## 5 Survivability Analysis
Consequently, the transition probability of the UAV from state C to S is expressed as  
\[\lambda=Pr(t>\frac{E_{ini}-E_{th}}{P})=exp\{-(\frac{E_{ini}-E_{th}}{\Lambda P} )^{k}\} \tag{14}\]  
2. R(S, F) = \(\mu\), which denotes the transition probability of the node from state S to state F. There are two situations in this process: (1) the energy of the UAV is exhausted; (2) the UAV fails to provide services due to faults or other accidents. 3. R(F, C) = \(\gamma\), which denotes the transition probability of the node from state F to state C. There are also two situations in this process: (1) the UAV finishes charging and restores to provide service; (2) the fault of the UAV is repaired. 4. R(C, F) = \(\xi\), which denotes the transition probability of the node from state C to state F. The process takes into account situations where the UAV malfunctions or other accidents prevent it from continuing to provide service. In order to analyze the survival performance of the network in DWCA-DCH, a parallel combination is carried out according to the single-node transition model. The initial state combination of the node is taken as the initial state of the network, and the state transition probability of the node is taken as the corresponding state transition probability of the network. Figure 3 shows the state transfer in the dual cluster head network, where \(S_{0}\), \(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\), \(S_{5}\), and \(S_{6}\) represent different dual cluster head combination states CC, SC, FC, SS, SF, FS, and FF, respectively. The first letter indicates the state of pCH, and the last letter indicates the bCH state. We denote \(P_{0}\), \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), \(P_{5}\), and \(P_{6}\) as steady-state probability of \(S_{0}\), \(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\), \(S_{5}\), and \(S_{6}\), respectively. According to the principle of equilibrium, the equilibrium equation set is given by:  
\[ \begin{split} S0&:(\lambda_{1}+\xi_{1})P_{0}=\gamma _{1}P_{2}\\ S1&:(\lambda_{2}+\xi_{2}+\mu_{1})P_{1}=\lambda_{1}P_{0}+ \gamma_{2}P_{4}\\ S2&:(\gamma_{1}+\lambda_{2}+\xi_{2})P_{2}=\mu_{1}P_{1 }+\xi_{1}P_{0}+\gamma_{2}P_{6}\\ S3&:\lambda_{2}P_{1}=\mu_{2}P_{3}\\ S4&:\mu_{2}P_{3}+\xi_{2}P_{1}=\gamma_{2}P_{4}\\ S5&:\lambda_{2}P_{2}=\mu_{2}P_{5}\\ S6&:\mu_{2}P_{5}+\xi_{2}P_{2}=\gamma_{2}P_{6}\\ \sum_{i=0}^{6}P_{i}=1\end{split}  \tag{15}\]  
where \(\lambda_{1}\), \(\mu_{1}\), \(\gamma_{1}\), and \(\xi_{1}\) are the pCH state transition probability from C to S, S to F, F to C, and C to F, respectively. Similarly, \(\lambda_{2}\), \(\mu_{2}\), \(\gamma_{2}\), and \(\xi_{2}\) are the corresponding transition probability of bCH. \(\lambda_{1}\) and \(\lambda_{2}\) can be deduced from Equations (8)-(14). The value of \(\mu_{1}\) consists of two parts: the probability resulting from energy consumption, which can be derived from Equations (8)-(14), and the probability resulting from UAV failure or other accidents in S state, denoted by \(\zeta\), which is set to a fixed value for pCH. Therefore, \(\mu_{1}\) can be calculated as  
\[\mu_{1}=exp\{-(\frac{E_{ini}-E_{th}}{\Lambda P})^{k}\}+\zeta \tag{16}\]  
In contrast, \(\mu_{2}\) is assumed to be adjustable, considering that the bCH is required to reduce the dwell time at the S state and, thus, it can quickly return to the C state by flying to the charging pile after tranferring to the F state, which ensures the availability of the bCH. The influence of \(\mu_{2}\) on system performance is evaluated through simulation. Furthermore, \(\gamma\) and \(\zeta\) represent the repairing rate and the failure rate, respectively, which are irrelevant to mobility and energy consumption. Therefore, we also assume them as adjustable parameters and they are evaluated through simulation. Figure 3: The state transfer among the dual cluster head network. According to Equation (15), the steady-state probability of each state can be obtained:  
\[