## 4 Joint Parameters Optimization
### _Initialization_
\begin{array}{l}E\left[\log_{2}\left(1+\frac{P_{l}n_{l}^{-a}}{\sum_{k=1}^{2}l _{k}+\sigma^{2}}\right)\right]\\ \stackrel{{(a)}}{{=}}\int_{0}^{\infty}\left[\log_{2}\left(1+ \frac{P_{l}n_{l}^{-a}}{\sum_{k=1}^{2}l_{l,k}+\sigma^{2}}\right)>t\right]dt\\ =\int_{0}^{\infty}\left[h>\left(2^{t}-1\right)\left(\sum_{k=1}^{2}l_{l,k}+ \sigma^{2}\right)r_{l}^{\alpha}P_{l}^{-1}\right]dt\\ \stackrel{{(b)}}{{=}}\int_{0}^{\infty}\exp\left[-\tau\sigma^{2}r_ {l}^{\alpha}P_{l}^{-1}\right]\exp\left[-\tau\;I_{l,1}r_{l}^{\alpha}P_{l}^{-1} \right]\\ \cdot exp\left[-\tau l_{l,2}r_{l}^{\alpha}P_{l}^{-1}\right]dt,\end{array} \] (A9)

where (a) is derived according to \(E\left(X\right)=\int_{0}^{\infty}P\left[X>x\right]dx\), (b) is obtained referring to \(h\sim\exp\left(\mu\right)\) and by setting \(2^{t}-1=\tau\) and \(\mu=1\). Substituting the following Laplace transform Equation (A10) into Equation (A9), and further substituting Equation (A9) into Equation (A8), we can then obtain the expression of the average rate of user type \(l\) in the network, which is expressed in Equation (7):

\[ \begin{array}{l}\mathcal{L}_{l,k}\left(\tau r_{l}^{\alpha}P_{l}^{-1}\right) \\ =\exp\left(-\pi\lambda_{k}\left(\frac{P_{k}}{P_{l}}\right)^{2/\alpha}r^{2}\tau ^{2/\alpha}\int_{\left(\beta_{k}/\tau\right)^{2/\alpha}}^{\infty}\frac{1}{1+x ^{2/\alpha}}dx\right).\end{array} \] (A10)

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