## 5 Outage Probability Analysis
\begin{array}{l}\int_{z_{k}}^{\infty}\int_{0}^{y_{k}}\cdots\int_{0}^{y_{k}}f_ {q_{1,l_{1}}}(y_{1})\cdots f_{\gamma_{k,l_{K}}}(y_{K})dy_{1}\cdots dy_{k-1}dy_ {k+1}\cdots dy_{k}dy_{k}\\ =\int_{z_{k}}^{\infty}\left[1+\sum_{s=1}^{K-1}(-1)^{s}\sum_{z_{1},<\cdots<\cdots,z_{k}}\sum_{i_{s},i_{s}=0}^{m_{k,1}-1}\cdots\sum_{i_{s},i_{s}=0}^{m_{k,i}-1} \cdots\sum_{i_{s,s}=0}^{m_{k,i}-1}\cdots\sum_{i_{s,s}=0}^{m_{k,i}-1}\left( \prod_{i=1}^{n}\prod_{j=1}^{l_{s}}\frac{F_{\gamma_{k,j}}}{\Gamma(i_{s,j}+1)} \right)\right.\\ \left.\quad\times y_{k}^{\sum_{s}+1}\sum_{j=1}^{l_{s}}\sum_{i_{s},i_{s}}e^{-y \sum_{s}+1}\sum_{j=1}^{l_{s}}\frac{1}{\Gamma(i_{s})}\right]f_{q_{k}}(y_{k})dy_{k} \\ =\left\{e^{-\sum_{j}^{\lambda_{s}}\frac{z_{k}}{\tilde{\tau}_{k,j}}}\sum_{i_{s }=0}^{m_{k,1}-1}\cdots\sum_{i_{s,k},i_{s}=0}^{m_{k,i}-1}\left(\prod_{j=1}^{l_ {s}}\frac{\rho_{i_{j}}^{-i_{j}}}{\Gamma(i_{s,j}+1)}\right)\left(\frac{z_{k}^{ \lambda_{s}}}{z_{k}^{\lambda_{s}}}\right)\right\}\\ \quad+\sum_{s=1}^{K-1}(-1)^{s}\sum_{z_{1},<\cdots<\cdots<z_{k}}\sum_{i_{s },i_{s}=0}^{m_{k,i}-1}\cdots\sum_{i_{s,s}=0}^{m_{k,i}-1}\cdots\sum_{i_{s,s}=0}^ {m_{k,i}-1}\cdots\sum_{i_{s,s}=0}^{m_{k,i}-1}\sum_{i_{s,s}=0}^{m_{k,i}-1} \cdots\sum_{i_{s,k}=0}^{m_{k,i}-1}\left(\prod_{j=1}^{l_{s}}\frac{F_{\gamma_{k,j}}^{-i_{j}}}{\Gamma(i_{s,j}+1)}\right)\\ \quad\times\left(\prod_{i=1}^{m_{k}}\prod_{j=1}^{m_{k}}\frac{\rho_{i_{j}}^{-i_{ j}}}{\Gamma(i_{s,j}+1)}\right)\left\{\left(\sum_{j=1}^{l_{s}}\frac{1}{\Gamma(i_{s })}\right)\Omega_{k_{s}}^{-\Lambda_{k_{s}}-1}\Gamma(\Lambda_{k_{s}}+1,\Omega_{k_{ s}}z_{k})\right.\\ \left.\quad-\left(\sum_{j=1}^{l_{s}}i_{j}\right)\Omega_{k_{s}}^{-\Lambda_{k_{ s}}}\Gamma(\Lambda_{k_{s}},\Omega_{k_{s}}z_{k})\right\},\end{array}  \tag{17}\]where \(u_{1}<\cdots<u_{s}\) denote all the possible subsets with size \(s\) in the set \(\{1,\cdots,k-1,k+1,\cdots,K\}\), and \(\Gamma(\cdot,\cdot)\) is an incomplete gamma function. Furthermore, \(\Omega_{k,s}=\sum_{j=1}^{l_{k}}(1/\rho_{k,j})+\sum_{\alpha=1}^{s}\sum_{j=1}^{l_ {k_{k}}}(1/\rho_{u_{s,j}})\), and \(\Lambda_{k,s}=\sum_{j=1}^{l_{k}}i_{k,j}+\sum_{\alpha=1}^{s}\sum_{j=1}^{l_{k}}i _{u_{s,j}}\). Substituting Equations (16) and (17) into Equation (15), a closed-form expression for \(P_{out}\) with the local CSI-based PPS is finally obtained as  
\[\begin{split} P_{out}&=1-\sum_{k=1}^{K}\left\{\prod_{ j=l+1}^{N_{k}}\left(e^{-\frac{i_{k}}{R_{j}}}\sum_{i=0}^{n_{k}-1}\frac{1}{\Gamma( +1)}\left(\frac{\alpha_{k}}{R_{j}}\right)^{i  
extremely closer to that for the BPS with global CSI as \(m_{k,2}\) and \(m_{k,3}\) increase to 3 (i.e., \(\eta_{k,1}\) increases), where it is noted that even though only CSI for one of three hops is used for the PPS, it achieves the similar outage performance to the BPS with global CSI. It is also observed that as \(K\) rises, better \(\eta_{k,l_{k}}\) is required to attain the similar outage performance to the BPS. In addition, as the transmit SNR \(P_{t}/\sigma^{2}\) decreases, \(P_{out}\) for \(l_{k}=1\) and \(S_{k}=\{1\}\) is extremely closer to that for global CSI. Figure 5 shows the outage probability for various transmit SNR \(P_{t}/\sigma^{2}\) with \(K=2,3\) and \(R_{t}=1,2\)\(\mathrm{bps}/\mathrm{Hz}\) when \(m_{k,n}=2,3\), \(N_{k}=3\) and \(l_{k}=2\) for all \(k\) and \(n\). In addition, \(\beta_{k,1}=1\), \(\beta_{k,2}=4\), and \(\beta_{k,3}=8\) for all \(k\). Then, for \(l_{k}=2\) and \(n=\{1,2\}\), \(\eta_{k,2}=0.975,0.992\) with \(m_{k,n}=2,3\) for all \(n\), respectively. Analogous to the observation in Figure 4, the PPS with higher \(\eta_{k,2}\) provides the similar outage performance to the BPS, but a small performance gap between the PPS and the BPS is observed for high transmit SNR since the BPS achieves more spatial diversity as all \(m_{k,n}\)'s increase to 3. Nevertheless, it is noteworthy that the outage performance of the PPS using local CSI with \(l_{k}=2\) and \(n=\{1,2\}\) is extremely closer to that of the BPS using global CSI for low and medium transmit SNR regimes.