## 5 Outage Probability Analysis
Figure 3: Outage probability for \(N_{k}=3\) for all \(k\) when \(K=2\) and \(R_{t}=1\)\(\mathrm{bps}/\mathrm{Hz}\): (**a**) \(P_{k,n}/\sigma_{k,n}^{2}=22\)\(\mathrm{dB}\) and \(m_{k,n}=1\) for all \(k\) and \(n\); (**b**) \(P_{k,n}/\sigma_{k,n}^{2}=16\)\(\mathrm{dB}\) and \(m_{k,n}=2\) for all \(k\) and \(n\). Figure 4: Outage probability for \(m_{k,1}=1\), \(N_{k}=3\), and \(l_{k}=1\) for all \(k\) when \(K=2,3\) and \(R_{t}=1\,\mathrm{bps}/\mathrm{Hz}\). Figure 5: Outage probability for \(m_{k,n}=2,3\), \(N_{k}=3\), and \(l_{k}=2\) for all \(k\) and \(n\) when \(K=2,3\) and \(R_{t}=1,2\,\mathrm{bps}/\mathrm{Hz}\). Figures 6 and 7 compare the outage performances of the proposed DF-based PPS and the conventional AF-based PPS with \(N_{k}=3\) for all \(k\), \(K=3\), and \(R_{t}=1\) bps/Hz when \(P_{t}/\sigma^{2}\triangleq P_{k,n}/\sigma_{k,n}^{2}\) and \(\beta_{k,n}=1\) for all \(k\) and \(n\), where it is noted that the average SNRs, \(\beta_{k,n}P_{k,n}/\sigma_{k,n}^{2}\), are set to be equal for all \(k\) and \(n\), i.e., \(\beta_{k,n}P_{k,n}/\sigma_{k,n}^{2}=P_{t}/\sigma^{2}\). By the traditional long-term routing process, the relays can be clustered relatively close together to establish the end-to-end communication from the source to the destination [33]. Thus, by the long-term routing process, there are \(N-1\) clusters when \(N_{k}=N\) for all \(k\), where each cluster consists of \(K\) relays. In addition, the long-term routing process results in the equivalent average SNRs for all the hops. Hence, in Figures 6 and 7, the setting of the same average SNR, \(P_{t}/\sigma^{2}\), for all \(k\) and \(n\) is done by considering the long-term routing process. The routing method can monitor and utilize a change in large-scale fading such as path-loss and shadowing, but not that in small-scale fading. Thus, the proposed DF-based PPS and the conventional AF-based PPS [16] are allowed to use instantaneous CSI for path selection to track a change in small-scale fading. The conventional AF  
Figure 6 shows the outage performances of the proposed DF-based PPS and the conventional AF-based PPS when \(m_{k,n}\) for all \(k\) and \(n\) are equal. In wireless channels, a line-of-sight component becomes stronger as the value of the Nakagami parameter \(m_{k,n}\) increases. Hence, the outage performances of both PPS become better as \(m_{k,n}\) rises. From the figure, it is observed that the proposed DF-based PPS works better than the conventional AF-based PPS in the low SNR regime, but both outage performances becomes similar for high SNR. The reason is that the noise propagation problem of AF relaying becomes severe for low SNR. As seen in Figure 6, the proposed PPS achieves no performance improvement in the high SNR regime, as compared with the conventional PPS, because the distributions of channels for all the hops are identical, and thus the one-hop CSI feedback for the both PPS works the same way. Therefore, in Figure 7, we assume unequal \(m_{k,n}\)'s for all \(n\) but equal for all \(k\), which results in non-identical channel distributions for hops on each path. In the figure, there are the three non-identical cases: \(m_{k,1}=1\), \(m_{k,2}=2\), \(m_{k,3}=3\); \(m_{k,1}=2\), \(m_{k,2}=1\), \(m_{k,3}=3\); \(m_{k,1}=3\), \(m_{k,2}=2\), \(m_{k,3}=1\). For the three cases, the proposed PPS provides the same outage performance because the end-to-end channel distributions on each path are identical for all the three cases, and one-hop CSI feedback is done by selecting a hop with the maximum SNR correlation coefficient. However, the conventional PPS has different outage performances for the three cases since one-hop CSI feedback is always carried out using the first hop. It is noted that the conventional PPS does not change the first hop for CSI feedback, while the proposed PPS can change a hop for CSI feedback based on the SNR correlation coefficient.