## 2 System Model and Problem Formulation
### Performance Metrics  
In this section, we will introduce the performance metrics used in this work. First, we define the UAV's successful transmission probability and then calculate the content caching hit rate, which is defined as the ratio between the number of successfully responded content requests and the total number of content requests from all vehicles. Let \(P_{U}\) represent the signal transmission power utilized by the UAVs, under the assumption that these UAVs share the spectrum resources with other UAVs, which results in interference from neighboring devices. When a user retrieves cached content from the UAV, the Signal-to-Interference-plus-Noise Ratio (SINR) can be formulated as  
\[SINR_{u,n}(t)=\frac{P_{U}h_{n,u}(t)|g_{n,u}|^{2}}{I_{other}^{u}(t)+\sigma^{2}}, \tag{14}\]where \(\sigma^{2}\) is the noise power, and \(|g_{n,u}|^{2}\) represents the Rayleigh channel gain from UAV \(u\) to vehicle \(n\). The interference from other UAVs is represented as  
\[l_{other}^{u}(t)=\sum_{v=1,v\
eq u}^{u}P_{U}h_{n,v}(t)|g_{n,v}|^{2}. \tag{15}\]  
The probability of successful transmission for vehicle \(n\) to receive the requested content through U2V communication is defined as the likelihood that the Signal-to-Interference-plus-Noise Ratio (SINR) of the received signal surpasses the decoding threshold. This is mathematically expressed as \(P_{u,n}^{\text{suc}}(t)=Pr[SINR_{u,u}\geq\theta]\), where \(Pr[A]\) denotes the probability of event \(A\)[47]. When vehicle \(n\) successfully retrieves the desired content via U2V communication, the successful transmission probability can be derived as follows. \[ \begin{split} P_{u,n}^{suc}(t)&=\text{Pr}[SINR_{u}( t)\geq\theta]\\ &=E\bigg{[}\text{Pr}\bigg{[}\frac{P_{U}h_{n,u}(t)|g_{n,u}|^{2}}{l_{ other}^{u}(t)+\sigma^{2}}\geq\theta\bigg{]}\bigg{]}\\ &\overset{\text{(a)}}{=}\int_{Z_{u}(t)}^{R_{u}}\text{Pr}\bigg{[} \frac{P_{U}[g_{n,u}|^{2}\eta L_{n,u}^{-\zeta}(t)}{I_{other}^{u}(t)+\sigma^{2}} \geq\theta\bigg{]}f_{R_{u}}(r_{u})dL_{n,u}(t)\\ &=\int_{Z_{u}(t)}^{R_{u}}\text{Pr}\Bigg{[}|g_{n,u}|^{2}\geq\frac{ \theta L_{n,u}^{\zeta}(t)(l_{other}^{u}(t)+\sigma^{2})}{P_{U}\eta}\bigg{]}f_{R_ {u}}(L_{n,u}(t))dL_{n,u}(t)\\ &=\int_{Z_{u}(t)}^{R_{u}}\text{exp}\{-\frac{\theta L_{n,u}^{ \zeta}(t)\sigma^{2}}{P_{U}\eta}\}E_{l_{other}^{u}(t)}\bigg{[}\text{exp}\{- \frac{\theta L_{n,u}^{\zeta}(t)I_{other}^{u}(t)}{P_{U}\eta}\}\bigg{]}f_{R_{u}} (L_{n,u}(t))dL_{n,u}(t)\\ &=\int_{Z_{u}(t)}^{R_{u}}\text{exp}\{-\frac{\theta L_{n,u}^{ \zeta}(t)\sigma^{2}}{P_{U}\eta}\}L_{l_{other}^{u}(t)}\bigg{(}\frac{\theta L_{n,u}^{\zeta}(t)}{P_{U}\eta}\bigg{)}f_{R_{u}}(L_{n,u}(t))dL_{n,u}(t)\\ \end{split}  \tag{16}\]  
wherein step (a) is derived based on the formula \(E[\chi]=\int_{0}^{\infty}\text{Pr}(\chi>x)\,dx\). The power gain \(|g_{n,u}|^{2}\) of a Rayleigh fading channel follows an exponential distribution. For an exponentially distributed random variable \(X\) with rate parameter \(\lambda^{\prime}\), the probability \(P(X\geq x)\) is given by \(e^{-\lambda^{\prime}x}\). In this context, we assume a unit mean for \(|g_{n,u}|^{2}\), which implies \(\lambda^{\prime}=1\). Therefore, by letting \(X_{\text{threshold}}=\frac{\theta L_{n,u}^{\zeta}(t)(l_{other}^{u}(t)+\sigma^ {2})}{P_{U}\eta 0}\), the probability \(\text{Pr}[|g_{n,u}|^{2}\geq X_{\text{threshold}}]\) is transformed to \(\text{exp}\{-X_{\text{threshold}}\}\). \(R_{u}\) represents the max communication distance between the UAV and vehicle and \(f_{R_{u}}(L_{n,u}(t))\) is the probability density function (PDF) of the distance from the UAV to the requesting vehicle, expressed as  
\[f_{R_{u}}(L_{n,u}(t))= \begin{cases}\frac{L_{n,u}(t)}{R_{u}^{2}},&L_{n,u}(t) \in[0,R_{u}],\\ 0,&\text{otherwise}.\end{cases}  \tag{17}\]  
Let \(z=\frac{\theta L_{n,u}^{\zeta}(t)}{P_{U}\eta}\), then the Laplace transform \(L_{l_{other}^{u}(t)}(z)\) of interference \(l_{other}^{u}(t)\) is derived by using the properties of stochastic geometry as follows [48].