# Interference Analysis for Vehicle-to-Vehicle Communications at 28 GHz
## 3 Analysis of PEP  
In the following, we derive the expression of PEP for aforementioned system model. We consider that a packet prior transmission is divided into \(M\) blocks, where the value of \(M\) depends on the speed of transmitting and receiving vehicles. Specifically, it can be written as \(M=\frac{\omega}{T_{c}\log(1+\psi)}\)[32], where \(\omega\) is the packet size, \(T_{c}\) refers to the duration of the coherence period which is a function of Doppler shift, and \(\psi\) is the required data rate. Assuming ideal coding and modulation in vehicular networks, the PEP is based on the outage effect of the wireless channel [32]. Thus, the PEP can be written as  
\[P_{e}=1-(1-P_{err})^{M}, \tag{8}\]  
Figure 1: System model for V2V communications. where  
\[P_{err}=\Pr\{\log(1+\frac{\lambda_{c}}{\lambda_{i}+1})\leq\psi\}=\Pr\{\lambda_{c} \leq(2^{\psi}-1)(1+\lambda_{i})\}=\int_{0}^{\infty}\int_{0}^{(2^{\psi}-1)(1+ \lambda_{i})}f_{\lambda_{c}}(\lambda_{c})f_{\lambda_{i}}(\lambda_{i})d\lambda_{ c}d\lambda_{i}. \tag{9}\]  
By replacing (6), applying variable transform and using identities ([36] (3.381), (8.970)), \(P_{err}\) can be simplified as  
\[P_{err}=\frac{\alpha_{i}2^{-(\mu_{i}+\mu_{c}+2)}\left(\bar{\lambda }_{c}\right)^{\left(-\frac{\alpha_{c}\beta_{c}}{2}\right)}}{\Gamma\left(\mu_{i }\right)\Gamma\left(\mu_{c}\right)\left(\bar{\lambda}_{i}\right)^{\left(\frac{ \alpha_{i}\beta_{i}}{2}\right)}}\sum_{n=0}^{\infty}\frac{n!c_{n,i}}{\left(\mu _{i}\right)_{n}}\sum_{s=0}^{n}\frac{\left(-1\right)^{s}2^{s}}{s!\left(\lambda _{i}\right)^{\left(\frac{\alpha_{i}\beta}{2}\right)}}\] \[\times\left(2\left(\bar{\lambda}_{c}\right)^{\left(\frac{\alpha _{i}}{2}\right)}\right)^{\left(\mu_{c}+r\right)}\times\int_{0}^{\infty}\lambda _{i}^{\left(\frac{\alpha_{i}\left(\mu_{i}+s\right)}{2}-1\right)}\exp\left(- \frac{\lambda_{i}^{\left(\frac{\alpha_{i}}{2}\right)}}{2\bar{\lambda}_{i}^{ \left(\frac{\alpha_{i}}{2}\right)}}\right)\] \[\times\gamma\left(\mu_{c}+r,\frac{\left\{\left(2^{\psi}-1\right) \left(1+\lambda_{i}\right)\right\}^{\frac{\alpha_{i}}{2}}}{2\left(\bar{ \lambda}_{c}\right)^{\left(\frac{\alpha_{i}}{2}\right)}}\right)d\lambda_{i}, \tag{10}\]  
where \(\gamma(. )\) denotes the incomplete gamma function [36]. In order to further simplify the above expression, we first apply the identity ([36] (8.354)). After performing some mathematical manipulations and using ([37] (2.2)), we get \(P_{err}\) as  
\[P_{err}=\sum_{n=0}^{\infty}\frac{n!c_{n,i}}{\left(\mu_{i}\right) _{n}}\times\sum_{s=0}^{n}\frac{\left(-1\right)^{s}2^{s}\left( \begin{matrix}n+ \mu_{i}-1\\ n-s\end{matrix} \right)}{s!\left(\bar{\lambda}_{i}\right)^{\left(\frac{\alpha_{ i}\beta}{2}\right)}\left(\frac{2^{\psi}-1}{2^{\psi}-2}\right)^{\left(\frac{ \alpha_{i}\left(\mu_{i}+s\right)}{2}\right)}\sum_{k=0}^{\infty}\frac{ktc_{k,c}} {\left(\mu_{c}\right)_{k}}}\] \[\times\sum_{r=0}^{k}\left(-1\right)^{r}2^{\left(r+1\right)}\left( \begin{matrix}k+\mu_{c}-1\\ k-r\end{matrix} \right)\frac{2^{-\left(\mu_{i}+\mu_{c}+\mu_{i}+1\right)}\left( \lambda_{c}\right)^{\left(-\frac{\alpha_{c}\beta_{c}}{2}\right)}}{\pi^{\left( \alpha_{i}-0.5\right)}\Gamma\left(\mu_{i}\right)\Gamma\left(\mu_{c}\right) \left(\bar{\lambda}_{i}\right)^{\left(\frac{\alpha_{i}\beta_{i}}{2}\right)}}\] \[\times\sum_{l=0}^{\infty}\left(\frac{2^{\psi}-2}{\alpha_{i}} \right)^{\left(\frac{\alpha_{c}\left(\mu_{c}+r+l\right)}{2}\right)}\times \frac{\left(-1\right)^{l}\left(2\bar{\lambda}_{c}^{\left(\frac{\alpha_{l}}{2} \right)}\right)^{-l}}{l!\left(\mu_{c}+r+l\right)\Gamma\left(-\frac{\alpha_{c} \left(\mu_{c}+r+l\right)}{2}\right)}\] \[\times G_{\alpha_{i},2+\alpha_{i}}^{2+\alpha_{i},\mu_{i}}\left( \frac{1}{16}\left(\frac{2^{\psi}-2}{2^{\psi}\bar{\lambda}_{i}-\bar{\lambda}_{i} }\right)\left| \begin{matrix}\Xi\left(\alpha_{i},\xi_{1}\right)\\ \Xi\left(2,0\right),\Xi\left(\alpha_{i},\xi_{2}\right)\end{matrix}