## 3 The Proposed EGR Approach for Highway Traffic Environment
### Temporal Link Quality Estimation Model  
In this section, we first calculate the link quality (temporal transmission probability) of each radio link between a sending vehicles and its neighboring vehicles that are moving in the same direction. Second, we estimate the remaining link lifetime of a specified link between a sending vehicle and its neighboring vehicles.  
_(a) Temporal Transmission Probability_  
We assume that there exists a sending vehicle \(N_{i}\) and a receiving vehicle \(N_{j}\) in the transmission range of the sender. Since the radio transmission range is affected by natural phenomena such as shadowing, multipath fading, noise, etc.; thus, the vehicle's radio transmission range is considered to be non-uniform in all directions. In the proposed model, we considered only the shadowing fading which is occurring due to the presence of some large transport vehicles or any other obstruction between vehicles \(N_{i}\) and \(N_{j}\). Let vehicle \(N_{i}\) wish to send some packets to the neighboring vehicle \(N_{j}\). The temporal received power \(P_{j,R}(d(t^{\prime}))\) at vehicle \(N_{j}\) at time \(t^{\prime}\) using (2) can be expressed as  
\[P_{j,R}\left(d(t^{\prime})\right)=P_{i,T}-\overline{P}(d_{0})-10\eta log_{10} \left(\frac{\sqrt{\left(\Delta x+\Delta v\Delta t\cos\theta\right)^{2}+\left( \Delta y+\Delta v\Delta t\sin\theta\right)^{2}}}{d_{0}}\right)+\chi_{\sigma} \tag{3}\]  
where \(P_{i,T}\) denotes the transmit power of the vehicle \(N_{i}\); \(\overline{P}(d_{0})\) denotes the mean path loss at reference distance \(d_{0}\), where the reference distance is generally assumed to be one; \(\eta\) is the path loss factor depending on the condition of the environment. The \(\chi_{\sigma}\) denotes different levels of cutter in the received power due to shadowing fading, and it is assumed to be a Gaussian random variable with zero mean and \(\sigma^{2}\) variance. The received threshold power of the vehicle \(N_{j}\) with the maximum transmission range \(R\) in a perfect environment can be expressed as  
\[P_{j,R}\left(R\right)=P_{i,T}-\overline{P}(d_{0})-10\eta log_{10}\left(\frac{R }{d_{0}}\right) \tag{4}\]  
The probability of successful transmission \(\mathcal{P}_{i,j}(d(t^{\prime}))\) between the sending vehicle \(N_{i}\) and the receiving vehicle \(N_{j}\) at time \(t^{\prime}\) can be expressed as  
\[\mathcal{P}_{i,j}\big{(}d(t^{\prime})\big{)}=\mathcal{P}\big{(}P_{j,R}\left(d (t^{\prime})\right)>P_{j,R}\left(R\right)\big{)} \tag{5}\]  
\[=\mathcal{P}\left(\begin{array}{c}\chi_{\sigma}>10\eta log_{10}\left(\frac{ \sqrt{\left(\Delta x+\Delta v\Delta t\Delta\cos\theta\right)^{2}+\left(\Delta y +\Delta v\Delta t\Delta\sin\theta\right)^{2}}}{R}\right)\end{array}\right) \tag{6}\]  
link between the sending vehicle and distant vehicles that will last longer is the purpose of this section. The residual link lifetime is the forthcoming time period in which the link remains active. The purpose of estimating the residual link lifetime during which an existing link between moving vehicles will continue to function properly under shadowing fading and to demonstrate how such a technique may be incorporated into a routing approach to achieve the required aim of the guaranteed delivery of the message with geocast routing for a highway traffic environment with links experiencing the adverse effect of shadowing fading. Let two vehicles \(N_{i}\) and \(N_{j}\) be connected at a time \(t^{\prime}\). The forthcoming temporal distance for both the vehicles remaining connected is \(R-d(t^{\prime})\). The uncertainty in the link-lifetime estimates can be introduced by temporal transmission probability \(\mathcal{P}_{i,j}(d(t^{\prime}))\), which represents variation in the link. The residual link-lifetime (\(T_{ll}\)) can be computed as  
\[T_{ll}=\frac{R-d(t^{\prime})}{|\Delta v|}\mathcal{P}_{i,j}\big{(}d\big{(}t^{ \prime}\big{)}\big{)} \tag{9}\]