## 3 Proposed Methodology
### Estimation of Link Lifetime
Therefore, we propose finding wireless link lifetimes for the SDVN architecture, in which global network information is known by the centralized controller. However, computing wireless link lifetimes using optimization is computationally complex, as the constraint in Equation (2) contains fourth-order decision variables. We observed that the optimizer iterates about 10-15 times in the process of finding the optimum solution for wireless link lifetime between two given nodes. The solution time will be even higher when the network size is high, which can cause an unnecessary delay in the process of finding the shortest or least delayed stable path. Therefore, we used a supervised machine learning approach to predict wireless link lifetimes in order to reduce the computational complexity, where a Deep Neural Network (DNN) is pre-trained to predict wireless link lifetimes. The design choice for DNN over other machine learning classifiers for wireless link lifetime prediction is discussed in Section 5. Note that, for the optimization approach, for each link, a new solution should be found. However, for the DNN, batch predictions can be used to achieve a low computational time for all of the links in the batch input to the DNN model. The prediction of wireless link lifetime using a DNN can be considered a regression problem. A DNN, which has an input layer, an output layer, and one or more hidden layers, is an Artificial Neural Network (ANN) [59]. DNN, once trained, finds a mathematical relationship between the input layer and the output layer [60]. In order to discover the mathematical link between the inputs and the outputs, the weights and biases of the neurons in the neural network are modified to reduce the loss in a user-defined loss function. Once trained, the computationally inefficient optimization-based approach can be replaced with the machine learning model. The architecture of the deep neural network used to predict the lifetime of a wireless link is shown in Figure 5.  
After considering the maximum value of the inequality in Equation (2), it can be expanded to yield the Equation given in Equation (4):  
\[ \begin{split} 0=0.25((\delta a_{x})^{2}+(\delta a_{y})^{2})(t_{ij})^{4}+(( \delta a_{x}\delta v_{x})+(\delta a_{y}\delta v_{y}))(t_{ij})^{3}+\\ (\delta a_{x}\delta x+\delta a_{y}\delta y+(\delta v_{x})^{2}+( \delta v_{y})^{2})(t_{ij})^{2}+2(\delta x\delta v_{x}+\delta y\delta v_{y})(t_ {ij})+\\ (\delta x)^{2}+(\delta y)^{2}-(D^{max}_{ij})^{2}\end{split}  \tag{4}\]  
As evident from Equation (4), when the maximum value of the optimization constraint is considered, the wireless link lifetime (\(t_{ij}\)) can be approximated as the root of a fourth-order polynomial consisting of coefficients composed using relative displacements, relative velocities, relative accelerations, and the maximum transmission distance. However, we consider the maximum value of the inequality only for identifying factors contributing to predicting wireless link lifetime, and the roots of Equation (4) do not directly reflect the wireless link lifetime; instead, this should be found using optimization via the constraint given in inequality (2) and objective Equation (3). Furthermore, note that the coefficients in Equation (4), corresponding to \(a\), \(b\), \(c\), \(d\), \(e\) of the standard fourth-degree polynomial equation \(ax^{4}+bx^{3}+cx^{2}+dx+e=0\), contain square terms of the differential displacement, velocity, etc. Hence, the solution to wireless link lifetime is best approximated by an eighth-degree polynomial combination of differential displacement, velocity, etc.