## 3 Proposed Joint Resource Optimization for NOMA-Enabled Vehicular Networks
### Proposed Optimal Resource Optimization
#### 3.2.1 Formulation (Phase A): Association, Channel Assignment and NOMA-OMA Switching
\[\forall b,u:\quad\sum_{c}\Bigl{(}x_{bcu}^{h}+x_{bcu}^{l}\Bigr{)}\leq a_{bu} \tag{4}\]  
Each RSU has a limited power budget, \(P^{max}\), and the maximum amount of power for a channel is also bounded by \(P^{max}_{ch}\). The following constraint Equation (5) limits the transmission power allocated to a VU on a certain channel by \(P^{max}_{ch}\). In the constraint, the real domain variables \(p^{h}_{bcu}\in\mathbf{P}^{h}\) and \(p^{l}_{bcu}\in\mathbf{P}^{l}\) are the decision variables corresponding to the transmit power allocation to the VU \(u\) associated with RSU \(b\) on channel \(c\) for the high-power and low-power user, respectively. \[\forall b,c,u:\quad 0\leq p_{bcu}^{h}\leq x_{bcu}^{h}\cdot P_{ch}^{\max},\quad 0 \leq p_{bcu}^{l}\leq x_{bcu}^{l}\cdot P_{ch}^{\max} \tag{5}\]  
The total amount of transmission power that an RSU can allocate to all VUs on a single channel is also bounded by \(P_{ch}^{max}\) (refer to Equation (6)). \[\forall b,c:\quad\sum_{u}\Bigl{(}p_{bcu}^{h}+p_{bcu}^{l}\Bigr{)}\leq p_{ch}^{ \max} \tag{6}\]  
The total amount of power that an RSU can use for transmission is bounded by \(P^{\max}\) as shown in Equation (7). \[\forall b:\quad\sum_{c}\sum_{u}\Bigl{(}p_{bcu}^{h}+p_{bcu}^{l}\Bigr{)}\leq p^{ \max} \tag{7}\]  
Although the accessibility matrix indicates whether a VU \(u\) can receive the beacon message from an RSU \(b\), having \(a_{bu}=1\) is not sufficient to ensure that VU \(u\) will receive quality service from RSU \(b\) while minimizing the power consumption. To this end, the proposed system further checks that, given the per-channel and per-RSU power budget, the S(I)NR threshold can be satisfied. It is worth noting that the VU receiving downlink service at low power will perform SIC, resulting in an interference-free signal. On the other hand, the VU receiving high power will decode its own data from the composite signal, meaning that the interference should be considered. As a result, we have the following constraint Equation (8) that can be derived by simply rearranging the well-known Shannon capacity formula. In the constraint, \(\eta_{\text{thr}}\), \(\sigma\) and \(\rho\) are the S(I)NR threshold, per-Hz noise and a very large number, respectively. \[ \begin{split}\forall b,c:\quad\sum_{u}g_{bcu}\cdot p_{bcu}^{h}+ \rho\Biggl{(}1-\sum_{u}x_{bcu}^{h}\Biggr{)}\geq\eta_{\text{thr}}\Biggl{(}\sum _{u}g_{bcu}\cdot p_{bcu}^{l}+\sigma\Biggr{)},\\ \frac{\sum_{u}g_{bcu}\cdot p_{bcu}^{l}}{\sigma}\geq\eta_{\text{ thr}}\cdot\sum_{u}x_{bcu}^{l}\end{split}  \tag{8}\]  
The use of \(\rho\) makes the constraint convex as follows. The original formulation of the first part of Equation (8) is \(\sum_{u}g_{bcu}\cdot p_{bcu}^{h}\cdot x_{bcu}^{h}\geq\eta_{\text{thr}}\Bigl{(} \sum_{u}g_{bcu}\cdot p_{bcu}^{l}+\sigma\Bigr{)}\), where \(g_{bcu}\) corresponds to the channel gain between the RSU \(b\) and VU \(u\) on channel \(c\). Due to the coupling of the two variables \(p_{bcu}^{h}\) and \(x_{bcu}^{h}\), the problem becomes non-convex and it cannot be solved with computer solvers. To make the original constraint convex, the \(\rho\) notation has been introduced. If there is a user assigned as a high-power user, the \(\rho\) term will become zero. If not, the value of \(\rho\) remains, allowing \(p_{bcu}^{h}\) to be zero. Finally, the objective function of the Phase A formulation is given in Equation (9) which is to minimize the total power consumption from all RSUs. Although the first phase does not determine the actual transmit power under QoS constraint, the decisions cannot be made without considering power minimization. \[\min_{\mathbf{X},\mathbf{P}}\quad\sum_{b}\sum_{c}\sum_{u}\Bigl{(}p_{bcu}^{h}+ p_{bcu}^{l}\Bigr{)}, \tag{9}\]  
where \(\mathbf{X}=\{\mathbf{X}^{h},\mathbf{X}^{l}\}\) and \(\mathbf{P}=\{\mathbf{P}^{h},\mathbf{P}^{l}\}\). It is worth noting that depending on the number of VUs associated with an RSU and their channel conditions, some channels are assigned to a single VU each, resulting in the realization of OMA.