# Resource Allocation for Reconfigurable Intelligent Surface Assisted Dual Connectivity
## 2 System Model
### Physical Channel Model  
We consider the time division duplexing (TDD) based system and we focus on downlink (DL) in this work. The direct channel gain \(h^{0}_{u,j}\) between the BS \(j\) and the user \(u\) is given as follows. We denote \(x^{0}_{u,j}\) as the binary association variable that is 1, if the direct link exists between BS \(j\) and the user \(u\) and 0, if it does not. The association \(x^{0}_{u,j}\) is based on the maximum received power and it is given as  
\[x^{0}_{u,j}=\begin{cases}1,\text{if }j=\arg\max_{j}\{P_{j}h^{0}_{u,j}\},\\ 0,\text{otherwise},\forall u\in\mathbf{U},\,\forall j\in\mathbf{J},\end{cases} \tag{1}\]  
where \(P_{j}\) is the transmit power of BS \(j\) and \(h^{0}_{u,j}\) is the direct LOS channel gain between BS \(j\) and the user \(u\) as expressed below.  
\[h^{0}_{u,j}=10^{-}\Bigg{(}\frac{PL(u,j)+v+\xi-G_{j}-G_{u}}{10}\Bigg{)} \tag{2}\]  
where \(PL(u,j)\) is the pathloss between user \(u\) and BS \(j\), \(v\) is the small scale fading loss, \(\xi\) is the loss due to log-normal shadowing, \(G_{j}\) is the transmit antenna gain, and \(G_{u}\) is the gain of  
\begin{table}
\begin{tabular}{l c} \hline \(h^{0}_{u,j}\) & Direct channel gain at user \(u\) from BS \(j\) \\ \hline \(\dot{h}^{r}_{u,j}\) & Channel gain at user \(u\) from BS \(j\) via RIS \(r\) \\ \hline \(\dot{l}^{0}_{u,j}\) & Direct link rate of user \(u\) from BS \(j\) \\ \hline \(\dot{l}^{r}_{u,j}\) & Link rate of user \(u\) from BS \(j\) via RIS \(r\) \\ \hline \(x^{0}_{u,j}\) & Binary user association variable of user \\  & \(u\) with BS \(j\) without RIS \\ \hline \(x^{r}_{u,j}\) & Binary user association variable of user \\  & \(u\) with BS \(j\) via RIS \\ \hline \(\alpha\) & Fairness parameter for the \(\alpha\)-Fair scheduler \\ \hline \(\dot{\delta}^{0}_{u,j}\) & User scheduling time fraction for user \(u\) by BS \(j\) without RIS \\ \hline \(\dot{\delta}^{r}_{u,j}\) & User scheduling time fraction for user \(u\) by BS \(j\) via RIS \(r\) \\ \hline \(\Gamma(.)\) & Spectral efficiency in \(bits/symbol\) \\ \hline \(\mu_{b}\) & Blockage density \\ \hline \(\omega^{0}_{u,j}\) & DL received SINR of user \(u\) from a BS \(j\) \\ \hline \(\omega^{r}_{u,j}\) & DL received SINR of user \(u\) from a BS \(j\) via RIS \(r\) \\ \hline \(p_{bs}\) & Number of BSs \\ \hline \(\rho_{user}\) & Number of users \\ \hline \(\Lambda\) & Utility function \\ \hline \(d^{r}_{u,j}\) & Binary association variable for DC user \\ \hline \(\lambda_{u}\) & Data rate of user \(u\) \\ \hline \(\beta_{r,n}\) & Reflection amplitude of \(n^{th}\) antenna element of \(r^{th}\) RIS \\ \hline \(\theta_{r,n}\) & Phase shift of of \(n^{th}\) antenna element of \(r^{th}\) RIS \\ \hline \(T_{\alpha}\) & Throughput \\ \hline \end{tabular}
\end{table}
Table 1: Mathematical notations.  
user antenna. The pathloss between \(u\) and \(j\) is determined by the appropriate radio access terminal. The direct link SINR of user \(u\) from the BS \(j\) based on [21] is given as follows.  
\[\omega_{u,j}^{0}=\frac{P_{j}^{c}h_{u,j}^{0}}{\sum\limits_{j^{\prime}\in\mathbf{I} \setminus j}P_{j^{\prime}}^{c}h_{u,j^{\prime}}^{0}+\sigma^{2}},\ \forall u\in\mathbf{U}, \tag{3}\]  
where, \(\sum\limits_{j^{\prime}\in\mathbf{B}\setminus b}P_{j^{\prime}}^{0}h_{u,j^{\prime}}^ {0}\) is the interference from the other BSs, \(P_{j}\) is the transmit power of BS \(j\), and \(\sigma^{2}\) is the noise power. The link rate and the scheduling of the system is presented next.