# Resource Allocation for Reconfigurable Intelligent Surface Assisted Dual Connectivity
## 3 RIS Assisted Dual Connectivity
The condition for user association with PBS and its associated RIS is given in the constraint (12). The user \(u\) is associated to the PBS \(j\) via the RIS \(r\) (if \(r=0\), without RIS) that has maximum SINR. Similarly, the condition for DC user association with its SBS \(k\) based on the constraint (18). The user is qualified to be a DC user if its SINR from both PBS \((\omega^{r}_{u,j})\) and SBS \((\omega^{q}_{u,k})\) via different RIS should greater thanthe threshold \(\tau\). The binary constraints of association variables are specified in (19) and (20), respectively. The positivity constraint on user scheduling time fraction through RIS \(r\) is given in (21). Since the set of RIS \(R\) also contains the value 0, this positivity constraint is ensured for the scheduling time fraction for the direct LOS link between BS \(j\) and user \(u\) also. The constraints on amplitude reflection coefficient \(\beta_{r,n}\) and the phase shift \(\theta_{r,n}\) of the \(n^{th}\) antenna element of \(r^{th}\) RIS are given in (22) and (23), respectively. The optimization problem presented in (14) is mixed integer non linear programming (MINLP) and is difficult to solve for all variables at once. Therefore, (14) is decomposed into subproblems that can be solved independently. Given the channel, \(f_{j}^{r,n}\) and \(g_{u,j}^{r,n}\), the maximum rate is achieved by selecting \(\theta_{r,n}=\angle h_{u,j}^{0}-\angle f_{j}^{r,n}g_{u,j}^{n}\) for each element \(n\)[22]. We assume a flat fading channel, i.e., the channel between BS \(j\) and every element \(n\) of the RIS \(r\) is the same. Therefore, the channel between BS \(j\) and RIS \(r\) is now denoted as \(f_{j}^{r}\). Similarly, the channel between RIS \(r\) and user \(u\) as \(g_{u,j}^{r}\) and the amplitude of the reflection coefficient is 1. The entire RIS is now focusing towards a single user \(u\). The modified SINR with the channel and the optimal phase shifts can be conclusively written as in [22] as  
\[\omega_{u,j}^{r}=\frac{P_{j}(\sqrt{h_{u,j}^{0}}+N\sqrt{f_{j}^{r}g_{u}^{r}})^{2 }}{\sum\limits_{j^{\prime}\in J\backslash j}P_{j^{\prime}}h_{u,j^{\prime}}^{0 }+\sigma^{2}},\ \forall u\in\mathbf{U},\forall r\in\mathbf{R},r\geq 1 \tag{24}\]  
where \(\sum\limits_{j^{\prime}\in I\backslash j}P_{j}h_{u,j^{\prime}}^{0}\) is the interference received by the user \(u\) from other BSs, \(P_{j}\) is the transmit power. Since the RIS parameters are optimized and the SINR is computed as in (24), the goal is now to determine the normal, RIS, and RIS-assisted DC users based on the computed SINR values. Given the SINR of \(u\) from BS \(j\) via RIS \(r\), the link rate of the given link \(l_{u,j}^{r}\) is computed as in (9), the global optimization problem is decomposed and formulated as user scheduling problem as below. \[\mathbf{P2} : \max\limits_{\theta_{u,j}^{r}}\sum\limits_{u\in\mathbf{U}}\Lambda_{ \alpha}(\lambda_{u})\] \[\mathrm{s.t}\left(\ref{eq:1}\right)-\eqref{eq:2}\]  
the problem in \(\mathbf{P2}\) is the maximization of \(\alpha\)-fair utility function given the association, SINR and the link rates. (25) is now a convex problem with respect to the user scheduling time fraction \(\delta_{u,j}^{r}\) based on [21] and it is solved using KKT conditions. The proposition for deriving optimal user scheduling time fraction \(\delta_{u,j}^{j}\) is presented next.