Cost-Sensitive and Load-Balancing Gateway Placement in Wireless Mesh Networks with QoS Constraints

Transcription

1 Zeng F, Chen ZG. Cost-sensitive and load-balancing gateway placement in wireless Mesh networks with QoS constraints. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 24(4): July 2009 Cost-Sensitive and Load-Balancing Gateway Placement in Wireless Mesh Networks with QoS Constraints Feng Zeng ( ), Student Member, CCF, and Zhi-Gang Chen ( ), Member, CCF School of Information Science and Engineering, Central South University, Changsha , China {fengzeng, czg}@mail.csu.edu.cn Received November 25, 2008; revised March 11, Abstract In wireless mesh networks (WMNs), gateway placement is the key to network performance, QoS and construction cost. This paper focuses on the optimization of the cost and load balance in the gateway placement strategy, ensuring the QoS requirements. Firstly, we define a metric for load balance on the gateways, and address the minimum cost and load balancing gateway placement problem. Secondly, we propose two algorithms for gateway placement. One is a heuristic algorithm, which is sensitive to the cost, selects the gateway candidates according to the capacity/cost ratio of the nodes, and optimizes the load balance on the gateways through scanning and shifting methods. The other is a genetic algorithm, which can find the global optimal solution. The two algorithms differ in their computing complexity and the quality of the generated solutions, and thus provide a trade-off for WMN design. At last, simulation is done, and experimental results show that the two algorithms outperform the others. Compared with OPEN/CLOSE, the average cost of gateway placement generated by our algorithms is decreased by 8% 32%, and the load variance on the gateways decreased by 77% 86%. For the genetic algorithm, the performance improvement is at the price of the increase of the CPU execution time. Keywords wireless mesh network, gateway placement, load balance, QoS constraint, genetic algorithm 1 Introduction A key technology, wireless mesh networks (WMNs), has emerged recently for future broadband wireless access [1]. In a WMN, a mobile client can access the Internet by multi-hop communication through a wireless backbone. In the architecture of WMNs shown in Fig.1, the middle layer is the wireless backbone, which consists of gateways and mesh routers. The gateways act as the communication bridges between the wireless backbone and the Internet. The wireless backbone has much impact on the quality of service (QoS) to mesh clients. In the wireless backbone, as most traffic is directed to/from the Internet, the gateways always become the performance bottlenecks [2]. Due to the existing unfairness in the wireless multi-hop communication, the mesh routers nearby gateways can get the better QoS [3]. But, to the mesh routers far from gateways, the required QoS cannot be guaranteed. Consequently, when designing a WMN, the location of gateways and the QoS guarantee to all of clients should be taken into account. This paper focuses on the design of wireless backbone, and aims to get the minimum-cost and load-balancing gateway placement solution, meanwhile ensuring QoS requirements of all clients. In WMNs, placing the gateways in a proper way is important in terms of the optimal throughput, load balancing on the gateways and satisfying QoS requirements of each terminal user. The more gateways placed, the better performance will be gained, but, the higher the cost. Fig.1. Architecture of WMNs. In the past several years, a lot of work has been done on the problem of gateway placement in WMNs. In [4], Wong et al. identified two main technical problems: Regular Paper Supported by the National Natural Science Foundation of China under Grant Nos and

2 776 J. Comput. Sci. & Technol., July 2009, Vol.24, No.4 gateway placement for minimizing communication delay and for minimum communication cost. For both the objectives, they developed a series of techniques and algorithms, using the same strategy: at each step they decide which of the candidate gateways will be eliminated from further consideration. Similar to [4], the work in [5] presented an algorithm to find the least latency gateways placement in wireless sensor networks. Many researchers try to find the placement strategy with a minimum number of gateways. In [6], Chandra et al. developed the algorithms to efficiently place gateways in the network, aiming to minimize the number of required gateways while guaranteeing users bandwidth requirements. In [7], Bejerano et al. presented a polynomial time approximation algorithm for partitioning the network nodes into a minimal number of disjoint clusters that satisfy multiple constraints and used a spanning tree rooted at each cluster head (i.e., gateway) for message delivery. In [8], Aoun et al. proposed a polynomial time near-optimal algorithm which recursively computes minimum weighted dominating sets, while consistently preserving QoS requirements across iterations. With the similar object as in [6 8], some various algorithms were proposed in [9, 10]. There are still some ideas for the cost-efficient placement of gateways. In [11], He et al. formulated the gateway placement problem as a linear program issue, and proposed two heuristic algorithms for the purpose of cost-efficient gateway placement. In [12], Prasad et al. proposed two approaches to minimizing the cost of gateway placement in WMNs. One is based on integer linear programming, and the other approach is an OPEN/CLOSE heuristic, which uses open and close methods to recursively update the solution to minimize the cost of gateway placement. But, OPEN/CLOSE fails to achieve the load balance among clusters. Most of the previous work puts emphasis on reducing the number of gateways in a WMN, while ensuring QoS requirements. But, the minimum number of gateways does not mean the minimum cost of gateway placement. In reality, the locations near the wired backbone can be easily accessed to Internet, and the cost to place gateways there will be lower than that to place them in other locations. Moreover, the gateways with various processing capability would charge for various prices. Consequently, the strategy of gateway placement should be sensitive to the installation cost. Load balance on the gateways is important to the network performance and the fair service among clients. In a WMN, the QoS on the clients around the overloading gateways could not be guaranteed. But, as to the lightly loaded gateways, they cannot make full use of their capacity to provide service for more clients. Hence, when designing the network, we should take into account the load balance on the gateways. Unfortunately, in the previous work, little analysis has been done to the load balance on the gateways. Different from previous work, this paper aims at the gateway placement solution with minimum cost and load balance on the gateways. First of all, we present the network model and problem description. Then, we introduce a heuristic algorithm for gateway placement in WMNs, which is sensitive to the cost and can achieve load balance on the gateways. For getting optimal solution, a genetic algorithm is proposed. At last, extensive simulations are done to compare our algorithms with other alternatives. The rest of this paper is structured as follows. In Section 2, we describe the network model and the problem. In Section 3, we propose a cost-sensitive algorithm for load-balancing placement of gateways in WMNs. In Section 4, a genetic algorithm is designed to find the optimal solution. Simulation results are presented in Section 5. Section 6 concludes this paper. 2 System Description In this section, we discuss the system model, the QoS constraints for gateway placement in WMNs, and the problem description. 2.1 System Model The wireless backbone in a WMN can be modeled as an undirected graph G(V, E), where V is the set of nodes, N = V is the total number of nodes. The nodes in V represent the placement locations, where a mesh router or a gateway can be installed. E is the set of edges, and two nodes are connected by an edge if and only if they are within transmission range of each other. In a WMN, due to the geography position and network environment, each node has various cost of gateway placement. If v V can be placed with a gateway, Cost(v) is the cost of gateway placement. Otherwise, the cost of gateway placement can be set to, the maximum integer in computer. The traffic on the mesh nodes are from two types of sources. One is the local traffic aggregated from its clients. The other is the relay traffic from other mesh routers. As to the node v V, the local traffic is denoted by L l (v), and the relay traffic is denoted by L r (v). For optimal performance, clustering ideology in [12] is adopted in this paper. A WMN would be logically divided into a set of disjoint clusters, covering all the nodes in the network. In each cluster, a gateway will be

3 Feng Zeng et al.: Cost-Sensitive and Load-Balancing Gateway Placement 777 placed to provide Internet access service for all members in the cluster. Consequently, the number of gateways is the same as the number of clusters, and all clusters make up a partition of the network. Definition 1 (Cluster). In a WMN modeled as G(V, E), a cluster is denoted as C(V, E ), which is a subgraph of G. The subgraph C(V, E ) should satisfy the following conditions: 1) there exists only one gateway node in C, which provides the Internet access service for all nodes inside cluster C; 2) the end-to-end service between gateway and every node in C should satisfy the QoS requirements, which are presented in the next subsection; 3) the traffic on all the nodes in cluster C should direct to the only gateway for the Internet access. The load on the gateway is equal to the total traffic inside cluster C. Definition 2 (Partition). In a WMN G(V, E), it is assumed that there are k clusters, denoted by C 1 (V 1, E 1 ), C 2 (V 2, E 2 ),..., C k (V k, E k ). We call these clusters a partition of the WMN, if they satisfy the following conditions: 1) V 1 V 2 V k = V ; 2) E 1 E 2 E k E; 3) V i V j =, E i E j =, 1 i, j k, i j. In the partition, each cluster has a gateway node. As to the cluster C i (1 i k), g i is assumed to be its gateway node. Then, GW = {g 1, g 2,..., g k } is a set of gateways in the WMN. As to the network design in a WMN, we first select some nodes as gateway nodes. Then, taking these gateways as cluster heads, clustering with QoS constraints is done. The selected nodes should be enough, so that the created clusters can make up a partition of the WMN. 2.2 Clustering with QoS Constraints When creating a cluster, we should take into account the end-to-end QoS requirements between the gateway and other members in the same cluster. In this paper, we use the routing tree rooted at the gateways to discuss the QoS requirements in the wireless backbone. A tree-based routing scheme would easily allow flows aggregation and would minimize overhead, ensuring an optimal utilization of bandwidth [7]. In order to ensure the QoS requirements, the routing trees rooted at the gateways should satisfy the following conditions: 1) the sum of local traffic on the nodes in the tree should be bounded to the maximum traffic that the gateway (root) can carry; 2) the distance (hop counts) between any node in the tree and the root should be bounded to a value of R; 3) the relay traffic on each node in the tree should be bounded to a value of L; 4) the degree of each node in the tree should be bounded to a value of D; 5) the total number of nodes in the tree should be bounded to a value of S. The above conditions would well ensure that the gateway provides QoS guaranteed service to the mesh nodes in the routing tree. All nodes in a routing tree constitute a cluster, and the gateway acts as cluster head, serving for other nodes inside the cluster. Condition 1) ensures that the gateway is capable of providing QoS guaranteed service for all mesh nodes in its own cluster. Condition 2) implies that the distance between any two nodes in a cluster is bounded, which is important to low delay and jitter in wireless communication. Condition 3) prevents a path in a tree from being overloaded. Condition 4) restricts the node density in the routing tree, which is helpful to reduce the contention for wireless channel. Condition 5) also contributes to the improvement of network performance. In our work, the routing trees are generated by means of breadthfirst search algorithm in the graph, which should begin with the selected gateway nodes and comply with the above conditions. Since the routing tree is not the emphasis of our work, the detail of generating the routing tree is not discussed further in this paper. 2.3 Problem Description In a WMN, the more gateways placed, the better performance will be gained, but, the higher price will be charged. From the economic point of view, we should reduce the cost of gateway placement, while ensuring the QoS requirements of terminal users. With a WMN G(V, E), the cost of gateway placement can be defined as (1). In the equation, GW is the set of gateways. Cost(G) = g i GW Cost(g i ). (1) In a cluster, the gateway provides the Internet access service for its clustermates. Therefore, the total load on the gateway g i (g i C i ) is equal to the total traffic in cluster C i, as shown in (2). Load(g i ) = L l (g i ) + L r (g i ) = v C i L l (v). (2) If the gateway is heavily loaded, its service to all members in the cluster will not be sure to satisfy the QoS requirements. On the other hand, if the gateway is lightly loaded, its capacity will not be able to fully provide service for more clients. Hence, on the whole, we should balance the load on the gateways. As to

4 778 J. Comput. Sci. & Technol., July 2009, Vol.24, No.4 a partition consisting of k clusters, the load variance on the gateways can be defined as (3), and VAR is the metric of load balance in a gateway placement strategy. VAR(g 1, g 2,..., g k ) = L = 1 k 1 k 1 1 i k 1 i k (Load(g i ) L) 2. (3) Load(g i ). (4) To formulate the gateway placement problem, we define a binary variable x ij, if the gateway g i and the mesh node v j belong to the same cluster, x ij takes the value of 1, otherwise x ij is 0. As to a WMN G(V, E), we can formulate the gateway placement problem to linear programming problem as follows. min Cost(G), VAR(g 1, g 2,..., g k ) s.t. 1) C i C j =, 1 i, j k, i j. 2) C i GW = {g i }, 1 i k. 3) v j V, x ij = 1. 4) v j V, v i GW v i GW (d ij x ij ) R. 5) v j V, L r (v j ) L. 6) v j V, δ(v j ) D. 7) v i GW, v j V x ij S. 8) v i GW, Load(v i ) = v j V (x ij L l (v j )). We have two goals, one is to minimize the cost, and the other is to minimize the load variance on the gateways. Conditions 1) and 2) denote the network is divided into k disjointed clusters, and each cluster has only one gateway connected to the wired Internet. Condition 3) denotes that each node has only one assigned gateway. Inequalities 4) 7) show the QoS requirements discussed above. In the equations, d ij is the shortest distance between v i and v j, and δ(v j ) is the degree of v j. 8) explains how to calculate the load on a gateway. As described above, the gateway placement problem is converted to a linear programming (LP) problem, which is NP hard [13]. In this paper, we propose a heuristic algorithm and a genetic algorithm to get the optimal solution, respectively. 3 Cost-Sensitive and Load-Balancing Gateway Placement Algorithm In order to efficiently find the solution for gateway placement in a WMN, we propose a cost-sensitive and load-balancing algorithm (CSLBA), which is sensitive to the cost of gateway placement, and try to find the minimum cost and load balancing gateway placement strategy. The algorithm has three key steps to get the optimal gateway placement strategy, including gateway selection, cluster building and partition optimizing. 3.1 Gateway Selection Definition 3 (R-matrix: R Hop Adjacent Matrix). To the graph G(V, E), R-matrix is an N N matrix, where N is the number of nodes in V, and R is the QoS parameter mentioned in the above section. R-matrix can be denoted by (5) and (6). In (6), d ij is the shortest distance between v i and v j in the graph G. r 11 r 12 r 1N r 21 r 22 r 2N R-matrix(G) =.... r N1 r N2 r NN { 1, dij R; r ij = 0, others.. (5) (6) In the placement of gateways, we expect the selected gateway nodes are capable of providing QoS ensured service for all mesh nodes, meanwhile the placement cost is as low as possible, and the load on the gateways can be well balanced. As far as the cost is concerned, the coverage capacity to cost ratio is a proper metric for measuring the capability of gateway candidates. To the node v i, the capacity/cost ratio for gateway placement can be denoted as p i in (7). p i = ( N ) min r ij, S j=1 Cost(v i ). (7) As to the node v i, its R hop adjacent nodes can be found in R-matrix. For QoS consideration, the quantity of nodes serviced by a gateway cannot exceed S. Therefore, the service coverage capacity of v i can be denoted by the small one in S and the number of its R hop adjacent nodes. The node s capacity/cost ratio is the metric for gateway selection in CSLBA. The nodes with high capacity/cost ratio have the high probability to be selected as gateways. 3.2 Cluster Building If a node is selected as a gateway, a routing tree rooted at the node will be built through breadth-first searching. The breadth-first search begins at the gateway node, and visits only the nodes which are not covered by the existing clusters. When a node is visited, and the QoS requirements are satisfied, the node joins

5 Feng Zeng et al.: Cost-Sensitive and Load-Balancing Gateway Placement 779 into the routing tree. The QoS requirements include the following: 1) the depth of tree should be less than or equal to R; 2) the relay load of every node in the tree should be bounded to L; 3) the degree of every node in the tree should be bounded to D; 4) the number of nodes in the tree would not be larger than S. R, L, D and S are given QoS parameters. When the searching ends, a routing tree with the above QoS constraints is created, so is the cluster. All the nodes in the routing tree constitute a cluster, and the gateway node acts as cluster head providing the Internet service only for its clustermates. 3.3 Partition Optimizing Doing gateway selection and cluster building repeatedly, we can get the initial partition of a WMN. In the initial partition, the load on various gateways may be not balanced. Let P = {C 1, C 2,..., C k } be the initial partition, and GW = {g 1, g 2,..., g k } (g i C i ) be the initial gateway set. The initial load variance on the gateways is VAR(P ). VAR(P ) = VAR(g 1, g 2,..., g k ). (8) The optimization starts with the initial partition, and repeatedly decreases the load variance on the gateways through scanning and shifting operations. The first step is to scan all the nodes and find the node with the largest load variance decrease on the gateways. To the node v i C i, if it can join into another cluster C j ensuring the QoS requirements, and the new partition P with VAR(P ) < VAR(P ), v i will leave its current cluster and join into C j. The second step is to shift a node from one cluster to another cluster. When shifting operation is done, the routing tree in the two clusters will be reconstructed, and the load on each gateway will be updated. The two steps will be done repeatedly until no such an operation can be found. The ultimate solution is deemed to be the best load-balancing gateway placement strategy. The optimization also tries to reduce the total cost of the gateway placement strategy. To a cluster C i, if there exists a node v i, and the following conditions are satisfied, v i can substitute g i for providing gateway service to the other clustermates. The updated cluster C i has v i as its new gateway with the placement cost decreased by Cost(g i ) Cost(v i ). The conditions are as follows: 1) a QoS-constraint routing tree rooted at v i can be built to cover all the nodes in cluster C i ; 2) the gateway placement cost of v i is less than that of g i. Scanning every node in each cluster for cost and load balance optimization, we can get a partition of a WMN, which shows a clustering strategy and gateway placement locations. The solution with the lowest gateway placement cost and the load variance on the gateways is the goal of our work. 3.4 Algorithm Description The key points in CSLBA have been explained in the above description. The pseudocode of CSLBA is shown in Fig.2. In the pseudocode, line 1 gets the R hop adjacent matrix, and based on the matrix and (7), nodes are selected as gateways in line 5. Gateway selection and cluster building (line 7) are done repeatedly until each node in the WMN is covered in some cluster, shown in lines 3 8. Lines 12 and 13 are done to gateway placement optimization for the cost and load balance. Lines 9 and 14 compute the cost and VAR of current partition, and the optimization is operated until no any improvement can be found. Input: original graph G and QoS requirements Output: the partition C 1, C 2,..., C k and the gateway set { GW CSLBA(G, R, D, T, S) 1 GetR Matrix(G, R, R matrix); 2 k = 0; GW = {}; 3 while (not all nodes covered) { 4 k++; C k = {}; 5 g k = SelectGateway(R matrix); 6 C k g k ; GW g k ; 7 BuildCluster(g k, C k, R, D, T, S); 8 Update R Matrix(R matrix); } 9 Calculate Cost Var(cost, var, C 1, C 2,..., C k ); 10 do{ 11 precost =cost; prevar = var; 12 Ajust Partition Load Balance(C 1, C 2,..., C k ); 13 Ajust Cluster Cost(C 1, C 2,..., C k ); 14 Calculate Cost Var(cost, var, C 1, C 2,..., C k ); 15 }while (cost < precost var < prevar) } Fig.2. Pseudocode of CSLBA. 3.5 Algorithm Example The example of CSLBA is shown in Fig.3. Fig.3(a) is the example WMN and its R hop adjacent matrix. In the graph, the gateway placement cost and local traffic are denoted beside every node. Based on (7) and the R-matrix, we find that R1 has the largest capacity/cost ratio. Therefore, the node R1 is the first node selected as a gateway node, and the cluster is built with

6 780 J. Comput. Sci. & Technol., July 2009, Vol.24, No.4 Fig.3. CSLBA example. (a) Original network and the R-matrix (given R = 2, S = 5). (b) Selected R1 as gateway to build cluster and the updated R-matrix. (c) Selected R13 as gateway to build cluster and the updated R-matrix. (d) Selected R2 as gateway to build cluster and do the load and cost adjustment to clusters. R1 acting as the cluster head. The R hop adjacent matrix is also updated for the uncovered nodes. That is shown in Fig.3(b). Then, R13 and R2 are selected as gateway nodes, and two clusters are created, shown in Figs. 3(b) and 3(c). So far, an initial partition of the WMN has been generated. At last, the cost and load balance optimization is done to the initial partition, and the optimal solution to gateway placement in the WMN is output, shown in Fig.3(d). 4 Genetic Algorithm for Gateway Placement The heuristic algorithm CSLBA has a high executive efficiency to get a low cost and load-balancing gateway placement strategy in a WMN. But, for an NP-hard problem, the final solution generated by CSLBA may be not the global optimal one, and CSLBA is not easy to simultaneously get the two goals: minimum cost and minimum load variance on the gateways. In contrast to heuristic and greedy algorithms, the genetic algorithms do well in multiple goal optimization searching [14]. But, they need a big iterative number to get the optimal solution. Consequently, attaching the high efficiency of CSLBA to algorithm design, we present a genetic gateway placement algorithm (GGPA) for cost-minimum and load-balancing placement of gateways. The motivation of GGPA is to efficiently find the near-optimal solution. We describe the genetic algorithm as follows, including the design of chromosome encoding and genetic operators. 4.1 Chromosome Encoding For a WMN G(V, E), a binary string with the length N(N = V ) is used to denote a chromosome

7 Feng Zeng et al.: Cost-Sensitive and Load-Balancing Gateway Placement 781 (individual), implying the solution for gateway placement in the WMN. In an individual, if the value of the i-th bit is 1, the node v i is a gateway node. Without enough bits of 1 in the strings, many binary strings with the length N could not be the valid individuals. Also, they are not the solutions to the gateway placement problem, because the clustering based on the gateways is not able to cover all nodes in the network. However, we can randomly select some uncovered nodes as gateways to update them to be the valid individuals. 4.2 Initial Population In order to get many initial gateway placement strategies different from each other in the population, we select a node as the gateway based on its selection probability. To the node v i, its selection probability is defined as (9). Pr(v i ) = max(c bound Cost(v i ), 1) N i=1 max(c bound Cost(v i ), 1). (9) C bound is an integer constant, a given parameter. If C bound is small, all nodes have the approximate selection probability. However, if C bound is large, the nodes with lower cost of gateway placement have the higher selection probability. In generating the initial population, except for the gateway selection, the methods of cluster building and partition optimization are the same as CSLBA. For the gateway selection based on probability, an initial population with all individuals different from each other would be generated. 4.3 Fitness Function To the individual I, the fitness function is defined as follows. Cost(I) VAR(I) f(i) = λ 1 e MAXCOST + λ 2 e MAXVAR. (10) In (10), λ 1 and λ 2 are accommodation coefficients used to adjust the optimization of the cost and load variance on the gateways. Cost(I) is the gateway placement cost in individual I, and MAXCOST is the maximum value of individuals cost in the population. VAR(I) is the load variance on the gateways in individual I, and MAXVAR is the maximum one of VARs in the population. 4.4 Selection In the selection operation, the better the fitness of the individual, the greater the chance that the individual will be selected into next generation population. In detail, the selecting probability of each individual is computed by (11), and the individual with the largest selecting probability will be selected directly into the next generation population. In (11), gs is the group size of current population. Pr(I i ) = f(i i) gs f(i j ) j=1 (1 i gs). (11) Furthermore, roulette wheel selection algorithm [14] is used to select some of the rest individuals into the next generation population. 4.5 Crossover In the operation, we use roulette wheel selection algorithm to select two individuals as parents, and these two individuals are denoted by I 1 and I 2. Then, we copy the clusters with small load variance on the gateways in solution I 1 to solution I 2, and use the clustering method in CSLBA to update solution I 2 to form a new individual. Similarly, we copy the clusters from solution I 2 to solution I 1 to form another new individual. The crossover steps are described as follows. 1) Using roulette wheel selection algorithm to select two individuals from current population, assumed to be I 1 and I 2, and the gateway set G 1 in I 1, G 2 in I 2. 2) In the solution I 1, for each g i G 1, if Load(g i ) L 1 < δ 1 (δ 1 is a constant, in the simulation, we set δ 1 = VAR(I 1 )/3), copy the cluster C i (g i C i ) to solution I 2. Then, cancel the clusters satisfying Load(g i ) L 2 > δ 2 (δ 2 = VAR(I 2 )/3) in I 2, and use the method in CSLBA to rebuild clusters covering those nodes, and form a new chromosome I 1. 3) Similar to step 2), in solution I 2, for each g i G 2, if Load(g i ) L 2 < δ 2, copy the cluster C i (g i C i ) to I 1. Then, cancel the clusters satisfying Load(g i ) L 1 > δ 1 in I 1, and rebuild clusters covering those nodes, and form a new chromosome I 2. 4) Let I 1 and I 2 join the next generation population. The idea of crossover is simply reflected in Fig.4. I 1 and I 2 have four clusters shown in Figs. 4(a) and 4(b) respectively, and we assume that the cluster 4 in I 1 and cluster 3 in I 2 are good clusters. In step 2), we copy the good cluster from one individual to another, as shown in Fig.4(c). Then, we cancel other bad clusters, and randomly or proportionally select a node as the gateway to build another cluster covering those uncover nodes, the new individual I 1 is generated in Fig.4(d). Using the similar methods, we can get another new individual I 2.

8 782 J. Comput. Sci. & Technol., July 2009, Vol.24, No.4 cost decreases. 2) Recursive DS: based on the similarity between gateway placement problem and dominating set problem, the algorithm recursively computes minimum weighted dominating sets. In each iteration, the algorithm computes a minimum dominating set of the graph resulting from the previous iteration. Finally, a near-optimal dominating set of the graph is found. The comparative metrics are the placement cost of gateways, the load variance on the gateways, the number of gateways and the execution CPU time. We also analyze the impact of QoS requirements on the performance of the above algorithms. The simulation is performed on a workstation with Pentium 4 CPU running at 3.06GHz with 512MB of RAMs. Fig.4. Idea of Crossover (given R = 2, S = 5). (a) I 1. (b) I 2. (c) Copy cluster 3 in I 2 to I 1. (d) I Mutation The mutation of an individual is simply to change one or several bits in the string. In this paper, the mutation steps are designed as follows. 1) With some low probability, probably select several individuals. 2) These selected individuals randomly have some of their bits flipped. 3) Use the methods in CSLBA to update these mutated individuals to denote the valid solutions of the gateway placement problem, and put the updated individuals into the new generation population. 4.7 End Condition of Algorithm In this paper, for less consideration of the real environment, we take a given number of generations as the end condition of the algorithm. Furthermore, if the real application is defined, we can set the end condition to be the allowed cost of gateway placement and the optimization extent of load balance on the gateways in WMNs. 5.1 Performance Comparison In the simulation, 1000 random topologies are generated, and each topology consists of a given number of nodes placed in an area of The communication radius is set to 1. The placement cost of each node is a random integer in [5, 20], and the local traffic is a random integer in [10, 50]. The QoS conditions are given: R = 4, D = 4, S = 20, and the relay traffic bound of each mesh router is set to ten times of its local traffic. To GGPA, the parameters are set to gs = 20, GEN = 30, λ 1 = 0.75, λ 2 = GEN is the number of iterations and gs is the population size. Fig.5 shows the average cost of results. CSLBA performs better than OPEN/CLOSE and Recursive DS in terms of the cost of gateway placement, and the genetic algorithm GGPA has the best performance. From statistic data, compared with OPEN/CLOSE and Recursive DS, the cost of gateway placement generated by CSLBA is decreased by 8.2% and 14.6% respectively, and to GGPA, the decrease is 32% and 40% respectively. 5 Simulations In this section, we compare simulation of the two algorithms proposed in this paper and the other two algorithms described in the literature: OPEN/CLOSE [12] and Recursive DS [8]. 1) OPEN/CLOSE: at first, gateway candidates are selected in descending order of their capacities to build the initial solution. Then, for reducing the cost of the solution, cluster adjustment is recursively done to the initial solution. The idea is to close one gateway and open a few other gateway candidates such that the total Fig.5. Cost of gateway placement.

9 Feng Zeng et al.: Cost-Sensitive and Load-Balancing Gateway Placement 783 Fig.6 shows the load variance (VAR) on the gateways in the gateway placement solutions generated by the algorithms. As far as the load balance on the gateways is concerned, the algorithms proposed in this paper have much better performance than the other two. Compared with OPEN/CLOSE and Recursive DS, CSLBA has the VAR decreased by 77% and 85% respectively, and GGPA has the VAR decreased by 86% and 91%. The experimental results validate that load balance on the gateways is well improved in our work. Fig.7 shows the number of gateways in the gateway placement solutions generated by these algorithms. The solutions generated by GGPA have a little less number of gateways than that of the other three. But, the decrease of gateway number is not the aim of this paper. We pay more attention to the optimization of the cost and load balance. 5.2 Impact of QoS Parameters on Performance In the simulation, various QoS conditions have some impact on the solution of gateway placement problem. Figs. 8, 9 and 10 show the impact of various R, D and S respectively. The increase of R, D and S leads to the decrease of the total cost. Fig.8. Impact of various R on performance. Fig.6. Load balance on the gateways. Fig.9. Impact of various D on performance. 5.3 Impact of Various Parameters on GGPA In GGPA, λ 1, λ 2 and GEN are the main parameters, we set λ 2 = 1 λ 1 in the simulation. Experiment shows that the increase of λ 1 leads to less cost of gateway placement, but VAR increases, shown in Figs. 11 and 12. The results also show that the larger GEN in GGPA will lead to better performance. 5.4 Executive Time Comparison Fig.7. Number of gateways. In the above simulation, the CPU execution time of

10 784 J. Comput. Sci. & Technol., July 2009, Vol.24, No.4 Table 1. Executive CPU Time (ms) Algorithm Node Recursive DS OPEN/CLOSE CSLBA GGPA Fig.10. Impact of various S on performance. From the data in Table 1, to the computing complexity, CSLBA is the lowest one among these algorithms, and the genetic algorithm is the highest. Compared with OPEN/CLOSE and Recursive DS, CSLBA has the execution time decreased by 72% and 53% respectively. The performance improvement in GGPA pays the price for the computing complexity. 6 Conclusion Fig.11. Impact of various λ 1 on the cost. In this paper, we focus on the gateway placement strategy with minimum cost and load balance on the gateways. First of all, a metric is presented to describe the variance of load on the gateways, and the gateway placement problem is addressed. Based on the previous work, routing tree scheme is used to discuss the QoS requirements of gateway placement in WMNs, and two algorithms are proposed to find the optimal solution. One is a heuristic algorithm CSLBA, which is sensitive to the cost of gateway placement. The key idea is to select the gateway nodes according to their capacity/cost ratio, and to improve the load balance through scanning and shifting methods. The other is a genetic algorithm GGPA, which has the advantage of global search for multiple goals, and can get the optimal solution at the price of computing complexity. Experimental results validate the performance of the two algorithms. Compared with other algorithms in the literature, the two algorithms have much better performance in terms of load balance on the gateways. As to the two algorithms, compared with each other, CSLBA has lower computing complexity, and GGPA has better quality. Thus, we provide a trade-off for network design. In our work, the change of topology and the traffic distribution is not considered. That would be possible direction for future work. Furthermore, it would be interesting to consider a decentralized version of the algorithms. References Fig.12. Impact of various λ 1 on VAR. each algorithm is recorded, and the average value is shown in Table 1. [1] Akyildiz I F, Wang X. A survey on wireless mesh networks. Communications Magazine, IEEE, 2005, 43(9): [2] Wu X, Liu J, Chen G. Analysis of bottleneck delay and throughput in wireless mesh networks. In Proc. IEEE MASS, Vancouver, Canada, October 9 12, 2006, pp

11 Feng Zeng et al.: Cost-Sensitive and Load-Balancing Gateway Placement 785 [3] Jun J, Sichitiu M L. Fairness and QoS in multihop wireless networks. In Proc. Vehicular Technology Conference, Orlando, Florida, USA, October 4 9, 2003, pp [4] Wong J L, Jafari R, Potkonjak M. Gateway placement for latency and energy efficient data aggregation. In Proc. the 29th Annual IEEE International Conference on Local Computer Networks, Tampa, FL, USA, November 16 18, 2004, pp [5] Youssef W, Younis M. Intelligent gateways placement for reduced data latency in wireless sensor networks. In Proc. ICC, Glasgow, Scotland, June 24 28, 2007, pp [6] Chandra R, Qiu L, Jain K et al. Optimizing the placement of integration points in multi-hop wireless networks. In Proc. ICNP, Berlin, Germany, October 5 8, 2004, pp [7] Bejerano Y. Efficient integration of multihop wireless and wired networks with QoS constraints. IEEE/ACM Transactions on Networking, 2004, 12(6): [8] Aoun B, Boutaba R, Iraqi Y et al. Gateway placement optimization in wireless mesh networks with QoS constraints. IEEE Journal on Selected Areas in Communications, 2006, 24(11): [9] Drabu Y, Peyravi H. Gateway placement with QoS constraints in wireless mesh networks. In Proc. the Seventh International Conference on Networking, Cancun, Mexico, April 13 18, 2008, pp [10] Vanhatupa T, Hännikäinen M, Hämäläinen T D. Performance model for IEEE s wireless mesh network deployment design. Journal of Parallel and Distributed Computing, 2008, 68(3): [11] He B, Xie B, Agrawal D P. Optimizing the Internet gateway deployment in a wireless mesh network. In Proc. Mobile Adhoc and Sensor Systems, Pisa, Italy, October 8 11, 2007, pp.1 9. [12] Prasad R, Wu H. Minimum-cost gateway deployment in cellular Wi-Fi networks. In Proc. IEEE Consumer Communications and Networking Conference, Las Vegas, NV,USA, January 7 10, 2006, pp [13] Hsiao P-H, Hwang A, Kung H T et al. Load-balancing routing for wireless access networks. In Proc. 20th Annual Joint Conference of the IEEE Computer and Communications Societies, Anchorage, USA, April 22 26, 2001, pp [14] Coley D A. An Introduction to Genetic Algorithms for Scientists and Engineers. Singapore: River Edge, NJ World Scientific Publishing Co., 1999, pp Feng Zeng received his B.Eng. degree in computer science in 2000, M.Eng. degree in computer application in 2005, both from Hunan University, China. He is currently a Ph.D. candidate in computer application at Network Computing and Distributed Processing Laboratory in School of Information Science and Engineering, Central South University, China. His main research interests include wireless mesh network, QoS routing and network calculus. He is a student member of China Computer Federation. Zhi-Gang Chen is a professor at School of Information Science and Engineering (ISE), Central South University (CSU), China. His research interests are in QoS mechanism for IP network, Web service, and wireless network. He is the supervisor of Network Computing and Distributed Processing Laboratory at School of ISE, CSU. He obtained his Ph.D. and M.Sc. degrees in computer science from CSU. He is a member of China Computer Federation.