## 3 Proposed Methodology
### Estimation of Link Lifetime  
We can identify the relative position, relative velocity, relative acceleration, and maximum transmission distance between two nodes as the factors affecting the link lifetime in a wireless communication channel. Let there be two nodes \(i\) and \(j\), and let \(x_{i}(t_{1}),v_{xi}(t_{1}),a_{xi}(t_{1})\) be the \(x\)-coordinate, velocity component in \(x\)-direction, and acceleration component in \(x\)-direction of the node \(i\), respectively, where \(t_{1}\) is the last timestep at which the status (position, velocity, and acceleration) were received. Let \(y_{i}(t_{1}),v_{yi}(t_{1}),a_{yi}(t_{1})\) be the y-coordinate, velocity component in \(y\)-direction, and acceleration component in \(y\)-direction of the node \(i\) at the last known time step \(t_{1}\), respectively. Position, velocity, and acceleration notation for the \(j\)th node should also be understood in a similar manner. Let \(D_{ij}^{max}\) be the maximum wireless transmission distance between the nodes \(i\) and \(j\). Let \(t_{ij}\) represent the lifetime of the wireless link between the \(i\)th and \(j\)th nodes. We can write an inequality as shown in Equation (1), which specifies that, at time step \(t_{1}\), the distance between the two nodes should not exceed the maximum wireless transmission distance (\(D_{ij}^{max}\)). If the inequality given in Equation (1) is violated, it should be noted that \(t_{ij}=0\) (a wireless link does not exist), and subsequent optimization or using machine learning to find the wireless link lifetime is not required. \[ \begin{split}(D_{ij}^{max})^{2}>&=(\delta x)^{2}+( \delta y)^{2}\\ >&=(x_{i}(t_{1})-x_{j}(t_{1}))^{2}+(y_{i}(t_{1})-y_{ j}(t_{1}))^{2}\end{split}  \tag{1}\]  
In the inequality given in Equation (1), \(\delta x\) and \(\delta y\) are the \(x\)-direction displacement difference and the \(y\)-direction displacement difference between nodes \(i\) and \(j\), respectively. Only in instances where the initial distance is less than or equal to the maximum wireless transmission distance (when the inequality given in Equation (1) is not violated), the optimization for finding the wireless link lifetime should be carried out. The optimization constraint given in Equation (2), written with the help of motion equation \(S=ut+1/2 at^{2}\)specifies that, at the end of the wireless link lifetime (\(t_{ij}\)), the distance between the two nodes is less than or equal to the maximum wireless transmission distance (\(D_{ij}^{max}\)):  
\[(D_{ij}^{max})^{2}>=(\delta x+\delta v_{x}(t_{ij})+1/2\delta a_{x}(t_{ij})^{2})^{ 2}+(\delta y+\delta v_{y}(t_{ij})+1/2\delta a_{y}(t_{ij})^{2})^{2}\]  
\[>=((x_{i}(t_{1})-x_{j}(t_{1}))+(v_{xi}(t_{1})-v_{xj}(t_{1}))(t_{ij})+1/2(a_{xi} (t_{1})-a_{xi}(t_{1}))(t_{ij})^{2})^{2}+\]  
\[((y_{i}(t_{1})-y_{j}(t_{1}))+(v_{yi}(t_{1})-v_{yj}(t_{1}))(t_{ij})+1/2(a_{yi} (t_{1})-a_{yj}(t_{1}))(t_{ij})^{2})^{2}\]  
In the inequality given in Equation (2), \(\delta v_{x},\delta v_{y},\delta a_{x}\), and \(\delta a_{y}\) are the \(x\)-direction velocity difference, \(y\)-direction velocity difference, \(x\)-direction acceleration difference, and \(y\)-direction acceleration difference between nodes \(i\) and \(j\), respectively. The objective is to find the maximum value of wireless link lifetime (\(t_{ij}\)) satisfying the constraint in Equation (2) if and only if the inequality in Equation (1) is true. Thus, finding link lifetime can be modeled as a non-linear optimization problem with the objective given in Equation (3). \[maximize\ t_{ij} \tag{3}\]  
Note that wireless link lifetime prediction is only possible in a scenario where global network information is known in the form of positions, velocities, and accelerations. Thus, in the vehicular network architecture, there should be a centralized server that collects the status data and makes network decisions accordingly. Note that knowing global network knowledge at each node in a traditional distributed vehicular ad hoc network is infeasible.