## 5 Simulation Results  
In this section, we provide numerical and simulation results to validate our analysis. The simulation setting follows the system model in Section 4 with \(R_{0}=2\) bit/s/Hz, \(\left\{\lambda_{k}\right\}_{k=1}^{K}=\lambda\), \(P=1\), \(N_{0}=1\) (hence, the received SNR will be lumped into the channel variances), and \(\left\{\rho_{k}\right\}_{k=1}^{K}=\rho\). Figure 4 illustrates the relation between the system outage probability and the average received SNR. As a benchmark comparison, we use Equation (6) of [27] with some appropriate modifications to obtain a lower bound on the outage probability of the system under independent fading channels. We first can see that when \(\rho=0\), the derived lower bound is exactly the same with the benchmark lower bound for the independent channels case. It validates our derived bounds on the system outage probability. Secondly, when \(\rho\) is small (close to zero) the analytic lower bound is relatively tighter than the upper counterpart, while when \(\rho\) is close to one, the analytic upper bound is relatively tighter than the lower one. This observation infers that the correlation between two channels of each relay has a significant impact on the relation given in Equation (6). Moreover, it is worth noting that when the channel correlation coefficients increase, the system outage probability decreases in low and intermediate SNR range, while it stays the same in the high SNR region. This fact can be further validated through Figure 5, in which we simulate the system outage probability versus the channel correlation coefficient for several average SNR values. Figure 5 shows that when the channel correlations approach one, the system outage probability decreases. One reasonable explanation may be described as follows. The end-to-end SNR at the destination via the relay \(R_{k}\) is upper bounded by the minimum between the two single-hop SNRs, i.e.,  
\(\gamma_{SR_{k}}\) and \(\gamma_{SR_{k}D}\). Under the independent channel scenarios, when \(\gamma_{SR_{k}}\) is large, \(\gamma_{R_{k}D}\) can be small and mainly decides the system performance. In other words, when one of the single-hop SNRs is large, the system performance can be sometimes good and sometimes bad. However, when the channels are strongly correlated, if \(\gamma_{SR_{k}}\) is large, \(\gamma_{R_{k}D}\) will also be large with a high probability. As a result, the system performance will be good with a high probability. This observation implies that the considered correlation form has a constructive effect on the system performance. Note that a similar observation for a single DnF relay system under the same correlated environment has been recently presented in [15; 28]. This is an important observation, which is in contrast to the existing results presented in [17; 18; 19; 20; 21], where channel correlations therein have a destructive effect on the system performance. In addition, it is noticed that an increase of the average SNR enlarges the threshold where the system outage probability starts decreasing. Particularly, in a high SNR range, for most values of \(\rho\), the effect of channel correlation on the system performance is negligible. To illustrate the effect of multiple relays on the system performance, we present Figures 6 and 7. In Figure 6, we simulate the system outage probability versus the number of relays at 20 dB average  
Figure 4: System outage probability versus average received SNR with \(K=4\). Figure 5: System outage probability versus average channel correlation with \(K=4\). SNR for independent channels and correlated channels with \(\rho=0.9\). The figure first shows that increasing the number of relays significantly improves the system performance, as expected. Moreover and importantly, it is observed that when the number of relays increases, the gap between the system performance under correlated channels and that under independent channels is enlarged. The reason is that the greater the number of relays, the greater the correlation coefficients we have.