## 3 Content Acquisition Process and Caching Algorithm Optimization
### Analysis of Success Probability of Content Acquisition
When \(s_{i}^{x}>s_{i}^{y}\), the condition of content success transmission is \(\frac{r^{v}+\frac{3r^{v}}{4}}{s_{i}^{x}-s_{i}^{y}}\cdot V^{a}\geq W\) and the feasible region is \(G3=\left\{(s_{i}^{x},s_{i}^{y})\Big{|}s_{i}^{x}-s_{i}^{y}\leq\frac{7r^{v}V^{a} }{4W}\right\},\) When \(s_{i}^{x}<s_{i}^{y}\), the condition of content success transmission is \(\frac{r^{v}-\frac{3r^{v}}{4}}{s_{i}^{y}-s_{i}^{x}}\cdot V^{a}\geq W\) and the feasible region is \(G4=\left\{(s_{i}^{x},s_{i}^{y})\Big{|}s_{i}^{y}-s_{i}^{x}\leq\frac{r^{v}V^{a} }{4W}\right\}.\) According to Formula (3), the probability that a vehicle on lane \(i\) can obtain content from another relay vehicle is derived as follows:  
\[ \begin{array}{ll}P_{i}^{two}&=p\{s_{i}^{x}>s_{i}^{y}\}\cdot\iint\limits_{G3 }f(s_{i}^{x},s_{i}^{y})\ d\sigma+p\{s_{i}^{x}<s_{i}^{y}\}\cdot\iint\limits_{G4 }f(s_{i}^{x},s_{i}^{y})\ d\sigma\\ &=\frac{1}{2}\cdot\iint\limits_{G3+G4}f(s_{i}^{x},s_{i}^{y})\ d\sigma\end{array}  \tag{13}\]  
The probability that the desired content fails to be cached by vehicles that are a one-hop distance away from the request vehicle is as follows:  
\[ \begin{array}{ll}P_{i,\tau}^{ns}&=\sum\limits_{k=1}^{+\infty}P_{i}\{K_{1}=k \}\ \cdot(1-f^{\tau}(N))^{k}\\ &=e^{-2\lambda_{i}D^{v}f^{\tau}(N)}-e^{-2\lambda_{i}D^{v}}\end{array}  \tag{14}\]  
Let \(K_{2}\) denote the number of vehicles that are a two-hop distance away from the request vehicle. The probability that the desired content manages to be cached by vehicles that are a two-hop distance away from the request vehicle is as follows:\[ \begin{array}{ll}P_{i,\tau}^{en}&=\sum\limits_{k=1}^{+\infty}P_{i}\{K_{2}=k\} \cdot[1-(1-f^{\tau}(N))^{k}]\\ &=1-e^{-2\lambda_{i}D^{\tau}f^{\tau}(N)}\end{array}  \tag{15}\]  
The desired content will be transferred to the request vehicle by a vehicle that is a two-hop distance away. According to Formulas (13-15), the probability that a request vehicle on lane \(i\) can obtain desired content from the vehicle that is a two-hop distance away from itself is derived as the following:  
\[ \begin{array}{ll}P_{i,\tau}^{eni}=P_{i,\tau}^{ns}\cdot P_{i,\tau}^{en}\cdot( P_{i}^{tuo})^{2}\end{array}  \tag{16}\]  
* Obtaining content from vehicles that are a one-hop or two-hop distance away from the request vehicle on the front opposite lane  
Let \(K_{3}\) denote the number of vehicles that are a one-hop or two-hop distance away from the request vehicle in the same lane. The probability that the desired content fails to be cached by those \(K_{3}\) vehicles is as follows:  
\[ \begin{array}{ll}P_{i,\tau}^{sa}&=\sum\limits_{k=1}^{+\infty}P_{i}\{K_{3}=k \}\cdot(1-f^{\tau}(N))^{k}\\ &=\ e^{-4\lambda_{i}D^{\tau}f^{\tau}(N)}-e^{-4\lambda_{i}D^{\tau}}\end{array}  \tag{17}\]  
Let \(K_{4}\) denote the number of vehicles with one-hop or two-hop distance away from request vehicle on front opposite lane, the probability that desired content manages to be cached at those \(K_{4}\) vehicles is as following  
\[ \begin{array}{ll}P_{i,\tau}^{pop}&=\sum\limits_{k=1}^{+\infty}P_{i+1}\{K_{4 }=k\}\cdot[1-(1-f^{\tau}(N))^{k}]\\ &=\ 1-\ e^{-2\lambda_{i}D^{\tau}f^{\tau}(N)}\end{array}  \tag{18}\]  
Assuming that the request vehicle needs to obtain the desired content from content source vehicle \(V_{c}\) on opposite lane \(i+1\), \(V_{c}\) chooses \(K^{s}\) vehicles that are a one-hop distance away from \(V_{c}\) to average cache content chunks. As depicted in Section 3.2, those \(K^{s}\) vehicles selected to cache content chunks will meet the request vehicle with a probability of 1. Assuming that the speed of any two vehicles, respectively, in the same and opposite lane are \(s_{i}^{x}\) and \(s_{i+1}^{y}\), the condition of content chunk success transmission between the two vehicles is \(\frac{2^{\tau}\cdot V^{sa}}{s_{i}^{y}+s_{i+1}^{y}}\leq\frac{W}{K^{s}}\) and the feasible region is \(\ G5=\{(s_{i}^{x},s_{i+1}^{y})\big{|}s_{i}^{x}+s_{i+1}^{y}\leq\frac{2^{\tau} \cdot V^{sa}\cdot K^{s}}{W}\}\).