 5 (2008) 219–231www.elsevier.com/locate/osn

Physical topology design for all-optical networksI

Huan Liua,∗, Fouad A. Tobagib

a Accenture Technology Labs, United Statesb Department of Electrical Engineering, Stanford University, United States

Received 8 February 2007; received in revised form 3 July 2007; accepted 13 February 2008Available online 29 February 2008

Abstract

When designing an all-optical network, the designers face a choice of laying down more fibers or increasing the numberof wavelengths. Although either choice could be used to support new connections, one increases the link cost and the otherincreases the node (wavelength equipments) cost. The tradeoffs between link and node cost are not well understood. Using theefficient physical topology design algorithm that we propose, we study this tradeoff. We use the asymptotic growth rate of theprovisioned capacity as a metric to compare various design alternatives. A higher asymptotic growth rate translates directly into ahigher deployment cost for large networks. Our study shows that taking fiber length into consideration can lead to lower capacityrequirement. We also find that a sufficiently large fiber-to-node ratio is necessary in order to minimize the asymptotic growth inthe provisioned capacity, increase capacity utilization and minimize the need for wavelength conversion. We study a real networkand find that its fiber-to-node ratio is too low. As a result, large provisioned capacity is required and less than 55% of the capacityis usable. By increasing the ratio, we can reduce the provisioned capacity and achieve close to 80% utilization.c© 2008 Elsevier B.V. All rights reserved.

Keywords: Physical topology design; All-optical networks; Optical switching

1. Introduction

Compared to the broadcast-based optical networkarchitectures [1–5], Wavelength Routed All-opticalNetwork (WRAN) utilizing the WDM technologypromises to greatly increase the transport capacity atmuch reduced cost. A connection, also known as alightpath [6], only occupies one wavelength on eachfiber link along the physical route used to connectthe two end nodes. Thus, the same wavelength onother fiber links could be reused for other lightpaths toincrease the utilization of the provisioned wavelengths.

I This is an extended version of a paper presented at BroadNets2006.

∗ Corresponding author. Tel.: +1 408 426 3330.E-mail addresses: [email protected] (H. Liu),

[email protected] (F.A. Tobagi).

1573-4277/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.osn.2008.02.003

Besides the increased utilization, WRAN hasmany other advantages. Since a lightpath is routedtransparently through the WRAN, that is, bypassingintermediate nodes without packet processing or costlyopto-electronic conversion, much of the queuing delayand electronic equipment cost can be eliminated.The cost savings in electronic equipment will besignificant [7], especially when the line speed is veryhigh. Furthermore, WRAN can be easily and cheaplyupgraded when the interface speed is increased. This isbecause optical switching is agnostic to the underlyingdata rate of an optical channel (up to a certain limitbecause the channel bandwidth limits the maximumdata rate), and thus, the intermediate optical switches donot have to be upgraded when the line speed increases.

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220 H. Liu, F.A. Tobagi / Optical Switching and Networking 5 (2008) 219–231

Given the high cost of deploying a WRAN, itis important to design the physical (fiber) topologyto minimize the total capital investment. The totalcost of deploying a WRAN is the sum of two costcomponents—the link cost and the node cost. The linkcost, i.e., the cost of laying down fibers to interconnectnodes, is a function of the total fiber length L . The nodecost, i.e., the cost of the all-optical wavelength switchis a function of the number of wavelengths W that areprovisioned on each fiber link (this is an approximationsince the node degree also affects node cost). In general,there is a tradeoff between L and W —more L willtranslate into less W and vice versa.

One can design the physical topology to use theminimum amount of fiber by connecting the nodes usinga minimum spanning tree. Even though the link (fiber)cost is at the minimum, the node (wavelength) cost willbe very high. Alternatively, one can connect all nodepairs using direct fibers. The node (wavelength) cost isat its minimum since W = 1. However, the link (fiber)cost will be very high. The optimum design with theminimum total (link and node) cost will be betweenthese two extreme solutions.

To pick the best topology with the minimum totalcost, one has to solve the problem of designinga physical topology to minimize the number ofwavelengths W required given a budget on L . Wewill present a comprehensive treatment of the designproblem, including both a mathematical problemformulation and an efficient heuristic algorithm. Usingthe design algorithm we propose, one can design thetopology with the minimum total (link and node)cost by repeatedly run the algorithm for different L ,comparing the resulting solutions based on the actualcost functions, and then picking the one with the lowesttotal cost. In order to be independent of the actualcost functions, we study the tradeoff between L andW . In addition, we also derive design principles andguidelines, which are unfortunately nonexistent as ofnow.

To evaluate the various design alternatives—someuse more fiber but fewer wavelengths and some useless fiber but more wavelengths—we propose to use theprovisioned capacity C = LW (capacity for short in thefollowing) as a metric. C essentially is the bandwidth-distance product. It is used to measure the amount ofnetwork resources that have to be provided for a givenset of lightpath demands. Since there is a cost associatedwith providing the network resources, naturally, it isdesirable to minimize the provisioned capacity. Notethat our use of the term “capacity” may be differentfrom the literature in other contexts, where some fixed

Fig. 1. 6 nodes lie in a line with an inter-nodal distance of 1.Physical topology (a) requires 9 wavelengths and physical topology(b) requires 8 wavelengths.

network resources are assumed given and the goal is tomaximize the capacity, i.e., the throughput.

C is a more fair metric compared to other metricssuch as MW , where M is the number of fiber links.Consider a sample network as shown in Fig. 1, where6 nodes lie on a straight line with an internodal distanceof 1. Let us assume that the traffic demands are uniformall-to-all, i.e., we need to establish a lightpath betweenevery pair of nodes. Two possible physical topologiesare depicted in the figure. Topology (a) is a lineartopology, and it requires 9 wavelengths. To see this, wejust need to consider link 3–4. Since there are threenodes on either side of this link, there are a total of9 lightpaths crossing this link; hence, 9 wavelengthsare needed. Topology (b) requires 8 wavelengths. Tosee this, we can consider either link 3–4 or 4–5. Sincethere are 4 and 2 nodes on either side of these twolinks respectively, there are a total of 8 lightpathscrossing these two links; hence, 8 wavelengths areneeded. Since both topologies have 5 edges, it maybe tempting to choose topology (b) because it requiresfewer wavelengths. However, the total fiber length intopology (b) is 7 units long, and therefore, 56 units ofcapacity is required. In contrast, topology (a) only uses45 units of capacity. The reason for the high capacity intopology (b) is because the demands between nodes 1,2 and 1, 3 are not routed along the direct line betweenthe two end nodes. In particular, both demands take adetour—going to node 4 first and then to the destinationnodes. Demand 1, 2 wastes 4 units of capacity anddemand 1, 3 wastes 2 units of capacity.

The capacity metric measures how efficiently theprovisioned network resources are utilized—highercapacity for the same set of lightpaths means thatthe provisioned capacity is less efficiently utilized.Unfortunately, the capacity metric does not directlyreflect the cost of deploying a WRAN. A design with ahigher capacity may have lower cost than a design witha lower capacity simply because of the differences in thecost functions for L and W .

To derive design guidelines independent of the actualcost functions, we focus on the asymptotic growth rate

H. Liu, F.A. Tobagi / Optical Switching and Networking 5 (2008) 219–231 221

of the capacity, which can be used to determine thelower-cost design alternative when N is large, whereN is the number of nodes in the network. Considertwo design alternatives. Let us assume design A uses1 unit of fiber, and let us also assume that the resultingcapacity can be expressed as a1 N e1 (as we will show inEq. (3) and in experimental results, C can be expressedas a power function of N ). Similarly, let us assumedesign B uses 2 units of fiber, and the resulting capacitycan be expressed as a2 N e2 . Let us assume e1 > e2.From the definition of capacity, we can see that DesignA uses a factor of c = 2 a1 N e1

a2 N e2 more wavelengths thanDesign B. When N is small, c is small (slightly morethan 2 a1

a2). If the cost of the fiber is a lot more than the

cost of the wavelengths, Design A will be the lower-cost solution. However, when N is large, c will be largebecause of the exponential term. At some point, thecost of the wavelengths will dominate, and Design Bbecomes the lower-cost solution. As long as e1 is greaterthan e2, even if e1 is only slightly larger than e2, DesignB will be a lower-cost solution when the network islarge enough. Hence, the exponent is an indicator of theactual cost for large-size networks.

Throughout this paper, we will assume that all fiberlinks will support the same number of wavelengths.This is a realistic assumption because it is necessaryto ensure the maximum interoperability betweenneighboring nodes. Having different W on each linkmay make sense currently because WDM systems areused primarily as transmission systems and there aremany opto-electronic conversions at a node. However,since there is no opto-electronic conversion in theWRAN, having different W not only will requireunsymmetrical optical cross connects, but also will limitthe flexibility in routing lightpaths.

1.1. Traffic model

We first consider WRANs that support full-meshconnectivity, i.e., there are T lightpaths to establishbetween every pair of nodes, where T is a constantthat is the same for all node pairs. Our designalgorithm could be easily generalized to other lightpathconnection patterns. We will present our results on non-uniform lightpath connections at the end.

1.2. Prior work

The physical topology design problem has beenstudied before in the literature. However, most ofthese works consider only the case where the cost is

proportional to the number of fiber links regardless oftheir lengths [8,9]. The work in [2] considered onlybroadcast-based optical networks, and the topology isrestricted to a tree. The work in [10] takes fiber lengthinto consideration. They considered a different problemwhere W is given and the goal is to minimize the totalcost of fiber. The proposed algorithm runs much slower.A problem instance with 100 nodes requires 11 hours.This is not suitable for a tradeoff study like ours, wherehundreds of thousands of problem instances have to besolved.

There have been several related studies on thewavelength requirements in an optical network. The firstwork on wavelength requirements to support full-meshconnectivity was reported in [11]. The authors foundthat the ensemble average number of wavelengths (W )required is only dependent on α, where α =

2MN (N−1)

isthe ratio between the number of edges in the networkand the number of edges required to fully connect allnode pairs (Recall that M is the number of links in thenetwork). The result only applies when the fiber linksare picked randomly, i.e., when short fibers and longfibers have an equal chance of being picked. If manyshort fibers are picked, then W could be much higher.Also, the work in [11] did not answer the question ofwhat α one should use in designing a network.

In [12], the authors gave an approximate equation forthe required number of wavelengths W . Unfortunately,the derivation of the equation was not shown. InSection 2, we give an equivalent derivation for a moreaccurate result that is a constant factor away from theone given in [12].

1.3. Organization of paper

The rest of the paper is organized as follows.In Section 2, we first give a lower bound and anapproximate equation for C . In Section 3, we presentour physical topology design algorithm. In Section 4,we evaluate our algorithm and show the tradeoffbetween L and W . In Section 5, we consider one real-life network and show how our design guideline couldbe applied. Lastly, we conclude in Section 6.

2. Lower bound and approximate equation

2.1. Lower bound

The most capacity-efficient way to establish alightpath between two nodes i and j is to lay downa direct fiber between the two nodes and use onewavelength on that fiber for the connection. The


capacity used for this connection is di j · 1 = di j ,where di j is the physical distance between the twonodes. Another way to establish the lightpath is byhopping through one or more other nodes and use onewavelength on each fiber link along the way. Hoppingthrough other nodes will take strictly more capacityunless the intermediate nodes lie on the direct (straight)line between node i and j . In the following, we say alightpath uses direct-line routing if the lightpath onlyhops through zero or more nodes along the directline. We say the lightpath uses non-direct-line routingotherwise.

Summing up the minimum capacity required foreach lightpath, we can derive the lower bound onthe provisioned capacity for full-mesh connectivity asfollows:

CL B = TN∑

i=1

N∑j=1

di j . (1)

Note that this lower bound holds regardless of therelative positions of the nodes (e.g., not necessarilyuniformly placed). It also holds even if di j is not theCartesian distance between the nodes. For example,there may be a physical constraint that forces a fiberlink not to be laid along the direct line (e.g., mountains,rivers). In such cases, di j denotes the actual length ofthe fiber that has to be laid down.

This lower bound cannot be achieved unless manyfiber links are laid down. If only a few fiber links areavailable, some lightpaths will not be able to use direct-line routing, either because there is no fiber link on thedirect line or because there is no wavelength left on thefiber links on the direct line.

2.2. Approximate equation

In [12], an approximate equation for W was given.Unfortunately, the derivation of the equation was notshown and the steps cannot be easily reproduced. In thissection, we give an equivalent derivation for a result thatis only a constant factor away from that in [12].

To be consistent with the rest of the paper, we assumethat the N nodes of a network are uniformly distributedin a square area, as opposed to a circular disk area usedin [12], with unit size as shown in Fig. 2. We notethat the choice of a square area is arbitrary, and anybounding area could be chosen. We also assume that thefiber links are uniformly distributed. The assumptionson uniform node placement and link placement aremade in order to derive an equation that approximates

Fig. 2. N nodes uniformly distributed in a square.

the average of that of a large ensemble of randomnetworks.

Our goal is to compute the number of lightpaths thatwill cross the cut in the middle of the square and alsothe number of fiber links that will cross the cut. Theminimum number of wavelengths required will simplybe a ratio of these two numbers.

Since we are considering the cut right in the middleof the square, there are exactly N/2 nodes on eitherside of the cut. Therefore, it is easy to see that thereare (N/2)2T lightpaths crossing the cut.

To compute the number of fibers crossing the cut,we first compute the number of nodes n in the rowright above the cut. Since the area of the square is 1,each side of the square is exactly 1 unit long. Eachnode will take up a square area of roughly 1/N witheach side of it being 1/

√N long. Dividing the length

of the row, the number of nodes in the row is thenn = 1/(1/

√N ) =

√N .

If the average node degree is d, there are nd edgesoriginating from these n nodes in the row. Among them,nh edges go to nodes in the same row, and the restn(d−h) go to nodes in the neighboring rows. Because ofthe uniform link placement assumption, the number ofedges staying in the same row should be proportional tothe number of nodes in the row, i.e., h/d = n/N . Sincethe number of edges going to the row below (across thecut) is half the number of edges leaving the row, it canbe expressed as:

12

n(d − h) =12

nd(N − n)/N =12

d(√

N − 1).

If the average fiber length L f (L f = L/M) is more thanthe average node distance Ln (we will explain shortlyhow to calculate Ln), some fiber links originating fromrows above could also cross the cut. Adding these fiberlinks, the number of edges crossing the cut will increaseby a factor of L f /Ln .

Dividing the number of lightpaths crossing the cutby the number of fiber links crossing the cut, we willget the minimum required number of wavelengths W asfollows:


W =N 2T/4

12 d(

√N − 1)L f /Ln

=1

2 − 2/√

N

N 3/2T Ln

d L f

= BN 3/2T Ln

d L f(2)

where B =1

2−2/√

Nis almost a constant, especially

when N is large.Since the total fiber length is L = Nd L f /2, we can

rearrange the terms in Eq. (2) to get the expression forthe provisioned capacity.

C = W Nd L f /2 =B

2N 5/2T Ln . (3)

If a circular disk area is assumed, the same derivationwill yield a result that differs from the one in [12]

by a constant of√

π

2 . We believe our result is moreaccurate because our result matches the lower boundfor the same network (nodes uniformly distributed in asquare) almost perfectly. The matching is not surprisingbecause, in the derivation of the approximate equation,we have implicitly assumed that each lightpath will berouted along the direct line, just like we did in derivingthe lower bound.

Eq. (3) shows that the capacity will scale as N 2.5

even though the number of lightpaths will only scaleas N 2. From the derivation, we can see that the extra0.5 factor in the exponent comes from the fact that thenumber of fiber links crossing the cut is on the orderof

√N , but the number of lightpaths crossing the same

cut is on the order of N 2. Alternatively, this extra 0.5factor can be also viewed as coming from the fact thatthe diameter of the network is on the order of

√N .

Since some lightpaths will have to cross the networkto reach their destinations, they will require an order of√

N more capacity. This is true whether the lightpathscross the network using few long fiber links or manyshort fiber links because only the fiber length comes inthe definition of the capacity, not the number of fiberlinks.

In the derivation, we have assumed that all Nnodes are uniformly distributed in the area in orderto approximate the average case. However, nodes inreal networks are almost never distributed evenly insidethe bounding area. But a change in this assumptionwill only affect the constant term. As long as thephysical topology is two-dimensional, the diameter ofthe network will be on the order of

√N , and some

lightpaths will have to use√

N more capacity. We will

Fig. 3. An example to show how to calculate Ln and L .

look at a real network in Section 5, where nodes are notuniformly distributed. We will see that our observationstill applies.

In addition to the non-uniform distribution of nodes,the bounding area of a real network is also seldomsquare. If we assume a different bounding area (suchas a circle), the same derivation will give a resultthat differs again only in the constant term. Even forirregular-shaped bounding area, the exponential termremains the same as long as it is a two-dimensional area(as opposed to a line). Since we are more interestedin the asymptotic growth rate of the capacity, we willassume a square area throughout this paper without lossof generality.

The average node distance Ln in Eq. (3) can bethought of as a scaling factor. Consider a topology, ifwe double the length of each fiber link (and thus pushthe nodes further apart), the topology still remains thesame. The only thing changed is that Ln is doubled.For an arbitrary network, Ln can be determined asfollows. We take the area A of the bounding box ofthe topology and divide it by the number of nodes N ,we will get the average area of each node to be A/N .To derive the average node distance, we assume thesenodes are uniformly located and each node will takeup a square area with each side being

√A/N long,

and then the distance between two neighboring nodeswill be Ln =

√A/N . If the fiber distance di j is not

the Cartesian distance (e.g., physical constraint forcessome fiber to be laid along non-direct lines), Ln canbe adjusted proportionally according to the actual fiberdistance. Again, using a different value of Ln will onlychange the constant term, not the exponential term. Forsimplicity of discussion, we let Ln = 1 without loss ofgenerality for the rest of this paper.

Let us look at a concrete example as shown in Fig. 3.We have two networks both having 4 nodes, but oneis in a 4 × 4 miles bounding area and the other is ina 2 × 2 miles bounding area. Both networks have theexact same topology and layout and, hence, should betreated the same. The network in Fig. 3(a) has area


A = 4 × 4 = 16 miles2; hence, the average nodedistance is Ln =

√A/N = 2 miles. The total fiber

length is L = 3 × 2 = 6 miles = 3Ln . Repeating thesame calculation for the network in Fig. 3(b), we haveA = 2 × 2 = 4 miles2, Ln =

√A/N = 1 mile. The

total fiber length is L = 3 × 1 = 3 miles = 3Ln . Aswe can see, both networks have the same fiber length ifexpressed in terms of Ln ; consequently, they have thesame capacity if they both have the same number ofwavelengths W . Ln removes the difference caused byscaling the fiber length, which enables us to capture theinvariant aspects of a topology.

Eq. (3) predicts two things. First of all, it predictsthat the capacity will scale linearly with T . We observein simulations that this is indeed true. Therefore, thispaper will not focus on the scaling as a function of T .In the following, we assume T = 1.

The second predication this equation presents is thatthe asymptotic growth rate of the capacity will remainthe same no matter how W and L scale. In other words,W can be traded off with L equally.

Even though the scaling as a function of T (the firstprediction) matches very well with our observations insimulation, given the simplicity of the analysis, there arefew reasons to believe that the second prediction is true.There are two factors that are not modeled by either thelower bound or the approximate equation. For the firstfactor (factor F1), we have assumed that each lightpathwill be routed along the direct line between the two endnodes. This is rarely the case even if each lightpath goesthrough the minimum number of fiber links. The reasonis because there simply may not be fiber links on thedirect line. The second factor (factor F2) not modeledis the utilization of the provisioned capacity. In general,it is not possible to even out the load on each fiber linksuch that the same number of wavelengths is used onevery link, i.e., some wavelengths cannot be utilizedbecause of topological constraints.

3. Physical topology design

The problem of designing a physical topologyto minimize W is clearly an NP-complete problembecause even if the topology is known, the problemof determining how many wavelengths are needed(the Routing and Wavelength Assignment Problem) isknown to be NP-complete [13].

In this section, we first formulate the physical topol-ogy design problem as an Integer Linear Programming(ILP) problem and then propose a practical heuristicalgorithm. The optimization problem takes the bud-get on fiber (in the form of a fixed fiber-to-node ratio

f = L/N ) as a constraint and designs a topology tominimize the number of wavelengths W required.

We consider the topology design problem bothwith and without the Wavelength Continuity Constraint(WCC), which requires each lightpath to be assigneda unique wavelength on each fiber link it traverses.The WCC constraint could limit the amount of usablecapacity. If a wavelength is used on a link, any otherconnections that must pass through this link cannot usethe same wavelength again.

3.1. ILP problem formulation

For brevity, we only show the formulation whichrelaxes the wavelength continuity constraint (WCC).The formulation can be easily modified if the WCCconstraint is enforced. We will use the followingnotations in the formulation:

• zml : This is the fiber link variable. zml = 1 if nodem is connected to node l via a fiber link. zml = 0 ifthere is no fiber link between node m and l.

• zi jml : This is the lightpath routing variable. zi j

ml = 1 ifthe lightpath between node i and j goes through thefiber link between node m and l.

• dml : This is a constant that specifies the fiber distancebetween node m and l, which could be longerthan the Cartesian distance (e.g., when physicalconstraints force a fiber link not to be laid along thedirect line).

First, we need to make sure that all demands(lightpaths) are routed. This is the same as a flowconservation constraint.

∑l

zi jml −

∑l

zi jlm =

1 if m = i−1 if m = j0 otherwise

∀m, i j. (4)

Second, we make sure we only use W wavelengthson each link.∑

i j

(zi jml + zi j

lm) ≤ W zml ∀ml. (5)

Note that we assume all lightpaths are bidirectional.Since link ml and lm are considered as separate linksmathematically, we need to add the lightpaths crossingboth links on the left-hand side. If we make W avariable, then this constraint is no longer linear. Wecould do a linear search for the right W , making Wa constant in each iteration. Alternatively, to makethe constraint linear, we can use the following two


constraints instead.∑i j

(zi jml + zi j

lm) ≤ W ∀ml (6)

∑i j

(zi jml + zi j

lm) ≤ Zzml ∀ml. (7)

Z is a large constant. The first constraint makes surethat the number of wavelengths on each link is fewerthan W and the second constraint makes sure that nolightpath passes through a fiber link if that fiber link isnot installed.

Since the fiber-to-node ratio f is given, we can onlyuse L = f N Ln = f N (since we assume Ln = 1)amount of fiber. When f = 1, roughly only N fiberlinks of length Ln can be added to connect all nodes.Therefore, f = 1 is the minimum required to guaranteea connected topology. If only short fiber links are used,f roughly corresponds to k, the average edge-to-noderatio. The following constraint limits the total fiberlength that can be used.∑

ml

zmldml < L = f N . (8)

The objective is to minimize the number ofwavelengths.

Objective: min W. (9)

3.2. Heuristic algorithm

The ILP formulations given above can only be usedto solve small problems exactly, e.g., for networks withless than 10 nodes. For larger problems, we proposean efficient heuristic algorithm. We compared ouralgorithm with the ILP for several problem instances,and found it produces near-optimal results. We alsocompared our algorithm with that in [10] and found thatour algorithm not only produces comparable results,but also runs several orders of magnitude faster. Forexample, for a network with 100 nodes, our algorithmtakes less than 0.05 s on a Sun Blade 1000 workstation,compared to 11 hours for the algorithm reported in [10].We call the algorithm TOPO F to denote that this is atopology design algorithm with the total fiber length asa constraint.

To understand what determines the number ofwavelengths needed, consider a set of nodes S and theremaining nodes S. Let X (S, S) denote the number offiber links that cross between the two sets of nodes.Since we want to establish a full-mesh connectivity,there are |S| × |S| lightpaths that will cross between

the two sets of nodes. Therefore, we can derive a lowerbound on W as follows:

WL B = maxS

|S| × |S|

X (S, S).

Such a lower bound was observed in [12,14], and it isalso called the flux of a graph in [14]. Note that WL B isthe maximum value among all possible sets of nodes. Soif only a few fiber links cross between two sets of nodes,WL B will be very high as the denominator is small. Todesign a topology that needs less capacity, we need tomake sure that the number of fiber links crossing anytwo sets of nodes is sufficiently large. This observationmotivates us to propose the following physical topologydesign algorithm.

The TOPO F algorithm proceeds in four steps.

• In the first step, a minimum spanning tree isestablished. This ensures that the topology isconnected.

• The second step tries to find the cut that achievesthe lower bound WL B , i.e., find the cut such that|S| × |S|/X (S, S) is the largest. To do so, we picka starting node s and initialize the set S to containonly the starting node (S = {s}). Then we add oneneighbor node into set S at a time until all nodes arein the set. When adding a node, we pick the neighbornode such that the cut X (S, S) is the smallest. Thereason we do so is because the smallest cut X (S, S)

will give the biggest WL B , which is what we arelooking for. When we have a new S (after adding anode), we compute |S| × |S|/X (S, S). If it is largerthan what we have seen before, we record it. To doa more exhaustive search, we try each node as thestarting node in turn and repeat the above process. Ifthere is a tie in the cut, we pick the cut for which wecan add a fiber link to bridge the cut and this fiberlink is the shortest among all cuts in the tie.

• At the end of the second step, we have identified aSmax such that |Smax| × |Smax|/X (Smax, Smax) is thelargest. Then, in the third step, we pick the shortestfiber link that can bridge the cut (one end of the fiberlink in Smax and the other end in Smax). If adding thefiber link will exceed the total fiber length budget, weproceed to step four. Otherwise, we add the fiber link,then return back to step two to find a new limitingcut. Note that we could have added a parallel fiber inthis step, even if a fiber link has already been addedbetween the two end nodes.

• In the last step, we check each remaining node pair inturn in increasing order of distance. If adding a fiberlink between the node pair will not violate the total


fiber length budget, the fiber link will be added to thetopology.

The TOPO F algorithm tends to use short fiber linksso that more fiber links could be added. Because thetopologies generated by the TOPO F algorithm havemany short fiber links, we call these topologies theshort-fiber topologies.

4. Numerical results

Using the physical topology design algorithm, wecan now study the tradeoff between L and W and seehow the provisioned capacity scales as a function of N .

We consider networks of practical sizes with upto a few hundred nodes. The backbone topologies ofmost Internet Service Providers (ISP) in the US havefewer than 100 nodes. Networks with more than a fewhundred nodes are less likely to be practical because ofthe difficulty involved in design and management. Ahierarchical network architecture might then be moreappropriate. Note that, unlike some recent work onunderstanding the Internet topologies [15,16], we arefocusing on the backbone fiber (physical) topology ofa single ISP. The backbone network is much smallercompared to the Internet, and it does not necessarilyfollow the characteristics of the Internet, such as apower-law distribution.

We consider a practical range of fiber-to-node ratiof , from 1 up to 4, to include networks that are currentlydeployed or to be deployed in the near future. The fiber-to-node ratio f of the current fiber networks of mostISPs is below 2 ( f < 2). Note that the fiber-to-noderatio is roughly the same as the edge-to-node ratio k ifshort fibers are used, and the edge-to-node ratio is halfof the average node degree, i.e., k = d/2.

For a network with N nodes, we assume these Nnodes are uniform-randomly distributed within a squareof area N (

√N on each side) to ensure Ln = 1.

To generate a network, we randomly place each oneof the N nodes at a point in the square with equalprobability. Once the network is generated, we thenapply the topology design algorithm.

To determine W after the physical topologyis designed, we use a Routing and WavelengthAssignment (RWA) algorithm to fit all lightpaths intoas few wavelengths as possible. Several heuristic RWAalgorithms have been proposed in the literature [17][18]. The algorithms we use are reported in a technicalreport [19], one with the Wavelength ContinuityConstraint (WCC) and one without. Since this paperfocuses on physical topology design and capacity

Fig. 4. Comparison of the results between the heuristic algorithm,the lower bound and the average and minimum of many randomtopologies. With WCC.

scaling, we will not discuss the details of the RWAalgorithms here.

In order for the experiments to be statisticallysignificant, we repeat the above procedure 1000 timesand then take the average before reporting the data. Inother words, for each N , we randomly generate 1000separate networks and then take the average result fromthe 1000 separate networks.

4.1. Heuristic algorithm and design approach evalua-tion

The results from the TOPO F algorithm (along withthe RWA algorithms) are very close to that fromsolving the ILP formulation directly for small-sizenetworks (<5 nodes). Unfortunately, because of thehigh computation time, we are not able to evaluate ourheuristic algorithm against the optimal for larger-sizednetworks. Instead, we will evaluate its performanceagainst the lower bound.

In Fig. 4, we plot the average capacity of short-fiber topologies (from the TOPO F algorithm) assumingf = 4, and the average lower bound (Eq. (1)) fromthe same set of networks. For comparison purpose, wealso plot the average and minimum capacity from 1000random topologies with an edge-to-node ratio k = 4.These random topologies are generated by randomlypicking a pair of nodes and then placing an edge acrossthem until the desired number of edges are generated.Since edges are picked without regard to their lengths,many long fiber links are used.

As shown in the figure, our topology designalgorithm can achieve capacity that is less than twiceof the lower bound. Considering that the lower boundis not achievable unless all lightpaths use direct-linerouting, we suspect that our algorithm is not too faraway from the optimal. Our physical topology design


Fig. 5. Wavelength requirement comparisons between randomtopologies and short-fiber topologies. With WCC.

algorithm can greatly reduce the required capacity, notonly compared to the average but also compared to theminimum of the 1000 random topologies.

Compared to random topologies, short-fiber topolo-gies with f = k use much less fiber at the cost of morewavelengths. In Fig. 5, we show the wavelength require-ment in random topologies with k = 4. For short-fibertopologies, we show two results. One is the wavelengthrequirement if we set f = k = 4; the other is the wave-length requirement if we set f to a value such that it willuse the same amount of fiber as in the random topolo-gies. We can see that the short-fiber topologies requiremuch fewer wavelengths if the same amount of fiberis used. This is because the TOPO F algorithm is con-scious about the fiber usage and, therefore, it is able toestablish more fiber links with the limited budget. Thisresult suggests that taking fiber length into considera-tion can lead to better designs.

4.2. Capacity scaling

In Fig. 6, we show the tradeoff between L and W .Clearly, W decreases as we increase L . Although hardto see in this figure, W increases by a larger proportion

Fig. 7. W as a function of N . With WCC.

as N gets larger when f is small compared to the casewhen f is large. This trend can be captured by thegrowth rate in W (or C). It is higher when f is smallcompared to the case when f is large.

In Fig. 7, we show W as a function of N (with theWCC constraint) in the short-fiber topologies. Note thatthe graph is on a log–log scale. Indeed, when f is small,W grows very quickly, as evident from the steeperslope, and close to 20 000 wavelengths are needed whenN = 500 and f = 1. This growth drops quickly as fis increased, resulting in a large reduction in W . Sincethe fiber-to-node ratio is fixed, the fiber length is a linearfunction of N , so the capacity will follow the same trendas the wavelengths.

The data in Fig. 7 appears to fall on a straight line,suggesting that W is a power function of N (Eq. (2) alsosuggests the same). Since C (Eq. (3)), W (Eq. (2)) and Lare all power functions of N in the form of aN e, we canuse the least square method to estimate the parametersa and e on a log–log plot. The results for short-fibertopologies are shown in Table 1. All curve fittings havea correlation coefficient of more than 0.99, confirmingthat a power function is a good fit.

Fig. 6. The tradeoff between L and W . With WCC.


Fig. 8. Number of wavelengths with or without the WCC constraint.

Fig. 9. Utilization ratio as a function of N for short-fiber topologies.With WCC.

In the following, we use the notation e(x) to denotethe exponent and a(x) to denote the constant in variablex , where x could be W , L or C . e(L), the exponent inL , is simply 1 because we have fixed the fiber-to-noderatio. e(W ), the exponent in W , and e(C), the exponentin C , again drop quickly as f increases. When f = 1,e(C) is more than 2.75. As f increases beyond 2.5, e(C)

quickly drops and it is between 2.6 and 2.64. Whenf is small, the topology is barely connected. Manylightpaths are forced to be routed through long detours.When f is large, there is a better chance for a lightpathto be routed close to the direct line. e(C) is smaller thanthat of the random topologies that we studied, whichis around 2.8, suggesting that our physical topologydesign algorithm is effective at reducing the capacityrequirement.

The constant a(L) of L is simply f , the fiber-to-node ratio. a(C), the constant in C , is almost thesame regardless of the parameter f . In other words,an increase in f (and therefore a(L)) would result ina corresponding decrease in a(W ), the constant in W .This result is consistent with our theoretical analysis(Eq. (3)).

Table 1Constant parameters from least square curve fitting for short-fibertopologies

f a eW L C W L C

1 0.27 1 0.27 1.76 1 2.761.5 0.15 1.5 0.23 1.73 1 2.732 0.12 2 0.23 1.69 1 2.692.5 0.10 2.5 0.24 1.66 1 2.663 0.09 3 0.26 1.64 1 2.643.5 0.08 3.5 0.28 1.61 1 2.614 0.07 4 0.28 1.60 1 2.60

With WCC.

The results in Table 1 are generated from networkswith up to 500 nodes, large enough to cover networks tobe deployed in the near future.

4.3. Effects of the WCC constraint

In Fig. 8, we plot the wavelength requirement as afunction of N for short-fiber topologies, for both thecase of with WCC constraint and without. When fis small ( f = 1) and N is large (N ≥ 100), thedifferences are noticeable. The differences are up to 9%for f = 1, and they are up to 7% for f = 2. In allother cases ( f > 2), the differences are very small,less than 5%. This suggests that the WCC constraintmakes little difference especially when the fiber-to-noderatio f is high enough. Therefore, the costly wavelengthconverters could be avoided by simply increasing f .This observation is consistent with that in [11], wherethey found that WCC has little effect on two-connectedtopologies.

4.4. Capacity utilization

In this section, we look at the effects of factor F2: theutilization of the provisioned capacity. Let wi denotethe number of wavelengths that are used on fiber linki , then the capacity utilization ratio can be defined as∑

i wiW M , i.e., the ratio between the sum of the number

of used wavelengths on each fiber link and the productof the number of wavelengths (W ) and the number offiber links (M). The capacity utilization ratio for short-fiber topologies (with the WCC constraint) is shown as afunction of N in Fig. 9. When f is small, the utilizationratio is very low. For example, when f = 1, at most70% utilization can be achieved and it is as low as40% when N is large. The situation quickly improvesas f gets bigger. Roughly 70% or 80% utilization ispossible when f > 2.5. The utilization ratio decreasesfor large N . This is because the average hop count of


lightpaths increases as N increases, making it less likelyto fully utilize the provisioned capacity. The decreasein the utilization ratio partly contributes to the higherasymptotic growth rate (the other contributor is factorF1, non-direct-line routing of lightpaths). We see asimilar trend on the utilization ratio when the WCCconstraint is relaxed.

In the figure, there is an anomaly. The utilizationratio is low when f is large and N is small. Thisis because of the “rounding effect:” when W issmall, adding one additional wavelength adds a largepercentage of capacity, which in turn greatly reduces theutilization ratio.

As mentioned in the beginning, we notice that Cscales almost perfectly linearly as T in our simulationstudy, as suggested by Eq. (3). It suggests that theincreased number of connection requests (lightpaths)cannot improve the capacity utilization. The lowcapacity utilization ratio seems to be a fundamentallimit of the topology. The only way to improvethe utilization ratio seems to be redesigning a bettertopology using a higher f .

Recall that wi denotes the number of wavelengthsthat is used on fiber link i . Let hi j denote the shortestpath (in number of physical hops) between node i and jin a given topology. Then we can define load as

∑i wi ,

i.e., the total number of wavelengths that are utilized.We can also define the shortest path load as

∑i j hi j ,

i.e., the load if all lightpaths are routed using the shortestpath. The difference between load and shortest pathload represents the extra number of wavelengths neededto route lightpaths along non-shortest paths in order tofully utilize the provisioned capacity. In Fig. 10, we plotthe exponent e from least square data fitting to a powerfunction (aN e) for C , the load and the shortest path loadas a function of f .

As shown in the figure, the gap between C andthe load is decreasing as f increases. This suggeststhat the capacity utilization keeps on improving as fincreases. In addition, the gap between the load andthe shortest path load is increasing. This suggests thatas f gets bigger, it becomes increasingly easier to findalternative paths to route a lightpath if the shortest pathis congested. Therefore, there is greater chance to evenout the routing of lightpaths onto all fiber links and thusreduce the required number of wavelengths W .

4.5. Non-uniform lightpath demands

We have so far only considered full-mesh connec-tivity because it makes the analysis easier. However, the

Table 2Asymptotic growth rate of the capacity for one particular set oflightpath demands in short-fiber topologies

f 1 1.5 2 2.5 3 3.5 4e(C) 1.78 1.74 1.65 1.64 1.62 1.58 1.59

Fig. 10. Growth exponent for C , the load and the shortest path loadas a function of f for short-fiber topologies. With WCC.

observation is not strongly dependent on the set of light-paths to support. Let us consider a different set of light-paths which is generated as follows. From each node,we randomly pick m other nodes and establish light-paths to them. The number of lightpaths generated ism N , i.e., the number of lightpaths is on the order of Ninstead of N 2 as in the full mesh. We pick m = 4 inour experiment so that the resulting W is large enoughto avoid the rounding effect.

We use the TOPO F algorithm to design the topologyand use curve fitting to determine the asymptotic growthrate of the capacity. The results are shown in Table 2.

e(C) should be 1.5 theoretically. But in simulation, itranges from 1.58 to 1.78. e(C) again decreases quicklyas f increases, which is consistent with our observationunder full-mesh connectivity.

5. Real-life networks

We have seen that increasing the fiber-to-node ratiof can reduce the capacity growth, increase the capacityutilization and avoid the use of wavelength converters.Therefore, as a design guideline, it seems to be a goodidea to have a high enough f . In this section, weexamine real-life networks and see how this designguideline can be applied. We studied several real-lifenetworks. Even though the bounding areas are notregular and the node placements are not uniform, theobservations we derive are very similar, therefore, weonly report our study on one real-life network here.

We consider the backbone network from Level 3 (anInternet Service Provider). According to the Rocketfuelproject [20], which maps ISP topologies as seen by


Fig. 11. Level3’s backbone network.

the electronic routers (the logical topology), Level 3attempts to establish a full-mesh connectivity, i.e., alightpath between every pair of nodes.

Even though the logical topology is a full mesh, theunderlying physical fiber topology is far from a fullmesh, as shown in Fig. 11. This fiber map could bedownloaded from their website directly [21].

This network has 56 nodes and 63 edges. The edge-to-node ratio is only k = 1.125. Only short fiber linksthat connect neighboring nodes are used, so it is a short-fiber topology. In fact, in most of the topologies westudied, long transcontinental fibers are almost neverused. The fiber-to-node ratio is f = 1.092, which isconsistent with our observation that the fiber-to-noderatio is roughly the same as the edge-to-node ratio forshort-fiber topologies.

Using our RWA algorithms, we determine that374 wavelengths are needed to support full-meshconnectivity with the WCC constraint. The capacityutilization ratio is only 55%.

We manually measure the distance between everynode pair and assume a direct fiber of that lengthcould be laid down to connect the node pair. Wethen apply the TOPO F topology design algorithm andRWA algorithms. Using the same total fiber length,the TOPO F algorithm designed a new topology with73 edges and the RWA algorithm found that only363 wavelengths are needed to support full-meshconnectivity. Our topology design algorithm is able todesign a topology that requires less capacity than that ofthe manually designed topology. This is a confirmationthat our algorithm performs well.

We also applied the TOPO F and RWA algorithmsfor different values of f . The results are shown inFig. 12.

Fig. 12. C and the utilization ratio as a function of f for Level3’snetwork. With WCC.

We can see that C rapidly decreases as f increases.The slope of decrease flattens out when f is at least2 or 3. At the same time, the utilization ratio quicklyimproves. When f is small, the utilization ratio is only50%, and it quickly improves to nearly 80%. Thisresult suggests that the current fiber-to-node ratio inLevel 3’s network is too low. It should be increased inorder to lower the capacity requirement and increase theutilization ratio.

6. Conclusion

We proposed an efficient algorithm to design thefiber topology with the goal of minimizing the numberof wavelengths under the constraint of a fixed fiber-to-node ratio. Using the algorithm, we studied the tradeoffbetween fiber (link cost) and wavelengths (node cost).Understanding this tradeoff allows network designers todesign cost effective transport networks.

We evaluated and compared designs under differentfiber-to-node ratios using the provisioned capacity asa metric, which is an indicator of how efficient the


provisioned resources are utilized. The asymptoticgrowth rate of the capacity not only captures thetradeoff between fiber and wavelengths independent ofthe network size, but it also indirectly translates into thedeployment cost regardless of the actual cost functionsfor fiber and wavelengths.

We showed that, compared to random topologiesand the lower bound, our physical topology designalgorithm is very effective not only at reducing thecapacity requirement but also at reducing the asymptoticgrowth rate of the capacity. We found that takingfiber length into consideration can reduce the capacityrequirement. We showed that having a large fiber-to-node ratio can greatly reduce the asymptotic growth rateand can lead to lower cost when N is large.

On studying several real-life topologies, we findthat most of them have too low a fiber-to-node ratio.By increasing it, we can greatly reduce the capacityrequirement and increase the capacity utilization.

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Physical topology design for all-optical networks