# Connectivity Analysis in VANETs with Dynamic Ranges
## 2 System Model  
We consider an unidirectional and uninterrupted two-way vehicle traffic highway for simplicity. The network consists of \(N\) vehicles, which are randomly distributed along the highway segment. We assume that of the \(N\) distributed vehicles, there are \(P_{k}\) probable big vehicles that block communication links between transmitter and receiver nodes. Each smart vehicle is equipped with a global positioning system (GPS) integrated into the on-board unit (OBU) from which the state of neighboring vehicles, such as location, velocity, and lane number information, is easily obtained and also provides inter-vehicle communication through basic messages (BMS) exchange. Big vehicles do not have the smart communication capability and just obstruct communications. Moreover, overtaking tendencies of vehicles and lane changes are not permitted. To proceed, we assume an observer position \(O\) where \(N\) vehicles with arrival rate \(\lambda(vehs/sec)\) follow a Poisson process. For each vehicle, an independent and identically distributed velocity of \(v_{i}\) is given for \(i=1,2,\ldots,N\) with rates of \(\lambda_{i}\). See details in [11]. A minimum safety distance \(s\) is maintained between vehicles at all times to prevent accidents as shown in Figure 1a, while a MAC protocol is assumed for simplicity of analysis. In Figure 1a, a free-flow traffic model is captured on the highway segment of length \(L\) where vehicle velocities \(V\) follow a Gaussian distribution [22]. A synchronized flow following GUE is considered in Figure 1b, to capture the traffic congestion effect. In the GUE case, the vehicle speeds are low, but the general flow is similar to a free-low traffic model and not in the extremely wide moving jam-where average speeds are lower than in free-flow traffic and tending towards a stop for some time. The inter-vehicle distance in this model is modeled as a Gaussian unitary ensemble of the random-matrix theory [19]. Thus, the limiting inter-vehicle distributions in \(a\) and \(b\) are the exponential and GUE as shown in Section 3.3. Therefore, this paper seeks to use a generalized mapping distribution that captures both extremes of free-flow and synchronized traffic flow models. Typically, V2V communications are prone to a dynamic environment modeled by small-scale fading channels. With extreme specific conditions captured by the Rayleigh, Nakagami-\(m\) and Weibull \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Value** \\ \hline \(P_{l}\) & Transmitted power \\ \hline \(P_{N}\) & Noise power \\ \hline \(\gamma_{th}\) & Signal-to-noise ratio (SNR) threshold \\ \hline \(\mathrm{G}(.)=\mathrm{G}_{b_{1},b_{2},\ldots,b_{m}}^{a_{1},a_{2},\ldots,a_{n}} \mathrm{F}(z)\) & G-Meijer function [21] \\ \hline \(\Phi(.)\) & Error function \\ \hline \(\
u\) & Traffic headway distribution \\  & mapping factor \\ \hline \(\rho\) & Vehicle density \\ \hline \(m\) & Nakagami-\(m\) fading factor \\ \hline \(C\) & Weibull factor \\ \hline \(\alpha\) & Pathloss exponent \\ \hline \(\dot{\alpha}=\frac{\alpha\mathrm{C}}{2}\) & Weibull fading pathloss exponent \\ \hline \(P_{k}\) & Big vehicle availability probability \\ \hline \(C\) & Weibull factor \\ \hline \(\tilde{P}_{SL}\) & Big vehicle pathloss \\ \hline \end{tabular}
\end{table} 
Table 1: Notations used throughout the paper. distributions. The small-scale fading to no-fading conditions vary between having no line-of-sight (NLOS) to LOS components. We also include another factor of big-vehicle obstructions, which has been shown to contribute immensely to path losses of 7 dB to 20 dB on highways [13]. A signal received by \(R_{1}\) from \(T_{1}\) would be affected by a small-scale fading channel, \(H\), in addition to other roadside shadowing effects, nearby vehicles, and big vehicle obstructions of \(B_{i}\). In Figure 1, we see that \(R_{1}\) in lane 1 and \(T_{1}\) in lane 2 are fully within each other's communication range, and they can connect and establish links between them.