## 2 System Model and Problem Formulation
### Vehicle Platooning
Subsequently, it is assumed that the channel fading is constant within a single subchannel and independent throughout all other subchannels. The channel gain of CV in \(b\) subchannel within a single coherent time t is modeled based on the occurrence of \(a_{i}^{t}\), large-scale fading and \(g_{i}^{t}\), small-scale fading as  
\[h_{i}^{t}[b]=a_{i}^{t}g_{i}^{t}[b] \tag{1}\]  
Furthermore, the Boolean factor \(\tilde{g}_{i,b}^{t}\) that features the allocation of specific subchannel \(b\) to the \(i\)th platoon at time slot \(t\) is desired by \(\tilde{g}_{i,b}^{t}\in\{0,1\}\). Indeed, CVs prefer the decision of utilizing the allocated spectrum either for communicating with the RSU or for CAM transmissions to the platoon members, i.e., for inter-platoon or intra-platoon communication, respectively. Subsequently, CVs' communication mode decision is construed by an addi  
Figure 1: An MEC-Aided multi-platoon architecture. tional Boolean factor \(\theta_{i}^{t}\in\{0,1\}\), where 0 symbolizes the decision of the allocated spectrum being utilized for inter-platooning and 1 symbolizes intra-platooning communication. Based on [PERSON]'s capacity theorem, the instantaneous rate attained in vehicular communication between CV and the RSU is defined as  
\[ \begin{split}&\mathcal{C}_{i,\mathfrak{L}}^{t}[b]=\log_{2}\Biggl{(}1+ \frac{\left(1-\theta_{i}^{t}\right)\xi_{i,b}^{t}\mathfrak{p}_{i}^{t}[b]h_{i, \mathfrak{L}}[b]}{I_{i}^{t}[b]+\sigma^{2}}\Biggr{)},\\ &\text{where,}\\ & I_{i}^{t}[b]=\sum_{i^{\prime}}\xi_{i^{\prime},b}^{t}\mathfrak{p} _{i}^{t}[b]h_{i^{\prime},\mathfrak{L}}[b],i\
eq i^{\prime}\end{split}  \tag{2}\]  
where the other platoons' interference is considered as noise; the CV's transmitting power level \(i\) on \(b\) bandwidth is given as \(\mathfrak{p}_{i}^{t}[b]\); the CV's channel gain from \(i\) to RSU in the bandwidth \(b\) is given as \(h_{i,\mathfrak{L}}[b]\); and the noise power and the RSUs locations are defined as \(\sigma^{2}\) and \(\mathfrak{L}\), respectively. The interfering channel from CV's \(i^{\prime}\) performs inter-platoon \(\left(\theta_{i}^{t}=0\right)\) or intra-platoon \(\left(\theta_{i}^{t}=0\right)\) modes of communication to the RSU, where \(i\in P\) and the total interference power is given as \(I_{i}^{t}[b]\). Consequently, the CVs' \(i\) and their Follower Vehicles' \(j\) instantaneous rates are defined as  
\[ \begin{split}&\mathcal{C}_{i,\mathfrak{L}}^{t}[b]=\log_{2}\Biggl{(}1+ \frac{\theta_{i}^{t}\xi_{i,b}^{t}\mathfrak{p}_{i}^{t}[b]h_{i,j}^{t}[b]}{I_{i}^ {t,t}[b]+\sigma^{2}}\Biggr{)},\\ &\text{where,}\\ & I_{i}^{t,t}[b]=\sum_{i^{\prime}}\xi_{i^{\prime},b}^{t} \mathfrak{p}_{i}^{t}[b]h_{i^{\prime},j}^{t}[b],i\
eq i^{\prime},j=\mathcal{V}_ {i}\end{split}  \tag{3}\]  
The channel gain from a CV \(i\) to its FV members in the bandwidth \(b\) is given as \(h_{i,j}^{t}[b]\); the channel interference created from CV \(i^{\prime}\)s vehicle to the CV \(j\)'s members, where the CV performs two modes of communication, are given as \(h_{i^{\prime},j^{\prime}}^{t}\); finally, the total interference power is given as \(I_{i}^{t,t}[b]\). As previously stated, the CVs are liable for establishing the timing accuracy while transferring the CAM messages to the RSU. In such scenarios, the Age of Information (AoI) of platoons is aggregated as \(AoI_{i}^{t}\) until the scheduling window \(t\) starts, i.e., the amount of time lapsed from the last effective V2I communication. The \(AoI_{i}^{t}\) progresses, following to  
\[AoI_{i}^{t+1}= \begin{cases}\Delta t,&\text{if }\left(1-\theta_{i}^{t} \right)\xi_{i,b}^{t}\cdot\mathcal{C}_{i,\mathfrak{L}}^{t}[b]\geqslant\mathcal{ C}_{i,\mathfrak{L}}^{min}\\ AoI_{i}^{t}+\Delta t,&\text{otherwise}\end{cases}  \tag{4}\]  
where the minimum capacity demanded by any reliable vehicular communication is represented as \(\mathcal{C}_{i,\mathfrak{L}}^{min}\), and whenever the RSU and CV successfully communicate the data, the \(AoI_{i}^{t}\) restarts from \(\Delta t\).