## 6 Performance Evaluation
### Sensor Node Position Estimation Accuracy  
Firstly, we evaluate the distance and closeness algorithms when selecting the most appropriate reference points. We simulate the same scenario for both algorithms together with the position distance algorithm to estimate node positions. Figure 11 shows the number of nodes within different position estimation error ranges. As can be seen, closeness has more nodes in the small error range while distance exceeds closeness in larger error ranges. Thus, the closeness algorithm outperforms the distance algorithm because it considers all possible combinations of intersection points and chooses the closest points within a combination set to be used as reference points. Therefore, the closeness algorithm is used in the next simulations in order to obtain reference points. The next simulation evaluated the time it takes for a sensor node to become qualified, as shown in Figure 12a. Both the position distance and hybrid algorithms showed similar performances to that with no validation, and they performed better than the circle limit algorithm. This can be explained in Figure 6, where the position distance algorithm has a smaller invalid area. This means that the probability of accepting a position packet is higher and, consequently, nodes become qualified faster. Additionally, the hybrid algorithm's  
Figure 11: Impact of reference point algorithms in sensor node accuracy. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Network Size & 1000 \(\times\) 1000 m\({}^{2}\) \\ Number of sensor nodes & 50, 75, 100, 125, and 150 \\ Number of sinks & 1 \\ Sink start position & (500, 500) \\ Sink avg speed & 10 m/s \\ Sink mobility model & Random waypoint \\ Sink beacon frequency & 5 s \\ Transmission range & 185 m \\ Position distance threshold & 50 m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation parameters. performance is similar to that of the position distance algorithm because it uses the same algorithm until the node is considered qualified. Figure 12b shows the average position estimation error, in terms of the distance between the estimated and actual position for all nodes. The position distance algorithm outperforms the circle limit algorithm because it validates and accepts more position packets. The hybrid algorithm starts to perform better than position distance over time because, after the sensor node is qualified, it starts running the circle limit algorithm, which results in better estimations. Although the algorithm with no validation accepts every position packet, it is outperformed by the other algorithms because it does not avoid situations depicted in Figure 5, resulting in inaccurate estimations. Although position distance shows better performance in terms of position estimation error when considering all nodes, the circle limit algorithm shows improved performance when we only consider qualified nodes, as shown in Figure 13a. This is because scenarios such as the one depicted in Figure 5b are impossible to occur in the circle limit algorithm, whereas position distance minimizes their occurrences. Hybrid outperforms position distance over time for the same reason that it starts running the circle limit algorithm after having qualified nodes, as discussed earlier. The algorithm with no validation is outperformed once again because it does not avoid situations depicted in Figure 5, which results in inaccurate estimations. Figure 13b shows the position estimation outlier errors for qualified nodes. The circle limit algorithm demonstrates superior performance by ensuring that each quadrant has only one position packet, leading to better estimations for qualified nodes. The hybrid algorithm shows improved results over time because it eventually switches to the circle limit algorithm. However, the algorithm with no validation and position distance exhibit similar performance, as they both do not completely avoid scenarios, such as the one shown in Figure 5b. Figure 12: (**a**) Number of qualified nodes.