An Electromagnetism-like mechanism algorithm for the router node placement in wireless mesh networks

Abstract

In this paper, we consider the problem of mesh router placement in a wireless mesh network (WMN). The latter is an emerging networking technology consisting of three kinds of nodes: mesh clients, mesh routers and gateways. Mesh routers form a backbone to forward data between client nodes and the external network. Therefore, the optimization of mesh routers positions strongly influences the performance of the WMN. Since this issue has already been proved as being computationally NP-hard to solve, the use of non-exact methods (such as heuristics and metaheuristics) is indispensable. In this sense, our current work consists to apply and adapt the electromagnetism-like mechanism (EM) metaheuristic to solve the router node placement issue. The idea is to consider a population of solutions encoded as particles subject to attractions and repulsions as in electromagnetic systems. Finally, we have evaluated our proposed approach by simulating different scenarios under various settings. The obtained results indicate that the proposed EM algorithm outperforms the existing particle swarm intelligence algorithm and genetic algorithm in defining near optimal positions for mesh routers with regard to coverage and connectivity.

Introduction

With the recent technological developments in wireless architectures and platforms, new solutions have been proposed to deal with the last miles Internet connectivity (Akyildiz et al. 2005; Gupta et al. 2017). Among all these solutions, wireless mesh network (WMN) is one of the most emerging networking paradigms, which offers interesting proprieties such as a low deployment cost. A typical WMN is built using three kinds of nodes: mesh clients (MC), mesh routers (MR) and gateways (MG). On the other hand, the WMN is organized in two tiers: access tier and backbone tier. The mesh clients form the access tier, whereas the interconnected MRs form the backbone tier to forward packets between Internet and the access tier. This kind of network is characterized by:

  • A low cost of deployment,

  • An easy network maintenance,

  • A robust, reliable and self-configured topology.

Wireless mesh networks can be involved in many application scenarios, including difficult environments (such as emergency situations, tunnels, oil rigs and battlefield surveillance), broadband home networking and community mesh network. Furthermore, some other attempts to deploy real-world testbeds have been tried such as "One hundred dollar laptop" project developed at MIT for schools in developing countries and "One laptop per child" project (Xhafa et al. 2012).

The network design strongly influences the performance of a WMN. Thus, designing a WMN consists of:

  • Defining the positions of mesh routers,

  • Selecting the appropriate mesh routers to act as gateways,

  • Designing an appropriate routing protocol.

In this paper, we have addressed the problem of router node placement in a wireless mesh network (WMN-RNP). Some previous studies have considered the WMN-RNP as a variant of a well-known problem in the literature called facility location problem (FLP) where the routers represent the facilities. Examples of some other similar issues are: sensor node placement in a WSN (Khalesian and Delavar 2016) and relay nodes placement in multi-hop wireless ad hoc networks (Yerra and Rajalakshmi 2014). Since the FLP issue has been proved as being computationally NP-hard to solve, the WMN-RNP is also NP-hard. In addition, we have considered, in the present work, a variant of the problem where candidate locations were not a priori defined. Therefore, the use of heuristic and metaheuristic algorithms is essential.

In this paper, we propose to adapt a metaheuristic-based approach to solve the WMN-RNP issue. Specifically, it consists of applying a metaheuristic inspired from the theory of physics called electromagnetism-like mechanism optimization algorithm (EM). It mimics the process of attraction and repulsion of sample points placed in a container to search for an optimal solution. The EM algorithm has been applied efficiently to solve several optimization problems, including job shop scheduling (Naderi et al. 2012), nurse scheduling (Maenhout and Vanhoucke 2007) and cell formation (Wei et al. 2012).

The main contributions of this paper are:

  • Formulating the WMN-RNP issue as a multi-objective problem.

  • Applying the Electromagnetism-like mechanism to solve the WMN-RNP issue.

  • Moreover, we present computational experiments on different generated network topologies. We show that the proposed approach performs mostly well under many different situations.

The remaining of this paper is organized as follows: we provide a summary of some related works in Sect. 2. In Sect. 3, we present the system model and the formulation of the problem. Our proposed algorithm is described in Sect. 4. Furthermore, we evaluate our scheme using simulations in Sect. 5. Finally, we summarize the main findings and the potential perspectives in the last section.

Related work

Many works have been proposed in order to solve the router node placement issue. This section gives a detailed presentation of some representative works.

Xhafa et al. (2011, 2012, 2015) have addressed the WMN-RNP issue using several metaheuristics. In Xhafa et al. (2011), Sakamoto et al. (2014), a simulated annealing algorithm has been used to determine the positions of mesh routers. The authors have considered the optimization of two parameters: the client coverage and network connectivity. However, the optimization was performed hierarchically (bi-level) in two stages. First, the algorithm tries to maximize the network connectivity defined as the size of the greatest sub-network component. Then, when the algorithm cannot improve the connectivity, the second stage is triggered. It consists of maximizing the client coverage, defined as the number of mesh clients covered by at least one mesh router, without deteriorating the best value of the network connectivity found in the first stage. In Xhafa et al. (2015), a Tabu search algorithm has been applied to tackle the same issue. Similarly, the same bi-level optimization model, which considers the optimization of two parameters: the connectivity and coverage, has been used. According to the simulation results, Tabu search algorithm has performed better than simulated Annealing. Xhafa et al. (2012) have used a hill climbing algorithm to hierarchically optimize the same parameters when placing the mesh routers.

Abdelkhalek et al. (2015) have applied a local search approach to solve the node placement problem in next generation networks (NGN). Specifically, a multi-objective variable length Pareto local search algorithm has been used to extend the existing network coverage by efficiently placing new nodes. Their optimization model considers: maximizing the coverage, minimizing the deployment cost, maximizing the total minimum bandwidth capacity and minimizing the total overlapping. Therefore, a solution to the considered issue was encoded as a variable size string.

In Lin (2013), Lin et al. (2013, 2014, 2016b), three new variants of the WMN-RNP have been formulated. In Lin (2013), Lin et al. (2016b), a PSO metaheuristic has been used to tackle the WMN-RNP with client mobility, whereas Lin et al. (2013) have considered an RNP with service priority constraint RNPSP, in which every mesh client is assigned a value representing its service priority. To solve this issue, a simulated annealing algorithm has been applied. Lin et al. (2014) have introduced another RNP variant which considers mobile clients with service priority constraint. A Bat inspired algorithm (BA) has been applied to solve this issue.

Finally, Benyamina et al. (2012) have adopted a different strategy to design and plan a WMN. The proposed scheme includes a number of issues such as router node deployment and gateway placement. A multi-objective particle swarm optimization (MOPSO) has been applied to address these issues. However, authors have considered different optimization parameters including deployment cost and network throughput. Table 1 addresses a summary of some representative works in the field of router node placement in WMN.

Table 1 Summary of some RNP representative works

System model and problem formulation

The aim of the router node placement problem, described in this paper, is to determine the positions of mesh routers that allow the optimization of two parameters: network connectivity and client coverage. In the following, we will describe the system model and the mathematical formulation of the WMN-RNP. Table 2 describes the symbols used in this work.

Table 2 Symbols used in this paper

System model

We consider a wireless mesh network consisting of three kinds of nodes: mesh clients (MC), mesh routers (MR), and gateways (GW). Therefore, let:

  • \(MC=\{c_1,c_2,...,c_m\}\) be the set of m mesh clients and \((x_i,y_i)\) be the position of mesh client \(c_i\) where \(i \in [1,m]\).

  • \(MR=\{r_1,r_2,...,r_n\}\) be the set of n mesh routers. We assume that all routers have the same coverage radius \(R=R_1=R_2=...=R_n\).

A link can be defined either between two mesh routers, or between a mesh router and a mesh client.

Definition 1

(link between two mesh routers). There is a link between a mesh router \(r_i\) and another mesh router \(r_j\) if the distance between them does not exceed two times the coverage radius R: \(d(r_i,r_j) \le 2 R\). Therefore, we define the set of links between routers \(E_{RR}\) as follows:

\(E_{RR}=\{(r_i,r_j) | (r_i \in MR) \wedge (r_j \in MR) \wedge d(r_i,r_j) \le 2R\}.\)

Definition 2

(link between a mesh router and a mesh client). A link between a mesh router \(r_i\) and a mesh client \(c_k\) exists if the client \(c_k\) is in the communication range of router \(r_i\): \(c_k \in \Gamma _i\). In other words, the corresponding link exists if: \(d(r_i,c_k) \le R\). Therefore, we define the set of links between routers and clients \(E_{RC}\) as follows:

\(E_{RC}=\{(r_i,c_k) | (r_i \in MR) \wedge (c_k \in MC) \wedge d(r_i,c_k) \le R\}.\)

Definition 3

(Wireless mesh network). Mathematically, a WMN G is defined as a graph G(VE) where V is the set of vertices and E is the set of edges. In this work, gateways are not considered since they are, generally, placed after defining the positions of mesh routers. Thus, \(V = MC \cup MR\) and \(E = E_{RR} \cup E_{RC}\).

Problem formulation

In this paper, we consider the following variant of the WMN-RNP issue. Given a set of a priori deployed clients, the aim is to define how and where to deploy a set of given mesh routers so that both client coverage and network connectivity will be optimized. Therefore, the goal is to determine the positions of the n mesh routers, denoted by \({(x_1,y_1 ),(x_2,y_2 )}\), \({...,(x_n,y_n)}\) corresponding to routers \({r_1,r_2 ,...,r_n}\) such that \((x_i,y_i )\in N^2\) and \(0 \le x_i \le W, 0 \le y_i \le H\). Thus, the problem studied in this paper considers the following two objectives:

Definition 4

(Client coverage) \(\phi ({{G}})\). The client coverage parameter is the number of clients covered by at least one mesh router in the graph G.

$$\begin{aligned} \phi (G)=\sum _{(i=1)}^m(Max_{j \in [1,n]} (\delta _{ij} )) \end{aligned}$$
(1)

where

$$\begin{aligned} \delta _{ij} = {\left\{ \begin{array}{ll} 1 &{} \text {if mesh client }c_i \text { is covered by mesh router } r_j,\\ 0 &{} \text {Otherwise.} \end{array}\right. }\nonumber \\ \end{aligned}$$
(2)

Definition 5

(Network connectivity) \(\psi ({{G}})\). The network connectivity is defined as the size, in terms of number of nodes, of the greatest subgraph component among the k components and is formulated as follows:

$$\begin{aligned} \psi (G) = Max_{i \in \{1,...,k\}} {|G_i |} \end{aligned}$$
(3)

The proposed EM algorithm

To solve the mesh router node placement issue formulated earlier, we adopt an approximation approach since it has been proved that the WMN-RNP issue is computationally hard to solve. Thus, a metaheuristic approach called electromagnetism-like mechanism (EM) algorithm has been applied. In this section, we give, first, an overview of the EM algorithm. Then, we explain how it can be used to solve the considered problem.

Overview of the Electromagnetism-like mechanism (EM) algorithm

The electromagnetism-like mechanism algorithm (EM) (Birbil and Fang 2003) is a population-based metaheuristic inspired from the attraction–repulsion mechanism of the electromagnetic theory. It has been proposed by Birbil and Fang (2003) to solve optimization problems in continuous space formulated as follows:

Minimizef(x)

Subject to \(x \in [l,u]\)

where \([l,u]:=\{x \in R^d | l_k \le x_k \le u_k, k=1,...,d\}\), d is the dimension of the problem.

Fig. 1
figure1

Flowchart of the EM algorithm

Basically, the EM algorithm consists of five steps as illustrated in Fig. 1. First, the population of p solutions is created and placed randomly in the search space. Each solution is encoded as a particle \(X^i\) characterized by an initial charge \(q^i, i=1,...,p\). A local search process is then applied to search for better solutions in the neighborhood of the best solution. The third stage concerns the computation of the charge of every particle according to the value of its objective function. Thus, the better the objective function value, the greater the charge of the corresponding particle. In the fourth stage, charges of particles are used to calculate the total force exerted on every particle. Finally, the particles are moved according to the value and direction of the total force. More details about this EM metaheuristic and its convergence analysis can be found in Birbil and Fang (2003).

The EM algorithm uses the two concepts of exploration and exploitation, like many other metaheuristics particularly PSO, as the key elements of the search process. However, it differs from PSO from many aspects. First, in PSO, the next movement of every particle is a function of its position, its best position and the global best position. In the EM algorithm, the next position of the particle is determined according to the total force exerted by the other particles in the search space. Another fundamental difference between PSO and EM algorithms is how did they implement the two mechanisms of exploration and exploitation. In PSO, the global and local best particles always attract the current particle to search in the surrounding. When a local minima is reached, it cannot be escaped until another different particle will find a more interesting solution. In EM, the best particle exerts attraction forces on all other particles. In addition, every particle i exerts a force \(F_{ij}\) on every other particle j according to the objective function value of every particle and the distance between them. When a local minima is reached, the other particles will exert repulsion forces to prevent a move in their direction and incite the best particle to explore other regions. Thus, this mechanism used by the EM algorithm to escape from local minima should be more efficient than the mechanism used by PSO.

EM algorithm applied to solve WMN-RNP problem

Motivations and similarities

In this section, we explain our choice of using a metaheuristic to solve the RNP issue and then we describe the main reasons of choosing, specifically, the EM algorithm among several other metaheuristics. Since the WMN-RNP issue has already been demonstrated as being computationally NP-hard to solve, the use of approximation techniques such as metaheuristics is the only reasonable way to solve the considered issue. On the other hand and as explained in the previous section, the EM algorithm provides interesting search capabilities: it uses an efficient mechanism to avoid local minima and can converge faster to near optimal solutions (Cuevas et al. 2017; Yurtkuran and Emel 2010; Lee and Chang 2010; De Castro and Von Zuben 2002) by maintaining a good balance between exploration and exploitation in the search space. Thus, the EM algorithm has shown good results when solving several NP-hard issues such as job shop scheduling (Naderi et al. 2012), nurse scheduling (Maenhout and Vanhoucke 2007) and cell formation (Wei et al. 2012). Furthermore, the EM algorithm requires only few parameters to be adjusted (Tsou and Kao 2006). Moreover, the similarities between the electromagnetic theory of physics and the treated problem make the EM algorithm more appropriate to solve the WMN-RNP issue. Table 3 gives an overview of some similarities between the two concepts: the electromagnetic theory and the router node placement problem.

Table 3 Similarities between electromagnetism theory and WMN-RNP elements

Objective function

The problem considered in this paper is bi-objective. It consists of, simultaneously, optimizing both client coverage and network connectivity. Thus, an objective function f has been defined to evaluate the quality of a solution. This function takes into consideration the two parameters as follows:

$$\begin{aligned} f(X^i )=\lambda \cdot \frac{\phi (G)}{m}+(1-\lambda ) \cdot \frac{\psi (G)}{n+m} \end{aligned}$$
(4)

where G is the graph corresponding to the solution \(X^i\), \(\lambda \) is a parameter that controls the balance between the two objectives and \(f(X^i ) \in [0,1]\). The goal of WMN-RNP is to maximize the value of \(f(X^i)\), whereas EM algorithm is applied when the problem considered needs to minimize a function. To this end, another function g has been defined, to evaluate the quality of every solution, as follows:

$$\begin{aligned} g(X^i )=1-f(X^i ) \end{aligned}$$
(5)

Therefore, the goal of defining the function g is to change the objective of addressing WMN-RNP from maximizing the function \(f(X^i)\), defined in Eq. 4, to minimizing \((1 - f(X^i))\) defined in Eq. 5.

Solution representation

A potential solution to the WMN-RNP issue consists of defining the positions of all mesh routers. In this paper, every solution is represented by an artificial particle with a charge \(q^i\) and exerting attraction/repulsion forces on other particles like real particles in electromagnetic systems. Thus, a solution in EM is represented by a particle \(X^i\) characterized by:

  • A charge \(q^i\).

  • An array of positions \(X^i=(x_1^i,y_1^i,x_2^i,y_2^i,...,x_n^i,y_n^i)\), where \((x_k^i,y_k^i)\) are the (x, y)-coordinates of the mesh router \(r_k, \forall k \in \{1,2,...,n\}\); \(0 \le x_k^i \le W\); \(0 \le y_k^i \le H\).

Algorithm

The pseudo-code of the EM algorithm defines five stages: Initialization, Local search, Compute charges, Compute forces and finally Move particles as illustrated in Fig. 2.

Fig. 2
figure2

Pseudo-code of the EM algorithm applied to WMN-RNP

(A) Initialization

During this phase, default values are assigned to the parameters and particles are created and placed randomly in the search space. In other words, for every particle (solution), a set of predefined number of mesh routers is randomly placed in the deployment field.

(B) Local search

The local search procedure represents the mechanism used to improve the quality of the solution by searching around the best particle. In this context, every iteration of the local search derives a new particle \(X^k\) from the best particle \(X^{best}\) as follows: for every router \(r_i (x_i^{best},y_i^{best})\) in \(X^{best}\) perform a random movement without exceeding a distance S:

$$\begin{aligned} {\left\{ \begin{array}{ll} x_i^k = (x_i^{best} + \lambda _1 \cdot S) \% H \\ y_i^k = (y_i^{best} +\lambda _2 \cdot S) \% W \end{array}\right. } \end{aligned}$$
(6)

where \(\lambda _1\) and \(\lambda _2\) are random variables uniformly distributed in [0, 1], and S is a parameter of the local search.

Table 4 Time complexity analysis

(C) Compute charges

Each particle \(X^i\) in the search space is characterized by a charge \(q^i\). In practice, the value of the charge \(q^i\) is proportional to the objective function value. The better the objective function value, the greater the corresponding charge value. The value of the charge is used later to compute the force exerted by \(X^i\) on the other particles of the system. Therefore, the value of the charge \(q^i\) is formulated as follows:

$$\begin{aligned} q^i=\exp (-n \cdot \frac{(g(X^i)-g(X^{best}))}{\sum _{k=1}^p (g(X^k)-g(X^{best}))}) \end{aligned}$$
(7)

where \(X^i\) is the i-th particle and \(g(X^i )=1-f(X^i)\) is the value of the objective function of the particle \(X^i\). \(X^{best}\) is the best particle and \(g(X^{best})\) is the objective function value of the best particle \(X^{best}\).

(D) Compute the total forces

For every particle \(X^j\), calculate the force \(F_{ij}\) exerted by any other particle \(X^i\) according to the attraction and repulsion mechanisms. Then, the total force \(F^i\) exerted on each particle \(X^i\) is calculated based on the superposition principle of the electromagnetic theory as follows:

$$\begin{aligned} F^i=\sum _{(j \ne i)}^p {\left\{ \begin{array}{ll} (X^j-X^i) \cdot \frac{(q^i q^j)}{||X^j-X^i ||^2} &{} \text { if}\quad g(X^j ) < g(X^i )\\ (X^i-X^j) \cdot \frac{(q^i q^j)}{||X^j-X^i ||^2} &{} \text { if }\quad g(X^j ) \ge g(X^i ) \end{array}\right. } \end{aligned}$$
(8)

where \(||X^j-X^i ||\) represents the distance between the two particles \(X^j\) and \(X^i\). According to Eq. 8, the force between two particles of the system depends on their charges and the distance separating them. These concepts of attraction and repulsion are the key elements of the EM algorithm convergence.

(E) Move particles

The final stage of the EM algorithm consists of moving every particle in the system in the direction of the exerted force by a distance, which is proportional to \(F^i\) as given in Eq. 9.

$$\begin{aligned} X^i=X^i+\lambda _3 \cdot \frac{F^i}{||F^i ||} \cdot RNG \end{aligned}$$
(9)

where \(\lambda _3\) is a random variable uniformly distributed between 0 and 1. RNG is an array whose components represent the allowed feasible movements toward the upper bound or the lower bound of the corresponding dimension. Furthermore, to keep the solution feasible, the total exerted force is normalized \((\frac{F^i}{||F^i ||})\).

Time complexity analysis

The time complexity of the elementary operations of the EM algorithm is presented in Table 4.

According to Fig. 2 presenting the EM pseudo-code applied to the WMN-RNP issue, the following steps are executed: In the first step, \(O(p \cdot n)\) operations are needed to create particles by randomly deploying routers since there are p particles with n routers each one, and \(O(n^2+n \cdot m)\) operations are needed to compute the objective function value of each particle. On the other hand, the time complexity of the local search phase is \(O(L.(n^2+n.m))\). It consists of creating a new particle from the best one and then computing its objective function value in order to measure its quality. This process is repeated L times. During the fourth step, charge of every particle is calculated with a time complexity of \(O(p^2)\). According to Eq. 7, O(p) operations are needed to compute the sum \(\sum _{k=1}^p (g(X^k )-g(X^{best}))\) and it is repeated for each particle. The force exerted by one particle on another one requires O(n) operations to be computed. Then, the total force exerted by all particles on one particle needs O(p.n). Thus, this phase is executed in \(O(p^2 \cdot n)\). The last phase needs \(O(p \cdot n)\) operations to move p particles. Therefore, the overall run-time complexity of the EM algorithm applied to WMN-RNP is \(O(p \cdot n+T \cdot (p \cdot (n^2+n \cdot m)+L \cdot (n^2+n \cdot m)+p^2+p^2 \cdot n+p \cdot n))\). With the above analysis, the time complexity of the proposed EM algorithm is \(O(T \cdot n \cdot ((p+L) \cdot (n+m)+p^2))\).

Results and discussion

The purpose of this section is to evaluate the performance of the proposed EM approach in solving the mesh router node placement. Furthermore, this proposed technique is compared to the existing PSO (Lin 2013; Lin et al. 2016b) and GA (Barolli et al. 2015) approaches. Therefore, we, first, introduce the parameter values, and then, we discuss the simulation results according to three metrics: client coverage, network connectivity and objective function value. We have implemented our approach and the existing PSO and GA techniques using C++ programing language based on the formulation presented earlier in Sect. 3 and using the following configuration. All the experiments were carried out on a Pentium CPU B950 @ 2.10 GHz with 4 Go of memory. In addition, the obtained results are an average of ten runs performed on randomly generated topologies.

Parameters

Table 5 summarizes the parameters and their values used during the simulations. In fact, in all simulations, we consider a rectangular simulation area of \(2 \mathrm{Km} * 2 \mathrm{Km}\). The positions of 2 to 36 mesh routers are defined to cover between 50 to 500 mesh clients that are randomly distributed in the deployment field. The rest of parameters are summarized in Table 5. In addition, PSO parameters are set to the same values as in Lin et al. (2016a).

Table 5 Parameter values

Results

Simulation results are illustrated and discussed in the following. In this context, we have evaluated our proposed approach by comparing its performances with the existing PSO algorithm described in Lin (2013), Lin et al. (2016a, b) and genetic algorithm (Barolli et al. 2015). PSO and GA approaches have been applied recently to address the issue considered in this paper. Furthermore, three performance metrics are used to evaluate these three approaches:

  • Client coverage (\(\phi \)): It represents the number of mesh clients covered by at list one mesh router.

  • Network connectivity (\(\psi \)): It is defined as the number of nodes of the biggest subgraph component.

  • objective function value: It is defined by aggregating the two previous objectives (coverage and connectivity).

On the other hand, we have evaluated the effectiveness of the proposed approach from many different aspects by varying:

  • The number of mesh clients,

  • The number of mesh routers,

  • The coverage radius of mesh routers.

Figures 3 and 4 illustrate the positions of mesh routers provided by the GUI interface of our system after applying EM and PSO approaches on the same network. In these figures, (+) symbols designate the positions of mesh clients a priori deployed and red circles (blue circles, respectively) designate the positions of mesh routers obtained after applying the EM algorithm (PSO algorithm, respectively).

Fig. 3
figure3

Example of mesh routers positions obtained after applying PSO

Fig. 4
figure4

Example of mesh routers positions obtained after applying EM

Convergence study

First of all, we have studied the convergence of the three algorithms. Therefore, a comparison between them is conducted according to two parameters:

  • Convergence speed: considered as the iteration number (Iter) since a stagnation was observed,

  • Convergence efficiency: considered as the value of the function f described in Eq. 4,

Ideally, the algorithm should converge quickly, but also, it should converge to an optimal/good solution. To this end, we have considered four network scenarios of different sizes as described in Table 6.

An example of the graphs provided by the GUI interface of our system is shown in Fig. 5.

Each result presented in Table 7 is an average of 10 runs. As shown in this table, PSO is the fastest algorithm in terms of convergence speed. However, according to the values of f in the table, it provides less interesting solutions when compared to GA and EM algorithms. EM algorithm, in turn, provides better solutions, compared to PSO and GA, with a reasonable convergence speed. On the other hand, GA algorithm needs more time (iterations) to reach its best solution which is often less interesting than the one provided by our proposed EM approach.

Evaluating the impact on coverage

Figures 6, 7 and 8 illustrate the impact of the number of clients, the number of routers and the radius of coverage, respectively, on client coverage. Figure 6 illustrates that the client coverage increases when increasing the number of clients in the deployment area. However, our proposed EM approach covers more clients than PSO and GA. For example, when the number of clients is 450, EM algorithm covers 39 (35, respectively) clients more than PSO (GA, respectively) which represents more than 8.5% (8%) of the total number of clients.

The same remarks are observed in Fig. 7 which illustrates the impact of the number of routers n on the client coverage. The obtained results demonstrate that when increasing the number of deployed routers, the total number of covered clients increases too. Though, EM outperforms the existing GA and PSO approaches. More precisely, when the number of routers exceeds ten, our proposed EM approach covers between 8 and 12 (4 and 10, respectively) more clients than PSO (GA, respectively).

Table 6 Network scenarios considered for convergence study
Fig. 5
figure5

Example of the convergence speed of GA, PSO and EM algorithms

Table 7 Convergence study between GA, PSO and EM algorithms
Fig. 6
figure6

Effect of the number of clients on coverage

Fig. 7
figure7

Effect of the number of routers on coverage

The radius of coverage, also, affects the coverage metric as illustrated in Fig. 8. The results illustrated in this figure show that the number of covered clients is proportional to the radius of coverage of every router. Thus, for all algorithms, when increasing the radius, the number of covered clients increases too. Moreover, our proposed approach outperforms PSO (GA, respectively), especially when the radius of coverage is between 100 and 250 m, which are realistic values when using WLAN standards such as IEEE 802.11. However, when the radius exceeds 350 m, mesh routers will cover a larger area, resulting in covering all mesh clients.

Fig. 8
figure8

Effect of the radius on coverage

Evaluating the impact on connectivity

The impact of the number of clients m on the network connectivity has been studied in Fig. 9. The results presented in this figure show, again, that the proposed EM technique provides better connectivity than PSO and GA. It is also observed that the connectivity is increased when increasing the total number of mesh clients.

Fig. 9
figure9

Effect of the number of mesh clients on connectivity

The same remarks are observed in Fig. 10 that show the evolution of the network connectivity when varying the number of mesh routers. In fact, when adding more routers, the network connectivity for the three algorithms is increased. It is obvious that when adding mesh routers, the number of subgraph components will be reduced by connecting some subgraphs to form bigger subgraphs. Consequently, the size of the biggest component may also be increased.

Fig. 10
figure10

Effect of the number of mesh routers on connectivity

In Fig. 11, we have measured the impact of varying the radius of MRs, from 50m to 500m, on the size of the biggest subgraph component in the network. When increasing the radius, MRs have more chance to cover more MCs and to connect to other MRs. Hence, the number of subgraph components will be reduced until forming one giant component, including approximately all MRs and MCs when the radius exceeds 300m.

Fig. 11
figure11

Effect of the radius on connectivity

Evaluating the impact on objective function value

We consider the same scenarios, but this time the aim is to measure the value of the function f (in Eq. 4) when varying the number of MCs, from 50 to 500 nodes, as shown in Table 8. For every scenario and for all algorithms, we compute the average value of f. The results show that when the number of MCs is increased, the value of f is slightly decreased even though the values of coverage and connectivity were increased. According to Eq. 4, when increasing the number of clients, the value of f will decrease even though the values of coverage and connectivity were slightly increased as already shown in Figs. 6 and 9. Again, the proposed EM scheme outperforms the existing PSO and GA approaches.

Table 8 Value of f under various mesh client numbers

On the contrary, when increasing the number of routers, the value of f increases too as shown in Table 9. In fact, our proposed approach improves the value of f by 9–15% (5– 13.5%, respectively) when deploying enough MRs (more than 14 MRs) compared to PSO (GA, respectively). The reason behind this increase is that varying the number of MRs has more impact on increasing coverage and connectivity than when varying MCs. On the other hand and according to Eq. 4, the value of n (which is increased in this experiment) is used only to normalize the connectivity, while the value of m (number of MCs) remains the same : 100 nodes. Thus, when increasing the number of routers n, \(\phi \) and \(\psi \) will be increased too and m remains the same. Since \(m \gg n\), the term \(\frac{\psi (G)}{(n+m)}\) will tend to \(\frac{\psi (G)}{m}\).

Table 9 Value of f under various mesh router numbers

The last scenario concerns the study of the impact of mesh routers’ radius on the value of f as shown in Table 10. It is observed from this table that the value of f is proportional to the radius value. Thus, more the mesh routers’ radius is greater, more the value of f is important. Our proposed EM approach performs, also, better than PSO and GA in this context.

Table 10 Value of f under various radius values

Conclusion

In this paper, we have proposed a new solution to deal with the optimal placement of mesh routers in order to improve the performance of a WMN in terms of client coverage and network connectivity. First, we have formulated the problem and discussed some relevant and representative works in the field of mesh router placement. Then, we have proposed to apply an algorithm called electromagnetism-like mechanism (EM), which mimics the attraction–repulsion mechanism of the electromagnetic theory based on Coulomb’s law, to determine the best positions of mesh routers. Finally, we have performed extensive simulations to study the performance of the EM algorithm under various parameter settings. The obtained results demonstrate that our proposed approach performs better than GA and PSO algorithms in terms of client coverage and network connectivity.

For future works, we plan to extend the proposed algorithm to support other variants of the router node placement problem such as mobility of clients, energy-harvested routers and consider clients’ demands. Furthermore, the proposed approach will be introduced in a joint design process, including rather than router node placement: gateway deployment, channel assignment, routing, and antenna placement.

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Correspondence to Lamri Sayad.

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Sayad, L., Bouallouche-Medjkoune, L. & Aissani, D. An Electromagnetism-like mechanism algorithm for the router node placement in wireless mesh networks. Soft Comput 23, 4407–4419 (2019). https://doi.org/10.1007/s00500-018-3096-y

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Keywords

  • Wireless mesh network
  • Router node placement
  • Facility location problem
  • Network design
  • Electromagnetism-like mechanism algorithm