# Evolutionary Game for Content Cache in a mm-Wave-Based Vehicular Fog
## 3 mm-Wave-Based Vehicular Communication Model
### Directional Antenna and Coverage  
Directional antenna increases radiation intensity toward a designated direction \((\theta,\phi)\) with beam-width of \(\theta_{w}\) compared to isotropic antenna that emits power in isotropic; \(\theta\) and \(\phi\) are angles in z-axis [0, \(\pi\)) and xy-axis domain [0, 2\(\pi\)), respectively. This directional antenna is useful for mm-Wave communications that suffer from high attenuation in the link path. Directional antenna gain is  
\[g(\theta,\phi)=\eta\frac{u(\phi,\theta)}{u_{0}}, \tag{5}\]  
where \(\eta\) is antenna efficiency (\(0<\eta\leq 1\)) and \(u_{0}\) is average power density toward omni-direction. Suppose that transmission power of all directions is almost constant, the average power density is \(u_{0}=1/4\pi P_{t}\) and directivity is \(D=4\pi u(\phi,\phi)/P_{t}\) if the loss \(\eta\) is negligible. Then directivity gain can be acquired by a following equation Equation (6). \begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Description** & **Unit** \\ \hline PL & Pathloss & dB \\ NF & Noise figure & dB \\ IL, CB & Implementation and cable loss & dB \\ S & Set of sectors & \{1, 2,..., \(|\) S \} \\ d & Distance between vehicles & meter \\ G\({}_{tr}\), \(G_{r}\) & Antenna gain for Tx or Rx & dBi \\ T & Threshold of target SNR & dB \\ \(P_{t}\) & Transmission power & dBm \\ \(\lambda\) & Vehicle density & – \\ \(L_{w}\) & Lane width of road & meter \\ \(L_{o}\) & Average vehicle length & meter \\ \(P_{b}\) & Beam blockage probability & [0, 1] \\ \(P_{c}\) & Connectivity probability & [0, 1] \\ \(\alpha\) & Pathloss exponent & 2, 4 \\ H & Channel response & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols for the mm-Wave V2X model. \[g(\phi,\theta)=\frac{|u(\phi,\theta)|}{1/4\pi\int_{0}^{2\pi}\int_{0}^{\pi}F(\theta )F(\phi)sin(\theta)d\theta d\phi} \tag{6}\]  
where the \(F(\phi)\) and \(F(\theta)\) are radiation intensity functions for azimuthal and elevation plane. The beam-forming achieved by a uniform linear array antenna (ULA) or uniform circular array antenna (UCA) can be modelled by a sectorized directional antenna following ITU-R reference [66] which covers 400 MHz to 70 GHz spectrum frequencies. Then the main lobe gain can be decided by beamwidth of the each sector antenna, \(\phi_{\mathrm{s}}\), and then number of sectors \(|S|\) is \(2\pi/\phi_{\mathrm{s}}\). The radiation intensity of the sectorized directional antenna can be considered by one of two different radiation intensity functions in the azimuthal plane, rectangular or exponential sectoral radiation while the elevation plane is assumed to be an exponential function. In rectangular sectoral radiation, the azimuthal power intensity is derived as  
\[F(\phi)= \begin{cases}0&\quad\text{if }\frac{\phi_{\mathrm{s}}}{2}-|\phi|\geq 0 \\ 1&\quad\text{if }\frac{\phi_{\mathrm{s}}}{2}-|\phi|<0.\end{cases}  \tag{7}\]  
Elevation power is  
\[F(\theta)=e^{-a\theta^{2}},\ a=-ln(0.5)\frac{4}{\theta_{3}^{2}}, \tag{8}\]  
where \(\theta_{3}\) is 3 dB beam-width of the antenna in the elevation plane (degrees). Consequently, the radiation intensity of an isotropic source, \(u_{0}\) can be calculated by Equations (6)-(8),  
\[u_{0}=\frac{\phi_{\mathrm{s}}\theta_{3}}{4\pi}\sqrt{\frac{\pi}{2.773}}e^{\theta _{3}^{2}/11.09}. \tag{9}\]  
The directivity gain of the rectangular radiation model is  
\[D(\phi_{\mathrm{s}})=\frac{38750}{\phi_{\mathrm{s}}\theta_{3}}e^{\theta_{3}^{ 2}/36400}, \tag{10}\]  
where \(D=u_{m}/u_{0}\) and maximum radiation intensity, \(u_{m}\) is 1. When applying the exponential function for sectoral radiation for the azimuthal function \(F(\phi)\), we can have the directivity gain as below,  
\[F(\phi)=e^{b\phi^{2}},\ b=ln(0.5)\frac{4}{\phi_{\mathrm{s}}^{2}},\ D(\theta_{3 })=\frac{36400}{\phi_{\mathrm{s}}\theta_{3}}e^{\theta_{3}^{2}/36400}. \tag{11}\]  
The directivity gain of both antennae was experimented on in our previous work [1].