## 3 Content Acquisition Process and Caching Algorithm Optimization
### Analysis of Success Probability of Content Acquisition
According to Formulas (3), (17), and (18), probability that a request vehicle on lane \(i\) can obtain desired content from a vehicle that is a one-hop or two-hop distance away from itself on front opposite lane is derived as the following:  
\[ \begin{array}{ll}P_{i,\tau}^{opp}=P_{i,\tau}^{sa}\cdot P_{i,\tau}^{opp} \cdot[\iint\limits_{G5}f(s_{i}^{x},s_{i+1}^{y})d\sigma]^{K^{s}}\end{array}  \tag{19}\]  
where \(K^{s}\) denotes the number of vehicles selected to cache content chunks, and \(V_{a}\) denotes the average transmission rate between vehicles. Based on what is discussed above, the probability that a vehicle on lane \(i\) can obtain desired content without the participation of an RSU is as follows:  
\[ \begin{array}{ll}P_{i,\tau}^{not\_,\tau su}=P_{i,\tau}^{self}+P_{i,\tau}^{ nei}+P_{i,\tau}^{opp}\end{array}  \tag{20}\]  
* Obtaining content directly from an RSU  
The authors of [22] indicated that the number of vehicles on the lane obeys the Poisson Point Process. Thus, the average driving distance of a vehicle in the coverage area of RSU was assumed to be \(R^{rst}\). Assuming that the RSU serve and the request vehicles within the coverage area of itself have an equal probability, the probability that RSU transfers desired content to a request vehicle is as follows:  
\[P^{r}=\frac{R^{max}}{\lambda_{m}^{rstu}\cdot W} \tag{21}\]  
Assuming that the average rate that an RSU transfers content to a vehicle is \(V^{per}\), the condition of request vehicle successfully obtains the desire content from an RSU is \(s_{i}\leq\frac{R^{max}.V^{per}}{W}\). According to Formulas (4) and (21), the probability that the request vehicle directly obtains the desired content from an RSU is derived as the following:  
\[P_{i,\tau}^{dir}=P^{r}\cdot F(\frac{R^{rstu}\cdot V^{per}}{W})=\frac{R^{max}}{ \lambda_{m}^{rstu}\cdot W}\cdot F(\frac{R^{rstu}\cdot V^{per}}{W}) \tag{22}\]  
Obtaining content from cluster with content cached by an RSU  
Assuming that \(R_{m}\) proactively caches content at cluster \(C_{m,i+I}\) and chooses \(k\) vehicles from \(C_{m,i+I}\) to respectively cache content chunks based on the caching optimization algorithm depicted in Section 3.2, the condition that RSU successfully transfers content chunks to a vehicle is \(s_{i}\leq\frac{R^{max}.V^{per}.k}{2W}\), and the prediction probability of those \(k\) vehicles constitute is set as \(\{P_{i+1,I}^{pre},P_{i+2,\tau}^{pre},\ldots,P_{i+k}^{pre}\}\). According to Formulas (3), (4), and (21), the probability that the request vehicle can successfully obtain the desired content from \(C_{m,i+I}\) is derived as the following:  
\[P_{i,\tau}^{indir}=[P^{r}\cdot F(\frac{R^{rstu}\cdot V^{per}\cdot k}{2W})]^{k} \cdot[\iint\limits_{G6}f(s_{i}^{x},s_{i+1}^{y})d\sigma]^{k}\cdot\prod_{j=1}^{k }P_{i+I,j}^{pre} \tag{23}\]  
where \(G6=\{(s_{i}^{x},s_{i+1}^{y})\Big{|}s_{i}^{x}+s_{i+1}^{y}\leq\frac{2^{\tau}.V^{ \prime}.k}{W}\}\) denotes the feasible region of \(s_{i}^{x}\), and \(s_{i+1}^{y},s_{i}^{x}\), and \(s_{i+1}^{y}\) denote the speed of request vehicle and the vehicle from \(C_{m,i+I}\). Because the number of vehicles on the lane obeys the Poisson Point Process [22], the probability that a vehicle is in the coverage area of an RSU is  
\[P^{in}=\frac{2\cdot R^{rstu}}{E} \tag{24}\]  
According to Formulas (22)-(24), the probability that a vehicle can obtain desired content with the participation of the RSU is derived as the following:  
\[P_{i,\tau}^{rstu}=P^{in}\cdot P_{i,\tau}^{dir}+(1-P^{in})\cdot P_{i,\tau}^{ indir} \tag{25}\]  
The average probability that a vehicle on lane \(i\) can successfully obtain desired content ranked \(\tau-th\) is derived as the following:  
\[P_{i}^{true}=\sum_{\tau=1}^{L}f_{\tau}\cdot[P_{i,\tau}^{not\_,rstu}+(1-P_{i, \tau}^{not\_,rstu})\cdot P_{i,\tau}^{rstu}] \tag{26}\]