# Resource Allocation for Reconfigurable Intelligent Surface Assisted Dual Connectivity
## 3 RIS Assisted Dual Connectivity  
We consider \(x^{r}_{u,j}\) as a binary association variable that indicates whether user \(u\) is associated with \(j\) via RIS \(r\) or not. The tuple \((u,r,j)\) in all the variables indicates the identification values of user, RIS, and BSs. When the value of \(r\) is 0, then there is no RIS participating in the link between the user \(u\) and BS \(j\). If \(r\) is not equal to 0, then the the user \(u\) is served by BS \(j\) via RIS \(r\). We denote \(d^{r}_{u,j}\) is the association variable between BS \(j\) and user \(u\) via RIS \(r\). When \(d^{r}_{u,j}=1\), then the user \(u\) receives SINR greater than the threshold \(\tau\) from two different BS via two different BSs. The DC association variable of user \(u\) with BS \(k\) via RIS \(q\) denoted as \(d^{q}_{u,k}\) is 1 only when its primary association \(x^{r}_{u,k}\) is 1 with some other BS \(j\) via RIS \(r\). Given the channel \((l^{0}_{u,i},f^{r,n}_{j},\xi^{r,n}_{u,j})\) and the SINR \((\omega^{r}_{u,j})\), the overall optimization problem for an \(\alpha\)-fair utility function is presented as follows. \[\mathbf{P1} : \max_{ \begin{subarray}{c}\beta_{r},\mu,\beta^{r}_{u,j},\omega^{r }_{u,j},\sum_{u\in\mathbf{U}}\Lambda_{\alpha}(\lambda_{u}),\\ \tau,\omega^{r}_{u,j},d^{r}_{u,j}\end{subarray} } \tag{14}\] \[\mathrm{s.t}\,\lambda_{u} = \sum_{r\in\mathbf{R}}\sum_{j\in\mathcal{I}}x^{r}_{u,j}\Bigg{(}l^{ r}_{u,j}\delta^{r}_{u,j}+\sum_{r\in\mathbf{R}}\sum_{k\in\mathcal{I}}d^{r}_{u,k}l^{ r}_{u,k}\delta^{r}_{u,k}\Bigg{)},\forall u\in\mathbf{U},\] (15) \[\sum_{u\in\mathbf{U}}\sum_{r\in\mathbf{R}}(x^{r}_{u,j}+d^{r}_{u,j })\delta^{r}_{u,j}\leq 1,\,\forall j\in\mathbf{I},\] (16) \[\sum_{r\in\mathbf{R}}\sum_{j\in\mathcal{I}}\bigg{(}x^{r}_{u,j}+d^ {r}_{u,j}\bigg{)}\leq 1,\,\forall u\in\mathbf{U},\] (17) \[x^{r}_{u,j}\text{ as in \eqref{eq:d_u_r_1},}\] \[d^{q}_{u,k} = \begin{cases}1,\text{if }k,q=\arg\max_{\begin{subarray}{c}k\in \mathcal{I},\{j\}\\ q\in\mathbf{R}\setminus\{r\}\end{subarray}}\{\omega^{q}_{u,k}\},\\ \omega^{r}_{u,j}\geq\tau,\omega^{q}_{u,k}\geq\tau,\\ 0,\text{otherwise},\forall u\in\mathcal{U},\end{cases} \] (18) \[x^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\] (19) \[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\] (20) \[\delta^{r}_{u,j}\geq 0,\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\] (21) \[\omega^{r}_{u,j}\text{ as in \eqref{eq:d_u_r_1},}\text{ as in \eqref{eq:d_u_r_1},}\] \[\beta_{r,n}\in[0,1],\] (22) \[\theta_{r,n}\in[0,\pi], \tag{23}\]  
where (14) is the maximization of the \(\alpha\)-fair utility function defined in (13) with respect to RIS parameters amplitude reflection coefficient \(\beta_{r,n}\), phase shift coefficient \(\theta_{r,n}\), the user scheduling time fraction \(\delta^{r}_{u,j}\), and the binary variables \(x^{r}_{u,j}\), and \(d^{r}_{u,j}\). The constraint in (15) specifies the actual data rate of the user \(u\). If \(u\) has the association only with its primary BS (PBS) via the RIS \(r\), then only \(x^{r}_{u,j}\) is 1. However, if \(u\) is a DC user via RIS \(r\), then \(d^{r}_{u,j}\) is also equal to 1. Thus DC user's data rate is computed from both the terms as in (15). However, if association is between the BS \(j\) and user \(u\) without RIS, then only \(x^{0}_{u,j}\) is 1. This is because the user \(u\) already has the strong direct LOS channel from BS \(j\). The constraint in (16) specifies that the sum of time fraction allocated to all the users by any BS \(j\) directly or via any RIS \(r\) can not exceed 1. Further, the constraint in (17) shows that the user \(u\) can be associated to one \(j\), i.e., if it associated to primary BS (PBS) \(j\), then \(x^{r}_{u,j}=1\) or if it associates to the secondary BS (SBS) \(j\), then \(d^{r}_{u,j}=1\). Both \(x^{r}_{u,j}\) and \(d^{r}_{u,j}\) will not be 1 at the same time for the same BS \(j\) and RIS \(r\).