## Appendix A  
If \(R_{\text{sec}}(z)\) is a concave function, its second derivative must be less than or equal to 0. To prove Lemma 1, let us first write the \(R_{\text{sec}}^{{}^{\prime}}\) function as follows:  
\[\begin{split} R_{\text{sec}}^{\prime}&=\beta_{1} \log_{2}\Biggl{(}1+\frac{P_{\text{BS}}\;z{|g_{\text{BS-HT}}}\;{g_{\text{HT- IR}}|}^{2}}{\sigma_{\text{IT}}^{2}}\Biggr{)}+k_{\text{HT}}\;\gamma_{2}\log_{2} \Biggl{(}1+\frac{P_{\text{active}}^{1}{|g_{\text{HT-IR}}|}^{2}}{\sigma_{\text{ IT}}^{2}}\Biggr{)}\\ &-\sum_{n=1}^{N}\Biggl{(}\beta_{1}\log_{2}\Biggl{(}1+\frac{P_{ \text{BS}}\;z{|g_{\text{BS-HT}}}\;{g_{\text{HT-ED}_{n}}|}^{2}}{\sigma_{\text{ ED}_{n}}^{2}+P_{\text{j}}{|g_{\text{j-ED}_{n}}|}^{2}}\Biggr{)}\Biggr{)}\\ &-\sum_{n=1}^{N}\Biggl{(}k_{\text{HT}}\;\gamma_{2}\log_{2} \Biggl{(}1+\frac{P_{\text{active}}^{1}{|g_{\text{HT-ED}_{n}}|}^{2}}{\sigma_{ \text{ED}_{n}}^{2}+P_{\text{j}}{|g_{\text{j-ED}_{n}}|}^{2}+P_{\text{BS}}{|g_{ \text{BS-ED}_{n}}|}^{2}}\Biggr{)}\Biggr{)}.\end{split} \tag{12}\]  
If the first derivative of the \(R_{\text{sec}}^{{}^{\prime}}\) is taken with respect to \(z\), the following equation can be expressed:  
\[\begin{split}\frac{\partial R_{\text{sec}}^{{}^{\prime}}}{ \partial z}&=\frac{P_{\text{BS}}{|g_{\text{BS-HT}}}\;{g_{\text{HT -IR}}|}^{2}\beta_{1}}{\ln 2\Bigl{(}\sigma_{\text{IT}}^{2}+P_{\text{BS}}\;z{|g_{\text{BS-HT}} }\;{g_{\text{HT-IR}}|}^{2}\Bigr{)}}-\frac{k_{\text{HT}}P_{\text{BS}}{|g_{ \text{BS-HT}}}\;{g_{\text{HT-IR}}|}^{2}\beta_{1}}{\ln 2\Bigl{(}\sigma_{\text{IT}}^{2}+P_{ \text{active}}^{1}{|g_{\text{HT-IR}}|}^{2}\Bigr{)}}\\ &-N\beta_{1}\frac{P_{\text{BS}}{|g_{\text{BS-HT}}}\;{g_{\text{HT -ED}_{n}}|}^{2}}{\ln 2\Bigl{(}\sigma_{\text{ED}_{n}}^{2}+P_{\text{j}}{|g_{\text{j-ED}_{ n}}|}^{2}+P_{\text{BS}}\;z{|g_{\text{BS-HT}}}\;{g_{\text{HT-ED}_{n}}|}^{2} \Bigr{)}}\\ &+Nk_{\text{HT}}\;\frac{{|g_{\text{BS-HT}}}\;{g_{\text{HT-ED}_{n} }|}^{2}P_{\text{BS}}\;{\beta_{1}}}{\ln 2\Bigl{(}\sigma_{\text{ED}_{n}}^{2}+P_{\text{j}}{|g_{ \text{j-ED}_{n}}|}^{2}+P_{\text{BS}}\;{|g_{\text{BS-ED}_{n}}|}^{2}+P_{\text{ active}}^{1}{|g_{\text{HT-ED}_{n}}|}^{2}\Bigr{)}}.\end{split} \tag{13}\]  
For convenience, all system parameters such as channel gains-noise power of EDs are considered equal to each other. The second derivative of \(R_{\text{sec}}^{{}^{\prime}}\) with respect to \(z\) is found as follows:  
\[\begin{split}\frac{\partial^{2}R_{\text{sec}}^{{}^{\prime}}}{ \partial z^{2}}&=-\frac{P_{\text{BS}}^{2}{|g_{\text{BS-HT}}}\;{g_{ \text{HT-IR}}|}^{4}\beta_{1}}{\ln 2\Bigl{(}\sigma_{\text{IT}}^{2}+P_{\text{BS}}\;z{|g_{ \text{BS-HT}}}\;{g_{\text{HT-IR}}|}^{2}\Bigr{)}^{2}}-\frac{k_{\text{HT}}P_{ \text{BS}}^{2}{|g_{\text{BS-HT}}}\;{g_{\text{HT-IR}}|}^{4}\beta_{1}}{\ln 2\; \gamma_{2}\beta_{2}\Bigl{(}\sigma_{\text{IT}}^{2}+P_{\text{active}}^{1}{|g_{ \text{HT-IR}}|}^{2}\Bigr{)}^{2}}\\ &+N\beta_{1}\frac{P_{\text{BS}}^{2}{|g_{\text{BS-HT}}}\;{g_{ \text{HT-ED}_{n}}|}^{4}}{\ln 2\Bigl{(}\sigma_{\text{ED}_{n}}^{2}+P_{\text{j}}{|g_{ \text{j-ED}_{n}}|}^{2}+P_{\text{BS}}\;z{|g_{\text{BS-HT}}}\;{g_{\text{HT-ED} _{n}}|}^{2}\Bigr{)}^{2}}\\ &+Nk_{\text{HT}}\frac{{|g_{\text{BS-ED}_{n}}}\;{g_{\text{HT-ED}_{ n}}|}^{2}P_{\text{BS}}^{2}\;{\beta_{1}}{|g_{\text{BS-HT}}|}^{4}}{\gamma_{2}\beta_{ 2}\ln 2\Bigl{(}\sigma_{\text{ED}_{n}}^{2}+P_{\text{j}}{|g_{\text{j-ED}_{ n}}|}^{2}+P_{\text{BS}}\;{|g_{\text{BS-ED}_{n}}|}^{2}+P_{\text{active}}^{1}{|g_{ \text{HT-ED}_{n}}|}^{2}\Bigr{)}^{2}}.\end{split} \tag{14}\]  
Since \(R_{\text{sec}}(z)=P(H_{1})(P_{d})R_{\text{sec}}^{{}^{\prime}}\) and \(R_{\text{sec}}^{1}=\left(R_{\text{s}_{1}}-R_{\text{ED}}^{1}\right)^{+}\), the total number of bits reaching EDs cannot be greater than \(R_{\text{s}_{1}}\). If it is larger, the system throughput is already considered 0. Therefore, by lengthy but simple algebraic operations, it is shown that \(\frac{\partial^{2}R_{\text{sec}}(z)}{\partial z^{2}}\leq 0\)[12; 45].