# SDN-Based Survivability Analysis for V2I Communications
## 5 Verification and Analysis of V2I Networks Survivability Model
### Instantaneous Evaluation Analysis of the V2I Network Survivability  
The V2I network survivability model is a CTMC, which has an expected reward at some instantaneous time. It is usually called an instantaneous expected value. The CSL property is defined as follows:  
\[R=?[I=T]. \tag{22}\]  
This expression represents the expected value of the V2I network at \(T\) instantaneous time. We first discuss the instantaneous expected value of the number of link repairs in V2I the network at different repair rates when the failure rate and number of vehicle failures are constant. The specific CSL formula is expressed as:  
\[R[\text{''Expected repair\_number''}]=?[I=T]. \tag{23}\]  
We set the number of link failures to be 10, \(\lambda_{1}\) and \(\lambda_{2}\) are set to be 0.1, the instantaneous time \(T\) is 20 s. Specific analytical values are shown in Figure 10. It shows that the instantaneous expected value of the V2I link repair increases with the increased repair rate \(\mu_{1}\) and \(\mu_{2}\). When \(\mu_{1}=0.8\) and \(\mu_{2}=0.9\), the expected value of repair numbers can reach the maximum value 30. The results show that each state achieves the target state to satisfy the requirements of the defined properties and the instantaneous expected reward. The V2I network can achieve maximum survivability by the instantaneous expected evaluation. Furthermore, by means of the instantaneous property formula, we also analyze the expected number of link failures in V2I the network under different failure rates. The specific CSL formula is expressed as:  
\[R[\text{''Expected failure\_number''}]=?[I=T]. \tag{24}\]  
Time \(T\) is also set as 20 s, the repair rate \(\mu_{1}\) and \(\mu_{2}\) as 0.2, and the number of vehicles as 10. The specific values are shown in Figure 11. It shows that, as the V2I link failure rate \(\lambda_{1}\) and the vehicle node failure rate \(\lambda_{2}\) gradually increase, the number of instantaneous failures in the V2I network becomes greater, and the expected value also increases. Thus, the instantaneous expected evaluation of network survivability is reduced. Figure 9: The maximum survival probability of the the V2I network with link failure rate \(\lambda_{2}\) (corresponding to \(\mu_{1}\) and \(\mu_{2}\)). ### Instantaneous Evaluation Analysis of the V2I Network Survivability  
The CTMC state model will obtain the cumulative expected reward after a period of time, which is the expected reward based on the time boundary, as defined below:  
\[R=?[C\leq T], \tag{25}\]  
where the formula represents the expected reward of V2I network over \(T\) time, and its expected value is calculated using time \(T\) as the boundary for cumulative rewards. Similar to the instantaneous expected value analysis, we first analyze the expected reward of the number of link repairs in the V2I network at  
Figure 11: The instantaneous expected failure number of the the V2I network with time (corresponding to \(\mu_{1}\) and \(\mu_{2}\)). Figure 10: The instantaneous expected repair number of the the V2I network with time (corresponding to \(\mu_{1}\) and \(\mu_{2}\)). different repair rates, when the failure rate and number of vehicle failures remain are fixed. The specific CSL formula is expressed as:  
\[R\{\textit{``Expected reward of repair\_num''}\}=?[C\leq T]. \tag{26}\]  
The specific numerical results are shown in \[R[\text{''Expected repair\_number''}]=?[I=T]. \tag{23}\]0 The number of link failures are set at \(n\) = 10, the \(\lambda_{1}\) and \(\lambda_{2}\) are set to be 0.1, instantaneous time \(T\) is 20 \(s\). The figure shows that the expected link repair number of V2I network is improved with the increased repair rate \(\mu_{1}\) and \(\mu_{2}\) in the \(T\) time boundary. This indicates that the number of V2I links available in the network increases. As a result of MPTCP, each vehicle seamlessly switches between its effective links, and the V2I network service quality is improved, so that the V2I network survival ability is enhanced, and the survivability is improved.