## 3 System Performance
### Op  
In HSTCN, OP is an important performance measure, which is defined as the probability that the instantaneous SNDR falls below a predefined threshold \(x_{0}\). Before deriving the OP of the system, what requires to be considered principally is to get the probability distribution function (PDF) of \(\lambda_{1i}\) and \(\lambda_{2i}\), respectively. According to [20], the PDF for \(\lambda_{1i}\) is given by  
\[f_{\lambda_{1i}}\left(\lambda_{1i}\right)=\sum\limits_{\xi_{1}=0}^{m_{1}-1} \ldots\sum\limits_{\xi_{M}=0}^{m_{1}-1}\Xi\left(M\right)\lambda_{1i}^{\Lambda _{1i}-1}e^{-\Delta_{1i}\lambda_{1i}},\] (  
By substituting (18) and (19) into (25), (25) can be rewritten as (26), which is at the top of this page,  
\[\mathrm{F}_{r1i}\left(x_{0}\right)= \begin{cases}\vspace{0.2 cm}\sum\limits_{i=1}^{ p(\Delta_{2i})}\sum\limits_{j=1}^{r_{i}(\Delta_{2i})}\sum\limits_{\xi_{1}=0}^{m_{1}-1} \cdots\sum\limits_{\xi_{M}=0}^{m_{1}-1}\frac{\Xi(M)\chi_{ij}(\Delta_{2i}) \overline{\lambda}_{ij}^{-j}}{(j-1)!}\left[\frac{(j-1)\overline{\lambda}_{ij}^ {\prime}(\Delta_{ii}-1)!}{\Delta_{ii}^{\Lambda_{ii}}}\right.\\ \left.-e^{-Y_{2}}\sum\limits_{r=0}^{j-1}\sum\limits_{s=0}^{p}\binom{r_{s}}{s }\frac{(j-1)\overline{\lambda}_{ij}^{r_{2}-s}\overline{\lambda}_{2i}^{-p}}{v! }\left(s+\Lambda_{1i}-1\right)!\left(\Delta_{1i}+Y_{1}\right)^{-s-\Lambda_{ii} }\right],x_{0}<\frac{1}{F_{1}}\\ 1,x_{0}\geq\frac{1}{F_{1}},\end{cases}  \tag{26}\]  
where \(Y_{1}=\frac{F_{1}x_{0}}{1-L_{1}x_{0}}\) and \(Y_{2}=\frac{x_{0}}{1-L_{1}x_{0}}\). With the same method, the CDF for \(\gamma_{r2i}\) can be derived as  
\[\mathrm{F}_{r2i}\left(x_{0}\right)= \begin{cases}\sum\limits_{\xi_{1}=0}^{m_{1} -1}\cdots\sum\limits_{\xi_{M-1}}^{m_{1}-1}\frac{\Xi(M)}{\Lambda_{ii}^{\Lambda_ {ii}}}\gamma\left(\Lambda_{1i},\frac{\Lambda_{ii}L_{2}x_{0}}{1-F_{2}x_{0}} \right),x_{0}<\frac{1}{F_{2}}\\ 1,x_{0}\geq\frac{1}{F_{2}}.\end{cases}  \tag{27}\]  
With the help of (13), we can get the CDF for \(\gamma_{r3i}\) as  
\[\mathrm{F}_{r3i}\left(x_{0}\right)=\int_{0}^{\infty}\mathrm{Pr}\left(\lambda_ {1i}\leq\frac{\lambda_{2i}F_{2}x_{0}+H_{2}x_{0}}{1-L_{2}x_{0}}\right)f_{ \lambda_{2i}}\left(\lambda_{2i}\right)d\lambda_{2i}. \tag{28}\]  
By substituting (18) and (19) into (28), (28) is rewritten as (29), which is at the top of next page, where \(Y_{3}=\frac{F_{1}x_{0}}{1-L_{3}x_{0}}\) and \(Y_{4}=\frac{x_{0}}{1-L_{3}x_{0}}\). \[\mathrm{F}_{r3i}\left(x_{0}\right)= \begin{cases}\sum\limits_{\xi_{1}=0}^{m_{1 }-1}\cdots\sum\limits_{\xi_{M-0}=0}^{m_{1}-1}\sum\limits_{j=1}^{r_{i}(\Delta_{ 2i})}\sum\limits_{j=1}^{r_{i}(\Delta_{2i})}\frac{\chi_{ij}(\Delta_{2i}) \overline{\lambda}_{ij}^{-j}}{(j-1)!}\left[\frac{(\Delta_{ii}-1)!(j-1)!\overline {\lambda}_{ii}^{\Lambda_{ii}}}{\Delta_{ii}^{\Lambda_{ii}}}\right.\\ \left.-e^{-\Delta_{ii}Y_{4}}\sum\limits_{s=0}^{\Lambda_{ii}-1}\sum\limits_{t=0 }^{s}\frac{(\Delta_{ii}-1)!}{s!}\binom{r_{s}}{s}\frac{Y_{4}^{s-s}Y_{4}^{s}(i+ j-1)!}{\Delta_{ii}^{\Lambda_{ii}}(\Delta_{ii}Y_{3}+1/\overline{\lambda}_{2i})} \right],x_{0}<\frac{1}{F_{3}}\\ 1,x_{0}\geq\frac{1}{F_{3}},\end{cases}  \tag{29}\]  
With the same steps of deriving (26), the CDF for \(\gamma_{r4i}\) is expressed as  
\[\mathrm{F}_{r4i}\left(x_{0}\right)= \begin{cases}\sum\limits_{i=1}^{r_{i}( \Delta_{2i})}\sum\limits_{j=1}^{r_{i}(\Delta_{2i})}\frac{\chi_{ij}(\Delta_{2i}) }{(j-1)!}\gamma\left(j,\frac{L_{4}x_{0}}{\lambda_{2}(1-F_{4}x_{0})}\right),x_{ 0}<\frac{1}{F_{4}}\\ 1,x_{0}\geq\frac{1}{F_{4}}.\end{cases}