we employ the common circular cell model [@baltzis_hexagonal_2011], where all the $N$ candidate-UEs are independently uniformly distributed (i.u.d.) in a circular cell having an inner radius of $R_{\min}$ and outer radius of $R_{\max}$, while the BS is located at the center point. We denote the locations in polar coordinates, i.e., $(r_k,\theta_k)$ is the location for the $k$-th candidate-UE and $(0,0)$ for the BS. Therefore, the probability density functions (PDFs) of $r_k$ and $\theta_k$ are given by (the subscript $k$ is omitted below for ease of notation) $$\label{eq_df_0}f_R(r)=\frac{2r}{R_{\max}^2-R_{\min}^2},\quad R_{\min}\leq r\leq R_{\max},$$ and $$f_{\Theta}(\theta)=\frac{1}{2\pi},\nonumber$$respectively. The cumulative distribution functions (CDFs) of $r_k$ and $\theta_k$ are written as $$\label{eq_df_1}F_R(r)=\frac{r^2-R_{\min}^2}{R_{\max}^2-R_{\min}^2},\quad R_{\min}\leq r\leq R_{\max},$$ and $$F_{\Theta}(\theta)=\frac{\theta}{2\pi},\nonumber$$respectively.

As a result, for $K^*$-RUS, the PDF of the distance from the $k$-th active-UE to the BS, i.e. $f_{d_{k}}^{\text{RUS}}(r)$, is given by $$f_{d_{k}}^{\text{RUS}}(r)=f_R(r),\quad R_{\min}\leq r\leq R_{\max},\quad k=1,\dots,K.$$Then, it is easy to obtain $$\label{eq_t123_1}\begin{split}
\sum_{i=1,i\neq k}^K\text{E}_{d_i}[d_i^{\alpha}]&=(K-1)\int_{R_{\min}}^{R_{\max}}r^{\alpha}f_R(r)dr\\
&=\frac{2(K-1)(R_{\max}^{\alpha+2}-R_{\min}^{\alpha+2})}{(\alpha+2)(R_{\max}^2-R_{\min}^2)}.
\end{split}$$Thus, $\tilde{\mathcal{R}}$ in can be rewritten as $$\label{eq_rus_0}\begin{split}
\tilde{\mathcal{R}}_{\text{RUS}}&=\left(1-\frac{K}{T}\right)K\\
&\times\int_{R_{\min}}^{R_{\max}}\log_2\left(1+\frac{1}{T_1r^{\alpha}+T_2r^{-\alpha}+T_3}\right)f_{d_k}^{\text{RUS}}(r)dr.
\end{split}$$Substituting and into , and then substituting and into , $\tilde{\mathcal{R}}_{\text{RUS}}$ can be reformulated as which is given on the next page.

$$\label{eq_rus_final}\begin{split}
&\tilde{\mathcal{R}}_{\text{RUS}}\\
&=\Gamma_1(T,M,K,R_{\min},R_{\max},P,c,\sigma_n^2,\rho,\alpha,\delta)\\
&=\left(1-\frac{K}{T}\right)K\\
&\times\int_{R_{\min}}^{R_{\max}}\frac{2r}{R_{\max}^2-R_{\min}^2}
\log_2\left(1+\frac{1-\rho}{\frac{\sigma_n^2}{Pc\phi M}\left(r^{\alpha}+\frac{2(K-1)(R_{\max}^{\alpha+2}-R_{\min}^{\alpha+2})}{(\alpha+2)(R_{\max}^2-R_{\min}^2)}\right)+\frac{\rho\psi}{M\phi^2-K\psi}\left(\frac{2(K-1)(R_{\max}^{\alpha+2}-R_{\min}^{\alpha+2})}{(\alpha+2)(R_{\max}^2-R_{\min}^2)}r^{-\alpha}+1\right)}\right)dr.
\end{split}$$

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We can see that is only related to the system parameters of $T,M,K,R_{\min},R_{\max},P,c,\sigma_n^2,\rho,\alpha\text{ 