## 5 Survivability Analysis  
In order to evaluate the performance of a UAV network based on the dual cluster head mechanism proposed above, we propose a survivability analysis method based on the Markov process (SAM-MP) to analyze the survivability of the UAV network. Since cluster head nodes are affected by various factors, the transition between their states is a random Markov process [19]. According to the availability and energy states of pCH and bCH, three states are introduced, namely cooperative state, selfish state, and failure state, to depict the UAV ad hoc network. The specific definitions are as follows:  
* Cooperative state (C), where UAVs can complete data acquisition, processing, receiving, and transmitting normally. * Selfish state (S), where the UAV's remaining energy is lower than the threshold \(E_{th}\). In this case, the UAV only receives data and no longer transmits data. In addition, the UAV will fly to the charging pile. * Failure state (F), when the energy of the UAV is exhausted or the UAV fails, it will enter into the failure state. In addition, the UAV is also in this state during charging or maintaining period. Since the UAV nodes can supplement the network through charging or maintenance, the state transfer among the states of the single node is shown in Figure 2. The transition probabilities between states are respectively expressed as:  
1. R(C, S) = \(\lambda\), which denotes the transition probability of the node from state C to state S. The transition happens along with the natural energy consumption process of UAVs. Therefore, \(\lambda\) is related to the energy consumption rate of UAVs. Since the energy consumption of UAV communication is far less than that of its flight, only the flight energy consumption of a UAV is considered in this paper. The following is the calculation process. Figure 2: The state transfer among different states of a single node. Assuming that the lifetime of the UAV follows the Weibull distribution [27], whose probability density function is  
\[f(t)=\frac{k}{\Lambda}(\frac{t}{\Lambda})^{k-1}e^{-(\frac{t}{\Lambda})^{k}},\ t\geq 0 \tag{8}\]  
where \(\Lambda\) is the scale parameter. \(k\) is the shape parameter and \(k>0\), the value of \(k\) can be adjusted according to the requirements of the actual scenario. \(t\) is the random variable representing the lifetime of the UAV. When the remaining UAV energy \(E_{rm}\) is less than the energy threshold \(E_{th}\), the state of the node transfers from C to S, which follows:  
\[E_{rm}=E_{ini}-E_{i}=E_{ini}-P\cdot t<E_{th} \tag{9}\]  
where \(E_{ini}\) and \(E_{i}\) are the initial and consumed energy, respectively. \(P\) is the horizontal flying power of UAV. Accordingly, the lifetime of UAV can be denoted by  
\[t>\frac{E_{ini}-E_{th}}{P} \tag{10}\]  
where the horizontal flying power \(P\) is given by [28; 29] as  
\[P=P_{p}+P_{I} \tag{11}\]  
where \(P_{p}\) is the parasitic power caused by the parasitic drag originating from the UAV's body friction from airlift drag, etc. It can be expressed as [29; 30]  
\[P_{p}=\frac{1}{2}\rho C_{D_{0}}5v^{3}+\frac{\pi}{4}M\rho c_{b}C_{D_{0}}\omega^ {3}\beta^{4}(1+3(\frac{v}{\omega\beta})^{2}) \tag{12}\]  
where \(\rho\) is the air fluid density in kg/m\({}^{3}\), \(C_{D_{0}}\) is the drag coefficient, \(S\) is the reference area (frontal area of the UAV-BS), \(M\) is the UAV rotor number, \(c_{b}\) is the blade chord, \(\omega\) is the angular velocity, \(v\) represents the velocity in the horizontal direction, \(\beta\) is the radius of the rotor spinning disk in m.  
\(P_{I}\) denotes the power for the lift generated by the wing to overcome the weight of the UAV-BS and the drag caused by the lift, and can be obtained by [28; 29]  
\[P_{I}=G\sqrt{\frac{\lambda^{\prime}-v^{2}}{2}},\ \lambda^{\prime}=\sqrt{v^{4}+ \frac{G}{\pi\rho\beta^{2}}},\ G=(W+w)g \tag{13}\]  
where \(W\) is the frame weight of the UAV in kg, \(w\) is the battery and payload weight in kg, given the gravity \(g\) in m/s\({}^{2}\).