# Connectivity Analysis in VANETs with Dynamic Ranges
## 3 Analytical Procedure
### Random Communications Ranges
#### 3.2.1 Rayleigh Fading Channel Effect
To simplify notation hereafter, the symbols \[\Lambda\] and \[\dot{\Lambda}\] are used to represent scaling factors of the CDFs above as follows: \[\Lambda=\frac{\gamma_{th}P_{N}}{P_{0}P_{1}}\dot{\Lambda}=\frac{\gamma_{th}P_{ N}P_{N}}{P_{0}P_{1}}\dot{\Lambda}=\frac{\gamma_{th}P_{N}P_{N}}{P_{0}P_{1}}\dot{ \Lambda}=\frac{\gamma_{th}P_{N}P_{N}}{P_{0  
\[f_{\gamma_{r}}(\gamma)=\left(\frac{m}{\gamma}\right)^{m}\frac{\gamma^{m-1}}{\Gamma(m )}\exp\bigg{(}-\frac{m\gamma}{\gamma}\bigg{)}, \tag{9}\]  
where \(\bar{\gamma}\) is the average received SNR, and \(m\) the fading parameter, gamma function \(\Gamma(\
u)=\int_{0}^{\infty}t^{\
u-1}e^{-t}dt\)[27]. We also follow the procedure used in the Rayleigh fading case to derive the distribution of the transmission range here. To decode the transmitted signal accurately, the received SNR, \(\gamma_{r}\geq\gamma_{th}\), where \(\gamma_{th}\) is dependent on receiver sensitivity. Thus;  
\[F_{\gamma_{r}}(\gamma) =1-\Pr(\gamma_{r}\geq\gamma_{th})=\Pr(\gamma_{r}\leq\gamma_{th})=1 -\int_{\gamma_{th}}^{\infty}f_{\gamma_{r}}(\gamma)\;d\gamma\] \[=1-\int_{\gamma_{th}}^{\infty}\left(\frac{m}{\gamma}\right)^{m} \frac{\gamma^{m-1}}{\Gamma(m)}\exp\bigg{(}-\frac{m\gamma}{\bar{\gamma}}\bigg{)} d\gamma=1-\frac{\Gamma(m,\frac{m\gamma_{th}}{\gamma})}{\Gamma(m)}. \tag{10}\]  
Given \(\gamma_{r}\) is dependent of transmission range, the distribution of \(R\) can be derived as well, i.e., CDF \(F_{R}(r)\) is given by;  
\[F_{R}(r) =1-\Pr(R\geq r)=1-\Pr(\gamma_{r}(r)\geq\gamma_{th})\] \[=\Pr(\gamma_{r}(r)\leq\gamma_{th})=\int_{\infty}^{\gamma_{th}} \left(\frac{m}{\bar{\gamma}(r)}\right)^{m}\frac{\gamma^{m-1}}{\Gamma(m)}\exp \bigg{(}-\frac{m\gamma}{\bar{\gamma}(r)}\bigg{)}d\gamma, \tag{11}\]  
and after substituting in the average bus and non-bus shadowed received SNRs (2) into (11) is;  
\[F_{R}^{\text{Nak}}(r)= \begin{cases}1-\frac{\Gamma(m,m\Delta r^{\alpha})}{\Gamma (m)},\;\text{w}/\text{BS}\\ 1-\frac{\Gamma(m,m\Delta r^{\alpha})}{\Gamma(m)},\;\text{BS}\end{cases} \] (12a) where \[\Delta=\frac{\gamma_{th}P_{N}}{P_{R}P_{0}}\] and \[\Delta=\frac{2\gamma_{th}P_{N}}{P_{0}P_{0}}\] and \[\Gamma(\
u,x)=\int_{x}^{\infty}t^{\
u-1}e^{-t}dt\]. Then related PDF, \[f_{R}^{\text{Nak}}(r)=\frac{\text{d}F_{R}^{\text{Nak}}(r)}{\text{d}r}\]. After some manipulation is given as; \[f_{R}^{\text{Nak}}(r)= \begin{cases}\frac{\alpha(m\Delta)^{m}r^{m-1}}{\Gamma(m )}\exp(-m\Delta r^{\alpha}),\;\text{w}/\text{BS}\\ \frac{\alpha(m\dot{\Delta})^{m}r^{m-1}}{\Gamma(m)}\exp(-m\dot{\Delta}r^{\alpha }),\;\text{BS},\end{cases} \] (13a) for ( 13a ) without bus and ( 13b ) with bus shadowing effects, respectively. For the random variable range \[R\], the expectation \[\mathbb{E}[R]\] and variance \[\forall ar[R]\] for a non-big vehicle shadowed channel are expressed as follows using results from [23]; \[\mathbb{E}[R]=\frac{\Gamma\big{(}\frac{1}{\alpha}+m\big{)}}{(m\Delta)^{\frac{ 1}{2}}\Gamma(m)}\text{ and }\mathbb{V}ar[R]=\frac{\Gamma(m)\Gamma\big{(}\frac{m+2}{ \alpha}\big{)}-\Gamma\big{(}\frac{m+1}{\alpha}\big{)}^{2}}{\big{(}(m\Delta)^{ \frac{1}{2}}\Gamma(m)\big{)}^{2}}.\] By replacing \[\Lambda\] with \[\dot{\Lambda}\] into \[\mathbb{E}[R]\] and \[\mathbb{V}ar[R]\] gives the expectation and variance of \[R\] in a corresponding big vehicle shadowed channel scenario.