## 4 Performance Evaluation
### Simulation Results  
We first find out the average estimated localization error, which is defined as the square root of the mean-square error (RMSE) that is due to difference between the estimated coordinate and the real coordinate of a normal node. As shown in Figure 7, the average estimated localization error for different number of beacon nodes with fixed number of total nodes \((N)\) is analyzed. In this simulation, the number of anchor nodes among the total number of nodes \(N\) is also fixed. From this figure, it is observed that the average estimated localization error decreases if number of beacon nodes increases. Besides, if more number of nodes \(N\) are deployed over the monitoring region, the average estimated localization error is also decreased. It is to be noted that the average localization error is more than 9 m when the number of beacon nodes is \(35\). Hence, in order to get more localization accuracy of the normal nodes, deployment of more beacon nodes is essential. Figure 6: Probability density functions of signal strength received from three beacon nodes. As discussed in Section 3, we use the path loss shadowing model as our propagation model given in Equation (3). In this model, \(X_{\sigma}\) is a random variable with standard deviation \(\sigma\), which affects the RSSI value and thereby influences the estimated average localization error. As shown in Figure 8, we simulated the percentage of deployed normal nodes with different standard deviation \((\sigma)\) to study the average estimated localization error. The number of anchor and beacon nodes in this experiment is fixed. It is noticed that the average estimated localization error is more for large value of the standard deviation. It is quite reasonable, as the large value of standard deviation means the degree of probability distribution is large, and therefore the average localization error is increased. Besides, the estimated localization error increases if percentage of deployed normal nodes is increased. It is to be noted that the average localization error is increased in Figure 8 due to the effect of standard deviation \((\sigma)\), only when 80\(\%\)\(\sim\)88\(\%\) of normal nodes out of the total number of deployed nodes is considered in the simulation. Figure 9 indicates how the average estimated localization error is affected for different percentage of normal nodes. This experiment is carried out for different communication range of the beacon nodes  
Figure 8: Effect of Standard Deviation (SD) on localization error for different number of normal nodes. Figure 7: Average estimated localization error for different number of beacon nodes. with fixed number of anchor nodes, which is equal to 9. From this figure, it is found that the average estimated localization error is reduced, if communication range of the beacon nodes is increased. This situation happens, since most of the normal nodes can receive enough beacon packets to calculate their location and thereby reducing the localization error. However, if percentage of normal node increases, average estimated localization error also increases, which is compatible with the results given in Figure 8.  
In Figure 10, the localization time for different number of network size \(N\) with different number of beacon nodes is shown. Here, the localization time increases with increase in number of beacon nodes. This situation arises as a normal node waits for \(T_{n}\) units and the communication time is also increased if number of beacon nodes increases. From Figure 10, it is interesting to note that the variation in localization time is much less although the node density changes. The analysis of average residual power for different communication range with different percentage of normal nodes is presented in Figure 11. In this experiment, first we measure the residual power of the beacon, anchor and normal nodes and then take the average of the total residual power.