# Connectivity Analysis in VANETs with Dynamic Ranges
## 3 Analytical Procedure
### Random Communications Ranges
#### 3.2.4 Lognormal Channel Shadowing Effect  
From [22], the shadowing is given by the lognormal distribution where the signal power is in dB. Thus, the received signal SNR dependent of the transmission range is lognormal distributed. The range, \(R\)'s distribution can be derived below. If the PDF of SNR \(\gamma_{r}\) is  
\[f_{\gamma_{r}}(\gamma)=\frac{1}{\sqrt{2\pi}\sigma\gamma}\exp\Big{(}-\frac{( \ln\gamma-\ln(r^{-\alpha}))^{2}}{2\sigma^{2}}\Big{)}, \tag{17}\]  
then the CDF of the range, \(R\) is derived as;  
\[F_{R}(r) =1-\mathbb{P}(R\geq r)=1-\mathbb{P}(\gamma_{r}(r)\geq\gamma_{th})\] \[=1-\int_{\gamma_{th}}^{\infty}f_{\gamma_{r}}(r)\ d\gamma=1-\int_{ \frac{\gamma_{th}P_{N}}{P_{N}}}^{\infty}\frac{1}{\sqrt{2\pi}\sigma\gamma}\exp \Big{(}-\frac{(\ln\gamma-\ln(r^{-\alpha}))^{2}}{2\sigma^{2}}\Big{)}d\gamma. \tag{18}\]  
Suppose the mean received SNR without bus shadowing effect is \(\frac{P_{0}P_{N}}{P_{N}}\). Let \(x=\frac{\ln\gamma-\ln(r^{-\alpha})}{\sigma}=\frac{\ln(\frac{\gamma-\ln(r^{- \alpha})}{\sigma}}{\sigma}\) and \(\frac{\text{d}\gamma}{\text{d}x}=\gamma\sigma\). Then by change of variable and simple manipulation;  
\[F_{R}^{\text{Log}}(r)=1-\int_{\ln\big{(}\frac{\gamma_{th}P_{N}}{P_{0}P_{N}r^{ -\alpha}}\big{)}^{\frac{1}{\sigma}}\big{\}}^{\frac{1}{\sigma}}\ \frac{1}{\sqrt{2\pi}}\exp\Big{(}-\frac{x^{2}}{2}\Big{)}dx=1-\text{Q}\bigg{(} \ln\Big{(}\frac{\gamma_{th}P_{N}}{P_{0}P_{N}r^{-\alpha}}\Big{)}^{\frac{1}{ \sigma}}\bigg{)}, \tag{19}\]  
where \(\text{Q}(x)=\int_{x}^{\infty}\frac{\exp(-\frac{x^{2}}{2})dx}{\sqrt{2\pi}}\). With a bus shadowing pathloss effect, the overall mean SNR is \(\frac{P_{0}P_{N}}{kP_{N}P_{N}}\) where \(k\) is the number of big vehicles each causing a pathloss of \(\hat{P}_{SL}\). Therefore, by the above procedure, the CDF of transmission range, \(R\) under big vehicle shadowing, is given by;  
\[\hat{F}_{R}^{\text{Log}}(r)=1-\text{Q}\bigg{(}\ln\Big{(}\frac{\gamma_{th}k \hat{P}_{SL}P_{N}}{P_{0}P_{N}r^{-\alpha}}\Big{)}^{\frac{1}{\sigma}}\bigg{)}. \tag{20}\]  
Shown as above, we write the CDF of \(R\) in a lognormal distributed shadowed channel as;\[F_{R}^{\text{Log}}(r)= \begin{cases}1-\text{Q}\big{(}\ln(\Lambda r^{\alpha})^{ \frac{1}{\varepsilon}}\big{)},\;\text{w}/\text{BS}\\ 1-\text{Q}\big{(}\ln(\Lambda r^{\alpha})^{\frac{1}{\varepsilon}}\big{)},\;\text {BS}\end{cases}  \tag{21a}\] \[F_{R}^{\text{Log}}(r)= \begin{cases}1-\text{Q}\big{(}\ln(\Lambda r^{\alpha})^{ \frac{1}{\varepsilon}}\big{)},\;\text{w}/\text{BS}\\ 1-\text{Q}\big{(}\ln(\tilde{\Lambda}r^{\alpha})^{\frac{1}{\varepsilon}}\big{)},\;\text{BS}\end{cases}  \tag{21b}\]  
where \(\Lambda=\frac{\gamma_{\text{H}}P_{N}}{P_{1}P_{0}}\) and \(\hat{\Lambda}=\frac{\gamma_{\text{H}}k\tilde{F}_{\text{GL}}P_{N}}{P_{1}P_{0}}\) as before. The PDF of \(R\), \(f_{R}(r)\) is derived as;  
\[f_{R}^{\text{Log}}(r)=\frac{d}{dr}\Big{\{}\text{F}_{R}^{\text{Log}}(r)\Big{\}} =\frac{d}{dr}\Big{\{}1-\text{Q}\big{(}\ln(\Lambda r^{\alpha})^{\frac{1}{ \varepsilon}}\big{)}\Big{\}}=-\frac{d}{dr}\Bigg{\{}\int_{\ln(\Lambda r^{ \alpha})^{\frac{1}{\varepsilon}}}^{\infty}\frac{\exp(-\frac{\varepsilon^{2}}{ 2})dx}{\sqrt{2\pi}}\Bigg{\}}.\]  
By using [PERSON] integral theorem on the last term yields;  
\[f_{R}^{\text{Log}}(r)=\frac{\alpha}{\sqrt{2\pi}\sigma r}\exp\Bigg{(}-\frac{ \left(\ln(\Lambda r^{\alpha})\right)^{2}}{2\sigma^{2}}\Bigg{)}. \tag{22}\]  
The PDF in both cases of shadowing effects (non-bus (23a) and bus (23b)), the \(f_{R}(r)\) is given by  
\[f_{R}^{\text{Log}}(r)= \begin{cases}\frac{\alpha}{\sqrt{2\pi}\sigma r}\exp \Big{(}-\frac{(\ln(\Lambda r^{\alpha}))^{2}}{2\sigma^{2}}\Big{)},\;\text{w}/ \text{BS}\\ \frac{\alpha}{\sqrt{2\pi}\sigma r}\exp\Big{(}-\frac{(\ln(\tilde{\Lambda}r^{ \alpha}))^{2}}{2\sigma^{2}}\Big{)},\;\text{BS}.\end{cases}