## 3 System Model  
Suppose that a set of UAVs \(\mathcal{S}\) monitors specific target areas (e.g., high-crime areas, festival venues, and so forth) during specific time periods. Let \(s_{i}\) denote the surveillance UAVs where \(\forall i\in\{1,\cdots,|\mathcal{S}|\}\). Consider that the UAVs are flying at altitude \(h_{i}\) and have corresponding circular surveillance areas \(a_{i}\), defined as \(a_{i}=r_{i}^{2}\pi\), where \(r_{i}\) is the radius of the surveillance area, \(\forall i\in\{1,\cdots,|\mathcal{S}|\}\). The radius is calculated as \(r_{i}=h_{i}\cdot\tan(FoV/2)\), where \(FoV\) is the field of view of the built-in UAV camera. In each unit period \(t\), the surveillance UAV stays in a place or moves to the next position. We assume that the moving UAV flies along a certain path that consists of a sequence of waypoints, (i.e., \(\mathbf{p}_{i}=\{p_{i}[1],p_{i}[2],\cdots,p_{i}[t]\}\), \(\forall i\in\{1,\cdots,|\mathcal{S}|\}\)). Each UAV has its own initial residual battery status, and it can be charged and discharged by charging and discharging processes, respectively. Furthermore, this process can be formulated with a queue, and its backlog evolves, as follows:  
\[Q_{i}(t+1)\triangleq\max\{0,Q_{i}(t)-\mu_{i}(t)\}+\lambda_{i}(t),\forall i\in \{1,\cdots,|\mathcal{S}|\}, \tag{1}\]  
where \(Q_{i}(t)\), \(\mu_{i}(t)\), and \(\lambda_{i}(t)\) stand for battery energy status (i.e., queue backlog size) at UAV \(s_{i}\), the amount of discharging energy at UAV \(s_{i}\), and the amount of charging energy at UAV \(s_{i}\), respectively, \(\forall i\in\{1,\cdots,|\mathcal{S}|\}\). The charging process is determined by the hardware and system specifications of the charger at each UAV, and the discharging process depends on the operation of each UAV. In this paper, each UAV has two operational modes in each unit period \(t\): (i) hovering during surveillance and (ii) round-trip traveling to a charging tower. Each mode has a different amount of energy expenditure to execute the associated tasks during the period \(t\). For the first action (hovering), the energy expenditure at UAV \(s_{i}\) is the combination of hovering and communication-related energy, i.e.,  
\[E_{i}^{hc}(t)\triangleq(P_{h}+P_{c})t, \tag{2}\]  
where \(P_{h}\) and \(P_{c}\) are the power consumption for hovering and communications, respectively. The blade profile is the power required just to turn the rotor blades. The induced power is the power that is required to overcome the induced drag of lift creation, which is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming toward it. The power consumption for hovering can be as follows [62; 63]:  
\[P_{h}\triangleq\underbrace{\frac{\delta}{8}\rho s\Lambda\Omega^{3}R^{3}}_{P_{ o}}+\underbrace{(1+k)\frac{W^{3/2}}{\sqrt{2\rho A}}}_{P_{i}}, \tag{3}\]  
where \(P_{o}\) and \(P_{i}\) represent the blade profile power and induced power, respectively. The relevant parameters are explained in Table 1. The communication-related energy is usually much lower than the hovering or propulsion energy (e.g., a few watts [64] versus hundreds of watts [65]). In this paper, we assume that the communication-related power is constant. For the second action (round-trip traveling), the UAV moves from one waypoint to another, in a straight line connecting the two waypoints. The total required expenditure of energy for round-trip traveling of a UAV \(s_{i}\) is as follows:  
\[E_{i}^{tr}(\Delta t_{tr},v(t))\triangleq\int_{0}^{\Delta t_{tr}}P_{p}(v(t))dt, \tag{4}\]where the round-trip UAV traveling time is denoted by \(\Delta t_{tr}\), and the instantaneous speed of traveling along the paths is denoted by \(v(t)\).