## 2 System Model and Dynamic Optimization Problem
### Availability Probability Calculated by Stochastic Geometry  
To control the interference in the vehicular network, the access control criterion is proposed in this section. Channel \(k\) is available to member \(n\) in the platoon when the following two conditions are satisfied simultaneously. Condition 1: Channel \(k\) is not occupied by the MU or the usage of channel \(k\) for member \(n\) in the platoon cannot cause strong interference with the MUs, which is guaranteed by the SIR threshold \(\theta\). This condition is written as:  
\[\left\{ \begin{array}{l}\text{the channel $k$ is not occupied by MUs, or}\\ \frac{P^{M}g_{k}^{M}\left(d_{k}^{M}\right)^{-\alpha}}{P^{V}g_{k}^{VM}\left(d^{ V}\right)^{-\alpha}}\geq\theta,\text{ when the channel $k$ is occupied by MUs,}\end{array} \right. \tag{5}\]  
where \(g_{k}^{M}\) and \(d_{k}^{M}\) represent the small-scale fading and the distance between the MBS and the corresponding MU that is active in channel \(k\). The symbols \(g_{k}^{VM}\) and \(d_{k}^{VM}\) represent the small-scale fading and the distance between the VU interferer and the MU that is active in channel \(k\). Condition 2: Channel \(k\) is not occupied by the MU or the ratio of the interference from the MBS to the interference from the other platoons is smaller than a threshold \(\beta\):  
\[\left\{ \begin{array}{l}\text{the channel $k$ is not occupied by MUs, or}\\ \frac{I_{nk}^{MV}}{I_{nk}^{MV}}\leq\beta,\text{ when the channel $k$ is occupied by MUs.}\end{array} \right. \tag{6}\]  
Condition 1 restricts the interference from VU to MU and Condition 2 restricts the interference from MBS to VU. Then, each platoon leader determines the set of available channels according to the above two conditions. Conditions 1 and 2 require information including the locations of MUs and MBS and the channel occupation of all MUs. Therefore, at the beginning of each time slot, the MBS broadcasts information on the geographical location and channel occupation of all MUs to each platoon leader. To decrease the overhead, this transmission of the control information is one-way; that is, only the MBS broadcasts the control information and there is no feedback. After receiving the control information, the platoon leader makes a decision regarding the set of available channels. The MBS allocates channels to the MUs by self-determination. Let \(b_{k}\) denote the channel allocation for the MUs in channel \(k\), which is defined as:  
\[b_{k}=\left\{ \begin{array}{ll}1,&\text{if channel $k$ is allocated to MU},\\ 0,&\text{otherwise}.\end{array} \right. \tag{7}\]  
Therefore, according to Equations (5)-(7), the availability probability denoted as \(\rho_{nk}\) can be derived as:  
\[\rho_{nk} =1-b_{k}+b_{k}\mathbb{P}\left(\frac{P^{M}g_{k}^{M}\left(d_{k}^{M }\right)^{-\alpha}}{P^{V}g_{k}^{V}\left(d_{k}^{V}\right)^{-\alpha}}\geq\theta \right)\mathbb{P}\left(\frac{I_{nk}^{MV}}{I_{nk}^{VV}}\leq\beta\right)\] \[=\rho_{nk}^{1}\rho_{nk}^{2}, \tag{8}\]  
where \(\rho_{nk}^{1}=P\left(\frac{p_{nk}^{M}\left(d_{k}^{M}\right)^{-\alpha}}{P^{V}g _{k}^{VM}\left(d_{k}^{VM}\right)^{-\alpha}}\geq\theta\right)\) and \(\rho_{nk}^{2}=P\left(\frac{I_{nk}^{MV}}{I_{nk}^{VV}}\leq\beta\right)\). Let the symbol \(z=\frac{g_{k}^{VM}}{g_{k}^{V}}\). As the small-scale fading of \(g_{k}^{VM}\) and \(g_{k}^{M}\) are independent and \(g\sim\exp(1)\), then, according to the distribution of random variables function, the probability density function of \(z\) can be derived as follows:  
\[f\left(z\right) =\int_{0}^{+\infty}g_{k}^{M}\exp(-g_{k}^{M}z)\exp(-g_{k}^{M})dg_{ k}^{M}\] \[=\frac{1}{\left(1+z\right)^{2}}. \tag{9}\]  
The distribution function of \(z\) is denoted as \(F_{z}(z_{0})=\int_{-\infty}^{z_{0}}f\left(z\right)dz\).