WDM Network Optimization by ILP Based on Source ... - CiteSeerX

WDM Network Optimization by ILP Based on Source Formulation Massimo Tornatore, Guido Maier, Achille Pattavina

Abstract—Efficient planning and optimization of wavelength division multiplexing networks is an important issue today. Integer Linear Programming (ILP) is the most used exact method to perform this task. In this paper we propose a new ILP formulation that allows to solve optimization with less computational effort compared to other ILP approaches. This formulation applies to multifiber mesh networks with or without wavelength conversion, when either the total fiber number or the total fiber length is the cost function to be minimized. After presenting the formulation we discuss the results we obtained by exploiting it in the optimization of two case-study networks.

I. I NTRODUCTION PTICAL networks based on Wavelength Division Multiplexing (WDM) are the main characters today of the transport infrastructure evolution towards high capacity and high reliability. They are being deployed at an extremely rapid rate for wide-area transport applications as well as in the metro and regional areas. These networks, based on switching and routing of optical circuits in space and wavelength switching domains, exploit the most advanced photonic technology that is available on the market and are therefore evolving with a similar fast pace as WDM transmission and switching equipment. Recently, on the switching equipment side, Optical Cross Connects (OXC) systems have become available, beside the more mature Optical Add-Drop Multiplexers. This opened up the road to the possibility of deploying complex WDM networks based on the mesh topology, while in the past single ring or overlaid multi-ring were the most used architectures for WDM. The increase in WDM complexity brought the need for suitable network planning strategies into foreground. Problems such as optimal dimensioning routing and resource allocation for optical connections must be continuously solved by new and old operators, to plan new installations or to update and expand the existing ones. These problems can no longer be manually solved in complex network architectures, as it usually happened in the earlier experimental WDM installations. Computer-aided planning tools and procedures are needed for the future which can achieve an efficient utilization of network resources in a reasonable computational time. Research on optical network since some years ago has been investigating design and optimization techniques in order to provide operators the most efficient and flexible procedures to solve the network design problem. The various proposed solutions can be classified into two main groups: heuristic methods

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M. Tornatore and G. Maier are with CoreCom, Via Ampere, 30 - 20131 Milan, Italy. E-mail:ft mastor, [email protected]. A. Pattavina is with Dept. of Electronics and Information, Politecnico di Milano, P.za Leonardo da Vinci, 32 - 20133 Milan, Italy. E-mail: [email protected]

and exact methods. The former return sub-optimal solutions that in many cases are acceptable and have the advantage of requiring a limited computational effort. The latter are much more computationally intensive and do not scale well with the network size, being even not applicable in some cases; however since they are able to identify the absolute optimal solution, they play a fundamental role either as direct planning tools or as benchmarks to validate and test the heuristic methods. The work we are presenting concerns exact methods to plan and optimize multifiber WDM networks. In particular we focus on Integer Linear Programming (ILP), a widespread technique to solve exact optimization: we propose a new formulation of the optimization problem that we named source formulation. Source formulation is equivalent to the well known flow formulation, but it allows a relevant reduction of the number of variables and of constraints, thus sensibly diminishing computation time and occupied memory. The paper summary is as follows. In section II we introduce our solution by presenting a short review of the literature regarding ILP application to WDM optimization. In section III the source formulation is presented and explained into details, in the two versions for network with or without wavelength conversion. Finally, in section IV results obtained by applying the source formulation to case-study networks are shown; using the case study the new formulation is then compared to the traditional flow formulation to point out the advantages of the method we are proposing. II. WDM NETWORK OPTIMIZATION BY INTEGER LINEAR PROGRAMING

Network design and planning is carried out with different techniques according to the type of traffic the network has to support. The connections requested by the nodes at a given time to a WDM network all together form the virtual topology (alias logical topology or offered traffic). In the simplified reference model we use in this work, each request is for a point-to-point optical circuit (lightpath) able to carry a given capacity from the source optical termination to the destination termination. It is understood that each node pair may request more than one connection if the total bandwidth needed exceeds the capacity of a single lightpath. Two different traffic types may be offered to a WDM network Static traffic: a known set of permanent connection requests is assigned a priori to the network, which must be able to satisfy all the requests together, starting from the idle state. Dynamic traffic: connection requests arrive at random

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time to the nodes of the network and connections are semipermanent (i.e. temporary with long duration). Though dynamic lightpath provisioning is becoming more and more important, in this work we wish to consider only the static situation, leaving for future development the extension to dynamic traffic conditions. Connections must be established by suitably configuring network switching resources and allocating network transmission resources: a lightpath, when established, is a sequence of WDM channels, one for each fiber it crosses. We assume that all the WDM channels carry the same capacity. Lightpaths are routed and switched by the OXCs of the network and the two lightpath terminations are located in the source and the destination OXCs. We assume that the channels composing the lightpath may have different wavelengths or may be all at the same wavelength, according to the availability of the wavelength conversion function in the transit OXCs. To simplify, we have considered two extreme cases Virtual Wavelength Path (VWP) network: all the OXC are able to perform full wavelength conversion, i.e. an incoming optical signal having any wavelength can be converted to an outgoing optical signal having any possible transmission wavelength; Wavelength Path (WP) network: no wavelength conversion is allowed in the whole network. In the WP case lightpaths are subjected to the so called wavelength continuity constraint, that is absent in the VWP case. WDM networks today are often designed in order to be resilient to failures that may occur to switching or transmission equipment. Many automatic protection techniques can be implemented in the WDM layer, such as, for example, path or link protection. The adoption of such techniques has an impact in complicating network design, since they require that planning has to be extended also to the spare capacity necessary to reroute connections in case of failure. Though automatic lightpath protection is very important today (given the high bit-rates that a WDM channel usually carries, e.g. 2.5 to 40 Gbit/s), this feature will not be covered in this work, for the reasons that will be explained later on. Static optimization of a WDM network can be so summarized: given a static traffic matrix, find the optimum values of a set of network variables that minimizes a given cost (or objective) function, under a set of constraints. The choice of variables, cost function and constraints greatly varies from case to case. In the past most of studies regarding WDM network planning were aimed at virtual topology optimization with singlefiber WDM links [1], [2]. The cost function to be optimized was either the number of wavelengths necessary to route the static traffic or the network load (the number of channels routed on the most loaded link of the network) [3]. The work we are proposing follows a new trend which overlooks the problem from a slightly different perspective. In recent times, in fact, many studies have been developed in which virtual topology optimization is accompanied by cost minimization of a multifiber physical network: the number of fibers per link is a variable of the problem to be minimized, while the amount of wavelengths per fiber is usually preset [4].

Finding the minimum network capacity that allows to route all the connections is the first result of the planning problem. The second goal of planning is the allocation of physical resources to the lightpaths that has to be set up, i.e. the definition of the sequence of WDM channels composing each lightpath. As a whole, solving static planning corresponds to find the optimal resources assignment for a preassigned traffic matrix over a preassigned physical topology (RFWA, routing fiber and wavelength assignment). We have already mentioned that several approaches have been proposed to perform optimal RFWA. Heuristic approaches can be in general described as follows. The static traffic matrix is considered as a set of connections that can be setup sequentially, starting from an idle network. Lightpaths demanded according to the static traffic matrix are routed in sequence, performing routing fiber and wavelength assignment upon each lightpath of the set according to given heuristic criteria. After all the lightpaths have been setup, the network is globally optimized by tearing down and rerouting some lightpaths again exploiting optimization heuristic criteria. The entire process can be iterated. This approach is defined “deterministic heuristic” [4], [5], [6]. Other methods called “stochastic heuristics” use such techniques as simulated annealing [7], [8] or genetic algorithms [9] to improve heuristic optimization by trying to avoid possible local-minima of the cost function. Since optical WDM network design is essentially an optimization problem, its exact solution, corresponding to the absolute minimum of the cost function, can be found by applying mathematical programming. Even if this powerful tool of operations research usually requires a greater computational effort compared to the heuristic methods, it is well suited for a simple modeling of WDM networks. In particular, a large number of WDM optimization problems can be solved by ILP. This is because models comprising cost functions of the type we have mentioned above and topological or wavelength continuity constraints give origin to set of linear equations; on the other hand, the fact that capacity is expressed in terms of number of WDM channels leads to the integer constraint on the variables. WDM network optimization by ILP has been widely studied in literature. ILP has been subject to considerable interest especially for its flexibility in modeling network environment that has allowed to face various design cases. We can subdivide research contributions in two groups according to which type of networks they are applied to WDM networks with single-fiber links; multifiber WDM networks. In the first group the problem consists in optimal routing and wavelength assignment (RWA) of the lightpaths. This is a NPcomplete problem, as it was demonstrated in Refs. [10], [11]. Two basic methods has been defined to model the RWA problem: flow formulation and route formulation [12]. In the former the basic variables are the flows on each link relative to each source-destination OXC pair; in the latter the basic variables are the paths connecting each source-destination pair. Both these two formulations have been employed to solve various sorts of problems and to investigate different aspects of WDM networks. For example, in Ref. [12] the optimization is carried on in order to emphasize the difference between WP and VWP

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scenarios; Ref. [3] studies the effects of imposing a constraint on the delay met by a message traveling along a lightpath; in Ref. [13] possible utilization of bounds derived from the two formulations by relaxation of the integer constraints are studied and compared. In all the works mentioned above only optimal RWA was performed without dimensioning, since the capacity of the physical topology was given as an input. In other cases also dimensioning was carried out, selecting the number of wavelengths [14], [12] or the total number of WDM channels in the network [15], [16] as cost functions. The average message delay has also been used sometimes as objective function in joint dimensioning-mapping problems [17], [18]. In optimization of multifiber WDM networks optimal allocation of fibers has also to be solved, thus complicating the problem of lightpath set up into routing fiber and wavelength assignment (RFWA). Solving RFWA becomes really challenging even with relatively small networks, especially because routing and wavelength assignment is coupled to dimensioning. In this case a new set of variables representing the number of fibers of each physical link must be considered in addition to the flow or the route variables defined above for the two formulations. This implies that RFWA scales from a multicommodity flow problem to a more complex localization problem. The choice of complex cost functions such as those comprising node or duct cost makes the ILP solution formidable even for very small size networks [8] (this is even worse in the case of non linear objective function that require integer non-linear programming [19]). When the problem becomes computationally impractical, route formulation becomes more useful than flow formulation. If it is acceptable that RFWA is performed in a constrained way, then the solution complexity of the route formulation can be controlled. For example, all the lightpaths can be constrained to be routed along the first s shortest paths connecting the source to the destination. Differently from the flow formulation, the complexity of which is strictly dependent on physical and virtual topologies, the size of the route formulation decreases with the number of paths that can be employed to route the lightpaths. Multifiber network optimization with route formulation and constrained routing has been studied in Refs. [17], [20], [21], [12]. Beside route formulation with constrained routing, other methods to control complexity have been proposed. A possibility is to stop the branch-and-bound algorithm (typically used to solve ILP problems) after finding the first or a pre-definite number of integer solutions. Ref. [8] shows that acceptable results (though quite far from the optimal solution) can be obtained when the branch-and-bound duration is fixed to 10 minutes. Ref. [22] proposed that the whole RFWA problem can be solved as a sequence of simpler problems (e.g. first routing, then fiber assignment, and so on). Other possible approaches are: exploitation of lagrangean relaxation [23], relaxation of integer constraints [17] and randomized routing [15]. Undoubtedly the massive need of computational resources (i.e. time and memory occupation) represents the main obstacle to an efficient application of ILP in optical networks design. Constraint routing and the other simplification techniques are able to overcome this limitation, but the solution they produce is only an approximation of the actual optimal network design.

The great advantage of ILP over heuristic methods is the ability to guarantee that the obtained solution is the absolute optimum value. Any of the above techniques aimed to reduce the computational burden implies that the ILP approach loses its added value, even if approximated solutions may be close to the exact one. Our work is aimed to find a new formulation of RFWA problem which is able to prune variable multiplicity without introducing any approximation, thus preserving the added value of mathematical programming. III. S OURCE FORMULATION OF THE ILP PROBLEM Let us consider a multifiber WDM network environment under static traffic, in which the number of wavelengths per fiber W is given a priori, while the fiber numbers per each physical link are variables of the problem. In ILP based on flow formulation 1 , dimensioning and RWA problems are simultaneously solved. This means that from the returned results we are able to evaluate the fiber distribution and to reconstruct the detailed RFWA of all the requested lightpaths. The latter feature is possible since each source-destination pair requiring connections is associated to a set of flow variables, as we have anticipated in section II. In the approach we are proposing, dimensioning and RWA are carried out in two steps. In the first step all the connections originating from each source OXC are allocated and the dimensioning of the network in terms of number of fibers per link is solved. This is why we called this approach source formulation. In the second step any lightpath is completely specified by finding out a RFWA coherent with the results of the first step. The first step, a localization problem, is difficult to solve. However, thanks to the source formulation, we are able to prune the flow variable multiplicity, thus reducing computational time and memory occupation compared to the flow formulation. The second step is carried out by exploiting the classical flow formulation (the total number of occupied WDM channels is selected as cost function) with the big difference that, for each physical link, the number of fibers is no more a variable, but a known term. Thus this step is merely a multicommodity flow problem: its computational time is negligible compared to the time required by the first step. In the following we refer to the first step as dimensioning step and to the second as RWA step Let us explain the details of the source formulation. Two different versions of it will be reported in the following, related to networks with or without wavelength conversion capability. A. Source formulation for VWP networks First we consider a VWP network, provided with full wavelength conversion as defined in II. The physical topology is modeled by the graph G = G (N ; A). Physical links are represented by the undirected edges l 2 A with jAj = L, while the nodes i 2 N = f1; 2; :::N g, with jN j = N , represent the OXCs. Each link is equipped with a certain amount of unidirectional fibers in each of the two directions; fiber direction is conventionally identified by the binary variable k. Finally, the virtual topology is represented by the set of known terms C i;j , 1 The route formulation case will be not considered any more from now on.

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each one expressing the number of connections that must be established from the source node i to the destination node j . Unidirectional point-to-point connections are considered (thus, in the general case, Ci;j 6= Cj;i). The variables in the source formulation are the following xl;k;i is the number of WDM channels on link l on fibers having direction k which have been allocated to lightpaths generated at node i; Fl;k is the number of fibers on link l in direction k. It should be noted that the flow variables x l;k;i are defined in such a way that all the traffic originating from the same node and traveling on the same link in the same direction is represented in an aggregated form, regardless of the destination. This is the main aspect that differentiates source to flow formulation. The following additional symbols are defined (l; k) identifies the set of fibers of link l that are directed as indicated by k; for sake of clarity, in the following we name (l; k) a “unidirectional link”; + Ii is the set of “unidirectional links” having the node i as one extreme and leaving the node; analogously, I i is the set of “unidirectional links” having the node i as a one extreme and pointing towards the node; Si = j Ci;j is the total number of requested connections having node i as source. Now we can detail the source formulation. The cost function to be minimized is the total fiber number

P

min

XF

l;k

(l;k )

Actually the source formulation can be very easily extended to solve optimization problems based on the length metric. The only change that must be made regards the cost function, which becomes min

XF

(l;k )

l;k pl

where pl is the geographical length of link l. The set of constraints is the following

X x l;k;i l;k 2I + X x

(

)

=

Si

X

i

2Ij+

(l;k )

l;k;i

=

2Ij

(l;k )

Xx i

xl;k;i

l;k;i W Fl;k

xl;k;i integer Fl;k integer

Ci;j

8 i;

(1)

8 (i; j ); j 6= i;

(2)

8 (l; k); 8 i; (l; k); 8 i; (l; k);

(3) (4) (5)

Constraint (1) is a solenoidality constraint which imposes that the total flow (number of lightpaths) generated by node i and exiting from it must be equal to the total number of connection requests having node i as source. Note that the solenoidality constraint is not applied on each node-pair (by which a connection is requested) but on the aggregated traffic relative to a source node: therefore it is not dependent on destinations.

Constraint (2) is again a solenoidality constraint. It corresponds to the following sequence. Let us take a node i. We express the flow conservation condition for each other node of the network j 6= i, considering only traffic having i as source node. This condition states that the total flow generated by i and escaping from j is given by the total flow generated by i and incident on j minus the number of requested connections having i as source and j as destination. The capacity constraint (3) allows to dimension the physical network capacity. In order to ensure a feasible resource allocation it imposes that on each link the sum of flows generated by all the nodes is smaller than the product of the number of fibers by the number of wavelengths per fiber. The remaining constraints (4 and 5) impose variable integrity. Let us now discuss the source formulation complexity for a VWP network. Table I shows the relations expressing the total number of variables and constraints as functions of the physical topology size and the number node pair requiring connection. The corresponding relations for the flow formulation are reported for comparison. In the table, C is the number of source-destination node-pairs requesting any connection, that is the number of node pairs of the virtual topology. For the source formulation only the dimensioning step is taken into account, since the RWA one, as stated in section III, has a negligible impact in the solution time. The number of variables of the source formulation grows with the product of the number of links by the number of nodes. In the flow formulation it grows instead with the product of the number of links by the number of pairs requesting connections. So from the variable number point of view source formulation should be more efficient than flow formulation under the condition C > N , that is presumably a common situation in real networks .In the worst case there’s at least one lightpath requested between each node pair, so we could set C = N (N 1). Source formulation in this case allows a reduction of the number of variables by the factor N compared to flow formulation. The reduction of the number of constraints is in the order of link number L. B. Source formulation for WP networks Source formulation can be extended to networks without wavelength conversion capability. ILP complexity in the WP case grows with the number of wavelengths per fiber W and constraints become more complicated because wavelength continuity has to be imposed on the lightpaths. Anyway the advantages of the source over flow formulation are still relevant. The cost function is the same as in the VWP case min

XF l;k

l;k

A new index 2 f1; 2; : : :W g must be added to identify the wavelength of the WDM channels, in order to impose the wavelength continuity constraint along a lightpath. Flow variables defined in the VWP case are transformed: xl;k;i; now indicates the number of WDM channels having wavelength which on the “unidirectional link” (l; k) carry lightpaths generated at node i. The known terms Si and Ci;j have to be split, originating the new variables si; and ci;j;.

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TABLE I C OMPARISON ON CONSTRAINT AND VARIABLE NUMBERS BETWEEN SOURCE AND FLOW FORMULATIONS.

formulation VWP source VWP flow WP source WP flow

constraints 2L +

variables

N LC 2L + N 2 (W + 1) N W C + 2L(W + 1) + C

2L(1 + N )

2

8L +

2L(1 + 2C )

N (N W + 2L W ) + 2L 2L(W C + 1) + W C

The set of constraints is modified as shown below

X X X

2Ii+

(l;k )

xl;k;i; = si;

si; = Si

2Ij

(l;k )

(6)

8 i;

(7)

xl;k;i;

X

=

(l;k )

X X

i

8 (i; );

2Ij+

(a)

xl;k;i;

ci;j; = Ci;j

ci;j;

8 (i; j; ); j 6= i; 8 (i; j ) i = 6 j;

xl;k;i; Fl;k

8 (l; k; );

xl;k;i; integer Fl;k integer si; integer ci;j; integer

8 (i; l; k; ); 8 (l; k); 8 (i; ); 8 (i; j; ); i 6= j ;

(8)

(9) (10)

The solenoidality constraints are split into the sets 6 and 8 in order to impose flow conservation independently for each wavelength. Also the capacity constraint 10 is modified. The new constraints 7 and 9 express the distribution of the total number of connections among the different wavelengths for a source node and for a source-destination pair respectively. Table I compares the complexity of source and flow formulations also in the WP case. In both the formulations the number of constrains and fibers linearly increases with W . It is important to notice that the increase of W in the WP scenario is accompanied not only by a growth of variable and constraint numbers, but also by the extension of the admissible fields of the variables. Though not visible form the table, this has a great impact on computational time and memory requirement. The advantage of the source formulation can be evaluated in a simple way by considering a fully-connected virtual topology (C = N (N 1)). Under such assumption the dominant term of the number of variables is 2W L N 2 and 2W L N 3 for source and flow formulation, respectively. As the number of constraints is concerned, the two dominant terms are W N 2 and W N 3, respectively. Last, we shall mention a limitation of the source formulation. Unfortunately, this formulation can not be extended to optimize path-protected WDM networks. In fact this feature requires to route lightpaths under the link-disjoint constraint, so

(b)

Fig. 1. Physical topologies of the two case-study networks: (a) NSFNET and (b) EON.

that a working lightpath can not share any physical link with its protection lightpath. The basic variables x l;k;i contains information concerning all the connections having the same source node aggregated together. No explicit reference can be inferred regarding lightpaths having the same source and the same destination: this makes the imposition of the link-disjoint constraint impossible. Anyway other protection techniques, such as link protection, could be planned using source formulation. In fact an approach to link protection consist in providing for each link (i.e for all its fibers) an alternative route in order to face link failure: such a feature doesn’t need information related to traffic destination node. A source formulation based model for link protection in both dedicated and shared cases is currently under study. IV. C ASE STUDY AND RESULT COMPARISON In this section we present and discuss the results we obtained by performing ILP optimization exploiting source formulation on two case-study networks. Two well-known networks have been considered: the National Science Foundation Network (NSFNET) and the European Optical Network (EON). Data regarding their physical topologies, represented in Fig. 1(a) and 1(b), were taken from Ref. [21]2 and Ref. [24], respectively. NSFNET has 14 nodes and 22 links, while EON has 19 nodes and 39 links. The virtual topologies are based on the static (symmetric) traffic matrices derived from real traffic measurements which are reported in the same references. The two virtual topologies comprises 360 and 1380 unidirectional connection requests for NSFNET and EON, respectively. Both VWP and WP cases have been analyzed. Table II shows the number of variables and constraints that are involved in the ILP problem applied to the two networks in the VWP case. Data are computed using the relations reported 2 The reported NSFNET topology is actually the NSFNET T1 backbone [3] with the addition of one extra link

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TABLE II

TABLE III

ILP VARIABLES AND CONSTRAINTS FOR NSFNET AND EON IN THE VWP CASE .

VWP NSFNET OPTIMIZATION : COMPUTATIONAL TIME .

Number of variables and constraints

network/formul. NSFNET/source NSFNET/flow EON/source EON/flow

8 10

4

7 10

4

5 10 4 10

4

3 10

4

source form. 27 m 28 m 36 s 6m 19 m

flow form 40 m 105 m 50 m 10 h 5h

TABLE IV VWP NSFNET OPTIMIZATION: MEMORY OCCUPATION .

source variables flow variables source constraints flow constraint

W 2 4 8 16 32

2 104

source form. 0,39 MB O.O.M 5 MB 47 MB 180 MB

flow form 1,3 MB O.O.M. 42 MB O.O.M. O.O.M.

1 104 0 0

2

4 6 8 10 12 14 Number of wavelengths, W

16

EON WP Number of variables and constraints

W 2 4 8 16 32

variables 570 4840 1560 53430

NSFNET WP

6 104 4

constraints 284 2552 517 13650

5 10

5

4 10

5

3 10

5

source variables flow variables source constraints flow constraints

2 105 1 10

5

0 0

2

4 6 8 10 12 14 Number of wavelengths, W

16

Fig. 2. ILP variables and constraints for NSFNET and EON in the WP case.

in table I for both source and flow formulation. They clearly show the advantage of the former. For the WP case, in Fig. 2 we have displayed the variable and constraint numbers as a function of the number of wavelength per fiber . The comparison between the curves shows that the simplification introduced by the flow formulation in both the networks is relevant. Moreover it increases linearly with the parameter , since the slope of the curves is more steep for the flow formulation. To solve the ILP problems we used the software tool CPLEX 6.5 based on the branch-and-bound method [25]. As hardware platform a workstation equipped with a 1 GHz processor was used. The available memory (physical RAM + swap) amounted to 460 MByte. Before the presentation of numerical results, it’s crucial to highlight the equivalence between the two formulations. Mathematically the flow to source formulation change is given by the following substitution: j l;k;i;j = l;k;i, in order to group in a single variable all the flow contributes generated by source

W

W

Px

x

i

node . Obviously the previous substitution has a significative effect on the ILP model: first of all the capacity constraints remain unchanged except for changes in the problem variables. Then we have to readapt the solenoidality constraint to match the new variable set, because solenoidality is not applied anymore on each node pair. The new constraints don’t distinguish between transit node and destination node: if a node is not the source node, the adapted solenoidality constraint guarantees the flow conservation for transit traffic and simultaneously traffic delivery for that subset of traffic addressed to that node. This equality is verified in all the network cases in which both formulations succeed in finding the optimum value: this value in fact results to be the same in the two formulations. We have shown above the advantage of source formulation versus flow formulation in terms of variable and constraint numbers. It is important to see how much this advantage impacts on the actual computational performance of ILP. Tables III and IV display computational time and memory occupation measurements of NSFNET optimization in the VWP case (s, m, and h stand for seconds, minutes and hours respectively, while MB stands for mega-byte). To clearly understand the reported data, a particular aspect of ILP must be clarified. The branch-and-bound algorithm progressively occupies memory with its data structure while it is running. When the optimal solution is found, the algorithm stops and the computational time and the final memory occupation can be measured. In some cases, however, all the available memory is filled up before the optimal solution can be found. In this cases CPLEX returns the best but non-optimal solution branch-and-bound was able to find and force the execution to quit. This cases are identified by the out-of-memory tag (O.O.M.) and the computational time measures how long it has taken to fill up memory. A final point to be clarified is that the reported measurements for the source formulation concern only the dimensioning step, since the RWA step is almost instantaneous, as stated in section III. From the table we can see that the out-of-memory event is more frequent with the flow

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TABLE V

source form. 2m 37 m 1.4 h 9h

flow form 35 m 4.25 h 17 h 37 h

12

25

10

20

8 15 6 10 4

percent difference 5

2

M

Absolute difference 0

formulation. Moreover source formulation always requires a smaller memory amount and a shorter run duration than flow formulation. The gap in run duration between the two tends to increase with the W parameter. This is probably due to the extension of the variable admissible field. NSFNET optimization in the WP scenario takes a very long time with the hardware we employed. In some cases it was too long to wait either for the optimal result or for a out-of-memory event. Thus in table V we have listed time necessary for the filling up of the first 7 MB of memory. The case with W = has proved to be too complex to be solved in a reasonable time and therefore it was omitted. The speed of the branchand-bound algorithm applied to the flow formulation decreases dramatically for high values of W . With the source formulation the speed does not decrease so much. We are now going to compare source and flow formulations on the basis of the final value of the cost function we obtained. In all the cases in which, for both the formulations, the branchand-bound ends up before an out-of-memory event, the final values obtained are coincident, thus proving the equivalence of source and flow formulation. In all the other cases the suboptimal value returned by the source formulation is the smallest one. The following parameters are introduced Msource (Mflow ): total fiber number returned by ILP based on the source (flow) formulation; M : absolute difference Mflow Msource ; M : percent relative difference Mflow Msource =Msource; We will show results concerning both the two case-study networks. The difference between the two formulations obtained in the VWP scenario are represented in Fig. 3a (absolute values) and in Fig. 3b (percent) as functions of the parameter W . The absolute difference is on average greater for the EON which has a larger number of nodes and links. Convergence between the two formulations occurs, for example, in the NSFNET case for W and W , in accordance with table IV. It should be noted that sub-optimal solutions with the flow formulation can be up to 18% worse than the corresponding solutions produced by the source formulation. In Fig. 4 M and M are displayed for NSFNET in the WP case. In this case the strong increase of variable and constraint number with W causes a relevant increase of the differences between the two formulations. Up to this point of the paper we have described multifiber WDM network optimization having the total number of fibers as cost function. This interpretation of network cost is called

32

100 (

)

=2

=8

Percent fiber difference

W 2 4 8 16

Total fiber difference M

NSFNET WP

WP NSFNET OPTIMIZATION : TIME REQUIRED TO FILL UP 7 MB OF MEMORY.

0 0

5

10 15 20 25 30 Number of wavelengths, W

35

Fig. 4. Source-flow comparison on the final number of fibers in the WP case for NSFNET.

hop metric and it models a situation in which all the fibers of the network have the same cost. However in real networks the cost of a link also depends on its geographical length, which for example determines the number of optical line amplifiers that must be installed. Measuring the cost of a fiber in this situation becomes much more complicated and the hop metric is not appropriate any more. Another simple alternative is the length metric, which assigns a cost to each fiber proportional to the geographical length of the link it belongs to. Though still not completely realistic (e.g. it does not take into account that the cost of the duct should be shared by all the fibers of a link), it could be useful in many situations (e.g. when the cost of optical line amplifiers is an important issue). Clearly, hop metric can be regarded as a particular case of length metric in which all the links have length 1 3 . We have tested source formulation based on length metric on NSFNET and EON in the VWP case. In a fashion similar to hop metric, the following parameters have been defined Psource(Pflow ): total fiber length returned by ILP based on the source (flow) formulation; P : absolute difference Pflow Psource ; P : percent relative difference Pflow Psource =Psource; Link lengths p l were assigned for the two networks according to Refs. [21], [24]. Fig. 5 displays the percent relative difference between the total fiber lengths obtained applying the source and the flow formulation. The same conclusions drawn for the hop metric can be extended to these new optimization experiments. Source formulation performs better in all the cases in which an outof-memory event occurs; otherwise, the results are coincident, but source formulation converges more rapidly (computational times are omitted for brevity). At the end of this section we would like to compare the ILP optimization by source formulation to optimization by a heuristic approach. Let us consider NSFNET and the hop metric. In Fig. 6 ILP and heuristic final results are displayed in the VWP (Fig. 6a) and WP case (Fig. 6b). The heuristic data were obtained by the deterministic approach published in the paper Ref.

)

100 (

3 Neither hop nor length metric take the node cost into account. Node-cost optimization issues are not covered by this paper.

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Source vs Flow

Source vs Flow

30

12

Percent fiber difference

Total fiber difference, M

M

NSFNET VWP EON VWP

10 8 6 4 2 0

NSFNET VWP EON VWP

25 20 15 10 5 0

0

10

20 30 40 50 Number of wavelengths, W

60

0

70

10

(a)


60

70

30

35

(b)

Fig. 3. Source-flow comparison on the final number of fibers in the VWP case, as absolute (a) and percent relative difference (b).

NSFNET VWP

NSFNET WP

400

400 350

source form. heuristic

300

Total fiber number, M


350

250 200 150 100 50

source form heuristic

300 250 200 150 100 50

0

0 0

5


30

35

0

5

(a)


(b)

Fig. 6. NSFNET total fiber number optimized by ILP source formulation and by a deterministic heuristic, in the VWP (a) and WP (b) cases.

Source vs Flow 5 Percent length difference

P

NSFNET VWP EON VWP

4 3 2

the absolute optimum (or comes closer to it when limitations on memory or computational time prevents branch-and-bound to converge). As the computational time is concerned, we have noticed heuristic and source-formulation ILP performance is similar for VWP networks. In the WP case, however, heuristic methods are much better than ILP, even when source formulation is adopted. It can be noticed in Fig. 6b that heuristic has = 32, been the only possible approach to obtain a result with given the hardware limitations of our workstation. Last, let us report for completeness the results that were obtained for EON by ILP exploiting the source formulation in terms of the final number of fibers. These data are displayed in Fig. 7 for the VWP case and were used to generate the curves of Fig. 3.

W

1 0 -1 0

10


60

70

Fig. 5. Source-flow comparison on the final total fiber length in the VWP case, as percent relative difference.

V. C ONCLUSIONS

[26]. The comparison shows that the results of the two techniques are quite close: the heuristic approach is able to provide good sub-optimal results, but only the exact approach allows to reach

We have presented and discussed a novel formulation, named source formulation, to solve static-traffic WDM network optimization by ILP. This formulation has been defined for multifiber networks with or without wavelength conversion capability supporting unidirectional unprotected optical connections. Thanks to the source formulation, we are able to substantially

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EON VWP 1600


1400 1200 1000 800 600 400 200 0 0

10


60

70

Fig. 7. EON total fiber number optimized by ILP source formulation.

prune the multiplicity of both variables and constraints compared to the well-known flow formulation. Exploiting source formulation we thus obtain a competitive optimization tool capable to solve ILP problems with relatively low computational time and memory occupation. The case-studies we have discussed in the paper proves the advantages of source formulation in several network-planning experiments.

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