## 4 Simulation Results  
In order to find the optimal relay position that minimizes the outage probability, we performed simulations according to the source and relay positions for the three relay scenarios shown in Figures 3-5. We considered a circular cell with radius 100 m and assumed 25 sub-channels: one was the target sub-channel, and the other 24 sub-channels produced the other-channel interferences to the target sub-channel. In the target sub-channel, there was one D2D group with 30 receivers randomly distributed in a uniform distribution over a circular area of radius 30 m. For the other 24 sub-channels, there was one interferer per sub-channel at a time, and the other-channel interferers were randomly distributed in a uniform distribution over the cell area. The central position of the D2D group on the target sub-channel was (0 m, 0 m). The source was placed from (\(-\)30 m, 0 m) to (0 m, 0 m), and the relay was located from (\(-\)30 m, 0 m) to (30 m, 0 m) on the \(x\)-axis. The detailed simulation parameters are shown in Table 1.  
Figures 9-11 show the outage probabilities with respect to the relay positions with different source locations for Scenarios 1, 2, and 3. The solid and dotted lines indicate the simulation and analytical results, respectively. For the analytical results, Equations (11), (19), (24), (29), (31), and (33) were used for calculating the outage probabilities. While only the interference from the nearest interferer was considered for the analytical results, the sum interference from the 24 other-channel interferers was used for the simulation results. Hence, the outage probabilities produced by the simulation were always greater than the corresponding outage probabilities calculated using the equations. There was no significant difference between the results by the analysis and by the simulation, and the equations were used to estimate the outage probability and to find the optimal relay position. The figures show that an appropriate relay position needs to be selected to reduce the outage probability. In addition, we could see that a better performance could be achieved by considering the direct transmission from the source as well as the transmission through the relay. The best result was generated with Scenario 3, where multiple candidates for the relay were considered, and one of the candidates that successfully received the packet performed the retransmission. If multiple relays transmit at the same time or consecutively, better results can be obtained, but the complexity may increase as well.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Cell radius (\(R\)) & 100 m \\ Group radius & 30 m \\ Number of sub-channels & 25 \\ Number of interferers per sub-channel & 1 \\ Number of receivers in D2D group & 30 \\ Target signal-to-interference ratio (SIR) at relay (\(\Gamma_{R}\)) & 10 dB \\ Target SIR at destination (\(\Gamma_{D}\)) & 10 dB \\ Interference effect to other sub-channels (\(\lambda_{m,m,\,m}\
eq n\)) & \(-\)30 dB \\ Source transmission power (\(P_{S}^{TX}\)) & 20 dBm \\ Relay transmission power (\(P_{R}^{TX}\)) & 20 dBm \\ Interferer transmission power (\(P_{O1,m}^{TX}\), \(P_{O2,m}^{TX}\)) & 20 dBm \\ Path loss exponent between two devices (\(\alpha\)) & 4 \\ Center position of D2D group & (0 m, 0 m) \\ Source position (\(x_{S},y_{S}\)) & (30\(\sim\)0 m, 0 m) \\ Relay position (\(x_{R},y_{R}\)) & (\(-\)30\(\sim\)30 m, 0 m) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters for Figures 9–12.  
Figure 11: Outage probability (Scenario 3).  
Figure 10: Outage probability (Scenario 2).  
Figure 9: Outage probability (Scenario 1).  
Figure 14: Outage probability (Scenario 2).  
Figure 13: Outage probability (Scenario 1).  
Figure 15: Outage probability (Scenario 3).