# Connectivity Analysis in VANETs with Dynamic Ranges
## 3 Analytical Procedure
### Connection Probability Using Dynamic Communication Ranges
#### 3.5.2 Case 2: Random Distributed Communication Range  
For this case, we use the random variables of range \(R\) and the separation distance \(D\) between two communicating vehicles. The connectivity probability \(P_{c}\) then can be derived by conditioning on \(R\) (using the law of total probability [26]). **Case 2.a: Nakagami-\(m\) distributed range. ** For Nakagami-\(m\) fading channel, the connectivity probability was given in Section 3.2.2 and here, we provide the closed-form expression for the mapped-inter-vehicle distance distribution. Later, Rayleigh distributed probability connectivity can be derived by simple substitution of the fading factor, \(m=2\). For mapped inter-vehicle distance and Nakagami-\(m\) distributed range PDF in (13), connectivity probability as in (33); where \(G(. )\) is the Meijer G function [22], \(\Gamma(. )\) function, and \(v=\beta+1\). See Appendix A, for proof. \[\mathrm{P}_{\mathrm{c}}^{\mathrm{Nak}}=\frac{\sqrt[2\pi]{(2\pi)^{v-\alpha}( \alpha v^{-1})^{2\beta+v}}}{\Gamma\big{(}\frac{2\beta+1}{v}\big{)}}\sum_{t=0}^ {m-1}\frac{(m\Lambda)^{t}}{t! }\Bigg{(}\frac{\alpha}{v}\Big{(}\frac{\Gamma( \frac{2\beta+1}{v})}{\rho}\Big{)}^{v}\Bigg{)}^{\frac{\alpha}{v}}G_{1,\frac{ \beta}{v}}^{\frac{\alpha}{v},1}\Bigg{[} \begin{smallmatrix}1\\ t+\frac{2\beta+1}{\pi},\ldots,1+t+\frac{\beta}{\beta}\end{smallmatrix} \Bigg{|} \frac{1}{m\Lambda}\Bigg{(}\frac{\
u\rho^{v}}{\alpha\left(\Gamma(\frac{2 \beta+1}{v})\right)^{v}}\Bigg{)}^{\frac{\alpha}{v}}\Bigg{]} \tag{33}\]  
Then, we write generally for exponential (\(\beta=0\)) and GUE (synchronised) inter-vehicle distance (\(\beta=1\)) distributions, is as (34). \[\mathrm{P}_{\mathrm{c}}^{\mathrm{Nak}}= \begin{cases}\sum_{t=0}^{m-1}\frac{\sqrt{a(2 \pi)^{1-\alpha}}}{t!}\left(\frac{m\Delta a^{\alpha}}{\rho^{\alpha}}\right)^{t} \mathrm{G}_{1,\mathrm{A}}^{\xi,1}\begin{bmatrix}1\\ 1\end{bmatrix}\begin{bmatrix}1\\ \frac{1}{m\Lambda}\left(\frac{\rho}{\alpha}\right)^{\alpha}\\ \end{bmatrix},\;\beta=0\;(\mathrm{Exp.}),\\ \\ \sum_{t=0}^{m-1}\frac{\sqrt{2}a(m\Lambda)^{t}}{\sqrt[4]{(2\pi)^{t}}t!}\left( \frac{\alpha\sqrt{\pi}}{\rho\sqrt{8}}\right)^{\alpha t}\mathrm{G}_{1,\frac{ \alpha}{\beta}}^{\frac{\alpha}{\beta}}\begin{bmatrix}1\\ 1\end{bmatrix}\begin{bmatrix}1\\ 1\end{bmatrix}\begin{bmatrix}1\\ \frac{1}{m\Lambda}\left(\frac{\rho\sqrt{8}}{\sqrt{\alpha\pi}}\right)^{ \alpha}\\ \end{bmatrix},\beta=1\;(\mathrm{GUE}).\end{cases}  \tag{34}\]  
Like before, replacing \(\Lambda\) with \(\dot{\Lambda}\) gives the connectivity probability for a bus-shadowed scenario. **Case 2.b: Weibull distributed range. ** The probability of connectivity of two adjacent vehicles with a generalised mapped inter-vehicle distance and Weibull distributed range is as (35); where \(\
u,\Lambda,\rho\) are as defined before, \(\dot{\alpha}=\frac{\alpha C}{2}\) and \(G_{p,d}^{m,n}[...]\) is the Meijer G function [32]. See Appendix B, for proof. \[\mathrm{P}_{\mathrm{c}}^{\mathrm{Weib}}=\sqrt[2]{\left(\frac{(2 \pi)^{\
u-\delta}(\dot{\alpha}\
u^{-1})^{2\beta+\
u}}{\left(\Gamma(\frac{2 \beta+1}{\
u})\right)^{2\
u}}\right)}\mathrm{G}_{1,\frac{\alpha}{\beta}}^{ \frac{\delta}{\beta},1} \begin{bmatrix}1\\ \frac{1}{2\dot{\alpha}+1},...,1+\frac{\beta}{\beta}\\ \end{bmatrix} \left(\frac{1}{\Lambda\Gamma\left(1+\frac{2}{C}\right)}\left( \frac{\
u\rho^{\
u}}{\dot{\alpha}\left(\Gamma(\frac{2\beta+1}{\
u})\right)^{ \
u}}\right)^{\frac{\alpha}{\beta}}\right)^{\frac{\zeta}{2}}\right] \tag{35}\]  
Following similar derivations as above and substituting \(\beta=0,1\) for exponential and GUE inter-vehicle distance distributions respectively, gives (36). \[\mathrm{P}_{\mathrm{c}}^{\mathrm{Weib}}=