## 3 SLD-GEDIR Protocol
### Segment Vehicles
If \(X\) is the random variable which represents number of vehicles present in the segment area, then the probability \(P_{area}^{NSE}\left(X=n\right)\) of the presence of \(n\) vehicles in the segment in a non-shadowing environment can be calculated as given by Equation (3):  
\[P_{Sarea}^{NSE}(X=n)=\frac{(\lambda\times S_{area})^{n}\times e^{-(\lambda \times S_{area})}}{n!} \tag{3}\]  
By substituting the value of \(S_{area}\) given in Equation (2), the Equation (4) is obtained:  
\[P_{Sarea}^{NSE}(X=n)=\left[\frac{\left[\lambda\left\{\left(R^{2}\times tan^{- 1}\left(\frac{L}{2M}\right)\right)-\left(\frac{L}{4}\times\sqrt{4R^{2}-L^{2}} \right)\right\}\right]^{n}}{n!}\right]\times e^{-\lambda\left[\left(R^{2}\times tan ^{-1}\left(\frac{L}{2M}\right)\right)-\left(\frac{L}{4}\times\sqrt{4R^{2}-L^{2 }}\right)\right]} \tag{4}\]  
By substituting \(n=0\), the probability \(P_{Sarea}^{NSE}\left(X=0\right)\) of presence of no vehicle in the segment area in a non-shadowing environment can be calculated as expressed in Equation (5):  
\[P_{Sarea}^{NSE}(X=0)=\left[\frac{\left[\lambda\left\{\left(R^{2}\times tan^{ -1}\left(\frac{L}{2M}\right)\right)-\left(\frac{L}{4}\times\sqrt{4R^{2}-L^{2}} \right)\right\}\right]^{0}}{0!}\right]\times e^{-\lambda\left[\left(R^{2}\times tan ^{-1}\left(\frac{L}{2M}\right)\right)-\left(\frac{L}{4}\times\sqrt{4R^{2}-L^{2 }}\right)\right]} \tag{5}\]  
\[P_{Sarea}^{NSE}(X=0)=e^{-\lambda\left\{\left(R^{2}\times tan^{-1}\left(\frac{ L}{2M}\right)\right)-\left(\frac{L}{4}\times\sqrt{4R^{2}-L^{2}}\right)\right\}}\]  
The probability \(P_{Sarea}^{NSE}\left(X\geq 1\right)\) of the presence of at least one vehicle in the segment area in a non-shadowing environment can be expressed as given in Equation (6):  
\[P_{Sarea}^{NSE}(X\geq 1)=1-e^{-\lambda\left\{\left(R^{2}\times tan^{-1} \left(\frac{L}{2M}\right)\right)-\left(\frac{L}{4}\times\sqrt{4R^{2}-L^{2}} \right)\right\}} \tag{6}\]1.2 Impact of Non-Circular Transmission Range on the Probabilistic Analysis of the Presence of Vehicles in a Segment Area (Shadowing Environment)  
The probabilistic analysis of the presence of vehicles in the segment area in vehicular traffic environments can be made more realistic by using a non-circular transmission range. The assumption of a non-circular transmission range is the basis for integrating the shadowing model with the segment area. The circular shape of the transmission range assumed for ideal conditions is transformed into an amoeba-like shape due to the shadowing effect in vehicular traffic environment. The actual transmission range varies in different directions due to the effect of shadowing [52] on the received power (_cf._ Figure 5), which is given in Equation (7):  
\[PW_{r}=PW_{t}\left\{10\log_{10}C-10\ \omega\log_{10}\left(\frac{d}{d_{0}} \right)-\tau\right\} \tag{7}\]  
where, \(C\) is a constant that represents channel attenuation and antenna characteristics, \(\omega\) denotes the path loss exponent, \(d\) and \(d_{0}\) represents the distance between vehicles and the reference distance for antenna respectively. The term \(\tau\) is a Gaussian random variable. As shown in Figure 5, a small incremental area \(da\) at distance \(r\) can be used to find the intersection area \(I_{area}\) between the circular transmission range and then amoeba-like transmission area. In \(I_{area}\) received power, \(PW_{r}\) always remains above the minimum required power \(PW_{min}\) to decode a signal. \(I_{area}\) can be calculated as given in Equation (8):  
\[S_{area}=\frac{\alpha}{360^{0}}\pi R^{2}-\frac{L}{4}\sqrt{4R^{2}-L^ {2}} \tag{1}\]0  
where \(PW_{r}(r)\) represents the received power in area \(da\) at distance \(r\). Log-normal distribution accurately models the variation in received power due to shadowing [53].