# Evolutionary Game for Content Cache in a mm-Wave-Based Vehicular Fog
## 3 mm-Wave-Based Vehicular Communication Model
### Connectivity Probability  
In this section, we derive probability of beam connectivity based on the model of vehicular fog that is established in Sections 3.2-3.5. Receive signal-to-interference-plus-noise ratio (SINR) of each vehicle is  
\[SINR=\frac{G_{0}P_{t}H_{0}d^{-\alpha}}{N_{0}+\sum_{n\in\Phi}G_{n}PH_{n}|X_{n}|^{- \alpha}}, \tag{20}\]  
where \(d\) is distance between a transmitter and receiver with path loss exponent \(\alpha\) at LOS, and \(H\) is channel fading modelled by Layleigh distribution. \(X_{n}\) is distance between the origin and interference nodes.  
The complementary cumulative distribution function (CCDF) of the SINR with target SINR \(T\) is referred to as connectivity probability, \(P_{c}(T)=\mathbb{P}(SINR\geq T)\).  
\[\mathbb{P}(SINR\geq T)=\mathbb{P}\bigg{(}H_{0}\geq\frac{Td^{-\alpha}(N_{0}+I_{ \Phi})}{G_{0}P_{t}}\bigg{)} \tag{21}\]  
\[=\int_{0}^{\infty}\mathbb{P}\bigg{(}H_{0}\geq\frac{Td^{-\alpha}}{G_{0}P_{t}} \bigg{)}f_{I_{\Phi}}(x)dx\]  
Laplace transform of the aggregate interference in 2D-PPP at \(s=\frac{Td^{-\alpha}}{G_{0}P_{t}}\) is,  
\[\mathcal{L}_{I_{\Phi}}(s)=\mathbb{E}\big{[}exp(-sI_{\Phi})\big{]}=\mathbb{E} \prod_{X\in\Phi}\mathbb{E}_{H}exp(-sG_{g}PH|X|^{-\alpha}) \tag{22}\]  
\[=\mathbb{E}\prod_{X\in\Phi}\frac{1}{1+sG_{S}P|X|^{-\alpha}}.\]  
Using the probability generating functional (PGFL) of PPP,  
\[\mathcal{L}_{I_{\Phi}}(s)=exp\bigg{(}-\lambda_{I_{\Phi}}\int_{\mathbb{R}^{2}} \frac{1}{1+(sG_{S}P)^{-1}|x|^{\alpha}}xdx\bigg{)}. \tag{23}\]  
Here we convert the 2D multi-lane model into 1-D non-homogeneous PPP model using the compression factor as follows [67].  
\[c(d,i)=\frac{1}{\sqrt{d^{2}+2\sqrt{d^{2}-(iL_{w})^{2}}+1}-d}, \tag{24}\]  
which maps positions in multi-lanes into ones of the central lane. The vehicle density of each coverage zone is \(\lambda_{gg}=p_{gg}\lambda\), \(\lambda_{Gg}=p_{Gg}\lambda\) and \(\lambda_{GG}=p_{GG}\lambda\). Accordingly, the PPP intensity of the transmitting vehicle on \(i\)-th lane becomes \(c(d,i)\lambda_{\{GG,Gg,gg\}}\).  
Therefore, the interference from the \(i\)-th lane can be calculated as  
\[\mathcal{L}_{g-GG}^{i}(s)=exp\bigg{(}-\int_{\mathbb{R}^{1}}\frac{c(d,i)p_{g,k |S}\lambda}{1+(sG_{S}P)^{-1}|x_{i}|^{\alpha}}dx_{i}\bigg{)}, \tag{25}\]  
where the \(p_{g,k|S}=p_{GG}\) and \(G_{S}=G_{t}G_{r}\) for example of the \(d_{3}\) coverage.  
Finally, total interference with \(N=N_{R}+N_{L}\) lanes that are right and left-side lanes from the origin node is,  
\[\mathcal{L}_{I_{\Phi}}(s,p_{g,k|S})=\mathcal{L}_{GG}^{0}(s)\times\prod_{i=1}^{ N}\mathcal{L}_{g}^{i}(s),\quad g\in\{GG,Gg,gg\}, \tag{26}\]where interferers are at LOS of each region. Thus, the interference at LOS and NLOS from the \(i\)-th lane can be denoted with the blockage probability, \(P_{b}(i)\) as follow.  
\[\mathcal{L}_{\mathcal{I}_{\Theta}}^{i}(\kappa,p_{g,k|S})=exp\bigg{(}-\int_{ \mathbb{R}^{1}}\frac{c(d,i)p_{g,k|S}\lambda}{1+\kappa|x_{i}|^{\alpha_{N}}}(1-P_ {b}(i))dx_{i} \tag{27}\]  
\[-\int_{\mathbb{R}^{1}}\frac{c(d,i)p_{g,k|S}\lambda}{1+\kappa|x_{i}|^{\alpha_{N} }}P_{b}(i)dx_{i}\bigg{)},\]  
where \(\kappa=(sGP)^{-1}\) and pathloss exponent of LOS or NLOS, \(\alpha^{L}=\)2 and \(\alpha^{N}=\)4 are given by [68]. In consequence, aggregate interference of all regions as connectivity can be  
\[P_{c}(k,S,T)=\prod_{P_{g,k|S}}\mathcal{L}_{\mathcal{I}_{\Theta}}(\kappa,p_{g,k |S}),\quad p_{g,k|S}\in\{p_{GG},p_{GG},p_{gg}\}. \tag{28}\]