## 3 System Model
### Caching Model  
The video content popularity reflects the statistical results of the video requested by vehicle users over a period. As a result, the time slot varying scale of video content popularity is much larger than the cache content update of the MECS. Furthermore, the time slot varying scale of the video content delivery process is much smaller than the cache content update of the MECS. Hence, the mobile video caching in the vehicular network can be modeled as a multi-time scale model, as shown in Figure 2.  
Figure 1: System model. Popularity variation time scale: We assume that one video popularity variation time slot is included \(K_{CN}\) caching placement time slot. The time scales of popularity variation can be defined as \(t^{x},x\in\{1,2,3,\ldots,K_{CG}\}\), which containing \(K_{CG}\) time slots, and define the length of the one MECS caching update period is \(\triangle t^{x}_{y}\), then the length of \(\triangle t^{x}\) is expressed as  
\[\triangle t^{x}=K_{CN}\triangle t^{x}_{y}. \tag{3}\]  
Caching placement time scale: The times scales of caching placement can be defined as \(t^{x}_{y},y\in\{1,2,3,\ldots,K_{CN}\}\), and caching video content of MECS is updated once every time slot of \(t^{x}_{y}\). Suppose \(\mathcal{D}\) is the length of service covered, the length of \(\triangle t^{x}_{y}\) at vehicles speed \(v\left(t^{x}_{y}\right)\) is calculated as  
\[\triangle t^{x}_{y}=\frac{\mathcal{D}}{v\left(t^{x}_{y}\right)}. \tag{4}\]  
Video delivery time scale: The rapid movement of the vehicle causes the geographic location and channel status to change, the transmission rate of the vehicle user is time-vary. The \(\triangle t^{x}_{y}\) is discretized into \(K_{SN}\) segments to simplify the model. Then the times scales of video delivery can be defined as \(t^{x}_{y,z},z\in\{1,2,3,\ldots,K_{SN}\}\). It can be considered that the communication rate in each content distribution period \(\triangle t^{x}_{y,z}\) remains unchanged. The length \(\triangle t^{x}_{y,z}\) of the time slot \(t^{x}_{y,z}\) is calculated as  
\[\triangle t^{x}_{y,z}=\frac{\triangle t^{x}_{y}}{K_{SN}}. \tag{5}\]  
The video requested by vehicle users (such as 4K/8K high definition movies, etc.) comes from a video content library \(\mathcal{F}=\{f_{1},f_{2},\ldots,f_{F}\}\), which containing \(F\) video files. Assumed that \(\mathcal{F}\) obeys [PERSON]'s law Zipf (1992), and the videos of \(\mathcal{F}\) are arranged in descending order of their content popularity. Using constant bitrate (CBR) technology to encode the videos of \(\mathcal{F}\) into a constant bitrate multiple bitrate video \(\mathcal{L}=\{\ell_{1},\ell_{2},\ldots,\ell_{L}\}\), its bitrate level is still arranged in descending order \(\ell_{1}>\ell_{2}>\ldots>\ell_{L}\), that is, \(\ell_{1}\) is the replicas of video with the highest bitrate. Without loss of generality, assume that the time lengths of the videos in \(\mathcal{F}\) are equal, denoted as \(\omega\), it can be considered that after coding using the CBR technology, all video copies of the same bitrate level in \(\mathcal{F}\) have the same size. The replicas size \(g_{j}\) of video \(f\) at the \(\ell_{j}\) bitrate level is expressed as  
\[g_{j}=\varpi\ell_{j}. \tag{6}\]  
Hence, the caching state matrix \(X(t^{x}_{y})\) of MECS at the time slot \(t^{x}_{y^{\prime}}\) is defined as  
\[X(t^{x}_{y})=\mathcal{F}\times\mathcal{L}=\left\{x_{f_{j},\ell_{j}}\left(t^{x }_{y}\right),f_{i}\in\mathcal{F},\ \ell_{j}\in\mathcal{L}\right\}_{F\times L}. \tag{7}\]  
Figure 2: Multi-time scale model. where \(F\) is the number of videos and \(L\) is bitrate level. The \(X(t_{y}^{x})\) can be further expressed as \(X\Big{(}t_{y}^{x}\Big{)}\in\{0,1\}_{F\times L}\). Then caching state variable of video \(f_{i}\) with bitrate level \(\ell_{j}\) at the time slot \(t_{y}^{x}\) is expressed as  
\[x_{f_{i},\ell_{j}}\Big{(}t_{y}^{x}\Big{)}= \begin{cases}1,\text{ if MECS cached the $\ell_{j}$ level of video $f_{i}$}\\ 0,\text{ otherwise}\end{cases} .