# Resource Allocation for Reconfigurable Intelligent Surface Assisted Dual Connectivity
## 3 RIS Assisted Dual Connectivity
Proposition 1: _Given the binary association of user \(u\) with BS \(j\) with RIS (\(r=0\)) or without RIS (\(r>1\)), the secondary association \(d_{u,j}^{r}\) based on the next maximum received SINR, the link rate of user \(u\) with BS \(j\) as a normal or DC user, the user scheduling time fraction allocated to the user \(u\) by BS \(j\) with or without RIS is given as follows._  
\[\delta_{u,j}^{r}=\frac{(x_{u,j}^{r}+d_{u,j}^{r})(l_{u,j}^{r})^{\frac{1-\alpha }{2}}}{\sum\limits_{u\in\mathbf{U}}\sum\limits_{r\in\mathbf{R}}(x_{u,j}^{r}+d_{u,j}^{ r})(l_{u,j}^{r})^{\frac{1-\alpha}{2}}},\ \forall j\in\mathbf{J}. \tag{26}\]  
Proof: Given the binary user association for the tuple (u,j,r), the optimization problem in (14) is convex with respect to the variable \(\delta_{u,j}^{r}\). The equivalent Lagrangian function of the (14) can be defined as follows. \[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]0 \[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]1  
where the \(\mathbf{Y}_{j}\), \(\mathbf{W}_{j}\), and \(\mathbf{Z}^{r}_{u,j}\) are intermediate Lagrangian variables. The necessary KKT conditions to compute the optimal solution for the given problem are as follows. \[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]2 \[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]3 (29) \[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]4  
where (28) is the first order necessary condition for optimal \(\delta^{r}_{u,j}\). (29) and (30) are the complementary-sackness conditions. Applying (28) in (27), we get  
\[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]5  
\[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]6  
The optimal \(\delta^{r}_{u,j}\) should satisfy the positivity condition \(\delta^{r}_{u,j}>0\) as given in (21) and should follow the complementary-sackness condition as in (30). Therefore, we consider \(\mathbf{Z}^{r}_{u,j}=0\). Using \(\mathbf{Z}^{r}_{u,j}=0\) in (32) and on further simplification,  
\[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]7  
substituting (33) in the constraint (19), we get,  
\[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]8  
Substituting (34) in the constraint (33), we get (26), and this completes the proof of _Proposition 1_. Similarly, the optimal user scheduling time fraction can be derived for \(\alpha=1\) and \(\alpha\rightarrow\infty\). The user scheduling time fraction of user \(u\) from BS \(j\) via RIS \(r\) for proportionally fair (PF) scheduler with \(\alpha=1\) is given as  
\[d^{r}_{u,j}\in\{0,1\},\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]9  
Further, when \(\alpha\rightarrow\infty\), the user scheduler with max-min fairness for RIS-assisted DC system is written as follows. \[\delta^{r}_{u,j}\geq 0,\,\forall u\in\mathbf{U},\,\forall j\in \mathbf{J},\,\forall r\in\mathbf{R},\]0The heuristic for solving the given optimization problem with the derived optimal user scheduling time fraction is presented next.