## 5 Simulations
### Simulation Results
#### 5.2.3 Case 3: NLOS Scenarios with and without Gaussian Noises  
For Case 3, we executed six different types of NLOS scenarios with different noise and NLOS conditions. We observed that a difference of 0.5 in \(\eta\) in the propagation constants can differentiate between an LOS and an NLOS condition, and a difference of two can differentiate a deep-NLOS condition from an LOS condition [6]. Therefore, in this simulation case, we use different different propagation constants for the relay-client and server-relay link to simulate NLOS conditions. The simulation parameters are similar to Case 2 (\(\eta_{c}=2.52\)), but \(\eta_{s}=3.02\) for an NLOS scenario and \(\eta_{s}=4.52\) for a deep-NLOS scenario.  
The results of all six scenarios are presented in Figure 10. The left figures present the NLOS scenarios without noise, so that the optimum position can be observed clearly (which are at (10,0) and (22,0)  
Figure 9: Case 2 simulation results showing 100 trials of two different noise conditions. The theoretical optimum (*) is at the center (0,0). (**a**) LOS scenario with 1 dBm spatio-temporal Gaussian noise in RSS; (**b**) LOS scenario with 1 dBm spatio-temporal Gaussian noise in RSS.  
for NLOS and deep-NLOS scenarios, respectively). As \(\eta_{c}\) and \(\eta_{s}\) values are different, the theoretical optimum is not equidistant from the server and the client nodes. The center figure presents the NLOS scenarios with 1 dBm noise. The right figure presents the NLOS scenarios with 2 dBm noise. We present only the first 10 trials of the actual simulation, so that the algorithm behavior can be observed without ambiguity.  
The results of the performance metrics for all of the simulation cases are presented in Table 1. One can infer that as the complexity increases, the success rate drops (and the error increases). Figure 11 presents the relay node path for a random run in each simulation case overlaid on the value of the objective function (Equation (10)). The results show promising convergence to the maximum (shown as *) of the objective function value.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Scenarios**} & **MAE** & **RMSE** & **Distance** & **Time** & **Success** & **Speed of** \\  & **(m)** & **(m)** & **Cost (m)** & **Cost (\#iter)** & **Rate (\%)** & **Convergence (\(\frac{m}{\#iter}\))** \\ \hline Case 1: LOS without noise & 1.7 & 2.1 & 65.6 & 49.1 & 88 & 1.3 \\ Case 2: LOS with 1 dBm noise & 2.3 & 3.2 & 120.7 & 69.0 & 81 & 1.7 \\ Case 2: LOS with 2 dBm noise & 4.7 & 7.2 & 128.3 & 98.4 & 76 & 1.3 \\ Case 3: NLOS with 1 dBm noise & 3.3 & 6.1 & 161.8 & 81.9 & 78 & 1.9 \\ Case 3: NLOS with 2 dBm noise & 6.4 & 8.6 & 179.8 & 90.9 & 72 & 1.9 \\ Case 3: deep-NLOS with 1 dBm noise & 3.7 & 5.14 & 210.4 & 89.2 & 75 & 2.3 \\ Case 3: deep-NLOS with 2 dBm noise & 6.9 & 9.5 & 210.1 & 95.5 & 65 & 2.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation results for all considered cases.  
Figure 10: Case 3 simulation results showing 10 trials each. Without noise (**left**), with 1 dBm noise (**center**) and with 2 dBm noise (**right**). The results of NLOS conditions are presented in the first row, and in the second row, the results of deep-NLOS conditions are presented. The theoretical optimum position is indicated as *.  
In Figure 12, the influence of sensor separation on the error for LOS condition with 2 dBm noise are presented, which shows that the higher the sensors' separation, the lower the error in reaching the optimum position. However, the physical size of wireless relay nodes restricts the feasible sensor separation distances.  
Figure 11: Path taken by the relay node and the objective function value for all scenarios. The theoretical optimum positions are indicated as *. (**a**) LOS, without noise; (**b**) LOS, with 1-dBm noise; (**c**) LOS, with 2-dBm noise; (**d**) NLOS, without noise; (**e**) NLOS, with a 1-dBm noise; (**f**) NLOS, with 2-dBm noise; (**g**) Deep-NLOS, without noise; (**h**) Deep-NLOS, with 1-dBm noise; (**i**) Deep-NLOS, with 2-dBm noise.