## 3 Methodology: Statistics of the Combined Received SNR
\begin{split} e(S_{K:M},\lambda_{1})&=\int_{S_{K:M} }^{\infty}dxf_{S_{K:M}}(x)\exp(\lambda_{1}x)\\ &=\frac{p_{k}}{\Gamma(m_{k})}\bigg{(}\frac{\eta_{k}}{S_{K:M}} \bigg{)}^{-m_{k}}\text{E}_{1-m_{k}}\bigg{(}S_{K:M}\bigg{(}-\lambda_{1}+\frac{ 1}{\eta_{k}}\bigg{)}\bigg{)}\\ &\text{or}\frac{p_{k}}{\Gamma(m_{k})}\bigg{(}\frac{\eta_{k}}{S_{K :M}}\bigg{)}^{-m_{k}}\bigg{(}S_{K:M}\bigg{(}-\lambda_{1}+\frac{1}{\eta_{k}} \bigg{)}\bigg{)}^{-m_{k}}\Gamma\bigg{(}m_{k},S_{K:M}\bigg{(}-\lambda_{1}+ \frac{1}{\eta_{k}}\bigg{)}\bigg{)}\\ &=\frac{p_{k}}{\Gamma(m_{k})}\eta_{k}^{-m_{k}}\bigg{(}-\lambda_{1 }+\frac{1}{\eta_{k}}\bigg{)}^{-m_{k}}\Gamma\bigg{(}m_{k},S_{K:M}\bigg{(}- \lambda_{1}+\frac{1}{\eta_{k}}\bigg{)}\bigg{)},\end{split}  \tag{13}\]  
respectively, where \(\Gamma(\cdot,\cdot)\) denotes an incomplete gamma function [46] (Section (8.350)), and \(\text{E}_{n}(x)=x^{n-1}\Gamma(1-n,x)\) denotes an exponential integral function [46] (Section (3.381)) [47]. Notably, \([c(z_{2})]^{M-K}\) and \([e(z_{2},-s_{1})]^{K-1}\) in (11) need to be derived in an integrable form. Hence, based on the derivation in Appendix A, \([c(z_{2})]^{M-K}\) in (11) can be obtained as (14). By using a similar approach, \([e(z_{2},-s_{1})]^{K-1}\) can be expressed in the summation form given in (15). Finally, for \(z_{1}>(K-1)z_{2}\), we can obtain \(\mathcal{L}_{s_{1}}^{-1}\Big{\{}[e(z_{2},-s_{1})]^{K-1}\Big{\}}\) in (11) as (16), and the detailed derivations are presented in Appendix B. \[ \begin{split}[c(z_{2})]^{M-K}=&\sum_{l=0}^{M-K} \binom{M-K}{l}\frac{(-1)!\,p_{k}^{M-K}}{\left(\Gamma(m_{k})\right)^{l}}[(m_{k}-1)!]^{l}\exp\biggl{(}-\frac{l}{\eta_{k}}z_{2}\biggr{)}\\ &\times\sum_{\begin{subarray}{c}i_{0}i_{1}\cdots i_{m_{k}-1}\geq 0 \\ i_{0}i_{1}+\cdots i_{m_{k}-1}=l\end{subarray}}\frac{l!}{i_{0}!i_{1}!\cdots i_{ m_{k}-1}!}\frac{m_{k}-1}{l!}\left(\frac{1}{\eta_{k}}\right)^{i_{j}}\left(\frac{z_{2}}{ \eta_{k}}\right)^{\sum_{j=0}^{m_{k}-1}j_{j}}\\ &\stackrel{{\text{\text{\tiny{@}}}}}{{=}}\sum_{l=0}^{M -K}\binom{M-K}{l}\frac{(-1)!\,p_{k}^{M-K}}{\left(\Gamma(m_{k})\right)^{l}}[(m_ {k}-1)!]^{l}\exp\biggl{(}-\frac{l}{\eta_{k}}z_{2}\biggr{)}\\ &\times\sum_{i_{1}=0}^{m_{k}-1}\sum_{i_{2}=0}^{m_{k}-1}\cdots\sum_{ i_{l}=0}^{m_{k}-1}\frac{1}{\prod_{j=1}^{K-1}i_{j}!}\left(\frac{z_{2}}{\eta_{k}} \right)^{\sum_{j=1}^{l}i_{j}}.\end{split}  \tag{14}\]  
\[ \begin{split}[e(z_{2},-s_{1})]^{K-1}=&\left[\frac{p _{k}}{\Gamma(m_{k})}\right]^{K-1}\eta_{k}^{-(K-1)m_{k}}[(m_{k}-1)!]^{K-1}\biggl{(} s_{1}+\frac{1}{\eta_{k}}\biggr{)}^{-(K-1)m_{k}}\exp\biggl{(}-(K-1)\biggl{(}s_{1}+ \frac{1}{\eta_{k}}\biggr{)}z_{2}\biggr{)}\\ &\times\sum_{\begin{subarray}{c}i_{0}i_{1}\cdots i_{m_{k}-1}\geq 0 \\ i_{0}i_{1}!\cdots i_{m_{k}-1}=K-1\end{subarray}}\frac{(K-1)!}{i_{0}!i_{1}!\cdots i _{m_{k}-1}!}\prod_{j=0}^{m_{k}-1}\left(\frac{1}{j!}\right)^{i_{j}}\biggl{(} \biggl{(}s_{1}+\frac{1}{\eta_{k}}\biggr{)}z_{2}\biggr{)}^{\sum_{j=0}^{m_{k}-1} j\cdot i_{j}}\\ &\stackrel{{\text{\tiny{@}}}}{{=}}\left[\frac{p_{k}}{ \Gamma(m_{k})}\right]^{K-1}\eta_{k}^{-(K-1)m_{k}}[(m_{k}-1)!]^{K-1}\biggl{(}s_ {1}+\frac{1}{\eta_{k}}\biggr{)}^{-(K-1)m_{k}}\exp\biggl{(}-(K-1)\biggl{(}s_{1} +\frac{1}{\eta_{k}}\biggr{)}z_{2}\biggr{)}\\ &\times\sum_{i_{1}=0}^{m_{k}-1}\sum_{i_{2}=0}^{m_{k}-1}\cdots\sum _{i_{k-1}=0}^{m_{k}-1}\frac{1}{\prod_{j=1}^{K-1}i_{j}!}\biggl{(}\biggl{(}s_{1 }+\frac{1}{\eta_{k}}\biggr{)}z_{2}\biggr{)}^{\sum_{j=1}^{K-1}i_{j}}.\end{split}  \tag{15}\]  
\[