## 5 Simulation Results and Discussions
### Performance of Different Algorithms  
As a comparison, we compare the performance of our algorithm to other three algorithms. Figures 2-4 show the system sum rate for different relay selection algorithms. The curve marked EORS is simulated by the exhaustive optimal way for relay selection, which guarantees a theoretical \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Cellular layout & one isolated cellular cell \\ System area & 200 m \(\times\) 200 m \\ UE distribution & randomly distributed \\ The exponent for D2D pathloss & 4 \\ D2D shadow fading std & 6 dB \\ Noise spectral density & \(-\)174 dBm/Hz \\ UE antenna gain & 0 dBi \\ Channel bandwith & 5 MHz \\ D2D node Tx power & 20 dBm \\ Cluster size & 4\(\sim\)12 \\ Realizations & 1000 \\ \hline \end{tabular}
\end{table} 
Table 1: Main simulation parameters. top bound of the system sum data rate. The curve marked GHRS is the result of a greedy heuristic algorithm for relay selection in [8], in which each NACK-UE selects the ACK-UE with best D2D link between them as the relay. The complexity for the greedy heuristic is \(\mathcal{O}(NM)\). Our algorithm has a higher complexity, but has a much better performance. The last one marked FSRS is the simulation result using the fixed single relay selection strategy in [6], where only one ACK-UE is selected as relay to participate in data retransmission. The selected relay, which can obtain the highest multicast rate amongst all ACK-UEs, multicasts data to all NACK-users. Figure 3: System sum data rate for different relay selection algorithms in the case of 4 NACK-Ues. Figure 2: System sum data rate for different relay selection algorithms in the case of 4 ACK-UEs. We plot the system sum date rate with different numbers of NACK-UEs and different numbers of ACK-UEs in Figures 2 and 3. Then, we observe that the system sum data rate goes up with both the number of NACK-UEs and the number of ACK-UEs increase. It is obvious as more NACK-UEs request for relays, rising D2D links lead to higher sum data rates. On the other side, as the amount of ACK-UEs increases, the probability of D2D links with better channel conditions being assigned to NACK-UEs enhances, an thus the performance of sum data rate is improved, even though the increase is slow as Figure 3 shows. This phenomenon is similar to the effect of multi-channel diversity. Definitely, ACK-UEs also contribute to the performance. In addition, we can see that the proposed algorithm CARS is relatively much superior to the FSRS, and with the increase of NACK-UEs and ACK-UEs, the superiority becomes more evident. This is because in the FSRS algorithm, the multicast rate likely suffers a bottleneck due to poorer D2D links between the selected relay and its recipients, and the possibility is greater for the cluster with more NACK-users; while the CARS choose to use multi-relay with good channel conditions capable for high multicast rates, which enjoys more benefits from great independent variations in D2D channels by exploiting multi-channel diversity with more relays. From the simulation results, we also find that the CARS performs better than the GHRS and the absolute performance gap is obviously in the case of 4 ACK-UEs shown in Figure 3 and expands with more NACK-UEs. In other words, when the number of ACK-UEs becomes smaller relative to NACK-UEs, the range of choice of relays becomes narrow, which makes the GHRS suffer a setback. Whereas the CARS can fully exploit the multi-channel diversity gain by adaptively selecting the optimal relays, which contributes to the efficient data delivery. Meanwhile, as the theoretical optimum, the EORS results in the highest system sum rate, but the superiority compared to CARS is quite small, especially when the number of ACK-UEs increases as Figure 3 shows.