## 3 Materials and Methods
### System Model and Problem Formulation
#### 3.1.1 AV2X Communication Scenario  
The AV2X scenario is depicted in Figure 1, in which vehicles go along four streets, each with two sides, and communicate with everything, including infrastructure, other vehicles, and devices; the infrastructure is situated on two sides in the crossroads. As a result, the signal-to-interference-plus-noise ratio between autonomous vehicle-relay-vehicle (AR) and the combined signal-to-interference-plus-noise ratio between autonomous vehicle-everything (AX) and relay-vehicle-everything (RX) can be stated as in [32]:  
\[SINR_{AR}=\frac{P_{A}H_{AR}}{I_{1}+N} \tag{3}\]  
\[SINR_{AX,RX}=\frac{P_{A}H_{AX}}{I_{2}+N}+\frac{P_{A}H_{RX}}{I_{3}+N} \tag{4}\]  
where \(H_{AR},\ H_{AX},\) and \(H_{RX}\) are the channel gain coefficient and the transmitted symbol between autonomous vehicle and relay-vehicle, autonomous vehicle and everything, and between relay vehicle and everything, respectively. \(I_{1}\), \(I_{2}\), and \(I_{3}\) represent the interference that occurs between autonomous vehicle and relay-vehicle link, autonomous vehicle and everything link, and between relay vehicle and everything link, respectively. Symbol \(N\) represents the noise power. Therefore, \(d_{RX}\) can be deduced from Equation (4) as follows:  
\[d_{RX}=\left[\frac{((I_{2}+N)SINR_{AR,AX}-P_{A}H_{AX})P_{Lo}(I_{3}+N)}{P_{A}( I_{2}+N)}\right]^{-\frac{1}{\alpha}} \tag{5}\]  
where the parameters \(P_{Lo}\) and \(\alpha\) are the path loss constant and the path loss exponent for the transmission links. Equation (5) will be used to optimize AV energy efficiency (\(EE\)) and achievable data rate (\(R\)), which are expressed as follows:  
\[EE=\frac{R_{RX}}{P_{A}+P_{o}}+\frac{R_{AX}}{P_{A}+P_{o}} \tag{6}\]  
\[R=R_{RX}+R_{AX} \tag{7}\]  
where \(R_{RX}\) and \(R_{AX}\) are the achievable data rate between relay-vehicle and everything and between autonomous vehicle and everything. \(P_{A}\) and \(P_{o}\) represent autonomous vehicle transmission power and internal circuitry power, respectively. Thus, \(R_{RX}\) and \(R_{AX}\) can be given as  
\[R_{RX}=Blog_{2}\Bigg{(}1+\frac{P_{A}\ H_{RX}}{\sum_{d=1}^{D}P_{D_{d}}\ H_{D_{d}X}+ \sum_{c=1}^{C}P_{C_{c}}\ H_{C_{c}X}+\sum_{v=1}^{V}P_{V_{v}}\ H_{V_{v}X}+\mathrm{N }}\Bigg{)} \tag{8}\]  
\[R_{AX}=Blog_{2}\Bigg{(}1+\frac{P_{A}\ H_{AX}}{\sum_{d=1}^{D}P_{D_{d}}\ H_{D_{d}X}+ \sum_{c=1}^{C}P_{C_{c}}\ H_{C_{c}X}+\sum_{v=1}^{V}P_{V_{v}}\ H_{V_{v}X}+\mathrm{N }}\Bigg{)} \tag{9}\]  
Symbols \(P_{D_{d}}\), \(P_{C_{c}}\) and \(P_{V_{v}}\) are the interfere transmission device (Dtx) transmission power, interfere transmission CUE transmission power, and interfere transmission vehicle (Vtx) transmission power, respectively. \(H_{D_{d}X}\), \(H_{C_{c}X}\), and \(H_{V_{v}X}\) are the channel gain coefficient between interfere transmission device (Dtx) and everything, interfere transmission CUE and everything, and interfere transmission vehicle (Vtx) and everything, respectively. As demonstrated in Equation (1), the major goal of the proposed approach is to maximize total energy efficiency (\(EE\)) and feasible data rate (\(R\)) under varied environmental conditions. As a result, the Lagrangian of the optimization problem stated in (1) is as follows:  
\[L\{d_{AR},\ I_{2},\ P_{A},\ \lambda_{1},\ \lambda_{2}\}=d_{RX}-\lambda_{1}(I_{2 max}-I_{2})-\lambda_{2}(P_{Dmax}-P_{D}) \tag{10}\]  
where the non-negative Lagrangian multipliers are \(\lambda_{1}\) and \(\lambda_{2}\). It is possible to determine the value of \(\lambda_{1}\) and \(\lambda_{2}\) for satisfying the constraint of the optimization problem for (\(d_{AR}\)) by considering the derivative of Equation (10) with respect to \(I_{2}\) and \(P_{A}\).