## Appendix A Derivation of \(ne^{+}(\tau)\) and \(ne^{-}(\tau)\) for the 1-D Vehicular Scenario  
We start with the following lemma, which derives the distribution of the relative speed between two nodes moving in the same direction. Let \(Z^{+}=V_{i}-V_{j}\) be a random variable modelling the relative speed between mobile nodes \(n_{i}\) and \(n_{j}\) moving in the same direction in a 1-D linear network topology (note that the variable \(Z^{+}\) is not dependent on the choice of \(i\) and \(j\) as the node speeds are independent and identically distributed random variables). If \(V_{i},V_{j}\sim\mathcal{U}(v_{a},v_{b})\), then the probability distribution of \(Z^{+}\) is given by  
\[f_{Z^{+}}(z)= \begin{cases}\frac{\Delta v+z}{\Delta v^{2}}&\text{if}-\Delta v \leq z<0\\ \frac{\Delta v-z}{\Delta v^{2}}&\text{if}-0\leq z\leq\Delta v\\ 0&\text{otherwise}\end{cases}  \tag{1}\]  
where \(\Delta v=v_{b}-v_{a}\). We can readily derive expression (1) by recalling that the probability distribution of the sum of two independent random variables is equal to the convolution of their individual probability distributions. Now, let us assume that nodes \(n_{i}\) and \(n_{j}\) have positions \(x_{i}(t)\) and \(x_{j}(t)\) at time \(t\). We introduce the random variable \(D^{+}(\tau)\), which describes the relative change, at time \(t+\tau\), of the distance between two randomly selected nodes that move in the same direction. In other words, \(D^{+}(\tau)=[x_{i}(t+\tau)-x_{j}(t+\tau)]-[x_{i}(t)-x_{j}(t)]\). It is straightforward to note that \(D^{+}(\tau)=\tau Z^{+}\), and we can obtain the probability distribution \(f_{D^{+}}(d)\) from (A1). Now, let us introduce the quantity \(\Gamma^{+}(\tau,d)\) that measures the probability that during the time interval \([t,t+\tau]\) the relative distance between two random nodes moving in the same direction changes by at least \(d\) meters. By definition, we can write that \[\Gamma(\tau,d)=\Pr\{\exists t^{\prime},t\leq t^{\prime}\leq t+\tau:D^{+}(\tau )=d\}\.\] (A2) For the sake of brevity in notation, let us introduce \(d_{m}(\tau)=\Delta v\cdot\tau\). Then, it holds that \(-d_{m}(\tau)\leq D^{+}(\tau)\leq d_{m}(\tau)\). To derive an analytical expression for (A2), we need to consider two separate cases: * \(-d_{m}(\tau)\leq d<0\), i.e., the first node is coming closer to the node in front. In this case, it holds that \[\Gamma(\tau,d)= \Pr\{D^{+}(\tau)\leq d\}=\] (A3) \[\int\limits_{-d_{m}(\tau)}^{d}f_{D^{+}}(x)dx =\frac{[d_{m}(\tau)+d]^{2}}{2d_{m}(\tau)^{2}}\.\]
* \(0\leq d\leq d_{m}(\tau)\), i.e., the first node is approaching the second node from behind. In this case, it holds that \[\Gamma(\tau,d)= \Pr\{D^{+}(\tau)\geq d\}=\] (A4) \[\int\limits_{d}^{d_{m}(\tau)}f_{D^{+}}(x)dx =\frac{[d_{m}(\tau)-d]^{2}}{2d_{m}(\tau)^{2}}\.\] The following lemma derives the analytical expression for \(ne^{+}(\tau)\). Lemma A2: _In a 1-D linear network topology, where nodes are randomly distributed on a two-lane road segment of \(L\) meters and node speeds are uniform independent and identically distributed random variables following a \(\mathcal{U}(v_{a},v_{b})\) distribution, we can write that_  
\[ne^{+}(\tau)=\rho\frac{\Delta v}{3}\tau\.\] (A5)  
Proof: Let us consider a content requester \(n\) with position \(x(t)\) at time \(t\). The number of contacts in the time interval \([t,t+\tau]\) between node \(n\) and nodes moving in its direction can be computed as the sum of the number of nodes behind \(n\), say \(ne^{+}_{l}(\tau)\), that will come as close as \(l_{tx}\) to \(n\) during the considered time interval, and the number of nodes in front of \(n\), say \(ne^{+}_{r}(\tau)\), that \(n\) will approach as close as \(l_{tx}\) during the same time interval. It is straightforward to observe that the only nodes that have a non-null probability of coming in contact with the node \(n\) are those that, at time \(t\), are located at a distance from \(n\) equal to either \([-(d_{m}(\tau)+l_{tx}),-l_{tx}]\) or \([l_{tx},(d_{m}(\tau)+l_{tx})]\).