Performance Analysis of WMNs by WMNPSODGA Simulation System Considering Weibull and Chisquare Client Distributions
Abstract
The Wireless Mesh Networks (WMNs) are becoming an important networking infrastructure because they have many advantages such as low cost and increased high speed wireless Internet connectivity. In our previous work, we implemented a Particle Swarm Optimization (PSO) based simulation system, called WMNPSO, and a simulation system based on Genetic Algorithm (GA), called WMNGA, for solving node placement problem in WMNs. Then, we implemented a hybrid simulation system based on PSO and distributed GA (DGA), called WMNPSODGA. In this paper, we analyze the performance of WMNs using WMNPSODGA simulation system considering Weibull and Chisquare client distributions. Simulation results show that a good performance is achieved for Chisquare distribution compared with the case of Weibull distribution.
1 Introduction
The wireless networks and devices are becoming increasingly popular and they provide users access to information and communication anytime and anywhere [2, 6, 7, 8, 9, 12, 17, 23, 24, 25, 28]. Wireless Mesh Networks (WMNs) are gaining a lot of attention because of their low cost nature that makes them attractive for providing wireless Internet connectivity. A WMN is dynamically selforganized and selfconfigured, with the nodes in the network automatically establishing and maintaining mesh connectivity among themselves (creating, in effect, an ad hoc network). This feature brings many advantages to WMNs such as low upfront cost, easy network maintenance, robustness and reliable service coverage [1]. Moreover, such infrastructure can be used to deploy community networks, metropolitan area networks, municipal and corporative networks, and to support applications for urban areas, medical, transport and surveillance systems.
Mesh node placement in WMN can be seen as a family of problems, which are shown (through graph theoretic approaches or placement problems, e.g. [4, 13]) to be computationally hard to solve for most of the formulations [29]. We consider the version of the mesh router nodes placement problem in which we are given a grid area where to deploy a number of mesh router nodes and a number of mesh client nodes of fixed positions (of an arbitrary distribution) in the grid area. The objective is to find a location assignment for the mesh routers to the cells of the grid area that maximizes the network connectivity and client coverage.
Node placement problems are known to be computationally hard to solve [10, 11, 30]. In some previous works, intelligent algorithms have been recently investigated [3, 5, 14, 15, 18, 19, 20, 26, 27].
In [21], we implemented a Particle Swarm Optimization (PSO) based simulation system, called WMNPSO. Also, we implemented another simulation system based on Genetic Algorithm (GA), called WMNGA [16], for solving node placement problem in WMNs.
In our previous work, we designed and implemented a hybrid simulation system based on PSO and distributed GA (DGA). We called this system WMNPSODGA. In this paper, we evaluate the performance of WMNs using WMNPSODGA simulation system considering Weibull and Chisquare client distributions.
The rest of the paper is organized as follows. The mesh router nodes placement problem is defined in Sect. 2. We present our designed and implemented hybrid simulation system in Sect. 3. The simulation results are given in Sect. 4. Finally, we give conclusions and future work in Sect. 5.
2 Node Placement Problem in WMNs
For this problem, we have a grid area arranged in cells we want to find where to distribute a number of mesh router nodes and a number of mesh client nodes of fixed positions (of an arbitrary distribution) in the considered area. The objective is to find a location assignment for the mesh routers to the area that maximizes the network connectivity and client coverage. Network connectivity is measured by Size of Giant Component (SGC) of the resulting WMN graph, while the user coverage is simply the number of mesh client nodes that fall within the radio coverage of at least one mesh router node and is measured by Number of Covered Mesh Clients (NCMC).
An instance of the problem consists as follows.

N mesh router nodes, each having its own radio coverage, defining thus a vector of routers.

An area \(W\times H\) where to distribute N mesh routers. Positions of mesh routers are not predetermined and are to be computed.

M client mesh nodes located in arbitrary points of the considered area, defining a matrix of clients.
It should be noted that network connectivity and user coverage are among most important metrics in WMNs and directly affect the network performance.
In this work, we have considered a biobjective optimization in which we first maximize the network connectivity of the WMN (through the maximization of the SGC) and then, the maximization of the NCMC.
In fact, we can formalize an instance of the problem by constructing an adjacency matrix of the WMN graph, whose nodes are router nodes and client nodes and whose edges are links between nodes in the mesh network. Each mesh node in the graph is a triple \({\varvec{v}}\,=\,< x, y, r>\) representing the 2D location point and r is the radius of the transmission range. There is an arc between two nodes \({\varvec{u}}\) and \({\varvec{v}}\), if \({\varvec{v}}\) is within the transmission circular area of \({\varvec{u}}\).
3 Proposed and Implemented Simulation System
3.1 WMNPSODGA Hybrid Simulation System
PSO Part
WMNPSODGA decide the velocity of particles by a random process considering the area size. For instance, when the area size is \(W\times {}H\), the velocity is decided randomly from \(\sqrt{W^{2}+H^{2}}\) to \(\sqrt{W^{2}+H^{2}}\). Each particle’s velocities are updated by simple rule [22].
DGA Part
Population of individuals: Unlike local search techniques that construct a path in the solution space jumping from one solution to another one through local perturbations, DGA use a population of individuals giving thus the search a larger scope and chances to find better solutions. This feature is also known as “exploration” process in difference to “exploitation” process of local search methods.
Selection: The selection of individuals to be crossed is another important aspect in DGA as it impacts on the convergence of the algorithm. Several selection schemes have been proposed in the literature for selection operators trying to cope with premature convergence of DGA. There are many selection methods in GA. In our system, we implement 2 selection methods: Random method and Roulette wheel method.
Crossover operators: Use of crossover operators is one of the most important characteristics. Crossover operator is the means of DGA to transmit best genetic features of parents to offsprings during generations of the evolution process. Many methods for crossover operators have been proposed such as Blend Crossover (BLX\(\alpha \)), Unimodal Normal Distribution Crossover (UNDX), Simplex Crossover (SPX).
Mutation operators: These operators intend to improve the individuals of a population by small local perturbations. They aim to provide a component of randomness in the neighborhood of the individuals of the population. In our system, we implemented two mutation methods: uniformly random mutation and boundary mutation.
Escaping from local optima: GA itself has the ability to avoid falling prematurely into local optima and can eventually escape from them during the search process. DGA has one more mechanism to escape from local optima by considering some islands. Each island computes GA for optimizing and they migrate its gene to provide the ability to avoid from local optima.
Convergence: The convergence of the algorithm is the mechanism of DGA to reach to good solutions. A premature convergence of the algorithm would cause that all individuals of the population be similar in their genetic features and thus the search would result ineffective and the algorithm getting stuck into local optima. Maintaining the diversity of the population is therefore very important to this family of evolutionary algorithms.
In following, we present our proposed and implemented simulation sistem called WMNPSODGA. We show the fitness function, migration function, particlepattern, gene coding and client distributions.
Fitness Function
Our implemented simulation system uses Migration function as shown in Fig. 1. The Migration function swaps solutions between PSO part and DGA part.
ParticlePattern and Gene Coding
In ordert to swap solutions, we design particlepatterns and gene coding carefully. A particle is a mesh router. Each particle has position in the considered area and velocities. A fitness value of a particlepattern is computed by combination of mesh routers and mesh clients positions. In other words, each particlepattern is a solution as shown is Fig. 2.
A gene describes a WMN. Each individual has its own combination of mesh nodes. In other words, each individual has a fitness value. Therefore, the combination of mesh nodes is a solution.
Client Distributions
3.2 WMNPSODGA Web GUI Tool
The Web application follows a standard ClientServer architecture and is implemented using LAMP (Linux + Apache + MySQL + PHP) technology (see Fig. 4). We show the WMNPSODGA Web GUI tool in Fig. 5. Remote users (clients) submit their requests by completing first the parameter setting. The parameter values to be provided by the user are classified into three groups, as follows.

Parameters related to the problem instance: These include parameter values that determine a problem instance to be solved and consist of number of router nodes, number of mesh client nodes, client mesh distribution, radio coverage interval and size of the deployment area.

Parameters of the resolution method: Each method has its own parameters.

Execution parameters: These parameters are used for stopping condition of the resolution methods and include number of iterations and number of independent runs. The former is provided as a total number of iterations and depending on the method is also divided per phase (e.g., number of iterations in a exploration). The later is used to run the same configuration for the same problem instance and parameter configuration a certain number of times.
WMNPSODGA parameters.
Parameters  Values 

Clients distribution  Weibull, Chisquare 
Area size  \(32.0 \times 32.0\) 
Number of mesh routers  16 
Number of mesh clients  48 
Number of migrations  200 
Evolution steps  9 
Number of GA islands  16 
Radius of a mesh router  2.0–3.5 
Replacement method  LDIWM 
Selection method  Roulette wheel method 
Crossover method  SPX 
Mutation method  Boundary mutation 
Crossover rate  0.8 
Mutation rate  0.2 
4 Simulation Results
We show simulation results in Figs. 6 and 7. We see that for both SGC and NCMC, the performance of Chisquare distribution is better than Weibull distribution. Thus, we conclude that the performance is better when the clients distribution is Chisquare distribution compared with the Weibull distribution.
5 Conclusions
In this work, we evaluated the performance of WMNs using a hybrid simulation system based on PSO and DGA (called WMNPSODGA). Simulation results show that the performance is better for Chisquare distribution compared with the case of Weibull distribution.
In our future work, we would like to evaluate the performance of the proposed system for different parameters and patterns.
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