Modeling the routing and spectrum allocation problem for flexgrid optical networks

Photon Netw Commun (2012) 24:177–186DOI 10.1007/s11107-012-0378-7

Modeling the routing and spectrum allocation problemfor flexgrid optical networks

L. Velasco · M. Klinkowski · M. Ruiz · J. Comellas

Received: 20 February 2012 / Accepted: 12 April 2012 / Published online: 27 April 2012© Springer Science+Business Media, LLC 2012

Abstract Flexgrid optical networks are attracting hugeinterest due to their higher spectrum efficiency and flexibil-ity in comparison with traditional wavelength switched opti-cal networks based on the wavelength division multiplexingtechnology. To properly analyze, design, plan, and operateflexible and elastic networks, efficient methods are requiredfor the routing and spectrum allocation (RSA) problem. Spe-cifically, the allocated spectral resources must be, in absenceof spectrum converters, the same along the links in the route(the continuity constraint) and contiguous in the spectrum(the contiguity constraint). In light of the fact that the conti-guity constraint adds huge complexity to the RSA problem,we introduce the concept of channels for the representation ofcontiguous spectral resources. In this paper, we show that theuse of a pre-computed set of channels allows considerablyreducing the problem complexity. In our study, we addressan off-line RSA problem in which enough spectrum needsto be allocated for each demand of a given traffic matrix.To this end, we present novel integer lineal programming(ILP) formulations of RSA that are based on the assignmentof channels. The evaluation results reveal that the proposedapproach allows solving the RSA problem much more effi-ciently than previously proposed ILP-based methods and itcan be applied even for realistic problem instances, contraryto previous ILP formulations.

Keywords Flexgrid optical networks · Off-line routing ·Spectrum allocation

L. Velasco (B) ·M. Ruiz · J. ComellasUniversitat Politècnica de Catalunya (UPC), Barcelona, Spaine-mail: [email protected]

M. KlinkowskiNational Institute of Telecommunications (NIT), Warsaw, Poland

1 Introduction

Spectrum-sliced flexgrid optical networks are gaining greatmomentum as a consequence of both the expected betterspectrum efficiency and flexibility compared with the rigidspectrum grid networks implementing the traditional WDMtechnology, and the maturity of the technology enabling theirdevelopment [1,2].

In flexgrid optical networks, the available optical spec-trum, for example, C-band, is divided into frequency slots ofa fixed (finer) spectral width in comparison with the currentITU-T WDM rigid frequency grid (50 GHz) [3]. Current pro-posals for the slot size are 25, 12.5 GHz, and even 6.25 GHz.Optical connections have allocated a number of these slots,which is a function of the requested capacity, the modulationtechnique applied, and the slot width.

As an analogy to the off-line routing and wavelengthassignment (RWA) problem in WSON, the off-line rout-ing and spectrum assignment (RSA) problem appears whendesigning or planning flexgrid optical networks. The RSAproblem was proved to be NP-complete in [4] and [5]. In theRSA problem, in addition to the spectrum continuity alongthe links of a given routing path where the same slots must beused in all links of the path, the spectrum contiguity must bealso guaranteed, which means that the allocated slots mustbe contiguous in the spectrum. As a consequence, it is crucialthat efficient methods are available to allow solving realisticproblem instances in practical times.

For illustrative purposes, Fig. 1a shows an example ofthe optical spectrum divided into frequency slots, each hav-ing same width. A guard band may be introduced to sepa-rate two spectrum adjacent connections in any optical link(Fig. 1b). We assume that once the requested frequencyresources have been allocated, the optical connection canbe used to convey single-carrier or multi-carrier modulated

123

178 Photon Netw Commun (2012) 24:177–186

Fig. 1 a Optical spectrumdivided into frequency slots. bOptical connections use a groupof spectrum contiguous slots. cChannel assignment

Frequency

Guard band

c.nnoCb.nnoCa.nnoC

s1 s2 s |S|Frequencyslots (S)

Optical spectrum(a)

(b)

Multicarrier modulatedsignals

Single-carrier modulatedsignal

(c)Channels

c1 C(a) c2 c)b(C 3 C(c)

Frequency

Frequency

slot width

signals. Nonetheless, the proper selection of the signal formatis out of the scope of this paper.

Due to the spectrum contiguity constraint, RWA problemformulations developed for WDM networks are not applica-ble for RSA in flexgrid optical networks, and they need tobe adapted to include that constraint. Several works can befound in the literature presenting integer linear programming(ILP) formulations of RSA [4–6] In [4], the authors addressthe planning problem of a flexgrid optical network, where atraffic matrix with requested bandwidth demands is given. Tosolve the problem, an ILP formulation that aims at minimiz-ing the spectrum used to serve the traffic matrix is proposed.The RSA formulation cannot be solved in practical times,and, therefore, the authors present a decomposition methodthat breaks the previous formulation into two sub-problems: ademand routing sub-problem and a spectrum allocation sub-problem. Since both sub-problems are solved sequentially,global optimality cannot be guaranteed. In [5], the authorsstudy the RSA problem by providing a different ILP formu-lation, but with the same objective as in [4]. Finally, in [6], theauthors formulate the RSA problem and propose an effectiveheuristic algorithm to obtain near optimal solutions.

All the above works show that the spectrum contigu-ity constraint increases significantly the complexity of ILPformulations of RSA by introducing a set of complicatedproblem constraints. In order to mitigate this obstacle, inthis paper, we propose the use of pre-computed sets ofspectrum contiguous frequency slots, which form so-calledchannels. Channels can be grouped as a function of thenumber of slots, for example, the set of channels C2 ={{1, 1, 0, 0, 0, . . ., 0, 0}, {0, 1, 1, 0, 0, . . ., 0, 0}, {0, 0, 1, 1, 0,

. . ., 0, 0}, . . .{0, 0, 0, 0, 0, . . ., 1, 1}} includes every channelcomposed of two contiguous slots, where each position is 1if a given channel uses that slot, and 0 otherwise. The useof demand-tailored channels (C(d) = Cnd , where nd is thenumber of slots requested by demand d) allows removing thespectrum contiguity constraint from mathematical formula-tions. Figure 1c shows the channels that were assigned to thethree optical connections in Fig. 1b.

Based on the use of channels, we propose novel ILP for-mulations for the RSA problem that remove the complexspectrum contiguity variables and constraints by means oftheir explicit representation in the set of input parameters.Exhaustive numerical experiments show that our formulationhighly outperforms those available in the literature [4–6].

In the context of flexgrid optical networks, the contribu-tion of this work is multi-fold. First, we propose a chan-nel assignment (CA) approach, which is the equivalent tothe wavelength assignment (WA) approach in WDM, for thespectrum allocation with the aim to mitigate the complex-ity of spectrum contiguity in the problem formulation. Sec-ond, we propose two ILP formulations that apply the channelassignment approach, namely (1) a link-path formulation and(2) a node-link formulation. As the obtained performanceresults show, our CA-based ILP formulations perform muchbetter than the ILP formulations existing in the literature.Third, a relaxed version of the RSA problem, which does notconsider spectrum continuity, is presented. The relaxed RSAformulation allows obtaining tight lower bounds in reallyshort computation times.

The remainder of this paper is organized as follows. InSect. 2, we present a definition of the off-line RSA problem

123

Photon Netw Commun (2012) 24:177–186 179

that is addressed in this paper and that is used as a scenariofor the comparison of ILP formulations. Then, we describethe notation used in all the ILP formulations presented inthis paper. Next, we present the adaptation to our off-lineRSA scenario of the ILP formulations from the literature.In Sect. 3, we present the proposed channels assignmentapproach together with our novel ILP formulations and arelaxed RSA formulation. In Sect. 4, first we compare theperformance of considered ILP formulations over a set ofsmall problem instances, and then we study the performanceof our CA-based ILP formulation over a realistic networktopology. Moreover, we analyze the goodness of the relaxedformulation. Finally, in Sect. 5, we draw the main conclu-sions of the paper.

2 Existing ILP formulations of RSA

In this section, we first provide a definition of the off-lineRSA problem together with a corresponding notation thatis used in ILP problem formulations. Then, we present twoalternative ILP formulations from the literature adapted forour problem. The adaptation concerns the modification of theobjective function with the aim to compare the performanceof these ILP formulations vs. our ILP formulation introducedin Sect. 3.

2.1 Off-line RSA problem statement

An off-line RSA problem can be formally stated as follows.Given:

• A flexgrid optical network, represented by a graphG(V, E), V being the set of optical nodes and E theset of fiber links connecting two nodes in V .

• An ordered set S of frequency slots in each link in E; S= {s1, s2, . . . , s|S|}. A guard band B (number of slots) isrequired between two spectrum contiguous allocations.

• A set D of demands to be transported. Each demand d isrepresented by a tuple (sd , td , bd , nd), where sd and td arethe source and the destination nodes, respectively, bd isthe requested bandwidth, and nd is the requested numberof slots. For more details on the relation between nd andbd , we refer to [2].

Output: The route over the flexgrid optical network and thespectrum allocation of every transported demand.Objective: Minimize the amount of rejected bandwidth.

Note that from the problem definition, demands can beserved or alternatively rejected. We choose that objectiveinstead of serving all demands to avoid infeasibility that mayappear when trying to serve large sets of demands over acapacitated network.

2.2 Notation

The following sets and parameters have been defined for allthe formulations in this paper. Topology:

V Set of nodes, index v.E Set of fiber links, index e.

E(v) Subset of fiber links incidents to location v.

Demands and paths:

D Set of demands, index d.{sd , td} Set of source and destination nodes of demand d.

bd Bandwidth of demand d in Gb/s.P(d) Set of predefined candidate paths for demand d.

Each path p consists of a set of links e ∈ E so thatnodes sd and td are connected.

P Set of all pre-computed paths, that is, P =⋃d∈D P(d). Index p.

δpe Equal to 1 if path p uses link e, 0 otherwise.

Spectrum:

S Set of frequency slots, index s.C Set of channels, index c. Each channel contains a sub-

set of contiguous frequency slots.B Guard band in number of slots for required between

two spectrum contiguous allocations.γcs Equal to 1 if channel c includes frequency slot s, 0

otherwise.C(d) Set of channels for demand d.

N Set of different sizes of channels to be requested bythe traffic demands, index n.

nd Number of slots to transport the requested bandwidthof demand d.

Others:

M A large positive constant.

The decision variables are:

xd Binary. Equal to 1 if demand d is rejected, 0 other-wise.

yp Binary. Equal to 1 if path p is selected, 0 otherwise.fd Positive integer containing the starting slot index for

demand d.

fd1d2 Binary. Equal to 1 if the starting slot index ofdemand d1 is smaller than that of d2, that is, fd1 <

fd2, and 0 otherwise.yps Binary. Equal to 1 if slot s is assigned to path p and

0 otherwise

123

180 Photon Netw Commun (2012) 24:177–186

ypc Binary. Equal to 1 if channel c is assigned to path pand 0 otherwise

ze Binary. Equal to 1 if link e is opened, 0 otherwisewdec Binary. Equal to 1 if demand d uses channel c in

link e, 0 otherwisewde Binary. Equal to 1 if demand d is routed through

link e, 0 otherwise

Using the above notation, the objective function of everysubsequent formulation of the RSA problem is the amountof un-served bandwidth:

� =∑

d∈D

xd · bd (1)

In the following subsections, two ILP formulations fromthe literature are adapted to solve the capacitated RSA prob-lem (i.e., with limited spectral resources). These formulationsare used as a reference in the comparison presented in Sect. 4.In this regard, both node-link and link-path formulations canbe used to solve the off-line RSA problem [7]. However, theformer, being the most complex, is used for network designproblems where a decision on whether a link is installed ornot needs to be taken. The latter, on the contrary, is usedwhen the network is already designed, so that a set of k dis-tinct paths can be computed beforehand for each demand. Inthis regard, for the sake of simplicity and without loss of gen-erality, in this section we use the link-path approach. Note,however, that similar conclusions can be obtained using anode-link formulation.

2.3 Starting slot assignment (SSA) formulation

This formulation, adapted from [4], consists in assigning thestarting slot to every demand to be transported together withavoiding slot overlapping of two demands whose paths shareat least one common link. Note that intermediate slots arenot explicitly assigned in this formulation. The formulationis as follows:

(SS A) min � (2)

subject to:∑

p∈P(d)

yp + xd = 1 ∀d ∈ D (3)

fd + nd · (1− xd) ≤ |S| ∀d ∈ D (4)

fd1d2 + fd2d1 = 1 ∀d1, d2 ∈ D : ∃p1

∈ P(d1) ∩ ∃p2 ∈ P(d2) ∩ (p1 ∩ p2 �= ∅) (5)

fd2 − fd1 < |S| · fd1d2 ∀d1, d2 ∈ D : ∃p1

∈ P(d1) ∩ ∃p2 ∈ P(d2) ∩ (p1 ∩ p2 �= ∅) (6)

fd1 − fd2 < |S| · fd2d1 ∀d1, d2 ∈ D : ∃p1

∈ P(d1) ∩ ∃p2 ∈ P(d2) ∩ (p1 ∩ p2 �= ∅) (7)

fd1 + nd1 · yp1 + B − fd2 ≤ (|S| + B) ·(1− fd1d2 + 2− yp1 − yp2

)

∀d1, d2 ∈ D ∩ ∀p1 ∈ P(d1) ∩ ∀p2 ∈ P(d2) :p1 ∩ p2 �= ∅ (8)

fd2 + nd2 · yp2 + B − fd1 ≤ (|S| + B) ·(1− fd2d1 + 2− yp1 − yp2

)

∀d1, d2 ∈ D ∩ ∀p1 ∈ P(d1) ∩ ∀p2 ∈ P(d2) :p1 ∩ p2 �= ∅ (9)

The objective of the SSA formulation (2) is to minimize theamount of un-served bandwidth. Constraint (3) either assignsone feasible path or blocks the demand. Constraint (4) guar-antees that the starting slot for each demand leaves enoughroom for the amount of slots that demand requests. Con-straints (5)–(7) manage starting slot ordering for all demandpairs such that any of their paths share at least one com-mon link. They compute whether the starting slot of one ofthe demands in the pair is lower than the starting slot of theother demand. Constraints (8)–(9) perform spectrum conti-nuity and non-overlapping spectrum allocation. They ensurethat the spectrum is assigned to all pair of demands and allpair of paths of the demands such that they share at least onecommon link and that paths are activated and do not overlap.

2.4 Slots assignment (SA) formulation

This formulation, adapted from [5], consists in explicitlyassigning slots to demands ensuring the spectrum contiguityconstraint. In contrast to [4], it makes use of variables andconstraints that depend on the frequency resources. A similarapproach can be found in [6].

The formulation uses constraint (3) and defines new con-straints:

(S A) min � (10)

subject to:Constraint (3)

∑

s∈S

yps = nd · yp ∀d ∈ D, p ∈ P(d) (11)

∑

d∈D

∑

p∈P(d)

δpe · yps ≤ 1 ∀e ∈ E, s ∈ S (12)

∑

d2 �=d1:p2∈P(d2)∩(p1∩p2 �=∅)s2∈[max(0,s1−B),min(|S|,s1+B]

yp2s2 ≤(1− yp1s1

) · M

∀d1 ∈ D,

p1 ∈ P(d1), s1 ∈ S (13)

− |S| · (yps − yp(s+1) − 1) ≥

∑

s2∈[s+2,|S|]yps2 ∀d ∈ D,

p ∈ P(d), s ∈ [1, |S| − 1] (14)

123

Photon Netw Commun (2012) 24:177–186 181

In the SA formulation, constraint (11) assigns therequested number of slots to each demand. Constraint (12)ensures that each slot is allocated to at most one path. Con-straint (13) reserves a guard band between spectrum alloca-tions belonging to two different demands. Finally, constraint(14) ensures spectrum contiguity by ensuring that if a slotis allocated for a demand and the next one is not, all theremaining slots in the spectrum will not be allocated to thatdemand.

The formulations presented in this section require a setof dedicated variables and constraints in order to representthe spectrum contiguity constraint. The next section presentsILP formulations that use channels, and, as a consequence,spectrum contiguity is ensured in the input data.

3 Proposed efficient RSA formulations

In this section, we first define channels and then proposenovel ILP formulations based on channel assignment thatremove spectrum contiguity from the problem, making useof a pre-computed set of candidate channels of contiguousspectrum as an input parameter to the problem. Finally, weprovide relaxed ILP formulations that provide tight lowerbounds of the problem in really short computation times.

3.1 Channels definition

The definition of channels can be mathematically formu-lated as follows. Let γcs be a coincidence coefficient, whichis equal to 1 whenever channel c ∈ C uses slot s ∈ S,

and 0 otherwise. Let us assume that a set of channels C(d)

is pre-defined for each demand d, which requests nd slots.Then, ∀c ∈ C (d) the spectrum contiguity constraint isimplicitly imposed by the proper definition of γcs such that∀si , s j : γcsi = γcs j = 1, si < s j ⇒ γcsk = 1,∀sk ∈{si , . . . , s j

},∑

s∈S γcs = nd . In this paper, we consider thateach set C(d) consists of all possible channels of the sizerequested by d that can be defined in S.

Since |C (n)| = |S| − (n − 1), the size of the com-plete set of channels C that needs to be defined is |C | =∑

n∈N [|S| − n + 1] < |N | · |S|.The algorithm in Table 1 computes C(d). Note that com-

putation of channels is trivial, and thus no additional com-plexity is added to the pre-computation phase.

To account for guard bands, without loss of generality, weconsider that they are included as a part of requested spec-trum (nd ).

Therefore, we can define the Routing and Channel Assign-ment problem as the problem that finds a route and assigns aproper channel to a set of input demands, so that the numberof active slots in the channel guarantees that the bandwidthrequested by each demand can be transported. Therefore,

Table 1 C(d) pre-computation

INPUT S, dOUTPUT C(d)

1: Initialize:C(d) ← 0[|S|−nd+1x |S|]2: for each i in [0, |S| − nd ] do

3: for each s in [i, i + nd − 1] do

4: C(d)[s] = 1

5: return C(d)

the Routing and Channel Assignment and the Routing andSpectrum Allocation problems are in fact the same prob-lem; a route and a set of contiguous slots are allocated foreach input demand. However, the complexity of the formeris much lower than that of the latter since the slots in thechannel are already contiguous in the spectrum.

3.2 Link-path channel assignment (LP-CA) formulation

The proposed link-path-based CA formulation is as follows:

(L P − C A) min � (15)

subject to:

∑

p∈P(d)

∑

c∈C(d)

ypc + xd = 1 ∀d ∈ D (16)

∑

d∈D

∑

p∈P(d)

∑

c∈C(d)

γcs · δpe · ypc ≤ 1

∀e ∈ E, s ∈ S (17)

Constraint (16), similar to constraint (3), either assignsone feasible path and channel or blocks the demand. In addi-tion, constraint (17), similar to constraint (12), ensures thateach slot in a link is assigned, at most, to one demand.

3.3 Node-link CA (NL-CA) formulation

For the node-link version, let us assume a modification inthe problem statement. The new objective is to minimize thenumber of links to be installed in order to transport all thebandwidth requested, that is, no demand blocking is allowed.The node-link CA formulation is as follows:

(N L − C A) min∑

e∈E

ze (18)

subject to:

123

182 Photon Netw Commun (2012) 24:177–186

∑

e∈E(v)

∑

c∈C(d)

wdec = 1 ∀d ∈ D, v ∈ {sd , td} (19)

∑

e∈E(v)

∑

c∈C(d)

wdec ≤ 2 ∀d ∈ D, v /∈ {sd , td} (20)

∑

e′∈E(v)

e′ �=e

wde′c ≥ wdec

∀d ∈ D, c ∈ C(d), v /∈ {sd , td}, e ∈ E(v) (21)∑

e∈E

∑

c∈C

γcs · wdec ≤ 1 ∀e ∈ E (22)

∑

d∈D

∑

c∈C(d)

wdec ≤ |S| · ze ∀e ∈ E (23)

The objective function (18) minimizes the number oflinks to be opened. Constraints (19)–(21) compute the routethrough the optical topology and assign a channel to thedemands. Constraint (19) ensures that only one channel isused to transport the demand in any link incident to sourceand destination nodes. Constraints (20) and (21) perform therouting in intermediate nodes. In addition, constraint (23)implements the spectrum continuity constraint in intermedi-ate nodes. Constraint (22) assures that each frequency slot isused by at most one lightpath. Finally, constraint (23) opensa link as soon as it is used by one or more demands.

3.4 Relaxed RSA formulations

In line with [4], to speed-up solving times, a lower boundon the objective function can be introduced, which is givenby the solution of the RSA problem with relaxed spectrumcontiguity and continuity constraints.

The relaxed link-path RSA model is as follows, whereconstraint (17) in the LP-CA formulation has been redefinedas constraint (25) to ensure that the capacity of each link isnot exceeded.

(relaxed link − path RS A) min � (24)

0 subject to:Constraint (3)

∑

d∈D

∑

p∈P(d)∩P(e)

yp·nd ≤ |S| ∀e ∈ E (25)

In the relaxed node-link RSA model, channel assignmenthas been removed, and then constraints (19)–(21) have beensimplified as constraints (27)–(29). In addition, constraints(22)–(23) have been joined into the new constraint (30) thatopens the links as soon as they are used and ensures that thecapacity of each link is not exceeded. The formulation is asfollows:

(relaxed node − link RS A) min∑

e∈E

ze (26)

subject to:

∑

e∈E(v)

wde = 1 ∀d ∈ D, v ∈ {sd , td} (27)

∑

e∈E(v)

wde ≤ 2 ∀d ∈ D, v /∈ {sd , td} (28)

∑

e′∈E(v)

e′ �=e

wde′ ≥ wde ∀d ∈ D, v /∈ {sd , td}, e ∈ E(v) (29)

∑

d∈D

nd · wde ≤ |S| · ze ∀e ∈ E (30)

The next section evaluates the performance of the pro-posed RSA models comparing them with the performance ofthe existing RSA formulations presented in Sect. 3.

4 Performance evaluation

In this section, we first analyze the complexity of above ILPmodels. Then, we compare exact link-path-based ILP mod-els in terms of their computational complexity. We do notprovide results of the optimization since they are exact RSAformulations and the results are the same. Next, in view thatthe CA formulation solves problem instances in short compu-tation times, real problem instances are generated and solvedusing the CA formulation. Finally, a network is designedusing the node-link-based CA formulation.

4.1 Complexity analysis

In Table 2, we present expressions to estimate the number ofvariables and constraints for the ILP models.

Regarding the link-path-based formulations, in contrastto the amount of variables, the amount of constraints greatlydiffers among the formulations; in SSA, it depends on thepower of the amount of demands times the power of k; in SA,it depends also on the size of the sets of slots and optical links,whereas it is proportional to only the size of the demand setor the size of E times the size of the slot set in our CA formu-lation. As an example, the number of constraints in the DTnetwork (14 nodes, 23 links) is presented in the next section,considering |S| = 30, |D| = 36, and k = 5, is 3.2e4, 1.15e4,and 7.3e2 for SSA, SA, and CA formulations, respectively. Incontrast, the relaxed RSA model has only 180 variables and56 constraints.

Note that the size of the node-link version of the CA for-mulation, 1.2e3 variables and 7.5e6 constraints, is the largestone as a consequence that routes are found by the model itselfinstead of being pre-computed. The relaxed node-link RSAmodel has 8.3e2 variables and 1.2e4 constraints.

123

Photon Netw Commun (2012) 24:177–186 183

Table 2 Size of the modelsFamily Formulation Variables Constraints

Link-path SSA (exact) O(|D|2 + k · |D|) O(k2 · |D|2)

SA(exact) O(k · |D| · |S|)O(2 · k · |D| · |S| + |E | · |S|)CA (exact) O(k · |D| · |C |) O(|D| + |E | · |S|)Relaxed RSA O( k · |D|) O(|D| + |E |)

Node-link CA(exact) O(|D| · |E | · |C |) O(|D| · |C | · |V | · |E |)Relaxed RSA O(|D| · |E |) O(|D| · |V | · |E |)

Table 3 Computation times to solve each link-path formulation (seconds)

Network(|V |, |E |) Formulation |S| = 30 |S| = 60

k = 1 k = 2 k = 3 k = 4 k = 5 k = 1 k = 2 k = 3 k = 4 k = 5

RING9 (9,9) SSA 7.70 1, 325.94 – – – 4.67 68.64 – – –

SA 9.14 1, 695.54 – – – 6.20 56.11 – – –

CA 0.07 0.08 – – – 0.19 0.32 – – –

BRASIL (10,12) SSA 10.42 2, 358.26 >6 h >6 h >6 h 9.27 189.44 >6 h >6 h >6 h

SA 3.16 1, 242.66 >6 h >6 h >6 h 3.28 84.04 >6 h >6 h >6 h

CA 0.24 1.94 2.65 5,69 9,18 0.24 0.36 0.41 0.52 0.91

ABILENE (12,15) SSA 9.84 53.26 3,198.35 >6 h >6 h 4.37 27.49 142.31 >6 h >6 h

SA 6.27 42.01 2,798.58 >6 h >6 h 1.47 19.44 88.32 5,856.34 >6 h

CA 0.25 0.26 0.29 4.90 19.71 0.14 0.24 0.33 0.52 0.58

DT (14,23) SSA 2.94 9.87 15.92 31.73 123.34 3.07 5.87 19.31 50.24 92.31

SA 1.80 4.35 6.48 13.51 27.65 2.60 4.86 16.48 30.28 64.28

CA 0.09 0.10 0.17 0.37 0.42 0.10 0.19 0.28 0.41 0.45

4.2 Performance comparison

The performance of the three formulations has been evalu-ated using four different optical topologies from [4,8], and[9] covering a wide range of mesh degrees.

From Table 2, the value of k and the size of the demandsand slots sets highly impact the complexity of the models.Consequently, a scenario consisting in 36 demands randomlygenerated and two different slot sets (|S| ∈ {30, 60}) wastested. The bandwidth of the demands was uniformly dis-tributed ranging from 1 . . . 4 slots. Finally, different sizesfor the set of pre-computed paths, k, was generated for eachdemand, ranging from 1 . . . 5, except for the ring topology;the range of k was limited as a consequence of the resultingcomputation times as explained below. All the formulationswere implemented in iLog-OPL and solved by the CPLEXv.12 optimizer [10] on a 2.4 GHz Quad-Core machine with8 GB RAM memory running Linux.

Table 3 shows the obtained computations times for thethree ILP formulations. We can see that computation timesincrease considerably with k for both SSA and SA, and,in particular, these formulations are rather impractical forhigher values of k. Note that in many problems, the values ofk in the order of 10 need to be used to ensure optimality [7].

Note also that as soon the size of the considered spectrumincreases (from 30 to 60), the effective load decreases andthus the problem becomes easier to be solved since it is easierto find free spectrum resources in the network. In contrast,as observed in Table 3, our CA formulation provides compu-tations times in the order of seconds for all tested scenarios.The computational efficiency is, at least, three orders of mag-nitude higher than when using the SSA or SA formulations,and it is a result of representing the spectrum contiguity con-straints by means of the pre-computed data. Now, the com-plexity of the problem is comparable to that of the RWA sinceonly spectrum continuity needs to be considered.

4.3 Performance evaluation of the proposed formulations inreal scenarios

In order to conduct additional and realistic experiments undermore stringent scenarios, we consider the 21-node and 35-link Spanish Telefónica optical network topology shown inFig. 2. We consider an optical spectrum of 800 GHz andslots of 12.5 GHz, that is, |S| = 64. For the demands, therequested bandwidth is 10, 40, or 100 Gbps (1, 2, or 4 slots).Both bandwidth and traffic distribution follow the uniformdistribution.

123

184 Photon Netw Commun (2012) 24:177–186

1

24

10

11

20

21

19

12

8

53

7

6

9

13

1814

1716

15

Fig. 2 The Spanish Telefónica optical network topology

Seven increasing traffic loads and 10 randomly generatedtraffic matrices per load were generated ranging from 5 to11 Tbps in steps of 1 Tbps. All traffic matrices but those withthe highest load could be completely served, that is, none ofthe demands was blocked. In contrast, some of the matriceswith total bandwidth equals to 11Tbps could not be com-pletely served. Note that these are the kind of studies thatnetwork operators are interested in, that is, to find the high-est amount of traffic that their networks can serve.

For each demand in each traffic matrix, k = 10, 15, and 20paths were pre-computed. Each resulting problem instancewas solved using the link-path CA formulation presentedin Sect. 4 in the conditions described above. The results interms of computation time as a function of the number ofdemands are shown in Fig. 3, where each point is the aver-age value of 10 instances with the same total bandwidth. Notethat although the graph shows a clear exponential behavior,computation times are still practical (few hours).

Finally, we tested the goodness to the proposed relaxedRSA formulations against the CA formulations in terms ofgoodness of the solutions and computation times. For thelink-path formulations, we generated a traffic matrix with250 demands and 12.5 Tbps of total bandwidth and solvedboth the relaxed RSA and the CA formulations. Results inTable 4 show that lower bounds provided by the relaxedformulation are really tight to the optimal value obtainedby solving the exact one; the relaxed formulation blockedone demand whereas the exact formulation blocked threedemands. Blocked bandwidth is shown also in Table 4.

Regarding the node-link formulations, in view of its largercomplexity, we generated small instances to compare exactand relaxed formulations. The results in Table 4 are fortraffic matrices with 300 Gbps of total bandwidth (only 12demands), where the same results were obtained by both for-

Com

puta

tion

Tim

e (h

ours

)

Number of demands

0

2

4

6

8

10

100 120 140 160 180 200 220

k=10

k=15

k=20

Fig. 3 Computation time in seconds as a function of the number ofdemands for several sizes of pre-computed paths

Table 4 Exact versus relaxed formulations

Family Formulation Computation time Optimal value

Link-path CA (exact) >6 h 60 Gbps (0.48 %)

Relaxed RSA 0.61 s 10 Gbps (0.08 %)

Node-link CA (exact) >6 h 14 links

Relaxed RSA 2.66 s 14 links

mulations, although computation times show that the exactformulation takes more than 400 times the computation timeof the relaxed formulation.

As a conclusion, the relaxed formulation can be used toobtain good lower bound in really short computation times.Note that in traffic scenarios where few or no demands areblocked, lower bounds are very closed to optimal solutions.

5 Concluding remarks

In this paper, a novel ILP formulation to solve the RSAproblem in flexgrid optical networks, named CA, has beenproposed. In general, two main spectrum-related constraintsneed to be considered for each demand: continuity and conti-guity. In the CA formulation, spectrum contiguity is ensuredby using a set of candidate spectrum contiguous channels,which is considered as an input parameter to the problem.As a result, RSA problems involving realistic optical net-works can be formulated in a compact way and solved inpractical times, much shorter than with the previous RSAformulations proposed in the literature.

It is important to highlight that our CA formulation is verycompact, when comparing with the previous ILP models, andit can be applied easily in other much more complex optimi-

123

Photon Netw Commun (2012) 24:177–186 185

zation problems involving the resolution of RSA. Below, wepresent some examples:

(i) in the study on recovery strategies that can be used inflexgrid optical networks, such as the one presented in[11]. In these problems, failure scenarios are definedand the RSA needs to be solved for each of them.There, values of k are as high as 10–20.

(ii) in multi-hour routing problems where the requestedbandwidth of each demand may vary in consecutivetime periods [12]. There, elastic spectrum allocationis performed for each demand and time period.

(iii) in (single-layer or multi-layer) flexgrid optical net-work design [13]. In these problems, the node-linkRSA formulation described in Sect. 4.3 needs to beused since the topology of the optical network is notknown and it is the subject of optimization.Our current experience in solving RSA shows that allthe above problems can be solved, although certainlyfor small network instances, using the CA formulation,in contrast to their intractability if the SSA or the SAformulations were used instead.

Acknowledgments The research leading to these results has receivedfunding from the European Community’s Seventh Framework Pro-gramme FP7/2007-2013 under grant agreement n◦ 247674 STRON-GEST project. Moreover, it was supported by the Spanish sci-ence ministry through the TEC2011-27310 ELASTIC project andby the Polish National Science Centre under grant agreement DEC-2011/01/D/ST7/05884.

References

[1] Jinno, M., Takara, H., Kozicki, B., Tsukishima, Y., Sone, Y.,Matsuoka, S.: Spectrum-efficient and scalable elastic optical pathnetwork: architecture, benefits, and enabling technologies. IEEECommun. Mag. 47, 66–73 (2009)

[2] Jinno, M., Kozicki, B., Takara, H., Watanabe, A., Sone, Y., Tanaka,T., Hirano, A.: Distance-adaptive spectrum resource allocationin spectrum-sliced elastic optical path network. IEEE Commun.Mag. 48, 138–145 (2010)

[3] ITU-T G.694.1: Spectral grids for WDM applications: DWDMfrequency grid. May 2002

[4] Christodoulopoulos, K., Tomkos, I., Varvarigos, E.: Elastic band-width allocation in flexible OFDM based optical networks. IEEEJ. Lightwave Technol. 29, 1354–1366 (2011)

[5] Wang, Y., Cao, X., Pan, Y.: A study of the routing and spectrumallocation in spectrum-sliced elastic optical path networks. In:Proceedings of IEEE INFOCOM 2011

[6] Klinkowski, M., Walkowiak, K.: Routing and spectrum assign-ment in spectrum sliced elastic optical path network. IEEE Com-mun. Lett. 15, 884–886 (2011)

[7] Pióro, M., Medhi, D.: Routing, Flow, and Capacity Design in Com-munication and Computer Networks. Morgan Kaufmann, Califor-nia (2004)

[8] Pavan, C., Morais, R., Ferreira, J., Pinto, A.: Generating realisticoptical transport network topologies. IEEE/OSA J. Opt. Commun.Netw. 2, 80–90 (2010)

[9] SNDlib http://sndlib.zib.de/[10] CPLEX http://www-01.ibm.com/software/integration/optimizat

ion/cplex-optimizer/[11] Sone, Y., Watanabe, A., Imajuku, W., Tsukishima, Y., Kozicki, B.,

Takara, H., Jinno, M.: Bandwidth squeezed restoration in spec-trum-sliced elastic optical path networks (SLICE). IEEE/OSA J.Opt. Commun. Netw. 3, 223–233 (2011)

[12] Velasco, L., Klinkowski, M., Ruiz, M., López, V., Junyent, G.:Elastic Spectrum Allocation for Variable Traffic in Flexible-GridOptical Networks. In: IEEE/OSA Optical Fiber CommunicationConference (OFC), March 2012

[13] Pedrola, O., Velasco, L., Castro, A., Fernández-Palacios, J., Care-glio, D., Junyent, G.: CAPEX study for grid dependent multi-layerIP/MPLS-over-EON using relative BV-WSS costs. In: IEEE/OSAOptical Fiber Communication Conference (OFC), March 2012

Author Biographies

L. Velasco received the B.Sc. degree in tele-communications engineering from Univers-idad Politècnica de Madrid (UPM) in 1989,the M.Sc. degree in physics from UniversidadComplutense de Madrid (UCM) in 1993, andthe Ph.D. degree from Universitat Politèc-nica de Catalunya (UPC) in 2009. In 1989, hejoined Telefónica of Spain and was involvedin the specifications and first office applica-

tion of the Telefónica’s SDH transport network. In 2003, he joinedUPC, where currently he is an Assistant Professor in the ComputerArchitecture Department (DAC) and a Researcher in the Optical Com-munications group (GCO) and the Advanced Broadband Communica-tions Center (CCABA). He has participated in various IST FP-6 andFP-7 European research projects such as EU DICONET, BONE, ISTNOBEL 2, e-Photon/ONe+, and STRONGEST. His interests includeplanning, CAPEX/OPEX issues, routing, and resilience mechanisms inmulti-layer networks.

M. Klinkowski is an Assistant Professor inthe Department of Transmission and OpticalTechnology (Z-14) at the National Instituteof Telecommunications (NIT) in Warsaw,Poland, and is a Collaborating Researcherat the Universitat Politecnica de Catalunya(UPC), Barcelona, Spain. He received theM.Sc. degree from the Warsaw Universityof Technology (WUT), Warsaw, Poland, in

1999 and the Ph.D. degree from UPC in 2008. His research inter-ests include optical networking with emphasis on network mod-eling, design, and optimization. He is currently involved in theresearch on Elastic Optical Networks (through grant NCN no. DEC-2011/01/D/ST7/05884) and the COST IC0804 action.

123

186 Photon Netw Commun (2012) 24:177–186

M. Ruiz received the B.Sc. degree in biol-ogy from Universitat de Barcelona (UB),Spain, in 2005 and the M.Sc. degree in statis-tics and operational research from UniversitatPolitFcnica de Catalunya (UPC), Barcelona,Spain, in 2009. He is currently workingtowards the Ph.D. degree with the Opti-cal Communication Group (UPC) and theAdvanced Broadband Communications Cen-

ter (CCABA). His research interests include characterization, designand reoptimization of next-generation multilayer optical networks.

J. Comellas received the M.S. (1993) andPh.D. (1999) degrees in TelecommunicationsEngineering from the Universitat Politecnicade Catalunya (UPC), Barcelona. Since 1992,he is a staff member of the Optical Commu-nications Research Group of UPC. His cur-rent research interests mainly concern opticaltransmission and IP over WDM network-ing topics. He has participated in different

research projects fund by the Spanish Government and the EuropeanCommission. He has co-authored more than 100 research articles in

national and international journals and conferences. He is associate pro-fessor of the UPC Signal Theory and Communications Department andsince 2009 is Vice Dean for International Affairs at the UPC Telecom-munications Engineering School of Barcelona (ETSETB). He is also astaff member of the CCABA (Advanced Broadband CommunicationsCenter).

123