## 3 Proposed Model and Problem Formulation
### Vehicle Communication Scenario  
The vehicular scenario is shown in Figure 2, where vehicles and autonomous vehicles are moving along a road and communicate with anything (AV2X). The scenario assumes that the infrastructures are located along the road with equal distance (\(L\)), and assumes that all infrastructure points have the same radio range (\(r_{I}\)) and all the vehicles have the same radio range (\(r_{V}\)) where \(r_{I}>r_{V}\), which reflects the fact that the capability of communication of the infrastructure is stronger than the capability of communication of vehicles [10]. Any pair of vehicles can directly communicate with each other only if this pair is within its radio range \(r_{V}\). Thus, the proposed model intends to find the best position of inter-vehicles to communicate with other vehicles or to relay the information to the infrastructure. The proposed model adopts two communication schemes; the first one is the communication between vehicles, and the second is the communication between vehicles (V) and everything (X) through the existence of the inter-vehicular between AV2X. The inter-vehicles are located at a distance equal to (\(q^{*}d_{AVX}\)), where \(q\) is the position weight of the inter-vehicles from the destination and \(d_{AVX}\) is the distance from the destination. The uplink communications between the AV2V and AV2X are assumed to be frequency-flat block-fading Rayleigh channels [31]. Therefore, the uplink data rate between autonomous vehicle-to-vehicle or relay-vehicle (\(R_{AVV1}\)) and autonomous vehicle-to-everything through relay-vehicle (\(R_{AVX}\)) are given by [32]:  
\[R_{AVV1}=B_{AVV1}\,log_{2}(1+SINR_{AVV1}) \tag{7}\]  
\[R_{AVX\to V1X}=B_{V1X}\,log_{2}(1+SINR_{AVX,V1X}) \tag{8}\]  
\(B_{AVV1}\) and \(B_{V1X}\) represent the bandwidth of the uplink channel between autonomous vehicle-relay-vehicle and relay-vehicle-everything, respectively. \(SINR_{AVV1}\) and \(SINR_{AVX,V1X}\) represent the signal-to-interference-plus noise between autonomous vehicle-relay-vehicle (AVV1) and between autonomous vehicle-everything (AVX) and relay-vehicle-everything (V1X), which can be expressed, respectively, as [32]:  
\[SINR_{AVV1}=\frac{P_{AV}|h_{AVV1}|^{2}}{I_{V1}+N_{o}B} \tag{9}\]  
\[SINR_{AVX,V1X}=\frac{P_{AV}|h_{AVX}|^{2}}{I_{1X}+N_{o}B}+\frac{P_{V}|h_{V1X }|^{2}}{I_{2X}+N_{o}B} \tag{10}\]  
where \(h_{AVV1}\), \(h_{AVX}\) and \(h_{V1X}\) are channel gain for the information of autonomous vehicle-relay-vehicle (AVV1), autonomous vehicle-destination and relay-vehicle-destination, respectively. \(P_{V}\) is the relay-vehicle transmission power. \(I_{V1}\), \(I_{1X}\) and \(I_{2X}\) are the interference occurred at the relay-vehicle from AVV1 link, at the destination due to AV2X link and at the destination due to V1X link. The interference occurred due to other transmitting devices or vehicles sharing the same spectrum. Therefore, \(I_{1X}\) and \(I_{2X}\) can be given as:  
\[I_{1X}=\sum_{k=1}^{K}P_{Ik}|h_{IkX}|^{2} \tag{11}\]  
\[I_{2X}=\sum_{r=1}^{R}P_{Ir}|h_{IrX}|^{2} \tag{12}\]  
where \(h_{IkX}\) and \(h_{IrX}\) are the fading co-channel interference at the destination due to AVX link and relay-vehicle-destination link, respectively. \(P_{Ik}\) and \(P_{Ir}\) are the interference transmission powers of the devices which share the AVX link and relay-vehicle-destination link, respectively. The fading channel coefficient between any source-destination \(h_{SD}\) can be expressed as [33]:  
\[|h_{SD}|^{2}=\frac{\left|h_{SD_{0}}\right|^{2}}{PL_{SD}} \tag{13}\]  
where \(h_{SD_{0}}\) and \(PL_{SD}\) are the complex Gaussian random variables to model fading phenomena and the pathloss of S-D link.