Optimization algorithms for WDM optical network dimensioning

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1 Optimization algorithms for WDM optical network dimensioning Andrea Concaro, Simone De Patre, Guido Maier, Massimo Tornatore CoreCom, Via Colombo Milan, Italy Department of Electronics and Information, Politecnico di Milano, Via Ponzio Milan, Italy tornator@elet.polimi.it Abstract In this paper 1 we study the particular approach to planning the optical transport network under static traffic which consists in improving an initial solution by lightpath rerouting. Several different heuristic strategies to carry out the rerouting process are proposed and described, including deterministic and stochastic algorithms. A performance comparison among them is presented, based on case-study examples and considering: final fiber-number (cost function) value, execution time and convergence behavior. Results of all the strategies are also compared with results obtained by integer linear programming, evaluating of the possibility to heuristically obtain good suboptimal solutions. The two cases of unprotected connections and dedicated path-protection have been considered, with and without wavelength conversion. I. INTRODUCTION In the last few years an intense research activity has been addressed to the problem of optical network dimensioning. Operators are more and more frequently challenged by design problems, both to plan new installations and to update the existing ones, by the continuous growth of the demand of bandwidth for new applications such as video and multimedia streams and advanced digital services. This is particularly true for the Metropolitan Area Networks (MANs), where traffic aggregation is low and thus the bandwidth requirements of the single applications have a high impact on network performance. On the other hand, planning problems increase in complexity as the topology of WDM networks evolves from ring to mesh, taking advantage of the constant improvement of optical transmission and switching technologies. The high connectivity of mesh Optical Transport Networks (OTNs), relying upon Optical Cross Connects (OXCs) for switching, improves the bandwidth provisioning service, but requires careful planning to avoid useless capital expenditure. Planning is further complicated by the need to equip the system with suitable protection resources. High-speed optical connections (at 10 Gbit/s or higher) are very vulnerable to failures: even a fewseconds outage means a huge waste of data. Survivability, that is the capability of the network of maintaining service 1 Work partially supported by MIUR, Italy, under FIRB Project TANGO and Project WONDER. continuity in presence of failures, from an attractive research topic has became an outstanding important planning issue for every OTN. Though dynamic traffic is becoming more and more important and probably will eventually become dominant as the GMPLS/ASON (Generalized Multi-Protocol Label Switching, Automatically Switched Optical Network) architecture spreads pervasively, present optical-network operators have still to provide mostly permanent or semipermanent optical circuits. In our paper we are thus going to deal with OTN dimensioning in a static-traffic scenario. Given the physical topology and the set of Lightpath Connections (LCs) that must be setup (LC-layer topology), network capacity dimensioning and resource allocation are solved simultaneously, minimizing a chosen cost function. An additional goal is to plan the spare capacity when network survivability against a single link failure has to be guaranteed. As a possible survivability technique, we have chosen to refer to Dedicated path-protection (DPP), which is the simplest and fastest survivability scheme, not requiring reconfiguration of transit OXCs upon a failure. Another advantage of DPP in a context of network planning under static traffic is that channel assignment is completely failure independent. In order to solve the planning problem outlined above, efficient dimensioning procedures are needed. The problem can be tackled by using either an exact or a heuristic approach. The former, namely the Integer Linear Programming (ILP), can find the optimal solution, but it can not be applied to large networks due to its enormous complexity, increasing exponentially with network size. The latter does not guarantee the optimality of the obtained solution, but it is generally simpler and faster. In this paper we wish to focus on the second approach, proposing a set of optimization strategies and comparing their performance by applying them to two realistic networkcases (NSFNET and EON). We are also going (when possible) to compare the results of the heuristics to those provided by the ILP solution of the same design experiments. As we will summarize later on in the paper, many studies have been published in the past on heuristic optimization /05/$ IEEE. 141

2 of the OTN. While most of such studies are focused on single specific methods, in this work we are proposing a flexible general scheme which is able to operate according to several heuristic methods, selecting one or another with just few changes in the input-parameter set. This approach allows us to easily compare the most common methods, showing differences and commonalities both in their algorithmic structure and in their performance. We also have the opportunity to observe and compare the computational time-complexity of the considered algorithms. Sec. II describes how OTN is modelled for planning. Previous literature on heuristic optimization is reviewed in Sec. III; the section also defines the ILP formulations that are adopted in this paper. Sec. IV introduces to our planning method and describes the heuristic strategies we have 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 the network is equipped with two independent sets of monodirectional fibers. Each fiber carries N λ wavelengths of the WDM multiplex: N λ is a pre-assigned design parameter, constant for every link of the network, and corresponding to a particular type of WDM transmission system that the operator has chosen to deploy. The nodes are equipped with Optical Cross Connect (OXC). They are able to switch an incoming optical channel (a wavelength on a fiber) to an outgoing channel. We will consider OXCs with or without wavelength converters. In the former case, a node can convert the wavelength of an incoming lightpath to a different outgoing wavelength; in the latter case, the node can not change the wavelength of the incoming lightpath. The LC-topology is known: connections are static and point-to-point (OXC-to-OXC) unidirectional. It is decided in advance wether the network is unprotected or if all the optical connections must be protected by DPP. In the unprotected case, a lightpath, i.e. a sequence of WDM channels along a path on the physical topology, must be allocated for each connection request. If all the nodes of the network are equipped with wavelength converters, we are considering a VWP (Virtual Wavelength Path) network. If no node is equipped with wavelength converters, we are considering a WP (Wavelength Path) network. In the first case wavelengths can be assigned to a lightpath link-bylink, with possible intermediate wavelength changes; in the second case we must assign a single wavelength to the entire lightpath (i.e. the lightpath must satisfy the wavelength continuity constraint). When DPP is adopted, two lightpaths must be set up per connection-request, composing a working and a protection pair (w/p pair): we have to route the w/p pair imposing that the two lightpaths must be link-disjoint (they can not share a common physical link). In this way, in case of a (single) link failure, it will always be possible to reroute the interrupted working traffic on the spare capacity. The variables of our model are the capacities of each physical link in terms of number of fibers and the cost function is the total number of fibers F that will be deployed to satisfy all the requests of the LC topology (with or without protection). We are thus seeking the Routing and Fiber and Wavelength Assignment (RFWA) solution for all the lightpaths (or the w/p pairs) resulting in the smallest possible F. The optimum RFWA is such that F is necessary and sufficient to setup the LC-topology. With heuristic approaches we are able to find RFWA solutions leading to values of F that are sufficient but not necessary, though hopefully not too much greater than the optimum F. III. OTN OPTIMIZATION IN LITERATURE As previously mentioned, this paper is mainly dedicated to OTN heuristic design. We will however present also some ILP-based results to benchmark the heuristic algorithms. Therefore in this section we are going to provide the standard ILP flow formulations for the unprotected and DPP scenarios, summarizing the ILP models presented in Refs. [8], [19], [24] to solve capacitated design in WDM networks. We will report only the case with full wavelength conversion: the extension of the models to WP can be carried on as explained in Refs. [22], [24], introducing a further term of complexity, function of N λ. Let us consider the physical topology, modeled by the graph G = G(N, A) 2. Physical links are represented by the undirected edges l A with A = L, while the nodes i N = {1, 2,...N}, with N = 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 (k =0for forward direction, k =1for backward direction). v c is the number of requested LCs having s c as source and d c as destination node, with c an index used to identify each source-destination node-couple requiring connectivity. The (integer) variables involved in the unprotected flow formulation are: x l,k,c, flow variable indicating whether a WDM channel on link l on a fiber having direction k has been allocated to one of the connections requested by node couple c; F l,k, capacity variable indicating the number of fibers on link l in direction k. The following additional symbols are also defined: (l, k) identifies a unidirectional link, i.e. the set of fibers of link l that are directed as indicated by k; 2 The following formulations require that the topology is at least 2-connected. 142

3 I + i is the set of unidirectional links having node i as one extreme and leaving the node; analogously, I i is the set of unidirectional links having node i as a one extreme and pointing towards the node. The cost function to be minimized is the total fiber number min F = min (l,k) F l,k The unprotected flow formulation is given by the following set of equations x l,k,c v c ifi=d c x l,k,c = v c ifi=s c i, c; (l,k) I + i (l,k) I 0 otherwise i (1) x i,k,c W l F l,k (l, k); (2) c (3) x l,k,c integer (l, k),c; (4) F l,k integer (l, k); (5) Constraint (1) is a solenoidality constraint. Let us consider the v c connections requested by c: the flow conservation condition for v c in each node i states that v c leaving i must be equal to v c incident on i. In the source (destination) node the flow balance is satisfied by adapting the constraint to the border condition: the total leaving (incoming) flow must be equal to v c. Constraint (2) ensures that the total number of WDM channels allocated to spare and working lightpaths on the unidirectional link (l, k) is bounded by the link capacity, given by the number of fibers F l,k multiplied by the number of wavelengths W. Constraints 4 and 5 enforce the integrity of the fiber number and flow unities. The definition of an ILP model in a WDM network with dedicated path protection is a well-known problem: to the set of constraints of the unprotected formulation, we must add constraints deriving from the link disjointness condition. These can be easily set, provided that the basic flow variable is enriched with a new index, increasing the description detail of the flows. If v c > 1, we add an auxiliary index t having values between 1 and v c ; the flow variables become: x l,k,c,t, boolean variable indicating whether a WDM channel on link l on a fiber having direction k has been allocated to the t-th connection requested by node couple c. We further introduce the following symbol: (c, t), identifying a single connection request. The set of constraints is the following x l,k,c,t x l,k,c,t = (l,k) I + i (l,k) I i 2 if i = s c 2 if i = d c i, (c, t); (6) 0 otherwise x l,k,c,t W F l,k (l, k); (7) (c,t) x l,k,c,t 1 l, (c, t); (8) k x l,k,c,t binary (l, k), (c, t); (9) F l,k integer (l, k); (10) Constraint (6) enforces the flow solenoidality, as in the unprotected case. A slight difference exists in the source (destination) node of the connection request (c, t), in which the outgoing (incoming) flow must be equal to 2. This is due to the fact that a w/p pair, instead of a single lightpath, is associated to the connection request (note that the distinction between working and protection lightpaths is irrelevant). Constraints concerning dimensioning (7) are simple extensions of the corresponding constraints in the unprotected case. 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 opposite directions. Solving the above set of equations for realistic networks is really difficult, since the number of variables and constraints tends to explode with the network size (OTN planning with capacitation is well-known to be a NP-hard problem). In order to keep computational time and memory occupation at reasonable levels we have actually obtained the results that will be presented in Sec. V by exploiting slightly different, but more efficient formulations compared to the standard flow formulation presented above. Such formulations, whose detailed description is outside the scope of this paper, are reported in Refs. [22], [23]. We will concentrate in the rest of the paper on the heuristic approach to OTN design, as an alternative to the ILP. We are going to consider some heuristic optimization strategies, all based on the concept of lightpath rerouting: given an initial RFWA solution (i.e. a feasible accommodation of all the requested lightpaths on the physical topology) we try to improve it by rerouting the optical connections on alternative paths, subject to the constraint that all the LC-requests must be satisfied. This kind of approach to the problem, despite being a very simple and direct solution method, can achieve good results, as we are going to verify by case-studies. We can find in literature a lot of examples of heuristic lightpath-rerouting. In [24] the total number of used wavelengths is minimized by rerouting lightpaths that crosses 143

4 the most-loaded links, while in [3] the connections, initially routed on shortest paths, are rerouted on alternative shortest paths, if this operation can decrease the load on the most congested link. In [1], where multifiber networks are considered, new configurations are iteratively searched by modifying the initial routing and/or wavelength assignment, trying to lower the cost of requested fibers. Ref. [17] considers an SDH-over-WDM network: the rerouting method is utilized to find a good LC-topology for the WDM layer, given the requests for SDH circuits. The SDH connection-requests are routed on a sequence of lightpaths (multi-hop routing) trying to minimize the number of LC-requests. In order to achieve this purpose, the optimization algorithm tries to reroute the SDH channels initially carried by a lightpath on the residual capacity of other lightpaths. All the above-cited techniques can be classified as deterministic heuristics, since rerouting attempts are carried out according to a fixed lightpath-sorting rule, accepting rerouting each time it leads to an improvement of the cost function. Due to this feature, deterministic methods are affected by the problem of getting trapped in local minima, which can not be overcome with a down-slope - only decision-rule. Other heuristic rerouting approaches do exist, which can be classified as stochastic and are able to avoid local-minima trapping. Simulated annealing [15] is a well-known general optimization method which stochastically simulates the slow cooling process of a physical system. The temperature of the system 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. A new configuration is constructed by imposing a random displacement. If the energy (the cost function) of this new state is lower than that of the previous one, the change is accepted unconditionally and the system is updated. If the energy is greater, the new configuration is accepted probabilistically. This procedure allows the system to move consistently towards lower energy states, yet still jumping out of local minima due to the probabilistic acceptance of some upward moves. In [21] simulated annealing is applied to OTN with the purpose of mapping a regular virtual topology on a given physical topology. In [20] simulated annealing is utilized to find a good virtual topology, and then a flow-deviation algorithm optimizes traffic routing. Simulated annealing and other stochastic algorithms (random sampling, local search, threshold accepting, tabu search and genetic algorithms) are applied in [10] to ATM network dimensioning. Other stochastic algorithms have been applied to optical networks, such as Monte Carlo [7] and genetic algorithms [2], [14]. Stochastic and deterministic heuristic approaches to OTN planning are compared in Refs. [2], [7], proving how a random component can often be useful to obtain good suboptimal results. Similar comparisons are presented in [16], where the objective of optimization is regular topology for packet-switched optical networks. IV. HEURISTIC PLANNING METHOD The heuristic approach to static-otn design we have developed is based on the rerouting concept. As previously mentioned, 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 been represented by a Multifiber Layered Graph (MLG). This is derived from the layered graph, introduced for mono-fiber networks [9], and extended in [18] to multifiber networks. A. Step one: greedy resource allocation Let us provide a synthesis overview of the first step, which is not the actual objective of this paper. A feasible RFWA solution is obtained with the following technique. Starting from the idle physical topology, all the connection requests of the LC-layer topology are set up in sequence one after the other until all have been satisfied. Each link initially contains a number of fibers so large to be considered infinite: in this way the existence of a solution is guaranteed. The connection requests of the LC-layer topology are initially sorted according to a balanced sorting rule: node-pairs with greatest topological distance and largest amount of connection-requests are served with the highest priority. The technique is greedy, as each requests is satisfied regardless of all the others, allocating resources to a lightpath (unprotected case) or a w/p pair (DPP case). In the unprotected case, RWFA is carried out on a single lightpath by adopting the prioritized multi-criteria approach, extensively described in [11]. The set of criteria adopted in this work was identified in [11] as the one leading in most of the cases to the best greedy allocation (with the lowest number of fibers). It comprises, with decreasing priority: Shortest Path Routing (SPR), First Fit fiber selection (FFF), First Fit wavelength assignment (FFW), Least-Loaded Routing (LLR) [4], [18]. The length metric adopted 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 always chosen according to this sorting. To implement the above heuristic RFWA criteria, suitable weights are assigned to the MLG arcs and the Dijkstra algorithm is used to find the route connecting the source to the destination on the MLG with the least total weight. We shall note that RFWA is performed in an unconstrained mode, that is all the possible routes on the MLG connecting source to destination are scanned in setting up a new lightpath. When DPP must be supported, the RFWA of the working lightpath is coupled to the RFWA of the protection lightpath 144

5 of the same w/p pair by the route-diversity constraint. There are two main techniques to find two link-disjoint paths connecting two nodes of a mesh topology. The simplest one is called two-step search. The shortest path is found (e.g. applying the Dijkstra algorithm) and it is allocated to the working lightpath. Then the shortest-path algorithm is run again to route the protection lightpath, which will be assigned, in this way, the second-shortest path. The two-step approach (greedy, because of the sequential computation of the two paths) in some special cases [12] may fail to find a w/p pair solution even if such solution actually exists. Bhandari [5], [6] proposed to overcome such a limitation by a one-step search, in which the two link-disjoint paths are not routed separately, but they are jointly routed by performing a suitable algorithm (the modified Dijkstra algorithm) in a directed graph. Besides being able to solve trap-networks, it also finds the actual minimum-length cycle connecting two nodes. We applied the one-step technique in the DPP case by adapting the Bhandari algorithm to the MLG. In particular, a set of rules has been added to the modified Dijkstra algorithm to control the edge inversion operation [6] in the MLG environment with and without wavelength conversion and to support LLR. B. Step two: rerouting procedures We are now going to discuss more in depth the second planning step. The basic purpose of this step is to modify the non-optimized network solution found by greedy resource allocation. The chosen cost function, which in our case is the total number of fibers, is decreased by rerouting some connections, under the constraint of preserving connectivity of the LC-layer topology. Qualitatively, some kind of iteration selects one specific fiber of the network at a time. Lightpaths or w/p pairs crossing the selected fiber are tentatively rerouted on the other fibers of the network. If free resources are enough to allow rerouting, the selected fiber is removed. Otherwise, everything remains unchanged. Iteration is used to reach at least a local minimum of the cost function, i.e. such that further improvement are impossible by applying the same heuristic procedure. The order by which fibers are selected and the iteration control depends on the particular optimization strategy adopted to solve the problem. In this work we have tested different heuristic strategies, belonging both to the deterministic and the stochastic class, which we are going to present in details. Since most strategies have been implemented as variations of a common procedure, it is convenient to describe them starting from their common elements. Let us define some basic actions that will be used many times in the procedures. In the following, we consider a fiber f on which a set S f of l f lightpaths (which can be working or protection in the DPP case) are routed. FTr - Temporary reroute f. Let us consider the unprotected case. The following steps are performed 1) The current RFWA of all the l f lightpaths is stored in a separate database 2) Each lightpath of S f is deallocated by freeing all the MLG arcs it occupies 3) Fiber f is disabled by preventing any subsequent occupation of the MLG arcs belonging to it (e.g. setting their weights to infinite) 4) A new RFWA on the residual MLG is attempted for each connection request having a lightpath belonging to S f In the DPP case, the above steps may be described in the same way, only considering in steps 1 and 2, instead of lightpaths, the w/p pairs with a lightpath crossing f. FBl - Block f. This conditional function returns NO if all the connection requests affected by a FTr function could be routed successfully on the residual MLG and YES otherwise. FRe - Remove f. All the MLG arcs belonging to f are permanently disabled, and f is removed from the list of the existing fibers. Moreover, a register SCS is set to, indicating that at least an FTr function has successfully terminated. FRs - Restore f. The RFWAs saved in step 1 of an FTr is restored on the MLG and arcs belonging to f are re-enabled for future usage. Fig. 1 shows how the above functions are combined to compose the core procedure. At the beginning of such procedure all the existing fibers are numbered from 1 to F. f is used as a local index to scan the existing-fiber set. Each fiber is processed by FTr and the subsequent functions FBl, FRe and FRs, provided that two conditional functions Y and X result to be. The definitions of such functions depend upon the specific heuristic strategy and will be given shortly below. Our spectrum of optimization strategies covers the following alternatives: Idle-Busy (IB), Busy-Idle (BI), Random Order (RO), Random Polarized (RP), Shortest Path (SP), Simulated Annealing (SA). The first four strategies have similar structures, which are all together represented by the flow-chart of Fig. 2. In Start the network solution found by the greedy planning step is considered. The first action performed is the removal of any residual empty fiber, by executing FRe on all the fibers having o f =0, where o f is the number of WDM channels of fiber f allocated to working or spare lightpaths. After this action, common to all the strategies, the procedure branches. One of the alternative branches is executed, according to the optimization strategy chosen for the particular planning experiment. In all the cases, residual empty fibers are again removed at the end; then, the result in terms of RFWA for all the lightpaths and network dimensioning is returned in 145

6 Core procedure Label fibers from 1 to F* f = 1 Y X(o f,k) Temporary reroute f Restore f YES Block f f > F* f = f + 1 Remove f NO Fig. 1. Flow chart of the core procedure. Start Remove f o f = 0 Strategy choice SP SA RO Random V IB BI RP k = 0 k = N k = 0 i = 0 SCS = k = k + 1 SCS = k = k + 1 i = i + 1 Core procedure k = N - 1 k = k - 1 Core procedure k = 1 SCS Eval p(k) Core procedure k = N - 1 Remove f o f = 0 k = V(i) Core procedure i = N - 1 Stop SCS 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 from those carrying only one lightpath and ending with those with just one WDM channel free. In the core procedure, the conditional function X(o f,k) is: o f k o f > 0 (Y, not needed, is set to the constant ). The strategy has the advantage of trying to eliminate first fibers that are easy to free, since FTr initially involves few connections. Beside that, saturation of the free capacity due to successful rerouting is gradually distributed in the k cycle, avoiding high blocking probability at the beginning. A fiber has the chance of being selected multiple times with different values of k. In fact, for each k increment, non-empty fibers having up to k (and not simply k) busy channels are considered. There are good chances that at the end of the k-cycle a suboptimal solution has been reached. Note that X(o f,k) prevents empty fibers from being selected: this further condition leaves more spare capacity for rerouting during the k-cycle, thus reducing blocking probability. The BI strategy, again deterministic, is almost the dual of the previous one: fiber selection-order is from the most to the least loaded. In the core procedure, the conditional function X(o f,k) is o f = k (Y is set to ): only fibers having exactly k busy channels are considered for each k increment. In order to compensate this restriction, the entire k cycle is repeated several times until FTr is no more possible for any fiber (FBl always detects a block): the optimization terminates when the register SCS is detected to remain F ALSE after the end of the inner cycle. The rationale of BI is that deallocation of very loaded fibers is attempted at the beginning, when chances of blocking are low due to a relatively high amount of unused capacity. The first stochastic strategy we have considered is the RO: fibers are considered for selection in a random order of occupation. At the beginning of the procedure, the function Random V randomly inserts the integer numbers from 1 to N λ 1 into the (N λ 1)-element vector V. The vector is then scanned by the index i, selecting a value k = V(i) per iteration. Fibers are selected according to the condition X(o f,k): o f = k (Y is set to ). As in the BI strategy, the entire k cycle is repeated many times until FTr is no more possible for any fiber (exploiting the check on SCS as termination condition). 146

7 The other stochastic strategy detailed in Fig. 2 is RP. Fiber selection occurs exactly as in IB, setting X(o f,k) to: o f k o f > 0. The selected fiber is however processed by the core procedure or skipped according to a random choice. Eval p(k) at the beginning of the k cycle sets a probability threshold p(k) given by the expression p(k) =p 0 +(1 p 0 ) k 1 N λ 2 where p 0 (0 <p 0 < 1) is a preset parameter. Inside the core procedure, 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 threshold p(k) increases with k, making the probability of processing p(k) low at the beginning of the k-cycle and steadily increasing with k up to 1 when k = N λ 1. The effect is that the deterministic order of fiber processing is perturbed by a random fluctuation with an amplitude high for idle and low for busy fibers. This strategy can be regarded as an hybrid between the deterministic IB and the stochastic approach. It should be noted that RP has also common aspects with simulated annealing, since p(k) plays the role of a system temperature: however, it should be actually considered a simulated melting, as the temperature increases. Another deterministic strategy is SP. Its flow-chart is not worth to be represented, since it is a mere iteration of IB. The only change to IB is a modification of the FBl condition adding a constraint on the length of the rerouted lightpaths. Let us suppose that c is a connection of the set of those that must be tentatively rerouted. The topological distance (the length of the shortest path, in hop) between source and destination is H c, while the minimum-length cycle between the two nodes has length (in hop) C c.ifn is the IB-iteration counter, going from 1 to n max +1, in the unprotected case, a possible newly-rerouted lightpath is constrained to have a length (in hops) less than or equal to H c + n 1, while in the DPP case, the sum of the lengths (in hops) of the newlyrerouted working and protection lightpaths is constrained to be less or equal to C c + n 1. The conditional function FBl in the SP strategy returns NO if all the connection requests affected by a FTr function could be carried out successfully on the residual MLG, subject to the above length constraints. Otherwise, YES is the result. The use of n allows to progressively relax the constraint up to a prefixed value n max, while controlling the number of iterations. The SP technique has the advantage of limiting the amount of resources that can be used by lightpath rerouting, thus slowing the saturation the unused capacity of the network. The SA strategy is the application of the well-known Simulated Annealing method to our planning problem. Since a flow-chart would be too complex to represent, we will explain the SA strategy in words. In SA, differently from all the other strategies, fibers can be not only removed but also added to the network: this gives the chance of escaping from local minima. The control temperature of the annealing process in our case is represented by the probability p of accepting fiber addition. p is initially set to a high value p 0 (0 p 0 1). Then a fiber-processing cycle begins. N F fibers are processed in each iteration, as described below. At the end of an iteration, if the current value of p is lower than a prefixed threshold p Th, the SA phase terminates; else, the new current value of p is set to p and the cycle iteration is repeated again, processing N F fibers. is the preset cooling-rate annealing parameter. The N F fibers processed in a cycle iteration are randomly chosen. Let us describe what happens when a given f fiber has been selected. First of all, the current network state S is saved and the current fiber number F is computed. Then, FTr is performed on f and the FBl condition is evaluated. If no block is detected, all the empty fibers of the network (including f) are permanently removed and the network state consequently updated. Processing begins again by randomly choosing a new fiber. If block is detected, a random choice is taken. With probability 1 p, f is left in place and no further action is taken: S is restored and a new fiber is selected. With probability p, instead, a fiberaddition procedure is performed. This latter comprises the following steps. First, one fiber is added to any link of the network. Then FTr is performed on f again: this time there can obviously be no block. After lightpath have been rerouted, all the empty fibers of the network (including f) are removed and the total number of fiber F is evaluated again. If F F >F T, being F T another preset annealing parameter, fiber addition is considered unacceptably large: S is restored, returning to the network state as it was before the beginning of processing of f, and processing of a new fiber begins. Else, the network state is updated by permanently accepting the added fibers and the newly-routed lightpaths before beginning to process another fiber. V. RESULTS We are now going to compare the various heuristic planning procedures and ILP optimization on the basis of numerical results. We have considered two well-known realistic networks for case-study: the National Science Foundation Network (NSFNET, 14 nodes and 44 links) and the European Optical Network (EON, 19 nodes and 78 links). Their physical topology is shown in Fig. 3: as it clearly appears, EON is much more densely-connected than NSFNET. The LC-layer topologies used for the planning experiments have been derived from the static (symmetric) traffic matrices based on real traffic measurements which are reported in Refs. [13], [19]. The two LC-layer topologies comprise 360 and 1380 unidirectional connection requests for NSFNET and EON, respectively. Stochastic strategies benefit from the advantage over the deterministic approach that re-executing the dimensioning 147

8 TABLE II FINAL FIBER NUMBER OBTAINED BY PLANNING NSFNET IN THE DPP CASE. (a) N λ IB BI RO RP SP SA ILP W P V W P (b) Fig. 3. Physical topologies of NSFNET (a) and EON (b) networks. TABLE I FINAL FIBER NUMBER OBTAINED BY PLANNING NSFNET IN THE UNPROTECTED CASE. N λ IB BI RO RP SP SA ILP W P V W P procedure different results may be obtained. We have exploited this property for NSFNET: being this network less connected than EON, the repetition of dimensioning does not take too much computational time. Therefore, results concerning RO, RP and SA have been obtained for the NSFNET by repeating the second planning step twice and considering the best result. The following values have been used for strategies requiring preset parameters (see Sec. IV- B): p 0 =0.2 for RP; n max =6for SP; p 0 =.5, =0.95, p Th =0.01 and N F =10for SA. A detailed list of the results is displayed for the NSFNET only by the two tables I and II. Results are given in terms of values of the cost function, which is the total number of fibers F deployed in the dimensioned networks, and they have been grouped by survivability feature (unprotected connections or DPP required for each connection). Inside each table, the WP and VWP cases are displayed, providing a row for each value of the number of wavelengths per fiber N λ. A bold number in a column indicates that the corresponding strategy reached the best result among all the considered heuristic optimization strategies. A dash tells that the corresponding heuristic strategy could not be applied due to memory (900 Mbyte) exhaustion. ILP optimization has been carried out by exploiting the commercial software CPLEX, implementing the branchand-bound algorithm. When a single value in normal type appears in the ILP column, the optimization ended, finding the optimum. In all the other cases, optimization was interrupted for memory exhaustion or after 3 days. In some of these cases, a single number in italic indicates that the only solution obtained was achieved relaxing all the integrity constraints on the link capacity. In some other cases, two values are reported: the one in italic is the lower bound estimated by CPLEX, while the one in normal type is the best non-relaxed solution found, not guaranteed to be the actual optimum. A dash with no numbers tells that CPLEX was not even able to setup the optimization session. Fig. 4(a) gives a synthetic overview of the strategy comparison. Each column, referring to a particular heuristic strategy, is subdivided in four rectangles. The hight of a rectangle referring to NSFNET is obtained by counting in table I the number of occurrences of a bold entry in the column of the strategy (similarly for rectangles referring to EON). The total hight of the bar is the total number of times the strategy was able to find the best heuristic F in the experiments carried out on both the networks. As expected, SA is on average the most reliable strategy. It is remarkable, however, that it has been overtaken by a deterministic strategy in some 148

9 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 as good as the best deterministic competitors: the comparison between these two proves that a deterministic random-choice polarization (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 of the cycle too quickly saturate the unused network capacity, preventing further rerouting. It should be noted that BI fiber selection is worse than random selection (RO). The improvement due to repetition for the stochastic strategies clearly appears by comparing the rectangles corresponding to the NSFNET and the EON. The lack of repetition makes stochastic strategies less performing than deterministic (e.g. SA vs. SP in the DPP-EON case). It can happen however that non-repeated SA is nevertheless the best strategy (e.g. in the unprotected-eon VWP case). Performance relations remain similar to those shown in the figure, if we count the number of times each strategy achieved the worst (instead of the best) result, with the only exception that IB comes out to be slightly worse than SP. Another synthetic view of the gathered data is given by Fig. 4(b), in which we show the differences between the heuristic approaches and the ILP method, in terms of F and for different values of N λ. We have plotted the DPP-NSF VWP case, since a complete set of ILP results was available. min ( max ) is the difference (in absolute terms) between the best (worst) heuristic F and F found by ILP. When two values appear in the ILP column, the non-relaxed result is employed. The differences are also plot in percent, using the ILP value of F for normalization. The best heuristic suboptima are never more than 11 fibers greater than the ILP result; the distance between the best and worst heuristic is also no greater than 3 fibers. The decreasing behavior of min and max seems not to be very meaningful, since it is not confirmed by the other experiments concerning EON and/or different protection and conversion conditions. The increasing trend of % min and % max is due to the descent of the optimum F, which is roughly inversely proportional to N λ (constant offered traffic). Averaging on all the experiments, including EON (considering only cases in which ILP returns an integer solution), we have: min = 8.2 and max =11. Moreover, if the two best heuristic strategies (SA and IB) are used, the maximum difference from ILP is 16%, while the mean difference is 5%: these data prove the validity of heuristic optimization as a practical design procedure for OTN. It should further noted that in some unprotected cases (WP NSFNET with N λ =32,WP EON with N λ =8, VWP EON with N λ =64), ILP was able only to find a worse integer solution than heuristics (these cases have been obviously excluded from the ILPheuristic comparison). Best performance count Absolute fiber difference Global comparison unp-nsf unp-eon DPP-NSF DPP-EON IB BI RO RP SP SA Heuristic strategy min max (a) DPP-NSF, VWP Number of wavelengths, N λ (b) % min % max 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 presented above can be completed by considering their computing times and their convergence behaviors. Fig. 5 displays the execution times of the second planning phase (heuristic optimization) for NSFNET, VWP, in the unprotected and DPP cases (all the strategies are run on a 1-GHz-clock computer). For all the strategies, there is a strong dependence of the execution time on N λ. By curve-fitting, we have verified that the execution time is O[N 2 λ ] for unprotected and O[N 3 λ ] for DPP. The difference is probably due to the complexity of the routing algorithms in the two cases when implemented on the MLG. Execution time in the VWP case resulted to be higher than in the WP case for all the strategies and for any value of N λ, probably due to the increased complexity of the MLG in the VWP case, in which the vertical arcs representing wavelength conversions must be added to the graph. A quantitative evaluation of complexity is however still under study and will probably give an explanation to these differences. A vertical comparison between the curves of the graphs points out the differences in computation time between the various strategies. Strategies having o f k in Percent fiber difference 149

10 500 unp-nsf, VWP 263 DPP-NSF, WP, N = 8 Execution time [min] IB BI RO RP SP SA Current number of fibers, F* IB BI RO RP Number of wavelengths, N 1000 (a) DPP-NSF, VWP Elapsed processing portion 268 (a) DPP-NSF, WP, N = 8 Execution time [min] IB BI RO RP SP SA Current number of fibers, F* SA Number of wavelengths, N (b) Fig. 5. Comparison between execution times in NSFNET VWP unprotected (a) and with DPP (b) Elapsed processing portion (b) Fig. 6. Convergence behavior of (a) IB, RO and RP and (b) SA in DPP-NSF (VWP case, N λ =8). the X condition (IB, SP and RP) have a higher execution time than the others (except SA), but achieve better results, showing the trade-off between accuracy and computational complexity. SA displays times similar to those of IB and RP: this is a consequence of the choice of the values given to the SA parameters (p 0,, p Th and N F ), taken precisely with the criterion of having execution times of the same order of magnitude for all the strategies. Finally, we shall point out that SP computation time is roughly 6 times that of IB, according to the assignment: n =6. Thus, SP and IB accuracy is similar, but the former requires much higher time than the latter. In Fig. 6 we show how the strategies convergence to their final solutions during their execution-time, considering as example the optimization of the NSFNET in the DPP and VWP case, with N λ =8. The horizontal coordinate has been normalized to the total execution duration of each strategy. The fastest converging strategy, IB, does most of the work at the beginning, when reallocating low-loaded fibers is easier; at the opposite, we have BI technique, which is more successful in deallocating fibers after some iterations. RP follows the behavior of IB but more slowly, due to the randomization of the reallocation acceptance decisions. SA was plotted separately in Fig. 6(b): its irregular trace results from the ability of adding fibers. It should be noted that SA has hit the best already in the middle of the execution, but it jumps out because a still high temperature, to return to it at the end of the process. VI. CONCLUSION For OTN, the possibility of obtaining an exact solution to the NP-hard problem of RFWA of static traffic demands with capacity dimensioning is severely limited by computational complexity. As we have shown, ILP applied to realistic network examples and solved with standard computing equipment in many cases does not provide any results or is not able to guarantee optimality: problems are exacerbated by wavelength continuity (WP cases), by dedicated path protection and by high values of N λ. The heuristic approach proves to be an acceptably reliable alternative when ILP does not make it. We have proposed a heuristic method based on finding a first greedy but 150

11 feasible solution which is subsequently improved by lightpath rerouting. We have presented six heuristic strategies to perform this task, the most accurate ones of which are able to contain the distance to the ILP solution below the 16% of this solution itself (5% on average). Result-analysis shows that the best compromise between computational complexity and reliability is achieved with the deterministic IB and the stochastic SA strategies. IB is however simpler to implement, it does not require special procedural parameters to be preset and it usually converges rapidly to the final solution. Stochastic heuristics gain a clear advantage over deterministic ones only when there is the possibility, as in SA, 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 it in most of the cases only with execution times much greater than those of IB. 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