## 3 System Model and SNR Distributions  
We consider a multi-hop parallel relay system (MRS), as shown in Figure 1[23], where \(K\) parallel independent paths are assumed to be available for data transmission from a source to a destination, and the \(k\)-th path between the source and the destination consists of \(N_{k}\) serial hops with \(N_{k}-1\) relays. In MRS, one of \(K\) paths can be selected for data transmission to achieve spatial diversity according to the selection combining rule, i.e., only a single path with the best end-to-end link quality between the source and the destination can be chosen for transmission. The path selection is assumed to be performed by the source using CSI reported by the nodes. The relays operate in the half-duplex DF mode, where the half-duplex mode means that each relay is not allowed to transmit and receive the signal at the same time. Then, for orthogonal transmission, relay \(n\) decodes a data signal received from the preceding node in the \(n\)-th time slot, and retransmits the decoded signal to the succeeding node during the \((n+1)\)-th time slot. It is assumed that data communication is available only between adjacent nodes due to high path-loss and shadowing, and there are no CSI feedback delay and error. Let the channel coefficient for the \(n\)-th hop of the \(k\)-th path be denoted as \(h_{k,n}\), and its magnitude be assumed to follow Nakagami-\(m\) distribution with the integer parameter \(m_{k,n}\) and the average power \(\beta_{k,n}\), where it is noted that Nakagami-\(m\) fading channels can generally represent line-of-sight as well as non-line-of-sight channel environments. It is also assumed that all the channels are independent. The received SNR for hop \(n\) of path \(k\) is then given as \(\gamma_{k,n}=P_{k,n}|h_{k,n}|^{2}/\sigma_{k,n}^{2}\), where \(P_{k,n}\) and \(\sigma_{k,n}^{2}\) denote the transmit power and noise power, respectively. The end-to-end SNR for DF relaying on path \(k\) is defined as the weakest of the received SNRs for each of \(N_{k}\) hops [23] and expressed as \(q_{k}\triangleq\min_{n=1,\cdots,N_{k}}\{\gamma_{k,n}\}\), which also denotes the end-to-end SNR  
Figure 1: A multi-hop WSN with parallel relays from a source to a destination. for global CSI, i.e, end-to-end perfect CSI. For the BPS, the path with the maximum end-to-end SNR among \(K\) paths is chosen as follows: \(k^{G}=\arg\max_{k=1,\cdots,K}\{q_{k}\}\). However, only local CSI (i.e., any subset of global CSI as a limited version) can be available at the source in practice because of limited CSI feedback, and then a path can be selected by the local CSI-based PPS as follows: \(k^{L}=\arg\max_{k=1,\cdots,K}\{q_{k,l_{k}}\}\), where \(l_{k}\) is the number of hops chosen for CSI feedback in path \(k\), and \(q_{k,l_{k}}=\min_{n=1,\cdots,l_{k}}\{\gamma_{\mathcal{R},n}\}\), which represents the end-to-end SNR for local CSI. For the simplicity of mathematical expressions, the subset of hops in path \(k\) is set to \(\{1,\cdots,l_{k}\}\), but it can be easily generalized to any subset of hops with size \(l_{k}\). Let \(\rho_{k,n}\triangleq P_{k,n}\beta_{k,n}/(\sigma_{k,n}^{2}m_{k,n})\). The probability density function (PDF) and the cumulative distribution function (CDF) of \(\gamma_{k,n}\) are respectively given by  
\[f_{\gamma_{k,n}}(x)=\frac{1}{\Gamma(m_{k,n})}\left(\frac{1}{\rho_{k,n}}\right) ^{m_{k,n}}x^{m_{k,n}-1}e^{-\frac{x}{\rho_{k,n}}}, \tag{1}\]  
\[F_{\gamma_{k,n}}(x)=1-e^{-\frac{x}{\rho_{k,n}}}\sum_{i=0}^{m_{k,n}-1}\frac{1} {\Gamma(i+1)}\left(\frac{x}{\rho_{k,n}}\right)^{i}, \tag{2}\]  
where \(\Gamma(\cdot)\) is a gamma function [30]. Using Equation (2), the CDF of \(q_{k,l_{k}}(x)\) is obtained as  
\[ \begin{array}{rcl}F_{q_{k,l_{k}}}(x)&=&1-\Pr\left\{\gamma_{k,1}>x,\cdots, \gamma_{k,l_{k}}>x\right\}\\ &=&1-e^{-\sum_{j=1}^{L_{k}}\frac{x}{\rho_{k,j}}}\left\{\sum_{i_{1}=0}^{m_{k,1} -1}\cdots\sum_{i_{k}=0}^{m_{k,k}-1}\prod_{j=1}^{L_{k}}\frac{1}{\Gamma(i_{j}+1 )}\left(\frac{x}{\rho_{k,j}}\right)^{i_{j}}\right\}.\end{array}