# Connectivity Analysis in VANETs with Dynamic Ranges
## 4 Discussion  
In this section, we discuss numerical results are presented for the dynamic range distribution and connectivity probability analysis. The simulation results were all obtained using MatLab 2023a [21]. General simulation parameters and their values are given in Table 4 (next page). Typical values for the mean speed and the standard deviation were adopted from [13,19]. For a road section of length \(L\), the simulation was conducted as follows: to construct a network of \(N\) vehicles in a free-flow or synchronized traffic, a random number of vehicles was generated each separated from one another with a minimum safety distance \(s\) and velocity value selected from the truncated Gaussian distribution as in (25). Thus, with \(N\), a total of \((N-1)\) communication links are formed. To simulate the big vehicle shadowing effect, vehicle nodes are inserted randomly between two randomly chosen vehicles in such a way that there is no link between them (rather, a path-loss value representing big vehicle obstruction effect is evaluated together with the received signal power). Due to the different fading environments, an average SNR \(\tilde{\gamma}\) and \(\tilde{\gamma}\) with or without big vehicle obstruction respectively for the two traffic scenarios above is computed for all \((N-1)\) links. With the inter-vehicle distance \(D\) and SNR threshold \(\gamma_{th}\) set, a random communication range, \(R\) is found. This procedure is repeated 10,000 times for all the fading channel environments.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Notation** & **Value** \\ \hline Mean speed & \(\mu\) & [70, 90, 110, 130, 150] (km/h) \\ \hline Speed std & \(\sigma\) & [21, 27, 33, 39, 45] (km/h) \\ \hline Road length & \(L\) & 4 km \\ \hline Inter-vehicle distance & \(D\) & [10–1000] (m) \\ \hline Lane width & \(W\) & 3 m \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Connectivity Probability** & **Free-Flow Traffic** & **Synchronised Traffic** \\ \hline Static range & \(1-e^{-\mu R}\) & \(\Phi\bigg{(}\frac{2\pi R}{\sqrt{\pi}}\bigg{)}-\frac{4\rho^{2}\pi}{\pi}\exp\left\{ -\frac{4\rho^{2}R}{\pi}\right\}\) \\ \hline Rayleigh range & \(\sqrt{a(2\pi)^{1-a}}\ G_{1,4}^{\mathrm{a.1}}\left[\frac{1}{2}-\gamma\right]^{1} \left|\frac{1}{4}\left(\frac{\rho}{\pi}\right)^{a}\right|=\frac{e\sqrt{\pi}e^{- \frac{2\rho^{2}R}{\pi}}}{2\sqrt{\Lambda}}\Phi\bigg{(}\frac{\rho}{2\sqrt{\pi}} \bigg{)}\) & \(\frac{a\sqrt{2}}{\sqrt{2(2\pi)^{2}}}\ G_{1,8}^{\mathrm{a.1}}\left[\frac{1}{2} \left(\frac{\rho\sqrt{8}}{\sqrt{\pi}}\right)^{a}\right]=\frac{(2\rho)^{3}}{ \sqrt{\pi(4\rho^{3}+\Lambda\pi)^{3}}}{\rho-\gamma_{4}=2}\) \\ \hline Nakagami-\(m\) range & \(\sum_{a=0}^{m-1}\frac{\sqrt{\pi(2\pi)^{2}}}{\pi}\bigg{(}\frac{m\Lambda x^{2}}{ \rho^{2}}\bigg{)}^{t}\ G_{1,4}^{\mathrm{a.1}}\left[\frac{1}{2}\left(\frac{ \rho}{\pi}\right)^{a}\right]\) & \(\sum_{t=0}^{m-1}\frac{\sqrt{\pi}a(m\Lambda x^{2})}{\sqrt{(2\pi)^{2}}}\left( \frac{\rho\sqrt{8}}{\sqrt{\pi}}\right)^{t}G_{1,8}^{\mathrm{a.1}}\left[\frac{1}{ 2}\left(\frac{\rho\sqrt{8}}{\sqrt{\pi}}\right)^{a}\right]\) \\ \hline Weibull range & \(\sqrt{a\left(2\pi\right)^{1-a}}\ G_{1,4}^{\mathrm{a.1}}\left[\frac{1}{2}\frac{ 1}{2}\frac{2}{2-\gamma}^{2}\right]^{1}\left[\left(\frac{\rho}{\pi\Gamma\left( 1+\frac{2}{\xi}\right)}\right)^{a}\right]\) & \(\frac{a\frac{4}{\sqrt{(2\pi)^{2}-a}}}{\sqrt{\pi}}\ G_{1,4}^{\mathrm{a.1}}\left[ \frac{1}{2}\left(\frac{\left(\frac{\rho\sqrt{8}}{\sqrt{\pi}}\right)^{a}}{ \Lambda\Gamma\left(1+\frac{2}{\xi}\right)}\right)^{a}\right]\) \\ \hline Lognormal range & \(\int_{0}^{\infty}\rho\mathbf{Q}\bigg{(}\frac{\ln(\Lambda x^{2})}{\sigma}\bigg{)} e^{-\mu x}dx\) & \(\frac{2\rho^{3}}{\pi^{2}}\int_{0}^{\infty}\mathbf{Q}\bigg{(}\frac{\ln(\Lambda x^{2})}{ \sigma}\bigg{)}x^{2}e^{\left(-\frac{4\rho^{2}\pi^{2}}{\pi}\right)}dx\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary Connectivity probabilities, \(\mathbb{P}_{\mathrm{c}}\) for fading environments.