## 3 Performance Analysis Framework
### Calculation of the MGF of the HSTS
)\) can be evaluated as in the following equation:  
\[f(a)=\frac{C^{a+1-m_{1}}\left(\beta_{1}-\delta_{1}\right)^{a+1-m_{1}}}{\left( s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1}-\delta_{1}\right)\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)\right)^{a+1}}\int\limits_{0}^{\infty}e^{ \frac{-\varsigma_{2}C(\beta_{1}-\delta_{1})t}{s\frac{E_{6}}{\sigma_{1}^{2}}+ \left(\beta_{1}-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right) }}\,t^{a}\left(1+t\right)^{-m_{1}}dt\]  
The MGF of the considered network can now be written, by using Equations (5), (9) and (16), as:  
\[M_{\gamma}(s) \triangleq \int\limits_{0}^{\infty}I_{2}f_{|h_{3}|^{2}}(z)dz\] \[= \int\limits_{0}^{\infty}I_{2}\lambda_{3}z^{m_{3}-1}e^{-\varsigma _{3}z}dz\]  
By putting Equation (16) in Equation (20), and after rearrangement of integrals, sums and manipulations, we finally write the MGF of our proposed system as:  
\[M_{\gamma}(s)=M_{1}-M_{2}+M_{3}+M_{4}+M_{5} \tag{21}\]  
In Equation (21):  
\[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]0 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]1 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]2\[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]3 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]4 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]5 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]6 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]7 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]8 \[\times\ \left(\frac{\left(s\frac{E_{6}}{\sigma_{1}^{2}}+\left(\beta_{1 }-\delta_{1}\right)\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)\right) \varsigma\mathcal{E}_{1}x}{s\frac{E_{6}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z \frac{E_{6}}{\sigma_{1}^{2}}\right)}+C(\beta_{1}-\delta_{1})}\right)^{-m_{1}}\]9 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]0 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]1 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]2 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]3 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]4 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]5 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]6 \[\times\ e^{\frac{-\varsigma_{2}\mathcal{E}_{1}x}{s\frac{E_{6}}{ \sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{6}}{\sigma_{1}^{2}}\right)}x^{e+m_ {2}+l-1}\left(1+\frac{\varepsilon}{x}\right)dx}\]7  
It can be noticed from Equation (21) that MGF contains finite and infinite integrals, which can be accurately/easily calculated by using numerical computing packages, such as MATLAB or Mathematica.