## 4 Simulation Results and Discussion
### Simulation for One-Hop Setup
#### 4.1.1 Parameter Design for Phase 1  
In phase 1, the more iterations the algorithm executes, the closer the estimated location of a sensor approaches its stopping point. Due to the RSS ranging error, the eventual stopping point may not be the true location. The task of phase 1 is mainly to bring the estimated location to the stopping point, and then the location estimation continues with phase 2 to bring further from the stopping point to the true location. Therefore, for the performance of phase 1, our interest is to investigate the estimation error by measuring the distance between the estimated location of a particular sensor and its eventual stopping point rather than its true location. In other words, we measure  
\[error=\sqrt{(x_{e,i}-x_{s,i})^{2}+(y_{e,i}-y_{s,i})^{2}}\]  
where \((x_{e,i},y_{e,i})\) is the estimated location and \((x_{s,i},y_{s,i})\) is the stopping location of \(Sensor_{i}\), \(i\in\mathbb{S}\). Starting with some preparation experiments, we were able to narrow down the value range of \(\tau_{1}\) and \(\delta_{1}\). We then executed Algorithm 1 ten times for each pair of (\(\tau_{1}\), \(\delta_{1}\)) combinations, and calculated the mean error and number of iterations by averaging over executions. In order to make the experiment more realistic, the initial guess of sensor locations are obtained randomly and independently for each piece of execution. Figure 4 shows the estimation error versus \(\delta_{1}\) for different \(\tau_{1}\) values. As can be seen from the figure, \(\tau_{1}\) value directly governs the estimation error where a smaller \(\tau_{1}\) allows the algorithm to terminate and give a closer location to the stopping point. In other words, setting a large value for \(\tau_{1}\) may give undesirable estimation location produced in phase 1. In the figure, using \(\tau_{1}=10\) gives a 10 m error indicating that the produced location estimate is still 10 m away to the stopping point. Whereas using an very small \(\tau_{1}=0.001\) gives a very small error of below 1 mm indicating that the produced location estimate almost reaches the stopping point. Notice that in the setup, the average distance between stopping points and corresponding actual positions of sensors is about 16.52 m. As a result, \(\tau_{1}\leq 0.1\) looks appropriate by giving a mean error less than 0.1 m for all cases of \(\delta_{1}\) values. \begin{table}
\begin{tabular}{c c} \hline Variable & Definition \\ \hline \(d\) & estimated distance between the transmitter and the receiver [m] \\ \(P_{TX}\) & transmitted power level [dBm] \\ \(P_{RX}\) & received power level [dBm] \\ \(G_{TX}\) & antenna gain of the transmitter [dBi] \\ \(G_{RX}\) & antenna gain of the receiver [dBi] \\ \(\lambda\) & signal wavelength [m] \\ \(n\) & path loss exponent \\ \(X_{\alpha}\) & Gaussian random variable with a standard deviation of \(\alpha\) \\ \hline \end{tabular}
\end{table} 
Table 1: The variables involved in Equation (8). Moreover, the figure also shows that the estimation error generally reduces as \(\delta_{1}\) increases for all the considered \(\tau_{1}\) settings. For this setup, the estimation errors reach their lowest values at around \(\delta_{1}=0.4\). The errors then jump to higher values as \(\delta_{1}\) increases further indicating diverging of the algorithm. In terms of the accuracy, it is the best choice to fix \(\delta_{1}=0.4\). While the previous results suggest that \(\tau_{1}\leq 0.1\) and \(\delta_{1}=0.4\) gives a very close location to the stopping point, it is also necessary to consider the convergence speed measured by number of iterations in executions. Figure 5 plots the number of executed iterations using different combinations of \(\tau_{1}\) and \(\delta_{1}\). Since \(\tau_{1}\) governs the termination of iterations, it can be expected that a smaller \(\tau_{1}\) gives a more strict constraint on termination, thus lengthens the converging process and results in more number of iterations.