## 5 Performance Analysis
Hence, we conclude that our proposal is more scalable than the deployment strategies found in the literature. In Figure 3, we plot the sensors positions using the Tabu Search approach. Compared to Figure 2 we can notice a clear concentration of the deployed sensors in the areas requiring high detection probability thresholds (top right, top left, and down right). Figure 4 illustrates the cumulative distribution function (cdf) obtained by all the deployment strategies of \(X\), where \(X\) is a random variable, which can take values in the set of \(\{\vartheta_{i,j}|\forall i,j\in\mathcal{A}\}\). Formally,  
\[\vartheta_{i,j}=(r_{(i,j)}-p_{(i,j)})\times 1_{\{r_{(i,j)}>p_{(i,j)}\}} \tag{10}\]  
\(\vartheta_{i}\) can take values in \([0,1]\). Each curve in Figure 4 indicates for each \(\delta\) value in the \(x\) axis the probability \(P\left(X\leq\delta\right)\). From the under figure, we can notice that our proposal Tabu Search, Diff-Deploy and DDA approaches provide the best performances compared to all other deployment algorithms. Moreover, we can observe that the Tabu Search approach satisfaction rate reaches very quickly \(100\%\) of the area units compared to the other approaches. This means that the not satisfied units receive a detection probability very close to the required detection probability thresholds. In figure 5, we illustrate the variation of the satisfaction rate and the number of deployed sensors at each Tabu Search iteration. A left y-axis represents the number of deployed sensors and a right y-axis shows the satisfaction rate. We remark that the number of sensors and the satisfaction rate oscillate, this behavior is due to the additional and suppression oriented stages. When we execute the suppression stage, the number of sensors is largely reduced consequently the satisfaction rate also is decreased. The suppression stage enables to the additional stage to generate different WSN topologies compared to the last one. Thereafter, we apply the additional stage in aim to deploy more sensors in areas under-covered. We remind that our aim is to minimize the number of deployed sensors with an additional stage and to  
Figure 4: cdf between desired and obtained detection probability  
reduce the gap between the generated and requested event detection probabilities. We can see in figure 5 that the number of sensors placed by the additional oriented stage is declining but the look of the satisfaction rate plot is constant (no variation). Besides, sometimes we have a growth curve of number of sensors. The main reason is the Tabu Search heuristic, it allows an impoverishment in the performance in order to do not be blocked in the local minima. The best solution among \(100\) Tabu Search iterations is obtained at the \(84^{th}\) iteration, in figure 5 is represented with a red vertical line. In order to compare the network connectivity between the different approaches, we run the second version of Tabu Search where the connectivity is ensured by construction. Tabu Search method deploys \(274\pm 2\) sensors and the satisfaction rate \(\theta\) is equal to \(99.01\pm 0.12\%\). With the same number of sensors (274), the satisfaction rate \(\theta\) of Random, Grid, Min_Miss, Max_Min_Cov, Max_Avg_Cov, Diff-Deploy, and DDA approaches are equal to \(79.51\pm 1.02\%\), \(88.84\%\), \(90.88\%\), \(86.64\%\), \(84.96\%\), \(100\%\) and \(100\%\) respectively. We can observe that Diff-Deploy and DDA reach \(100\%\) in satisfaction rate. Figure 6 shows the cumulative distribution function (cdf) of random variable X defined in equation 10. We calculate the number of connected components in all deployment topologies generated by the different deployment strategies. The network is connected if the graph of connectivity contains only one connected component. Figures 7 and 8 illustrate the graph of connectivity of Tabu Search and Diff-Deploy. We can see clearly that only our proposal Tabu Search approach provides a connected graph.