# SDN-Based Survivability Analysis for V2I Communications
## 4 V2I Network Survivability Model  
According to the V2I definition of vehicle network survivability in Section 3, we now establish the survivability model of the vehicle under the V2I link failure with the specific design shown in Figure 3. To simplify the state diagram, we set \"\(S_{0}\), \(S_{1}\), \(S_{2}\),..., \(S_{n}\),..., \(S_{3N}\)\" states in the diagram to represent the state \"0, 1, 2, 3,..., \(n\),..., \(3N\)\" in the definition of survivability, respectively. The model is a CTMC with \(N\) vehicles on the road and their failure probability. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Detailed Description** \\ \hline \(n\) & failure number \\ \(N\) & vehicle number \\ \(\lambda_{1}\) & V2I link failure rate \\ \(\lambda_{2}\) & Vehicle failure rate \\ \(\mu_{1}\) & V2I link repair rate \\ \(\mu_{2}\) & vehicle repair rate \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter description. When any one of the V2I links fails, through the SDN control center and the MPTCP protocol, these three links of the vehicle node can be seamlessly switched to continue transmitting data. Moreover, we further define that there are \(3N+1\) states in the survivability model of the whole vehicle network, in which the state \"0\" means no V2I link fault, \"1\" means that a V2I link fault occurs, and the subsequent digital state analogizes the corresponding V2I link fault number in turn, i.e., the maximum number of link failures in the whole network is \(3N\). At the same time, with the occurrence of a V2I link failure, in order to ensure the continuous operation of the V2I network, the network will respond to the corresponding timely repair function. According to the survival definition, the repair rate of a V2I link failure is set as \(\mu_{1}\). The repair rate after two V2I link failures is \(2\mu_{1}\). Similarly, the repair rate after n V2I link failure is \(n\mu_{1}\), and the repair rate after 3N V2I link failures is \(3N\mu_{1}\). In this model, the number of failures corresponding to different states is repaired to different degrees. The transient probability of the survivability model is the probability at any t time. It reflects the short-term survivability of vehicle networks. It is expressed as \(P_{i}(t)\). When the initial time \(t=0\), we have the probability \(P_{i}(t)=P_{i}\). According to the state transition, the following transient probability equations can be obtained from:  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]  
\[\frac{dP_{n}(t)}{dt}=\lambda_{1}P_{n-1}(t)+(n+1)\mu_{1}P_{n+1}(t)-(\lambda_{1 }+n\mu_{1})P_{n}(t),\ for\ n=1,2,\ldots,3N-1, \tag{2}\]  
\[\frac{dP_{3N}(t)}{dt}=\lambda_{1}P_{3N-1}(t)-3N\mu_{1}P_{3N}(t), \tag{3}\]  
where \(t\)\(\rightarrow\)\(\infty\), and they will tend to steady state. However, the steady state probability \(P_{n}=P_{n}(t)\) is an important measure reflecting the quality of the network operation. Let the steady state probability be \(P_{i}\) in the \(i\)-th state, it can be obtained from:  
\[\lambda_{1}P_{0}=\mu_{1}P_{1}, \tag{4}\]  
\[(\lambda_{1}+n\mu_{1})P_{n}=\lambda_{1}P_{n-1}+(n+1)\mu_{1}P_{n+1},\ for\ n=1,2, \ldots,3N-1, \tag{5}\]  
\[\lambda_{1}P_{3N-1}=3N\mu_{1}P_{3N} \tag{6}\]  
where \(\sum\limits_{n=0}^{3N}P_{n}=1\), by calculating the above Equations (4)-(6), the steady state probability in the \(i\) state can be obtained as follows:  
\[P_{i}=\frac{\rho^{i}}{i!\sum\limits_{n=0}^{3N}\rho^{n}\frac{1}{n!}}, \tag{7}\]  
where \(\rho=\frac{\lambda_{1}}{\mu_{1}}\), if the expected number of failures in the time range from 0 to \(t\) is calculated as follows:  
\[R_{failure\_number}(t)=\int_{0}^{t}\lambda_{1}P_{i}(x)dx. \tag{8}\]  
Figure 3: The survivability model of the V2I link failure. The above discussions are related to the survivability model of direct failure of the V2I link, however, in the actual process, the internal fault cause of the vehicle node itself may also indirectly lead to the V2I link failure.