Wireless broadband networks are being increasingly

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1 A Multi-objective Optimization Model For Planning Robust and Least Interfered Wireless Mesh Networks Djohara Benyamina, Abdelhakim Hafid NRL, University of Montreal, Canada {benyamid, Michel Gendreau CIRRELT, University of Montreal, Canada Abstract A wise network planning becomes the most important phase in determining the network efficiency. In this paper we consider the wireless mesh network (WMN) planning problem where no much work has been done. We propose a new multi-objective optimization model for planning WMNs, where the two conflicting objectives, namely network deployment cost and network channels interferences, are simultaneously minimized while guaranteeing endusers full coverage and robust topologies. We also propose a novel performance metric to evaluate the network interference level and a population-based optimization heuristic to solve our model, whereby many WMN planning solutions are provided to the end-planner to choose among. We use realistic-size instances (up to 81) mesh nodes to test our multi-objective optimization model, and discuss the impact of the key parameters on the characteristics of the solutions. Key Words: Wireless Mesh Network, Planning problem, Multiobjective optimization, Meta-heuristic search algorithm, Robustness. I. INTRODUCTION Wireless broadband networks are being increasingly deployed in a multi-hop wireless mesh network (WMN) configuration. Wireless Mesh architecture is considered a first step towards providing high bandwidth network coverage. Its basic building block is the infrastructure which is composed of fixed nodes interconnected via wireless links to form a multihop ad-hoc network. Nodes in a WMN are essentially routers and gateways. They act as classical access point to mesh clients. They also interconnect with each other through pointto-point wireless links. Gateways do have extra functionalities which make them more expensive than routers. Simultaneous communication is allowed thanks to the use of multi radios (interfaces) over orthogonal channels. However, since the number of available orthogonal channels is limited, interferences happen thus degrading the network performance. Major research efforts have focused on developing planning network solutions for cellular networks and WLANs. However, these solutions strongly differ from those of planning WMN [1]. Planning a WMN basically involves choosing the installation locations and the type of network nodes and deciding on a judicious channel/node interface assignment, while guaranteeing users coverage, wireless connectivity and traffic flows at a minimum cost. Furthermore, it is desirable to plan fault-tolerant networks, i.e. multiple paths to the gateways. The two main criteria in WMNs planning are the deployment cost and the network performance in terms of throughput. The more nodes are deployed (the higher the cost is) the better the throughput is expected. Therefore, it is not possible to improve one objective without worsening the other one. It is then legitimate to ascertain that modeling such problem using single objective optimization models doesn t accurately capture the true nature of the problem and therefore, optimal solutions obtained from solving it are very limited. Up to date, there has been no attempt to model WMN planning problems using multi-objective approaches. Most of related work on the problem of performance improvement assumes a priori fixed topologies [2, 3, 4, 5]. Some of the main drawbacks can be found in [6]. Other studies (e.g., [7 and 8]), consider topologies where gateways are fixed a priori, while the studies in [9, 10] attempt to optimize the number of gateways given a fixed layout of mesh routers. Just recently, the authors in [1, 11, 12] propose WMN planning schemes where the locations of routers and gateways are not fixed. In [1, 11], the problem is formulated as singleobjective ILP and solved for only small size networks because of the exponential number of constraints and variables. Authors in [12], however, were able to solve it for real-size networks by using heuristics. Nonetheless, it must be noted that the proposals in [1, 11, 12] do consider the interference aware model in their formulations, which is seen as a restriction that is difficult to overcome in real-world networks. Network reliability is an important feature to consider when planning WMNs even if they have a relatively stable topology. Planning fault-tolerant networks is not addressed by authors in [1, 11, 12]. Our contribution differs from the above in that we plan the WMN from scratch to meet the QoS requirements (by favoring cost-effective, robust, and lower interference level topologies) while satisfying the planner s objectives. A multiobjective approach is needed to provide different solutions and leave the network planner make the final decision. A somewhat more expensive solution that provides a better throughput might be preferable than a cheaper solution with low throughput, or vice versa. We propose a new modeling approach to address WMN planning problems, where the two conflicting objectives, deployment cost and channels interferences are simultaneously optimized (minimized). In order to define the channels interferences objective, we devise a new network performance metric that computes the network interference level. We also consider robustness in our incremental planning design. Most of current contributions use exact methods to solve such problems (e.g., CPLEX). This is acceptable as long as the

2 2 size of the problems is small. However, when confronted with real-world large-sized problems, they prove fruitless as the search space is huge and the problem is NP hard. This necessitates the use of a (meta-) heuristic optimization algorithm in order to better explore the search space. Such heuristics necessarily return a set of near-optimum solutions that are non-dominated with each other, i.e., none is better than the rest with respect to all objectives. The paper is organized as follows: Section II is the core of this work. It describes the mathematical formulation of the problem solution. We also propose a new network performance metric and a new approach thatt ensures network robustness. In Section III, we propose a population-based multi-objective optimization algorithm to solve the mathematical model presented in section II. Section VI summarizes our experimental results. Finally, we conclude the paper in Section V. II. PROBLEM DEFINITION AND FORMULATION Let I be the set of positions of traffic concentrations in the service area (Traffic Spots: TSs) and L the set of positions where mesh nodes can be installed (Candidate Locations, CLs) The planning problem aims at: Selecting a subset S L of CLs where a mesh node should be installed so that the signal level is high enough to cover the considered TSs. Defining the gateway set by selecting a subset G L of CLs where the wireless connectivity is assured so that all traffic generated by TSs can find its way to reach a node in G. Maintaining the cardinalities of G and S small enough to satisfy the financial and performance requirements of the network planner. In order to describe the problem formally we introduce the following notation - partially similar to that in [1]. Given n TSs and m CLs, let I={1,..,n} and L={1,..,m}. In the following, unless otherwise stated, i and j belong to I and L respectively. The cost associated to installing a mesh node j is denoted by c j, while p j is the additional cost required to install a gateway at location j. d i is the traffic generated by TS i. U jl is the traffic capacity of the wireless link between CLs j and l. v j is the capacity of the radio access interface of CL j. The coverage and connectivity parameters are given respectively by the binary variables a ij and b ij. a ij takes the value 1 whenever TS i is covered by a mesh node in CL j. b jl indicates whether CLs j and l can be connected via a wireless link. We define other decision variables (see Figure 1) in our formulation including: x ij takes the value 1 if TS i is assigned to CL j and g j is equal to 1 if a gateway is installed in CL j. We suppose initially that the mesh nodes operate using the same number of radios R, each with k channels, (k>r) and k C, where C ={1,..,c} and c can be at most 12 orthogonal channels if IEE802.11a is used. Extra installation variables are added: =1 if a mesh node is installed in CL j and is assigned channel q, q k, =1 if a wireless link from CL j to CL l using channel q, q k exists. Finally, we define the flow variables and F j. the first variable denotes the traffic flow routed from CL j to CL l using channel q, the second is the traffic flow on the wired link between a gateway j and the backbone network. Figure. 1. WMN planning problem A. Our Initial Model Our first interference-aware formulated as follows: Subject to: x ij a ij * t j i I, j L (4) (5),,, (6),,, (7),, (10), (12) (13),,,,,,, (14), F j R + j,l L (15) optimization model is The objective function (1) computes the total cost of the network including installation cost c j and additional gateway installation cost p j. Constraint (3) is used to make sure that a given TS i is assigned to only one CL j. Inequality (4) implies that a TS i is assigned to an installed and covering mesh node j. Constraint (5) defines the flow balance for each mesh node j. Constraint (6) limits link interferences; we assume that the IEEE MAC protocol is used. The set of links that cannot be active simultaneously with the link y jl named N jl are all links within one or two hops using the same channel q. Constraint (7) stipulates that a flow on a link cannot exceed the traffic capacity of that link. Constraint (8) denotes that the aggregated demand received by mesh node j does not exceed the capacity of the radio access interface. Constraint (9) implies that the flow routed to the wired backbone is different from zero only when the mesh node installed is a gateway. M is a very large number to limit the capacity of the installed gateway. Constraint (10) forces a link between CL j and CL l using the same channel q to exist only when the two devices are installed, wirelessly connected and tuned to the same channel q. Constraint (11) ensuress that a device can be a (1) (3) (8) (9) (11)

3 3 gateway only if it is installed. Constraint (12) prevents a mesh node from selecting the same channel more than once to assign it to its interfaces. Constraint (13) states that the number of links emanating from a node is limited by the number of its radio interfaces. It also states that if a channel is assigned only once to a mesh node, it is a sufficient condition for its existence. B. Our Muti-objective Model Instead of just limiting interferences, as defined in constraint (6), it would be more effective to have it as an objective to be optimized altogether with the deployment cost. For this purpose, we propose a novel performance metric defined below. 1) Interference Level Metric: A Balanced Channel Repartition (BCR) is defined as follows: ϕ q = max O q - O c q,c C, q c, where O k =,, Our single-objective optimization model is then transformed into a bi-objective model where the network interference level, the second objective, is then defined as follows: Min 2) Robustness Consideration: Since theree must be at least r node disjoint paths from each node in S to some gateway in G, the number of the flows originating from node j S must be at least 2, as shown in Equation (16). The idea we use to ensure node disjoint paths is that for a node j, there can be at most one incoming flow originating from node l S (Equation 17). It is not hard to see that this condition is sufficient and necessary to guarantee vertex disjoint paths. j L (16) j L-G (17) Note that (17) automatically prevents the flows from forming cycles at the intermediate nodes on the way to the gateway(s). We consider one node failure scenario where shared path/segment protection scheme is more appropriate [14]. In such protection policy, protection resources may be shared by many backup paths/segments (i.e., a backup path is used when the primary path fails). However, they can be defined only when the failure occurs to know which path or segment to activate. In this case, we do not need to allocate channels to each backup path/segment thus saving more resources and reducing channel interferences. This implies that constraints (16) and (17) need to be rewritten by replacing with v jl to consider a wireless link without assigning a channel. The connectivity is guaranteed by constraint (5). v jl is defined by Inequality (18). v jl b jl ( t j +t l ) j,l L When a single node failure occurs, an alternative backup segment is activated. The segment is constructed by selecting a minimum number of nodes in the neighborhood of the failed node so that the flow can find its way to the gateway. The channel with the minimum value of ϕ is assigned to the first link while the others (if they exist) restoree the failed node channels. (2) (18) Figure 2: Backup segment activation following a failure at node 6. As shown in Figure 2, failure at node 6 leads to the activation of backup segment {(3,2), (2,7)}. The link (2,7) is activated by assigning a channel to it. If node 1 fails, then the backup segment {(2,7), (7,9)} is activated. The two segments are sharing the same resources on link (2,7) which makes this type of protection adequate only for single node failure. We consider the constraint (5) a soft constraint while the remaining constraints are considered hard constraints. The WMN planning system attempts to optimize the two objectives and satisfy all hard and soft constraints as defined above. III. PLANNING PROBLEM RESOLUTION The rationale behind our planning is the minimization of network interference level in order to maximize network throughput, and the minimization of total deployment cost by selecting a minimum number of routers/gateways and choosing their positions so that robust network connectivity is ensured, while providing full coverage to all mesh clients. Multi-objective problems have a set of Pareto-optimal solutions. Each solution represents a different optimal trade- to be non-dominated off between the objectives and is said since it is not possible to improve one criterion without worsening another. We propose a multi-objective approach based on Particle Swarm optimization technique to solve our planning problem. A. PSO: Particle Swarm Optimization Kennedy and Eberhart proposed [13] an evolutionary population-based heuristic for optimization problems. It models the dynamic movement or behavior of the particles in a search space. By sharing information across the environment over generations, the search processs is accelerated and is more likely to visit potential optimal or near-optimal solutions. PSO has been extended to cope with multi-objective problems which mainly consist of determining a local best and global best position of a particle in order to obtain a front of optimal solutions. One of the well-known multi-objective techniques based on PSO algorithm is MOPSO [14]. It is able to generate almost the best set of non-dominated solutions close to the true Pareto front. The main algorithm is given below. Algorithm 1: MOPSO Main Input : Swarm at iteration t S t, MaxArchiveSize, MaxIteration Output: Repository REP Step 0 : Initialization of Swarm Initialize S at iteration t=0 for each i S 0 do for each dimension d do Initialize position i, save pbest i, initialize velocity Specify lowerbound i and upperbound i end for Step 1 : Evaluation of Particles in S

4 4 Step 2 : Update REP for each i S t do compvector(i,rep) search_insert(s,rep) Step 3 : Generate hypercubes (adapive grid) : make_hyper(maxcube) Step 4 : Update Swarm: for each i S t do for each dimension d Update_velocity i, Update_position i end for Step 5 : Boundary check Step 6 : Update pbest Step 7 : if t > MaxIteration then Stop t=t+1 and GO TO Step 1 The following are the phases involved in the resolution of the proposed ILP. In continuous optimization problems, getting the initial position and velocity is more straightforward because random initialization can be used. However, since the WMN planning problem is a constrained optimization problem, the initial positions must represent feasible solutions. Thus, they need to be designed carefully. A position in the search space represents a set of assignments that is a solution to the problem. In our case, each position provides information about user connectivity (x ij ), device installation, device connectivity, gateway existence (g j ), link flows, and gateway/backbone link flows (Fj). For each particle in the swarm, we assign a Boolean value to the variables x ij,,, and g j and a real value to the flow variables. We consider a feasible solution, a solution that satisfies all hard and soft constraints. During the search, only non-feasible solutions that violate some soft constraints can be included in the population. This increases the likelihood of a non-feasible solution to mutate and provide a feasible one in later generations. B. Algorithm Description The most challenging part in our solution process is finding an initial set of feasible solutions that satisfy a considerable number of constraints (3 to 17). Building such initial solutions requires four main design stages, namely coverage insurance, connectivity augmentation, two-disjoint-path connectivity and gateway assignment. First of all, we assume that our WMN is represented by a grid-like graph, where each mesh node, if installed, establishes a wireless communication with its eight cell corners neighbors. This assumption increases the chances of selecting a candidate neighbor among the eight with which a wireless link will be set up in the channel assignment procedure. Coverage ensurance is handled by assigning each TS i to a CL j. We start by selecting randomly a mesh node from the set of CLs that cover that TS i. A mesh node is installed at this location only if it has not yet been selected (as shown in Figure 3.a). By applying the same procedure to all TSs, we are sure to obtain a small set, S 1, of mesh nodes that provide full coverage to all TSs. At this stage, obviously, constraints (3), (4) and (7) are satisfied. This initial set S 1 ={ j L, CL j covers TS i, i I } contains vertices of a disconnectedd graph as shown in Figure 3.b. The following stage, connectivity augmentation, consists of augmenting the set S 1 by adding new mesh nodes. We augment S 1 so that the graph becomes connected and near- robust. The algorithm consists of increasing the number of neighbors for all mesh nodes. To every mesh node being either internal, edge or corner node a different augmentation procedure is applied. We force each node, regarding its location, to reach its minimum node-degree d. (a) TSs locations (b) TSs are assigned to LCs Figure 3: A Particle position example Figure 4: Initial set S1 in step1, S1 is augmented in steps 2 and 3 and gateways are selected We assign the value of d equal to 2, 3 and 4 to corner, edge and internal nodes respectively. The augmentation process stops when all nodes are visited and each one has reached at least its minimum degree. The final graph is connected and almost bi-connected. A cut-node is a node that its removal disconnects the graph. The following stage (3 rd stage) consists of eliminating every cut-node, if it exists, by adding a neighbor node at its right or bottom side; Figure 4 illustrates the case. In the gateway assignment last design stage, gateway determination is based on random node selection from among the final augmented set S 1 while satisfying constraint (11). A WMN is represented by an undirected graph G(V,E), called a connectivity graph. Each node v represents either a router (which can be an access point (AP) or a relay added when augmenting S 1 ), or a gateway. The neighborhood of v, denoted by N(v), is the set of nodes residing in its transmission range. A bidirectional wireless link exists between v and every neighbor u in N(v) and is represented by an edge (u,v). The maximum degree of G denoted by is bounded by the number of radio interfaces as given by constraint (13). For computational purposes, we use an adjacency matrix to represent the connectivity graph, which is symmetric. We assign channels randomly but taking constraint (12) as a restriction to avoid the case where two radio interfaces of the same node use the same channel and we implement Edmonds-Karp s max flow algorithm [15] to assign a value on each link y jl using channel q to route a flow. All remaining constraints are satisfied. Our iterative algorithm (Algorithm 2) consists, for each particle in the swarm, of placing a subset S 1 of APs to cover all TSs, augmenting S 1 to construct the 2-connectivity matrix, and assigning channels and flows. The resulting feasible solution (topology) satisfies both soft and hard constraints. The initial subset S 1 (before being augmented) is mutated from generation to generation to provide different feasible solutions.

5 5 Algorithm2: Planning Resolution Input : MaxGeneration,pMut, Output : Repository REP Construct_Initial_Soft&Hard_feasible_solutions () t=0 while (t<maxgeneration) for each particle in the swarm S1 Mutate(S1,pMut) S Augment(S1) Y1 Construct_near-biconnectivity_matrix() Y1 Eliminate-cut-point(Y1) Y Assign_channels(Y1) G Select_getways() Compute-flows() Construct_New_Particle() endfor Compute_Velocities Evaluate_Particles REP Insert_feasibleNonDominated_Solutions() Update_ParticuleBest t++ endwhile IV. EXPERIMENTATIONS AND RESULTS ANALYSIS We study the performance of our algorithm over grid graphs as stated in section IV.B and under many deployment scenarios. Unless stated otherwise, our standard settings are: n=150, m=49, a uniform traffic demand of the Mesh Clients (MCs) d i =2Mb/s, u jl =54Mb/s for all j,l L, vj=54mb/s for all j L, and M=128 Mb/s. The positions of the n TSs are randomly generated. The installation cost of a mesh node is c j =200 for all j L and additional installation cost of a gateway is P j =8*c j, j G. The algorithm is coded in the Java programming language and all the experiments were carried out on a Pentium M 1.5GHz machine. We use three radios interfaces (R=3) with k=11 channels each. Initially, we fix the runs to 100 generations with a population size and archive size of 50 and 20 particles respectively. For each of the following deployment scenarios, we report results on 10 runs. A. Effect of mutation factor variation (fmut) We performed three different experiments by varying the value of the mutation factor (fmut=0.25, 0.50, 0.75). For the sake of comparison, we display the three Pareto fronts of the experiments on the same graph in Figure 5. TABLE I. NON-DOMINATED SOLUTIONS WHEN m VARIES. m NR NG NL Cost Interf NR-NUMBER OF ROUTERS,NG-NUMBER OF GATEWAYS, NL-NUMBER OF LINKS.COST-TOTAL DEPLOYMENT COST, INTERF.-NETWORK INTERFERENCE LEVEL Figure 5: Impact of mutation factor Figure 6: The Pareto fronts for 3values of m Mutating at a rate of 50% of the population leads to the best Pareto front of optimal solutions when compared to optimal solutions under fmut=0.25 and fmut=0.75. Therefore, we take fmut=0.5 as our standard setting for the remaining experiments. B. Effect of number of candidate locations m variation In this study, m is gradually increased while all other parameters are maintained fixed. The number of mesh clients, n, is set to 200. Table I reports the characteristics of the nondominated solutions we obtained. As we are dealing with multi-objective optimization, only the cheapest solutions are compared. A higher number of candidate locations leads to an increase in the number of routers and gateways even for the same number of users. The first reason is the fact of increasing the number of CLs increases the probability of a MC not being connected to an AP through a multi hop wireless path, which leads to installing more nodes. The second reason, is to satisfy robustness requirements. The resultss reported in Table 1 show that the planner has to pay more for robust network with higher throughput. Notice that the difference of cost is related to adding routers rather than gateways in some instances. Figure 6 shows that the best front is obtained when m is 49. C. Effect of changing the number of radio interfaces R. We also study how our algorithm would behave when R varies from 2 to 5, each with 11 channels. Notice that an increase in the number of radio interfaces R leads to an increase of network links. Hence, a smaller number of network devices are required as shown in Figure 7.a. We change the second objectivee function to ignore BCR effect. We consider the maximization of throughput as our second objective together with interference aware model represented by constraint (6). The results are depicted in Figure 7.b. (a) Figure 7: Impact of the number of radio interfaces (a): with BCR, (b): without BCR, (a) (b) (b) Figure 8: Impact of changing traffic demands: (a)) Non robust topologies. (b) Robust

6 6 F. Effect of changing additional cost of gateways As we can see from Figure 12, the number of gateways decreases sharply; this is expected. On the other hand, the number of routers increases. The reason behind this is to satisfy connectivity and robustness constraints. Figure 9: Impact of varying traffic demand Figure 10: Impact of varying R Figure 11:Impact of varying n Figure 12: Impact of changing the cost As can be seen in Figure 7 (a and b), the number of routers together with gateways decreases the more we add radio interfaces. However, when BCR is not considered (Figure 7.b), the number of these mesh nodes sharply increases when we pass from 4 radio interfaces to 5. This can be caused by the high level of interferences which leads to look for alternative paths to route the traffic forcing more gateways/routers to be installed. Thus, our planning approach (Figure 7.a) tends to be more efficient. When more radio interfaces are added, the number of links and mesh nodes decreases accordingly leading to less interference (higher throughput) and cost-effective topologies respectively. Regarding the Pareto front, notice that increasing the number of radios increases the number of non-dominated solutions offered to the network planner and provide the best Pareto front (when R=5) as shown in Figure 10. D. Impact of demand variation. When the traffic demand d i is more than 3Mb/s there can hardly exist a feasible solution. On the other hand, when we do not consider robustness constraints, we obtain a set of optimal solutions even for d i =4, 5 Mb/s and more (Figure 8.a). We envision that planning a robust network under the standard setting cannot be guaranteed to support a large traffic demand if the value of m, R, u jl and v jl are not changed accordingly. This means that if a solution is found, given a traffic demand d i, it ensures that the topology is robust. The results reported when d i =2, 3 Mb/s (Figure 8.b) show that when demand increases the number of gateways increases accordingly to satisfy connectivity constraintss by creating new routing paths. More relays than APs are added in order to connect these APs to newly added gateways. Notice also that when the demand increases, the number of possible non- for decision dominated solutions offered to network planner making decreases (see Figure 9). E. Effect of changing the number of mesh clients n We further study the case when n varies. Naturally, when n increases (i.e., more users need to be covered and connected) then more routers need to be connected to the backbone. This is achieved by increasing the number of gateways as shown in Figure 11. VI CONCLUSION In this work, we have considered the WMN planning problem and have shown that the optimization of such problem is naturally multi-objective. A spectrum of alternative trade-offs solutions is provided to the engineer allowing a flexible decision making. Our multi-objective WMN planning model is unique and quite different from other WMN planning models found in the literature. The primary difference being our consideration of mesh clients coverage in the minimization of total cost deployment simultaneously with the minimization of total interference level of the network. A new metric BCR to measure network interference level is proposed and near-optimal population-based search algorithm is used to solve this problem. The solutions generated are guaranteed to be fault tolerant in the presence of single node failure where shared protection schemes are applied to save scare resources. Following the strand of this planning framework, we will next investigate the issues of rate adaptation and scalability. REFERENCES [1] E. Amaldi, A. Capone, M. Cesana, F. Malucelli. 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