# Resource Allocation for Reconfigurable Intelligent Surface Assisted Dual Connectivity
## 2 System Model
### RIS Channel Model and Scheduling  
The physical channel model of RIS based system consists of two channel links. One is the channel between BS \(j\) and RIS \(r\) denoted as \(f_{r,j}^{n}\) and the other one is between RIS \(r\) and user \(u\), denoted as \(g_{u}^{n,r}\). The superscript \(n\) implies the \(n^{th}\) reflective element. The overall channel between user \(u\) and BS \(b\) via RIS \(r\) is the cascading of the above two channels along with the reflective parameter matrix of of RIS (\(\Theta\)), i.e., \(f_{r,j}^{m}\Theta g_{u}^{n,r}\). The SINR of user \(u\) from BS \(j\) via RIS \(r\) is denoted by \(\hat{\omega}_{u,j}^{r}\) as given below  
\[\omega_{u,j}^{r}=\frac{P_{j}(h_{u,j}^{0}+f_{r,j}^{n}\Theta g_{u}^{n,r})}{\sum \limits_{j^{\prime}\in\mathbf{I}\setminus j}P_{j^{\prime}}h_{u,j^{\prime}}^{0}+ \sigma^{2}},\ \forall u\in\mathbf{U},\ \forall r\geq 1 \tag{7}\]  
where, \(h_{u,j}^{0}\) is the direct channel gain between BS \(j\) and user \(u\), \(\Theta\) is the diagonal matrix containing phase shifts (\(\theta_{r,n}\)) and reflection coefficient values (\(\beta_{r,n}\)) of each element \(n\) of RIS \(r\). The \(\sum\limits_{j^{\prime}\in\mathbf{B}\setminus b}P_{j^{\prime}}^{0}h_{u,j^{\prime}}^ {0}\) is the interference from the other BSs, \(P_{j}\) is the transmit power of BS \(j\), and \(\sigma^{2}\) is the noise power. The interference from the other RIS is negligible because the received signal is assumed to be coherently combined and beamformed towards the intended receiver \(u\). When the diagonal matrix is expanded in terms of \((\theta_{r,n})\) and \((\beta_{r,n})\), the SINR equation is expressed as  
\[\omega_{u,j}^{r}=\frac{P_{j}\bigg{(}h_{u,j}^{0}+\sum\limits_{n=1}^{N}f_{j}^{r,n }\beta_{r,n}e^{j\theta_{r,n}}g_{u}^{r,n}\bigg{)}}{\sum\limits_{j^{\prime}\in I \backslash J}P_{j}^{c}h_{u,j^{\prime}}^{0}+\sigma^{2}},\ \forall u\in\mathbf{U},\forall r\geq 1 \tag{8}\]  
where \(N\) is the total number of antenna elements in RIS, \(\beta_{r,n}\), and \(\theta_{r,n}\) is the reflection coefficient and phase shift of \(n^{th}\) antenna element of \(r^{th}\) RIS, respectively. Similar to (4), if the user \(u\) is associated with BS \(j\) via RIS \(r\), then the link rate would be a function of \(\omega_{u,j}^{r}\) and is given as follows.  
\[l_{u,j}^{r}=\frac{\Gamma(\omega_{u,j}^{r})\,SC_{OFDM}\,SY_{OFDM}\mathbf{C}}{T_{sc}},\ j \in J,\ c\in\mathbf{C}\,. \tag{9}\]  
Given the link rate \(r\) as in the above equation, the actual data rate of the user \(u\) in the RIS system is expressed as  
\[\lambda_{u}=\sum\limits_{j\in I}\sum\limits_{r\in\mathbf{R}}x_{u,j}^{r}g_{u,j}^{r} l_{u,j}^{r},\forall u\in\mathbf{U}, \tag{10}\]  
where the user association \(x_{u,j}^{0}\) as in (12), link rate \(l_{u,j}^{r}\) as in (9), \(\delta_{u,j}^{r}\) is the user scheduling time fraction for user \(u\) by BS \(j\) via RIS \(r\). The \(\delta_{u,j}^{r}\) from BS \(j\) to user \(u\) via RIS \(r\) is given as  
\[\delta_{u,j}^{r}=\frac{(l_{u,j}^{r})^{\frac{1-\alpha}{\alpha}}}{\sum\limits_{u \in\mathbf{U}}\sum\limits_{r\in\mathbf{R}}x_{u,j}^{r}(l_{u,j}^{r})^{\frac{1-\alpha}{ \alpha}}},\ \forall j\in\mathbf{J}. \tag{11}\]  
where \(\alpha\) is the fairness parameter, and \(x_{u,j}^{r}\) is the user association between the BS \(j\) and user \(u\) via RIS \(r\). This is expressed as follows.  
\[x_{u,j}^{r}=\begin{cases}1,\text{if}\ j=\arg\max_{j,r}\{\omega_{u,j}^{r}\}, \\ 0,\text{otherwise},\forall u\in\mathbf{U},\,\forall j\in\mathbf{J},\,\forall r\in\mathbf{R}.\end{cases} \tag{12}\]  
The performance metrics considered in RIS-assisted DC systems are considered next.