## Conflicts of Interest: The authors declare no conflict of interest. ### _Asymptotic Analysis of System SOP_  
In this section, the asymptotic analysis of the system SOP is derived. Particularly, when \(\overline{\gamma}_{r}\rightarrow\infty\), the asymptotic expression of the SOP can be determined from (31) as:  
\[SOP^{\infty}=SOP_{1}^{\infty}+SOP_{2}^{\infty}.\] (A1)  
Thus, the asymptotic expression of the first term of (A1) can be obtained by applying [47; 48] as follows:  
\[SOP_{1}^{\infty}=\frac{\xi^{2}r^{\kappa+\beta-2}exp(-K)}{2\pi r^{-1}(\Gamma( \alpha)\Gamma(\alpha))}\sum_{l=0}^{\infty}\sum_{d=1}^{3r}\frac{K^{l}}{{(l!)}^{ 2}}\mathcal{H}\Big{(}\frac{(\alpha\beta)^{r}\Theta}{\eta r^{2r}\overline{ \gamma}_{r}}\Big{)}^{\psi_{4,l}},\] (A2)  
where  
\[ \begin{array}{l}\prod_{\begin{subarray}{c}g=1\\ g\
eq t\end{subarray}}^{3r}\Gamma\big{(}\psi_{3,j}-\psi_{3,t}\big{)}\prod_{i=1} ^{2}\Gamma\big{(}1-\psi_{3,j}+\psi_{4,t}\big{)}\\ \mathcal{H}=\frac{g\
eq t}{\prod_{t=3}^{r+2}\Gamma\big{(}\psi_{3,j}-\psi_{4,t} \big{)}\prod_{t=3r+1}^{3r+1}\Gamma\big{(}1-\psi_{3,j}+\psi_{4,t}\big{)}}.\end{array} \]  
In addition, the asymptotic expression of the second term of (A1) can be derived by applying [45] as:  
\[ \begin{array}{l}SOP_{2}^{\infty}=\omega\eta exp(-K)\sum_{l=0}^{\infty}\sum_ {q_{1}=0}^{m_{h}-1}\cdots\sum_{q_{N_{h}}=0}^{m_{h}-1}\frac{(Kq)^{l}\Xi(N_{h}) \Delta_{H}^{-\lambda}}{(l!)^{2}(2\pi l)^{2}}\int_{L_{1}}\frac{\Gamma(\lambda+ s)\Gamma(-s)\Gamma(s+1)}{\Gamma(1-s)}(\Delta_{H}C)^{-s}ds\times\\ \int_{L_{2}}\Gamma\big{(}\frac{t}{r}-s\big{)}\frac{\Gamma\big{(}\xi^{2}+t \big{)}\Gamma(\alpha+t)\Gamma(\beta+t)\Gamma(t-\frac{t}{r}+1)}{\Gamma(\xi^{2} +1+t)\Gamma\big{(}\frac{t}{r}+1\big{)}}\bigg{(}\frac{\Theta^{1/r}M}{(\eta \overline{\gamma}_{r})^{1/r}}\bigg{)}^{-t}dt.\end{array} \] (A3)  
By using the identity stated in Reference [44] (Equation (1.2)), (A3) can be further expressed as:  
\[ \begin{array}{l}SOP_{2}^{\infty}=\omega\eta exp(-K)\sum_{l=0}^{\infty}\sum_ {q_{1}=0}^{m_{h}-1}\cdots\sum_{q_{N_{h}}=0}^{m_{h}-1}\frac{(Kq)^{l}\Xi(N_{h}) \Delta_{H}^{-\lambda}}{(l!)^{2}(2\pi l)}\int_{L_{1}}\frac{\Gamma(\lambda+s) \Gamma(-s)\Gamma(s+1)}{\Gamma(1-s)}(\Delta_{H}C)^{-s}\times\\ H_{3,4}^{4,1}\bigg{(}\frac{\Theta^{1/r}M}{(\eta\overline{\gamma}_{r})^{1/r}} \bigg{|}\begin{array}{c}(-1,1/r)\\ (-s,1/r)\end{array}\big{(}\xi^{2},1\big{)}\begin{array}{c}(\alpha,1)(\beta,1)\\ (\alpha,1)(\beta,1)\end{array}\bigg{)}ds.\end{array} \] (A4)  
According to Reference [44], the asymptotic value of the Fox's H-function in (A4) can be expressed as the residue of the closest pole to the left of the integration path \(l\) when the argument tends to zero. Therefore, by utilizing the identity detailed in Reference [50] (Equation (1.8.4)), (A4) can be expressed as:  
\[ \begin{array}{l}SOP_{2}^{\infty}=\omega\eta exp(-K)\sum_{l=0}^{\infty}\sum_ {q_{1}=0}^{m_{h}-1}\cdots\sum_{q_{N_{h}}=0}^{m_{h}-1}\frac{r(Kq)^{l}\Xi(N_{h}) \Delta_{H}^{-\lambda}}{(l!)^{2}(2\pi l)}\bigg{(}\frac{(\eta\overline{\gamma}_{ r})^{1/r}}{\Theta^{1/r}M}\bigg{)}^{-r}\times\\ \int_{L_{1}}\frac{\Gamma(\lambda+s)\Gamma(-s)\Gamma(s+1)}{\Gamma(1-s)}\frac{ \Gamma\big{(}\xi^{2}+s\big{)}\Gamma(\alpha+t)\Gamma(\beta+s)\Gamma(1+t-s)}{ \Gamma(\xi^{2}+1+s\sigma r)\Gamma(s+1)}\bigg{(}\frac{M_{H}\xi\Theta^{1/r}M}{( \eta\overline{\gamma}_{r})^{1/r}}\bigg{)}^{-s}ds.\end{array} \] (A5)  
By applying the identity stated in Reference [44] (Equation (1.2)), (A5) can be solved as:  
\[\begin{array}{l}SOP_{2}^{\infty}=\omega\eta exp(-K)\sum_{l=0}^{\infty}\sum_ {q_{1}=0}^{m_{h}-1}\cdots\sum_{q_{N_{h}}=0}^{m_{h}-1}\frac{r(Kq)^{l}\Xi(N_{h}) \Delta_{H}^{-\lambda}}{(l! )^{2}(2\pi l)}\bigg{(}\frac{(\eta\overline{\gamma}_{ r})^{1/r}}{\Theta^{1/r}M}\bigg{)}^{-r}\times\\ H_{4,6}^{5,2}\bigg{(}\frac{\Theta^{1/r}M}{(\eta\overline{\gamma}_{r})^{1/r}} \bigg{|} \begin{array}{c}(1,1)(-1,1/r)\big{(}\xi^{2}+1,1)(1,1/r)\\ (\lambda,1)(1,1)\big{(}\xi^{2},r\big{)}\begin{array}{c}(\alpha,r)(\beta,r)(0,1 )\end{array}\bigg{)}.\end{array}