## 3 System Model
### Communication Model  
We assume that the access system of the vehicular network is based on orthogonal frequency division multiple access (OFDMA). Therefore, every vehicle linked to RSU or MBS is assigned an orthogonal subchannel. Thus, we do not consider interference among different links. For simplicity, the channel gains are assumed to remain constant during one video delivery period. They have the same distribution, so it is sufficient to concentrate on one vehicle user to study the performance of interest. The transmission rate between vehicle and RSU at time slot \(t_{y,z}^{x}\) can be calculated by  
\[\mathcal{R}_{MEC}\Big{(}t_{y,z}^{x}\Big{)}=W_{r}\left(\log_{2}\left(1+\frac{P _{r}\Big{|}h_{r}\Big{(}t_{y,z}^{x}\Big{)}\Big{|}^{2}}{\delta^{2}}\right) \right), \tag{17}\]  
where \(W_{r}\) is the channel bandwidth allocated by RSU to vehicle users, \(P_{r}\) is the signal transmit power of RSU, \(\delta^{2}\) is the noise power, \(h_{r}\Big{(}t_{y,z}^{x}\Big{)}\) is the RSU channel gain at time slot \(t_{y,z}^{x}\), it can be expressed as  
\[\Big{|}h_{r}\Big{(}t_{y,z}^{x}\Big{)}\Big{|}^{2}=G_{r}\Big{|}d_{1,u}\Big{(}t_ {y,z}^{x}\Big{)}\Big{|}^{-\epsilon}\Big{|}h_{0}\Big{(}t_{y,z}^{x}\Big{)}\Big{|} ^{2}, \tag{18}\]  
where \(\epsilon\) is the path loss coefficient, \(h_{0}\Big{(}t_{y,z}^{x}\Big{)}\) is a complex Gaussian distributed random variable for Rayleigh channel fading, \(|.|\) is absolute operation, \(\Big{|}h_{0}\Big{(}t_{y,z}^{x}\Big{)}\Big{|}^{2}\sim CN(0,1)\), \(G_{r}\) is the antenna gain coefficient of RSU, \(d_{1,u}\Big{(}t_{y,z}^{x}\Big{)}\) represents the distance between the vehicle user and RSU. If the MECS does not cache any bitrate video copies of the video requested by the vehicle user, MBS will take over the service of the vehicle user through the backhaul network, and distribute video copies with the highest bitrate under the constrained backhaul bandwidth to the VUs. The transmission rate between the vehicle user and MBS is expressed as  
\[\mathcal{R}_{MBS}\Big{(}t_{y,z}^{x}\Big{)}=B_{0}\left(\log_{2}\left(1+\frac{P _{bs}\Big{|}h_{bs}\Big{(}t_{y,z}^{x}\Big{)}\Big{|}^{2}}{\delta^{2}}\right) \right), \tag{19}\]  
where \(B_{0}\) is MBS allocates bandwidth to each vehicle user, \(P_{bs}\) is the MBS transmit power, MBS channel gain at the time slot \(t_{y,z}^{x}\) is calculated as  
\[\Big{|}h_{bs}\Big{(}t_{y,z}^{x}\Big{)}\Big{|}^{2}=G_{bs}\Big{|}d_{0,u}\Big{(}t _{y,z}^{x}\Big{)}\Big{|}^{-\epsilon}\Big{|}h_{0}\Big{(}t_{y,z}^{x}\Big{)} \Big{|}^{2}, \tag{20}\]  
where \(G_{bs}\) is the antenna gain coefficient of MBS, \(d_{0,u}\Big{(}t_{y,z}^{x}\Big{)}\) represents the distance between the vehicle user and MBS. Due to \(\mathcal{R}_{MBS}\) is both limited by the wireless access network and the backhaul bandwidth \(C_{BM}\), when the number of vehicle users \(N\Big{(}t_{y,z}^{x}\Big{)}\) connected to MBS at time \(t_{y,z}^{x}\) does not reach the maximum number of bearer users \(N_{Max}\) of MBS, the principle of fairness will be taken into consideration. The bandwidth resource required for caching update will allocate a size of \(C_{MBS}\) backhaul bandwidth to each vehicle user. \(N_{Max}\) is defined as  
\[N_{Max}=\bigg{\lfloor}\frac{C_{BM}}{C_{MBS}}\bigg{\rfloor}, \tag{21}\]  
where \(\lfloor.\rfloor\) is the rounding down operation. When the number of vehicle users connected to MBS exceeds \(N_{Max}\), the backhaul resources will be average allocated to users, and the transmission rate \(\mathcal{R}_{Back}\left(t_{y,z}^{x}\right)\) can be expressed as  
\[\mathcal{R}_{Back}\left(t_{y,z}^{x}\right)=\left\{ \begin{array}{ll}C_{MBS},N \left(t_{y,z}^{x}\right)\leq N_{Max}\\ \frac{C_{BM}}{N\left(t_{y,z}^{x}\right)},N\left(t_{y,z}^{x}\right)>N_{Max} \end{array} .\right. \tag{22}\]  
Hence, the service state probability can be expressed as  
\[P_{Ar}\left(U\left(t_{y,z}^{x}\right),N\left(t_{y,z}^{x}\right)\right)= \begin{pmatrix} U\left(t_{y,z}^{x}\right)\\ N\left(t_{y,z}^{x}\right)\end{pmatrix}