# Connectivity Analysis in VANETs with Dynamic Ranges
## 3 Analytical Procedure
### Random Communications Ranges
#### 3.2.1 Rayleigh Fading Channel Effect  
We assume that a slow fading channel is constant over a frame transmission period and only varies in the next data transfer period. For a received signal, the signal-to-noise ratio SNR, \(\gamma\) is Rayleigh distributed with probability density function, PDF given by [23];  
\[f_{\gamma_{r}}^{\text{Ray}}(\gamma)=\frac{1}{\tilde{\gamma}}\exp\left(-\frac{ \gamma}{\tilde{\gamma}}\right), \tag{3}\]  
where \(\tilde{\gamma}=\frac{P_{t}}{P_{N}}\) and \(P_{N}\) is the noise power in the channel. Thus the average received SNR Rayleigh channel both with and without bus shadowing effect is expressed as (using range in (2))  
\[\tilde{\gamma}=\frac{P_{t}P_{0}}{P_{N}R^{\alpha}}\text{, and }\tilde{\gamma}=\frac{P_{t}P_{0}}{k\hat{P}_{SL}P_{N}R^{ \alpha}}.\]  
By considering the transmission range for a received signal, the average SNR for non-bus and bus shadowing effects are respectively given as;  
\[f_{\gamma_{r}}(\gamma)=\frac{\exp\left(-\frac{\gamma}{\tilde{\gamma}}\right) }{\tilde{\gamma}}\text{, and }\hat{f}_{\gamma_{r}}(\gamma)=\frac{\exp\left(-\frac{ \gamma}{\tilde{\gamma}}\right)}{\tilde{\gamma}}. \tag{4}\]  
The received power in both cases is due \(P_{t}\) transmitted through channel, and the transmission radius for each vehicle is a random variable whose distribution can be derivedfrom the received SNR's distribution above as follows. To successfully decode a received signal, the SNR must be greater than the receiver sensitivity (a threshold set according to coding and modulation schemes, [24]), \(\gamma_{th}\), i.e., \(\gamma_{r}\geq\gamma_{th}\). Thus, the cumulative distribution function CDF, \(F_{\gamma_{r}}(\gamma)\) of the received SNR, is expressed as;  
\[F_{\gamma_{r}}(\gamma) =1-\Pr(\gamma_{r}\geq\gamma_{th})=\Pr(\gamma_{r}\leq\gamma_{th})\] \[=1-\int_{\gamma_{th}}^{\infty}f_{\gamma_{r}}(\gamma)\;d\gamma=1- \int_{\gamma_{th}}^{\infty}\frac{1}{\tilde{\gamma}}\exp(-\frac{\gamma}{\tilde {\gamma}})\;d\gamma=1-\exp(-\frac{\gamma_{th}}{\tilde{\gamma}}), \tag{5}\]  
where average SNRs \(\tilde{\gamma}\) and \(\tilde{\gamma}\) are given above for both bus and non-bus shadowing effects, respectively. Now, if connectivity is to be established between adjacent vehicles as per the channel effects, then their geometrical spatial distance of separation must be less than the communication or transmission range of either vehicles, i.e., \(d\leq\min(R_{1},R_{2})=R\)[25]. Thus, it can be shown that since the received SNR, \(\gamma_{r}\) is a function of transmission range \(R\), the CDF of \(R\) can be expressed as follows  
\[F_{R}^{\text{Ray}}(r) =1-\Pr(R\geq r)=1-\Pr(\gamma_{r}(r)\geq\gamma_{th})\] \[=1-\int_{\gamma_{th}}^{\infty}\frac{1}{\tilde{\gamma}(r)}\exp(- \frac{\gamma}{\tilde{\gamma}(r)})\;d\gamma=1-\exp(-\frac{\gamma_{th}}{\tilde{ \gamma}(r)}), \tag{6}\]  
such that \(f_{\gamma_{r}}\) is a function of the transmission range, \(r\) and the PDF, \(f_{R}(r)\) is simply obtained by differentiating \(F_{R}(r)\) w.r.t. \(r\). On substituting for the bus and non-bus shadowing related average received SNRs, \(\tilde{\gamma}_{r}\) and \(\tilde{\gamma}_{r}\) defined as before, the CDFs are given as follows respectively;  
\[F_{R}^{\text{Ray}}(r)= \begin{cases}1-\exp\big{(}-\frac{\gamma_{th}P_{N}r^{a}}{ P_{I}P_{0}}\big{)},\text{WBS}\\ 1-\exp\big{(}-\frac{\gamma_{th}k\tilde{P}_{S1}P_{N}r^{a}}{P_{I}P_{0}}\big{)}, \text{BS},\end{cases} \] (7a) for ( 7a ) without bus-shadowing (WBS) and ( 7b ) with bus shadowing (BS) effect, respectively.