# SDN-Based Survivability Analysis for V2I Communications
## 4 V2I Network Survivability Model
For this situation, we establish a V2I survivability model based on vehicle node failure, as shown in \[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]0 It involves \(n\) states, where \(n\) indicates that the vehicle has no failure number, and the total number of failures at the vehicle nodes is not greater than the total number of vehicles. According to the survivability definition in Section 3, we set the failure rate of each vehicle node as \(\lambda_{2}\), meanwhile, for n vehicles, the possible failure rate is \(n\lambda_{2}\). For \(n-1\) vehicles, the possible failure rate is \((n-1)\)\(\lambda_{2}\), and so on. The corresponding failure rate for each state until the \"0\" state is \(\lambda_{2}\). At the same time, the repair rate of timely recovery after the vehicle failure is \(\mu_{2}\). Let the transient probability of any \(t\) time of the survivability model be \(P_{i}\) (\(t\)). Then, according to the state transition, the transient probability equations of each state can be expressed as:  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]1  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]2  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]3  
Hence, when the time \(t\) tends to infinity, transient states tend to be stationary. The steady state probability is \(P_{n}=\underset{t\rightarrow\infty}{lim}P_{n}(t)\). Let the steady state probability be \(P_{i}\) in the \(i\)-th state, we have:  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]4  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]5  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]6  
where \(\underset{n=0}{\overset{N}{\sum}}P_{n}=1\). After calculating the above equations, the steady state probability of the survival model \(i\) state \(P_{i}\) can be obtained from:  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]7  
If there is \(k\) number of faults in the V2I network, for a period of time between 0 and \(t\), the expected survivability rate can be calculated as follows:  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]8  
\[\frac{dP_{0}(t)}{dt}=\mu_{1}P_{1}(t)-\lambda_{1}P_{0}(t), \tag{1}\]9 The V2I link survivability model under vehicle node failure. For practical applications, V2I vehicle networks sometimes have both internal and external faults, that is, a link failure between the vehicle and the infrastructure and a failure of the vehicle node itself. In view of this problem, we can combine these two survivability models to establish a more perfect comprehensive survivability model, as shown in \[\frac{dP_{n}(t)}{dt}=\lambda_{1}P_{n-1}(t)+(n+1)\mu_{1}P_{n+1}(t)-(\lambda_{1 }+n\mu_{1})P_{n}(t),\ for\ n=1,2,\ldots,3N-1, \tag{2}\]0 In this model, we take into account the survivability of the above two fault situations when they occur alone but also the survivability for the case in which these two faults occur simultaneously. In Figure 5, the set state is represented by \((n,j)\), and the parameter n indicates a failure-free number of vehicle nodes. At the same time, the parameter \(n\) is less than or equal to the maximum number \(N\) of vehicles. The parameter \(j\) represents the number of link failures for \(j\leq 3N\). An initial state is represented by \((n,0)\). The digital characters \"\(n\)\" and \"\(0\)\" in this state indicate that the vehicle node and V2I link are in normal condition (no failure), respectively.