# Evolutionary Game for Content Cache in a mm-Wave-Based Vehicular Fog
## 3 mm-Wave-Based Vehicular Communication Model
### Beam Direction and Content Cache  
To classify beam directions, we divide sectors based on road formation. When the vehicle length, e.g., \(L_{\varphi}=4.5\) m and number of lanes (\(N_{L}\)) with each width (\(L_{w}\)) are given, the minimum sector size \(\theta\) can be \(\tan^{-1}(d/L_{\varphi})\) with beam range, \(d=N_{L}L_{w}\) (\(d\leq d_{3}\)).  
Figure 4: Blockage model in multilane highway. Some vehicles in \(d_{1}\) range region can block signals from an origin in other lanes of \(d_{2}\) and \(d_{3}\); colored vehicles blocks signals for two vehicles on \(d_{3}\) regions.  
The transmitting origin vehicle chooses randomly one of those beam sectors for content data delivery. Figure 5 shows there are two vehicles that have requested contents among eight neighbor vehicles. In a vehicular ICN, vehicles cache contents to improve network performance. Furthermore, each vehicle can cache coded blocks of content instead of an entire content in order to overcome the limitation of local storage and increase availability.  
Caching probability \(p_{k|S}\) is a proportion of sectors \(k=[1,|S|]\) that contain coded caches over total sectors. For example, the \(p_{k|S}\) is \(1/8\) and \(1/4\) when the \(k=1\) and \(k=2\) in the example of Figure 5a,b, respectively.  
The beam directions of vehicles in the vehicular fog depend on coded cache distribution for a content. For this, antenna gain in Figure 2 can be different according to number of coded caches, \(k\). Suppose that only one of neighbour has coded blocks in the Figure 5a, the probability to get interference from neighbour nodes in range \(d_{3}\) is \(P(G_{t}G_{r}|k=1)=\frac{1}{|S|}\) as the cache and client nodes beam to each other. Meanwhile, \(P(g_{t}G_{r}|k=1)=1-\frac{1}{|S|}\) for other sectors not chosen. For \(k=2\), as the client node has two adjacent caches, there are two beam-forming cases of Figure 5a,b. First, the probability to face to each other is  
\[P(G_{t}G_{r}|k=2)=\frac{1}{2|S|}. \tag{18}\]  
Otherwise the client chooses the other cache,  
\[P(G_{t}g_{r}|k=2)=\frac{1}{|S|}\left(1-\frac{1}{2}\right). \tag{19}\]  
Table 2 shows four cases of beam-forming gain according to antenna directions when total \(k\) sectors have coded cache. Probability of each gain is denoted as \(p_{GG}\), \(p_{GG}=P(G_{t}g_{r}|k)+P(g_{t}G_{r}|k)\) and \(p_{gg}\). Accordingly, effective antenna gain of the node \(i\) can be expressed as a discrete random variable, \(G_{g}(p_{ggk|S}),~{}g\in\{GG,Gg,gg\}\) in following sections.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Gain** & \(P_{e}\) & **Range** \\ \hline \(G_{t}\)\(G_{r}\) & \(\frac{1}{|S|}\frac{1}{k}\) & \(d_{3}\) \\ \(G_{t}\)\(g_{r}\) & \(\frac{1}{|S|}(1-\frac{1}{k})\) & \(d_{2}\) \\ \(g_{t}\)\(G_{r}\) & \((1-\frac{1}{|S|})\frac{1}{k}\) & \(d_{2}\) \\ \(g_{t}\)\(g_{r}\) & \((1-\frac{1}{|S|})(1-\frac{1}{k})\) & \(d_{1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Probability of beam-forming gain with cache sectors, \(k\).  
Figure 5: Beam diversification toward coded caches in sectors. In (**a**), beams from adjacent client nodes can interfere in the single cache node. In contrast, the client node can choose one of cache nodes randomly which can decrease probability of inter-beam interference as shown in (**b**).