Optimization algorithmsfor WDM optical network dimensioning

Andrea Concaro∗, Simone De Patre∗, Guido Maier∗, Massimo Tornatore†∗CoreCom, Via Colombo 81 - 20133 Milan, Italymaier@corecom.it†Department of Electronics and Information, Politecnico di Milano, Via Ponzio 34-35 - 20121 Milan, Italy

tornator@elet.polimi.it

Abstract— In this paper1 we study the particular approachto planning the optical transport network under static trafficwhich consists in improving an initial solution by lightpathrerouting. Several different heuristic strategies to carry outthe rerouting process are proposed and described, includ-ing deterministic and stochastic algorithms. A performancecomparison among them is presented, based on case-studyexamples and considering: final fiber-number (cost function)value, execution time and convergence behavior. Results ofall the strategies are also compared with results obtained byinteger linear programming, evaluating of the possibility toheuristically obtain good suboptimal solutions. The two casesof unprotected connections and dedicated path-protection havebeen considered, with and without wavelength conversion.

I. INTRODUCTION

In the last few years an intense research activity has beenaddressed to the problem of optical network dimensioning.Operators are more and more frequently challenged bydesign problems, both to plan new installations and toupdate the existing ones, by the continuous growth of thedemand of bandwidth for new applications such as videoand multimedia streams and advanced digital services. Thisis particularly true for the Metropolitan Area Networks(MANs), where traffic aggregation is low and thus thebandwidth requirements of the single applications have ahigh impact on network performance. On the other hand,planning problems increase in complexity as the topologyof WDM networks evolves from ring to mesh, taking ad-vantage of the constant improvement of optical transmissionand switching technologies. The high connectivity of meshOptical Transport Networks (OTNs), relying upon OpticalCross Connects (OXCs) for switching, improves the band-width provisioning service, but requires careful planningto avoid useless capital expenditure. Planning is furthercomplicated by the need to equip the system with suitableprotection resources. High-speed optical connections (at 10Gbit/s or higher) are very vulnerable to failures: even a few-seconds outage means a huge waste of data. Survivability,that is the capability of the network of maintaining service

1Work partially supported by MIUR, Italy, under FIRB ProjectTANGO and Project WONDER.

continuity in presence of failures, from an attractive researchtopic has became an outstanding important planning issuefor every OTN.

Though dynamic traffic is becoming more and moreimportant and probably will eventually become dominantas the GMPLS/ASON (Generalized Multi-Protocol LabelSwitching, Automatically Switched Optical Network) ar-chitecture spreads pervasively, present optical-network op-erators have still to provide mostly permanent or semi-permanent optical circuits. In our paper we are thus goingto deal with OTN dimensioning in a static-traffic scenario.Given the physical topology and the set of Lightpath Con-nections (LCs) that must be setup (LC-layer topology),network capacity dimensioning and resource allocation aresolved simultaneously, minimizing a chosen cost function.An additional goal is to plan the spare capacity whennetwork survivability against a single link failure has tobe guaranteed. As a possible survivability technique, wehave chosen to refer to Dedicated path-protection (DPP),which is the simplest and fastest survivability scheme, notrequiring reconfiguration of transit OXCs upon a failure.Another advantage of DPP in a context of network planningunder static traffic is that channel assignment is completelyfailure independent.

In order to solve the planning problem outlined above,efficient dimensioning procedures are needed. The problemcan be tackled by using either an exact or a heuristic ap-proach. The former, namely the Integer Linear Programming(ILP), can find the optimal solution, but it can not beapplied to large networks due to its enormous complexity,increasing exponentially with network size. The latter doesnot guarantee the optimality of the obtained solution, but itis generally simpler and faster.

In this paper we wish to focus on the second approach,proposing a set of optimization strategies and comparingtheir performance by applying them to two realistic network-cases (NSFNET and EON). We are also going (when possi-ble) to compare the results of the heuristics to those providedby the ILP solution of the same design experiments. Aswe will summarize later on in the paper, many studieshave been published in the past on heuristic optimization

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of the OTN. While most of such studies are focused onsingle specific methods, in this work we are proposing aflexible general scheme which is able to operate accordingto several heuristic methods, selecting one or another withjust few changes in the input-parameter set. This approachallows us to easily compare the most common methods,showing differences and commonalities both in their algo-rithmic structure and in their performance. We also havethe opportunity to observe and compare the computationaltime-complexity of the considered algorithms.

Sec. II describes how OTN is modelled for planning.Previous literature on heuristic optimization is reviewedin Sec. III; the section also defines the ILP formulationsthat are adopted in this paper. Sec. IV introduces to ourplanning method and describes the heuristic strategies wehave compared. Case-study analysis is reported in Sec. V,while the major results of this work are summarized in Sec.VI.

II. NETWORK MODEL

We consider multifiber optical networks. Each link of thenetwork is equipped with two independent sets of mono-directional fibers. Each fiber carries Nλ wavelengths of theWDM multiplex: Nλ is a pre-assigned design parameter,constant for every link of the network, and correspondingto a particular type of WDM transmission system that theoperator has chosen to deploy. The nodes are equipped withOptical Cross Connect (OXC). They are able to switchan incoming optical channel (a wavelength on a fiber)to an outgoing channel. We will consider OXCs with orwithout wavelength converters. In the former case, a nodecan convert the wavelength of an incoming lightpath to adifferent outgoing wavelength; in the latter case, the nodecan not change the wavelength of the incoming lightpath.

The LC-topology is known: connections are static andpoint-to-point (OXC-to-OXC) unidirectional. It is decidedin advance wether the network is unprotected or if allthe optical connections must be protected by DPP. In theunprotected case, a lightpath, i.e. a sequence of WDMchannels along a path on the physical topology, must beallocated for each connection request. If all the nodes ofthe network are equipped with wavelength converters, weare considering a VWP (Virtual Wavelength Path) network.If no node is equipped with wavelength converters, we areconsidering a WP (Wavelength Path) network. In the firstcase wavelengths can be assigned to a lightpath link-by-link, with possible intermediate wavelength changes; in thesecond case we must assign a single wavelength to the entirelightpath (i.e. the lightpath must satisfy the wavelengthcontinuity constraint). When DPP is adopted, two lightpathsmust be set up per connection-request, composing a workingand a protection pair (w/p pair): we have to route the w/ppair imposing that the two lightpaths must be link-disjoint

(they can not share a common physical link). In this way, incase of a (single) link failure, it will always be possible toreroute the interrupted working traffic on the spare capacity.

The variables of our model are the capacities of eachphysical link in terms of number of fibers and the costfunction is the total number of fibers F that will be deployedto satisfy all the requests of the LC topology (with orwithout protection). We are thus seeking the Routing andFiber and Wavelength Assignment (RFWA) solution for allthe lightpaths (or the w/p pairs) resulting in the smallestpossible F . The optimum RFWA is such that F is necessaryand sufficient to setup the LC-topology. With heuristicapproaches we are able to find RFWA solutions leading tovalues of F that are sufficient but not necessary, thoughhopefully not too much greater than the optimum F .

III. OTN OPTIMIZATION IN LITERATURE

As previously mentioned, this paper is mainly dedicatedto OTN heuristic design. We will however present also someILP-based results to benchmark the heuristic algorithms.Therefore in this section we are going to provide the“standard” ILP flow formulations for the unprotected andDPP scenarios, summarizing the ILP models presented inRefs. [8], [19], [24] to solve capacitated design in WDMnetworks. We will report only the case with full wavelengthconversion: the extension of the models to WP can be carriedon as explained in Refs. [22], [24], introducing a furtherterm of complexity, function of Nλ.

Let us consider the physical topology, modeled by thegraph G = G(N ,A) 2. Physical links are represented bythe undirected edges l ∈ A with |A| = L, while thenodes i ∈ N = {1, 2, ...N}, with |N | = N , representthe OXCs. Each link is equipped with a certain amountof unidirectional fibers in each of the two directions; fiberdirection is conventionally identified by the binary variable k(k = 0 for forward direction, k = 1 for backward direction).vc is the number of requested LCs having sc as source anddc as destination node, with c an index used to identify eachsource-destination node-couple requiring connectivity.

The (integer) variables involved in the unprotected flowformulation are:

• xl,k,c, flow variable indicating whether a WDM chan-nel on link l on a fiber having direction k has beenallocated to one of the connections requested by nodecouple c;

• Fl,k, capacity variable indicating the number of fiberson link l in direction k.

The following additional symbols are also defined:

• (l, k) identifies a “unidirectional link”, i.e. the set offibers of link l that are directed as indicated by k;

2The following formulations require that the topology is at least2-connected.

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• I+

i is the set of “unidirectional links” having node i asone extreme and leaving the node; analogously, I−i isthe set of “unidirectional links” having node i as a oneextreme and pointing towards the node.

The cost function to be minimized is the total fiber number

min F = min∑(l,k)

Fl,k

The unprotected flow formulation is given by the followingset of equations

∑

(l,k)∈I+i

xl,k,c −∑

(l,k)∈I−i

xl,k,c =

⎧⎨⎩

vc if i = dc

−vc if i = sc

0 otherwise∀ i, c;

(1)

∑c

xi,k,c ≤ Wl · Fl,k ∀ (l, k); (2)

(3)

xl,k,c integer ∀ (l, k), c; (4)

Fl,k integer ∀ (l, k); (5)

Constraint (1) is a solenoidality constraint. Let us considerthe vc connections requested by c: the flow conservationcondition for vc in each node i states that vc leaving i mustbe equal to vc incident on i. In the source (destination) nodethe flow balance is satisfied by adapting the constraint to theborder condition: the total leaving (incoming) flow must beequal to vc. Constraint (2) ensures that the total number ofWDM channels allocated to spare and working lightpaths onthe unidirectional link (l, k) is bounded by the link capacity,given by the number of fibers Fl,k multiplied by the numberof wavelengths W . Constraints 4 and 5 enforce the integrityof the fiber number and flow unities.

The definition of an ILP model in a WDM network withdedicated path protection is a well-known problem: to theset of constraints of the unprotected formulation, we mustadd constraints deriving from the link disjointness condition.These can be easily set, provided that the basic flow variableis enriched with a new index, increasing the descriptiondetail of the flows. If vc > 1, we add an auxiliary index thaving values between 1 and vc; the flow variables become:

• xl,k,c,t, boolean variable indicating whether a WDMchannel on link l on a fiber having direction k hasbeen allocated to the t-th connection requested by nodecouple c.

We further introduce the following symbol:

• (c, t), identifying a single connection request.

The set of constraints is the following∑

(l,k)∈I+i

xl,k,c,t −∑

(l,k)∈I−i

xl,k,c,t =

⎧⎨⎩

2 if i = sc

−2 if i = dc

0 otherwise∀ i, (c, t); (6)

∑(c,t)

xl,k,c,t ≤ W · Fl,k ∀ (l, k); (7)

∑k

xl,k,c,t ≤ 1 ∀ l, (c, t); (8)

xl,k,c,t binary ∀ (l, k), (c, t); (9)

Fl,k integer ∀ (l, k); (10)

Constraint (6) enforces the flow solenoidality, as in theunprotected case. A slight difference exists in the source(destination) node of the connection request (c, t), in whichthe outgoing (incoming) flow must be equal to 2. This is dueto the fact that a w/p pair, instead of a single lightpath, isassociated to the connection request (note that the distinctionbetween working and protection lightpaths is irrelevant).Constraints concerning dimensioning (7) are simple exten-sions of the corresponding constraints in the unprotectedcase. Constraint (8) stems from link-disjointness condition:no more than one lightpath associated to connection request(c, t) can exist on the same link, in both the oppositedirections.

Solving the above set of equations for realistic networks isreally difficult, since the number of variables and constraintstends to explode with the network size (OTN planning withcapacitation is well-known to be a NP-hard problem). Inorder to keep computational time and memory occupation atreasonable levels we have actually obtained the results thatwill be presented in Sec. V by exploiting slightly different,but more efficient formulations compared to the “standard”flow formulation presented above. Such formulations, whosedetailed description is outside the scope of this paper, arereported in Refs. [22], [23].

We will concentrate in the rest of the paper on the heuris-tic approach to OTN design, as an alternative to the ILP. Weare going to consider some heuristic optimization strategies,all based on the concept of lightpath rerouting: given aninitial RFWA solution (i.e. a feasible accommodation of allthe requested lightpaths on the physical topology) we try toimprove it by rerouting the optical connections on alternativepaths, subject to the constraint that all the LC-requests mustbe satisfied. This kind of approach to the problem, despitebeing a very simple and direct solution method, can achievegood results, as we are going to verify by case-studies.

We can find in literature a lot of examples of heuristiclightpath-rerouting. In [24] the total number of used wave-lengths is minimized by rerouting lightpaths that crosses

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the most-loaded links, while in [3] the connections, initiallyrouted on shortest paths, are rerouted on alternative shortestpaths, if this operation can decrease the load on the mostcongested link. In [1], where multifiber networks are consid-ered, new configurations are iteratively searched by modify-ing the initial routing and/or wavelength assignment, tryingto lower the cost of requested fibers. Ref. [17] considers anSDH-over-WDM network: the rerouting method is utilizedto find a good LC-topology for the WDM layer, given therequests for SDH circuits. The SDH connection-requests arerouted on a sequence of lightpaths (multi-hop routing) tryingto minimize the number of LC-requests. In order to achievethis purpose, the optimization algorithm tries to reroute theSDH channels initially carried by a lightpath on the residualcapacity of other lightpaths.

All the above-cited techniques can be classified as de-terministic heuristics, since rerouting attempts are carriedout according to a fixed lightpath-sorting rule, acceptingrerouting each time it leads to an improvement of thecost function. Due to this feature, deterministic methodsare affected by the problem of getting trapped in localminima, which can not be overcome with a “down-slope”-only decision-rule.

Other heuristic rerouting approaches do exist, which canbe classified as stochastic and are able to avoid local-minimatrapping. Simulated annealing [15] is a well-known generaloptimization method which stochastically simulates the slowcooling process of a physical system. The temperature of thesystem is lowered by small steps until the system “freezes”and no further changes occur. To apply simulated annealing,the system is initialized with a particular configuration. Anew configuration is constructed by imposing a randomdisplacement. If the energy (the cost function) of this newstate is lower than that of the previous one, the changeis accepted unconditionally and the system is updated. Ifthe energy is greater, the new configuration is acceptedprobabilistically. This procedure allows the system to moveconsistently towards lower energy states, yet still “jumping”out of local minima due to the probabilistic acceptance ofsome upward moves. In [21] simulated annealing is appliedto OTN with the purpose of mapping a regular virtualtopology on a given physical topology. In [20] simulatedannealing is utilized to find a good virtual topology, andthen a flow-deviation algorithm optimizes traffic routing.Simulated annealing and other stochastic algorithms (ran-dom sampling, local search, threshold accepting, tabu searchand genetic algorithms) are applied in [10] to ATM networkdimensioning. Other stochastic algorithms have been appliedto optical networks, such as Monte Carlo [7] and geneticalgorithms [2], [14]. Stochastic and deterministic heuristicapproaches to OTN planning are compared in Refs. [2],[7], proving how a random component can often be usefulto obtain good suboptimal results. Similar comparisons are

presented in [16], where the objective of optimization isregular topology for packet-switched optical networks.

IV. HEURISTIC PLANNING METHOD

The heuristic approach to static-OTN design we havedeveloped is based on the rerouting concept. As previouslymentioned, it is divided into two steps:

Step 1 feasible RFWA-solution evaluation;Step 2 improvement by rerouting.

In the whole design procedure the network state has beenrepresented by a Multifiber Layered Graph (MLG). This isderived from the layered graph, introduced for mono-fibernetworks [9], and extended in [18] to multifiber networks.

A. Step one: greedy resource allocation

Let us provide a synthesis overview of the first step, whichis not the actual objective of this paper. A feasible RFWAsolution is obtained with the following technique. Startingfrom the idle physical topology, all the connection requestsof the LC-layer topology are set up in sequence one afterthe other until all have been satisfied. Each link initiallycontains a number of fibers so large to be considered infinite:in this way the existence of a solution is guaranteed. Theconnection requests of the LC-layer topology are initiallysorted according to a “balanced” sorting rule: node-pairswith greatest topological distance and largest amount ofconnection-requests are served with the highest priority. Thetechnique is greedy, as each requests is satisfied regardlessof all the others, allocating resources to a lightpath (unpro-tected case) or a w/p pair (DPP case).

In the unprotected case, RWFA is carried out on a singlelightpath by adopting the prioritized multi-criteria approach,extensively described in [11]. The set of criteria adoptedin this work was identified in [11] as the one leading inmost of the cases to the best greedy allocation (with thelowest number of fibers). It comprises, with decreasingpriority: “Shortest Path Routing” (SPR), “First Fit” fiberselection (FFF), “First Fit” wavelength assignment (FFW),“Least-Loaded Routing” (LLR) [4], [18]. The length metricadopted for routing is the number (minimum hop - mH)of the crossed links. With the “First Fit” criterion, fibers(wavelengths) are sorted in the same way in all the links(fibers) of the network and the first available is alwayschosen according to this sorting. To implement the aboveheuristic RFWA criteria, suitable weights are assigned to theMLG arcs and the Dijkstra algorithm is used to find the routeconnecting the source to the destination on the MLG withthe least total weight. We shall note that RFWA is performedin an unconstrained mode, that is all the possible routes onthe MLG connecting source to destination are scanned insetting up a new lightpath.

When DPP must be supported, the RFWA of the workinglightpath is coupled to the RFWA of the protection lightpath

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of the same w/p pair by the route-diversity constraint. Thereare two main techniques to find two link-disjoint pathsconnecting two nodes of a mesh topology. The simplestone is called two-step search. The shortest path is found(e.g. applying the Dijkstra algorithm) and it is allocated tothe working lightpath. Then the shortest-path algorithm isrun again to route the protection lightpath, which will beassigned, in this way, the second-shortest path. The two-stepapproach (greedy, because of the sequential computation ofthe two paths) in some special cases [12] may fail to find aw/p pair solution even if such solution actually exists. Bhan-dari [5], [6] proposed to overcome such a limitation by aone-step search, in which the two link-disjoint paths are notrouted separately, but they are jointly routed by performinga suitable algorithm (the modified Dijkstra algorithm) in adirected graph. Besides being able to solve trap-networks,it also finds the actual minimum-length cycle connectingtwo nodes. We applied the one-step technique in the DPPcase by adapting the Bhandari algorithm to the MLG. Inparticular, a set of rules has been added to the modifiedDijkstra algorithm to control the edge inversion operation[6] in the MLG environment with and without wavelengthconversion and to support LLR.

B. Step two: rerouting procedures

We are now going to discuss more in depth the secondplanning step. The basic purpose of this step is to modify thenon-optimized network solution found by greedy resourceallocation. The chosen cost function, which in our caseis the total number of fibers, is decreased by reroutingsome connections, under the constraint of preserving con-nectivity of the LC-layer topology. Qualitatively, some kindof iteration selects one specific fiber of the network at atime. Lightpaths or w/p pairs crossing the selected fiber aretentatively rerouted on the other fibers of the network. Iffree resources are enough to allow rerouting, the selectedfiber is removed. Otherwise, everything remains unchanged.Iteration is used to reach at least a local minimum of the costfunction, i.e. such that further improvement are impossibleby applying the same heuristic procedure.

The order by which fibers are selected and the itera-tion control depends on the particular optimization strategyadopted to solve the problem. In this work we have testeddifferent heuristic strategies, belonging both to the determin-istic and the stochastic class, which we are going to presentin details. Since most strategies have been implementedas variations of a common procedure, it is convenient todescribe them starting from their common elements. Let usdefine some basic actions that will be used many times inthe procedures. In the following, we consider a fiber f onwhich a set Sf of lf lightpaths (which can be working orprotection in the DPP case) are routed.

• FTr - Temporary reroute f . Let us consider the unpro-tected case. The following steps are performed

1) The current RFWA of all the lf lightpaths isstored in a separate database

2) Each lightpath of Sf is deallocated by freeing allthe MLG arcs it occupies

3) Fiber f is disabled by preventing any subsequentoccupation of the MLG arcs belonging to it (e.g.setting their weights to infinite)

4) A new RFWA on the residual MLG is attemptedfor each connection request having a lightpathbelonging to Sf

In the DPP case, the above steps may be described inthe same way, only considering in steps 1 and 2, insteadof lightpaths, the w/p pairs with a lightpath crossing f .

• FBl - Block f . This conditional function returns NO ifall the connection requests affected by a FTr functioncould be routed successfully on the residual MLG andYES otherwise.

• FRe - Remove f . All the MLG arcs belonging to f arepermanently disabled, and f is removed from the listof the existing fibers. Moreover, a register SCS is setto TRUE, indicating that at least an FTr function hassuccessfully terminated.

• FRs - Restore f . The RFWAs saved in step 1 of anFTr is restored on the MLG and arcs belonging to fare re-enabled for future usage.

Fig. 1 shows how the above functions are combined tocompose the core procedure. At the beginning of suchprocedure all the existing fibers are numbered from 1 to F ∗.f is used as a local index to scan the existing-fiber set. Eachfiber is processed by FTr and the subsequent functions FBl,FRe and FRs, provided that two conditional functions Yand X result to be TRUE. The definitions of such functionsdepend upon the specific heuristic strategy and will be givenshortly below.

Our spectrum of optimization strategies covers the follow-ing alternatives: Idle-Busy (IB), Busy-Idle (BI), RandomOrder (RO), Random Polarized (RP), Shortest Path (SP),Simulated Annealing (SA).

The first four strategies have similar structures, which areall together represented by the flow-chart of Fig. 2. In Startthe network solution found by the greedy planning step isconsidered. The first action performed is the removal ofany residual empty fiber, by executing FRe on all the fibershaving of = 0, where of is the number of WDM channelsof fiber f allocated to working or spare lightpaths. After thisaction, common to all the strategies, the procedure branches.One of the alternative branches is executed, according tothe optimization strategy chosen for the particular planningexperiment. In all the cases, residual empty fibers are againremoved at the end; then, the result in terms of RFWA forall the lightpaths and network dimensioning is returned in

145

f = f + 1f > F*

Label fibersfrom 1 to F* Yf = 1 X(of,k) Temporary

reroute f

Block f

Remove f

Restore f

TRUE

FALSE YES

NO

FALSE

TRUE

FALSE

TRUE

Core procedure

Fig. 1. Flow chart of the core procedure.

FALSE

k = 0

k = k + 1

k = N - 1

Coreprocedure

Removef | of = 0

Stop

Start

TRUE

FALSE

SCS = FALSE

k = 1

Coreprocedure

k = N

k = k - 1

SCS

Removef | of = 0

k = 0

k = N - 1

Coreprocedure

TRUEFALSE

Eval p(k)

FALSE

Coreprocedure

i = 0

i = i + 1

SCS

FALSE

Random V

SCS = FALSE

i = N - 1

k = k + 1

k = V(i)

TRUETRUE

FALSE

Strategychoice

IB BI RP

RO

TRUE TRUE

SP SA

Fig. 2. General flow chart with the various heuristic strategies (IB, BI.RO and RP detailed here).

Stop.The IB strategy [18] is deterministic. Fibers are selected

for rerouting from the idlest to the busiest, beginning fromthose carrying only one lightpath and ending with thosewith just one WDM channel free. In the core procedure,the conditional function X(of , k) is: of ≤ k ∧ of > 0(Y , not needed, is set to the constant TRUE). The strategyhas the advantage of trying to eliminate first fibers that areeasy to free, since FTr initially involves few connections.Beside that, saturation of the free capacity due to successfulrerouting is gradually distributed in the k cycle, avoidinghigh blocking probability at the beginning. A fiber hasthe chance of being selected multiple times with differentvalues of k. In fact, for each k increment, non-empty fibershaving up to k (and not simply k) busy channels areconsidered. There are good chances that at the end of thek-cycle a suboptimal solution has been reached. Note thatX(of , k) prevents empty fibers from being selected: thisfurther condition leaves more spare capacity for reroutingduring the k-cycle, thus reducing blocking probability.

The BI strategy, again deterministic, is almost the dualof the previous one: fiber selection-order is from the most

to the least loaded. In the core procedure, the conditionalfunction X(of , k) is of = k (Y is set to TRUE): onlyfibers having exactly k busy channels are considered foreach k increment. In order to compensate this restriction,the entire k cycle is repeated several times until FTr is nomore possible for any fiber (FBl always detects a block): theoptimization terminates when the register SCS is detectedto remain FALSE after the end of the inner cycle. Therationale of BI is that deallocation of very loaded fibers isattempted at the beginning, when chances of blocking arelow due to a relatively high amount of unused capacity.

The first stochastic strategy we have considered is theRO: fibers are considered for selection in a random order ofoccupation. At the beginning of the procedure, the functionRandom V randomly inserts the integer numbers from 1 toNλ − 1 into the (Nλ − 1)-element vector V. The vector isthen scanned by the index i, selecting a value k = V(i)per iteration. Fibers are selected according to the conditionX(of , k): of = k (Y is set to TRUE). As in the BI strategy,the entire k cycle is repeated many times until FTr is nomore possible for any fiber (exploiting the check on SCSas termination condition).

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The other stochastic strategy detailed in Fig. 2 is RP.Fiber selection occurs exactly as in IB, setting X(of , k) to:of ≤ k∧of > 0. The selected fiber is however processed bythe core procedure or skipped according to a random choice.Eval p(k) at the beginning of the k cycle sets a probabilitythreshold p(k) given by the expression

p(k) = p0 + (1 − p0)k − 1

Nλ − 2where p0 (0 < p0 < 1) is a preset parameter. Inside the coreprocedure, the Y condition is defined as follows:Y - x < p(k), where x a random variable uniformly

distributed between 0 and 1.The strategy is random and polarized since the thresholdp(k) increases with k, making the probability of processingp(k) low at the beginning of the k-cycle and steadilyincreasing with k up to 1 when k = Nλ−1. The effect is thatthe deterministic order of fiber processing is perturbed by arandom fluctuation with an amplitude high for idle and lowfor busy fibers. This strategy can be regarded as an hybridbetween the deterministic IB and the stochastic approach.It should be noted that RP has also common aspects withsimulated annealing, since p(k) plays the role of a systemtemperature: however, it should be actually considered a“simulated melting”, as the temperature increases.

Another deterministic strategy is SP. Its flow-chart is notworth to be represented, since it is a mere iteration of IB.The only change to IB is a modification of the FBl conditionadding a constraint on the length of the rerouted lightpaths.Let us suppose that c is a connection of the set of thosethat must be tentatively rerouted. The topological distance(the length of the shortest path, in hop) between source anddestination is Hc, while the minimum-length cycle betweenthe two nodes has length (in hop) Cc. If n is the IB-iterationcounter, going from 1 to nmax +1, in the unprotected case,a possible newly-rerouted lightpath is constrained to have alength (in hops) less than or equal to Hc + n − 1, while inthe DPP case, the sum of the lengths (in hops) of the newly-rerouted working and protection lightpaths is constrained tobe less or equal to Cc + n − 1. The conditional functionFBl in the SP strategy returns NO if all the connectionrequests affected by a FTr function could be carried outsuccessfully on the residual MLG, subject to the abovelength constraints. Otherwise, YES is the result. The use ofn allows to progressively relax the constraint up to a prefixedvalue nmax, while controlling the number of iterations. TheSP technique has the advantage of limiting the amountof resources that can be used by lightpath rerouting, thusslowing the saturation the unused capacity of the network.

The SA strategy is the application of the well-knownSimulated Annealing method to our planning problem. Sincea flow-chart would be too complex to represent, we willexplain the SA strategy in words. In SA, differently fromall the other strategies, fibers can be not only removed but

also added to the network: this gives the chance of escapingfrom local minima. The control temperature of the annealingprocess in our case is represented by the probability p ofaccepting fiber addition. p is initially set to a high value p0

(0 ≤ p0 ≤ 1). Then a fiber-processing cycle begins. NF

fibers are processed in each iteration, as described below.At the end of an iteration, if the current value of p is lowerthan a prefixed threshold pTh, the SA phase terminates; else,the new current value of p is set to p · ∆ and the cycleiteration is repeated again, processing NF fibers. ∆ is thepreset cooling-rate annealing parameter.

The NF fibers processed in a cycle iteration are randomlychosen. Let us describe what happens when a given f fiberhas been selected. First of all, the current network stateS is saved and the current fiber number F ∗ is computed.Then, FTr is performed on f and the FBl condition isevaluated. If no block is detected, all the empty fibers ofthe network (including f ) are permanently removed and thenetwork state consequently updated. Processing begins againby randomly choosing a new fiber. If block is detected, arandom choice is taken. With probability 1 − p, f is leftin place and no further action is taken: S is restored and anew fiber is selected. With probability p, instead, a fiber-addition procedure is performed. This latter comprises thefollowing steps. First, one fiber is added to any link ofthe network. Then FTr is performed on f again: this timethere can obviously be no block. After lightpath have beenrerouted, all the empty fibers of the network (including f )are removed and the total number of fiber F ∗∗ is evaluatedagain. If F ∗∗−F ∗ > FT , being FT another preset annealingparameter, fiber addition is considered unacceptably large: Sis restored, returning to the network state as it was before thebeginning of processing of f , and processing of a new fiberbegins. Else, the network state is updated by permanentlyaccepting the added fibers and the newly-routed lightpathsbefore beginning to process another fiber.

V. RESULTS

We are now going to compare the various heuristicplanning procedures and ILP optimization on the basisof numerical results. We have considered two well-knownrealistic networks for case-study: the National Science Foun-dation Network (NSFNET, 14 nodes and 44 links) andthe European Optical Network (EON, 19 nodes and 78links). Their physical topology is shown in Fig. 3: as itclearly appears, EON is much more densely-connected thanNSFNET. The LC-layer topologies used for the planningexperiments have been derived from the static (symmetric)traffic matrices based on real traffic measurements which arereported in Refs. [13], [19]. The two LC-layer topologiescomprise 360 and 1380 unidirectional connection requestsfor NSFNET and EON, respectively.

Stochastic strategies benefit from the advantage over thedeterministic approach that re-executing the dimensioning

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(a)

(b)

Fig. 3. Physical topologies of NSFNET (a) and EON (b) networks.

TABLE IFINAL FIBER NUMBER OBTAINED BY PLANNING NSFNET IN

THE UNPROTECTED CASE.

Nλ IB BI RO RP SP SA ILP2 380 380 380 380 380 380 3744 199 198 198 200 199 197 188-192

W 8 110 107 109 110 109 107 95-100P 16 59 60 61 60 60 59 49-55

32 36 38 37 36 36 36 24-4164 27 31 26 26 28 25 -

128 21 20 24 20 26 19 -2 380 380 380 380 380 380 3744 198 199 198 198 198 199 188-189

V 8 106 106 104 106 106 102 96W 16 58 60 58 56 58 54 51P 32 35 36 36 35 34 34 30

64 22 29 28 25 28 23 21128 20 25 22 23 26 15 15

procedure different results may be obtained. We have ex-ploited this property for NSFNET: being this network lessconnected than EON, the repetition of dimensioning doesnot take too much computational time. Therefore, resultsconcerning RO, RP and SA have been obtained for theNSFNET by repeating the second planning step twice andconsidering the best result. The following values have beenused for strategies requiring preset parameters (see Sec. IV-B): p0 = 0.2 for RP; nmax = 6 for SP; p0 = .5, ∆ = 0.95,pTh = 0.01 and NF = 10 for SA.

A detailed list of the results is displayed for the NSFNETonly by the two tables I and II. Results are given in terms

TABLE IIFINAL FIBER NUMBER OBTAINED BY PLANNING NSFNET IN

THE DPP CASE.

Nλ IB BI RO RP SP SA ILP2 995 995 995 995 995 994 9834 505 507 505 505 506 505 492

W 8 263 263 262 263 263 261 -P 16 140 141 140 139 140 139 -

32 78 80 77 79 78 78 -64 45 48 46 46 46 45 -

128 34 35 35 34 34 35 -2 995 994 994 995 992 992 9834 505 506 504 504 503 503 492

V 8 256 259 257 256 256 256 247-248W 16 134 136 137 134 135 135 125P 32 73 74 73 73 73 73 65

64 41 42 43 41 42 42 38128 33 34 33 33 33 33 30

of values of the cost function, which is the total numberof fibers F deployed in the dimensioned networks, andthey have been grouped by survivability feature (unprotectedconnections or DPP required for each connection). Insideeach table, the WP and VWP cases are displayed, providinga row for each value of the number of wavelengths perfiber Nλ. A bold number in a column indicates that thecorresponding strategy reached the best result among all theconsidered heuristic optimization strategies. A dash tells thatthe corresponding heuristic strategy could not be applied dueto memory (900 Mbyte) exhaustion.

ILP optimization has been carried out by exploiting thecommercial software CPLEX, implementing the branch-and-bound algorithm. When a single value in normal typeappears in the ILP column, the optimization ended, findingthe optimum. In all the other cases, optimization was in-terrupted for memory exhaustion or after 3 days. In someof these cases, a single number in italic indicates that theonly solution obtained was achieved relaxing all the integrityconstraints on the link capacity. In some other cases, twovalues are reported: the one in italic is the lower boundestimated by CPLEX, while the one in normal type is thebest non-relaxed solution found, not guaranteed to be theactual optimum. A dash with no numbers tells that CPLEXwas not even able to setup the optimization session.

Fig. 4(a) gives a synthetic overview of the strategycomparison. Each column, referring to a particular heuristicstrategy, is subdivided in four rectangles. The hight of a rect-angle referring to NSFNET is obtained by counting in tableI the number of occurrences of a bold entry in the column ofthe strategy (similarly for rectangles referring to EON). Thetotal hight of the bar is the total number of times the strategywas able to find the best heuristic F in the experimentscarried out on both the networks. As expected, SA is onaverage the most reliable strategy. It is remarkable, however,that it has been overtaken by a deterministic strategy in some

148

of the experiments. The best deterministic strategy is IB,followed by SP (which overtakes IB in the DPP-EON case).The other two stochastic strategies, RO and RP, are not asgood as the best deterministic competitors: the comparisonbetween these two proves that a deterministic random-choicepolarization (in favor of less loaded fibers) is an advantage.The BI approach resulted to be inappropriate in all the cases:clearly, too many lightpaths rerouted at the beginning ofthe cycle too quickly saturate the unused network capacity,preventing further rerouting. It should be noted that BIfiber selection is worse than random selection (RO). Theimprovement due to repetition for the stochastic strategiesclearly appears by comparing the rectangles correspondingto the NSFNET and the EON. The lack of repetition makesstochastic strategies less performing than deterministic (e.g.SA vs. SP in the DPP-EON case). It can happen howeverthat non-repeated SA is nevertheless the best strategy (e.g.in the unprotected-EON VWP case). Performance relationsremain similar to those shown in the figure, if we count thenumber of times each strategy achieved the worst (insteadof the best) result, with the only exception that IB comesout to be slightly worse than SP.

Another synthetic view of the gathered data is given byFig. 4(b), in which we show the differences between theheuristic approaches and the ILP method, in terms of F andfor different values of Nλ. We have plotted the DPP-NSFVWP case, since a complete set of ILP results was available.∆min (∆max) is the difference (in absolute terms) betweenthe best (worst) heuristic F and F found by ILP. When twovalues appear in the ILP column, the non-relaxed result isemployed. The differences are also plot in percent, usingthe ILP value of F for normalization. The best heuristicsuboptima are never more than 11 fibers greater than theILP result; the distance between the best and worst heuristicis also no greater than 3 fibers. The decreasing behavior of∆min and ∆max seems not to be very meaningful, sinceit is not confirmed by the other experiments concerningEON and/or different protection and conversion conditions.The increasing trend of %∆min and %∆max is due tothe descent of the optimum F , which is roughly inverselyproportional to Nλ (constant offered traffic). Averaging onall the experiments, including EON (considering only casesin which ILP returns an integer solution), we have: ∆min =8.2 and ∆max = 11. Moreover, if the two best heuristicstrategies (SA and IB) are used, the maximum differencefrom ILP is 16%, while the mean difference is 5%: thesedata prove the validity of heuristic optimization as a practicaldesign procedure for OTN. It should further noted that insome unprotected cases (WP NSFNET with Nλ = 32, WPEON with Nλ = 8, VWP EON with Nλ = 64), ILP wasable only to find a worse integer solution than heuristics(these cases have been obviously excluded from the ILP-heuristic comparison).

IB BI RO RP SP SA0

5

10

15

20

25

30

35Global comparison

unp-NSFunp-EONDPP-NSFDPP-EON

Bes

t pe

rfor

man

ce c

ount

Heuristic strategy

(a)

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140

DPP-NSF, VWP

∆min

∆max

%∆min

%∆max

Absolu

te fib

er

diffe

rence P

erc

ent fib

er d

iffere

nce

Number of wavelengths, Nλ

(b)

Fig. 4. (a) Number of times each strategy gives the best result.(b) Comparison between ILP and heuristic solutions.

The comparison between the heuristic strategies presentedabove can be completed by considering their computingtimes and their convergence behaviors. Fig. 5 displays theexecution times of the second planning phase (heuristic op-timization) for NSFNET, VWP, in the unprotected and DPPcases (all the strategies are run on a 1-GHz-clock computer).For all the strategies, there is a strong dependence of theexecution time on Nλ. By curve-fitting, we have verifiedthat the execution time is O[N2

λ] for unprotected and O[N3λ]

for DPP. The difference is probably due to the complexity ofthe routing algorithms in the two cases when implementedon the MLG. Execution time in the VWP case resulted tobe higher than in the WP case for all the strategies and forany value of Nλ, probably due to the increased complexityof the MLG in the VWP case, in which the vertical arcsrepresenting wavelength conversions must be added to thegraph. A quantitative evaluation of complexity is howeverstill under study and will probably give an explanation tothese differences. A vertical comparison between the curvesof the graphs points out the differences in computation timebetween the various strategies. Strategies having of ≤ k in

149

0

100

200

300

400

500

0 20 40 60 80 100 120 140

unp-NSF, VWP

IBBIRORPSPS A

Exe

cutio

n tim

e [m

in]

Number of wavelengths, N

(a)

0

200

400

600

800

1000

0 20 40 60 80 100 120 140

DPP-NSF, VWP

IBBIRORPSPS A

Exe

cutio

n tim

e [m

in]

Number of wavelengths, N

(b)

Fig. 5. Comparison between execution times in NSFNET VWPunprotected (a) and with DPP (b).

the X condition (IB, SP and RP) have a higher executiontime than the others (except SA), but achieve better results,showing the trade-off between accuracy and computationalcomplexity. SA displays times similar to those of IB andRP: this is a consequence of the choice of the values givento the SA parameters (p0, ∆, pTh and NF ), taken preciselywith the criterion of having execution times of the sameorder of magnitude for all the strategies. Finally, we shallpoint out that SP computation time is roughly 6 times thatof IB, according to the assignment: n = 6. Thus, SP andIB accuracy is similar, but the former requires much highertime than the latter.

In Fig. 6 we show how the strategies convergence to theirfinal solutions during their execution-time, considering asexample the optimization of the NSFNET in the DPP andVWP case, with Nλ = 8. The horizontal coordinate has beennormalized to the total execution duration of each strategy.The fastest converging strategy, IB, does most of the workat the beginning, when reallocating low-loaded fibers iseasier; at the opposite, we have BI technique, which ismore successful in deallocating fibers after some iterations.

255

256

257

258

259

260

261

262

263

0 0.2 0.4 0.6 0.8 1

DPP-NSF, WP, N = 8

IB

BI

RO

RP

Cur

rent

num

ber

of f

iber

s, F

*

Elapsed processing portion

(a)

254

256

258

260

262

264

266

268

0 0.2 0.4 0.6 0.8 1

DPP-NSF, WP, N = 8

S A

Cur

rent

num

ber

of f

iber

s, F

*

Elapsed processing portion

(b)

Fig. 6. Convergence behavior of (a) IB, RO and RP and (b) SAin DPP-NSF (VWP case, Nλ = 8).

RP follows the behavior of IB but more slowly, due to therandomization of the reallocation acceptance decisions. SAwas plotted separately in Fig. 6(b): its irregular trace resultsfrom the ability of adding fibers. It should be noted that SAhas hit the best already in the middle of the execution, butit jumps out because a still high temperature, to return to itat the end of the process.

VI. CONCLUSION

For OTN, the possibility of obtaining an exact solution tothe NP-hard problem of RFWA of static traffic demands withcapacity dimensioning is severely limited by computationalcomplexity. As we have shown, ILP applied to realisticnetwork examples and solved with standard computingequipment in many cases does not provide any results or isnot able to guarantee optimality: problems are exacerbatedby wavelength continuity (WP cases), by dedicated pathprotection and by high values of Nλ.

The heuristic approach proves to be an acceptably reliablealternative when ILP does not make it. We have proposeda heuristic method based on finding a first greedy but

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feasible solution which is subsequently improved by light-path rerouting. We have presented six heuristic strategiesto perform this task, the most accurate ones of which areable to contain the distance to the ILP solution below the16% of this solution itself (5% on average). Result-analysisshows that the best compromise between computationalcomplexity and reliability is achieved with the deterministicIB and the stochastic SA strategies. IB is however simpler toimplement, it does not require special procedural parametersto be preset and it usually converges rapidly to the finalsolution. Stochastic heuristics gain a clear advantage overdeterministic ones only when there is the possibility, as inSA, of adding fibers beside pruning. In practice, however,the problem of local-minima trapping is not so important,and it can be conjectured that SA would be able to solve itin most of the cases only with execution times much greaterthan those of IB. Finally, we have shown that in same casesconstrained rerouting (SP strategy) can improve IB accuracyat the cost of an execution-time extension.

As a final remark we shall point out that a theoretical anal-ysis of complexity of the compared strategies is currentlyunder study.

REFERENCES

[1] M. Alanyali and E. Ayanoglu. Provisioning Algorithmsfor WDM Optical Networks. IEEE/ACM Transactions onNetworking, 7(5):767–778, Oct. 1999.

[2] M. Ali, B. Ramamurthy, and J. S. Deogun. Routing andWavelength Assignment (RWA) with Power ConsiderationsIn All-Optical Wavelength-Routed Networks. In GlobalTelecommunications Conference - Globecom’99, pages 1433–1437, 1999.

[3] S. Baroni and P. Bayvel. Wavelength Requirements inArbitrarily Connected Wavelength-Routed Optical Networks.IEEE Journal of Lightwave Technology, 15(2):242–251, Feb.1997.

[4] S. Baroni, P. Bayvel, R. J. Gibbens, and S. K. Korotky.Analysis and design of resilient multifiber wavelength-routedoptical transport networks. IEEE Journal of LightwaveTechnology, 17:743–758, May 1999.

[5] R. Bhandari. Optimal Physical Diversity Algorithms andSurvivable Networks. In Computers and Communications,1997. Second IEEE Symposium on, pages 433–441, 1997.

[6] R. Bhandari. Survivable networks, algorithms for diverserouting. Kluwer Academic Publishers, 1999.

[7] E. Bouillet and T. E. Stern. Monte Carlo Techniques forDesign of Wavelength-Routed All-Optical Networks. InGlobal Telecommunications Conference. GLOBECOM ’99,volume 1b, pages 549–552, 1999.

[8] B. V. Caenegem, W. V. Parys, F. D. Turck, and P. M.Deemester. Dimensioning of survivable WDM networks.IEEE Journal on Selected Areas in Communications, pages1146–1157, Sept 1998.

[9] I. Chlamtac, A. Farago, and T. Zhang. Lightpath (wavelength)routing in large WDM networks. IEEE Journal on SelectedAreas in Communications, 14(5):909–913, June 1996.

[10] T. Cinkler. Heuristic Algorithms for Configuration of theATM-Layer over Optical Networks. In Communications,1997. ICC ’97 Montreal, Towards the Knowledge Millennium.1997 IEEE International Conference on, volume 3, pages1164 –1168, 1997.

[11] A. Dacomo, S. D. Patre, G. Maier, A. Pattavina, and M. Mar-tinelli. Design of static resilient WDM mesh networks withmultiple heuristic criteria. In Proceedings, IEEE INFOCOM2002, volume 3, pages 1793–1802, Jun 2002.

[12] D. A. Dunn, W. D. Grover, and M. H. Gregor. Comparisonof k-shortest paths and maximum flow routing for networkfavility restoration. IEEE Journal on Selected Areas inCommunications, pages 88–89, 1994.

[13] A. Fumagalli, I. Cerutti, M. Tacca, F. Masetti, R. Jagannathan,and S. Alagar. Survivable Networks Based on OptimalRouting and WDM Self-Healing Rings. In Proceedings, IEEEINFOCOM ’99, 1999.

[14] R. Inkret, B. Mikac, and I. Podnar. A Heuristic Approach toWavelength Assignment in All-Optical Networks. In Mediter-ranean Electrotechnical Conference, 1998. MELECON 98,volume 2, pages 759 –763, 1998.

[15] S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi. Optimisa-tion by Simulated Annealing. Science, 220(4598):671–680,13 May 1983.

[16] O. Komolafe, D. Harle, and D. Cotter. Next generationoptical network design and modelling, chapter : “A studyon the efficacy of regular virtual topology design heuristicsfor optical packet switching”. (A. Bianco and F. Neri editors)Kluwer Academic Publisher, 2003.

[17] V. R. Konda and T. Y. Chow. Algorithm for Traffic Groomingin Optical Networks to Minimize the Number of Transceivers.In IEEE Workshop on High Performance Switching andRouting, pages 218–221, 2001.

[18] G. Maier, A. Pattavina, L. Roberti, and T. Chich. A heuristicapproach for the design of static multifiber WDM networks:principles and applications. Optical Network Magazine,3(5):52–66, Sep./Oct 2002.

[19] Y. Miyao and H. Saito. Optimal design and evaluation of sur-vivable WDM transport networks. IEEE Journal on SelectedAreas in Communications, 16:1190–1198, Sept 1999.

[20] B. Mukherjee, D. Banerjee, S. Ramamurthy, and A. Mukher-jee. Some Principles for Designing a Wide-Area WDMOptical Network. IEEE/ACM Transactions on Networking,4(5):684–696, Oct. 1996.

[21] B. Mukherjee, S. Ramamurthy, D. Banerjee, and A. Mukher-jee. Some Principles for Designing a Wide-Area OpticalNetwork. IEEE/ACM Transactions on Networking, 4:684–696, Oct. 1994.

[22] M. Tornatore, G. Maier, and A. Pattavina. WDM networkoptimization by ILP based on Source Formulation. InProceedings, IEEE INFOCOM ’02, June 2002.

[23] M. Tornatore, G. Maier, and A. Pattavina. Variable aggre-gation in the ILP design of WDM networks with dedicatedprotection. In TANGO Project Workshop, Jan. 2004.

[24] N. Wauters and P. M. Deemester. Design of the Optical PathLayer in Multiwavelength Cross-Connected Networks. IEEEJournal on Selected Areas in Communications, 14:881–891,June 1996.

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