## 2 System Model  
To set up the system model, a large area containing the number of users can be considered. The terrestrial base stations are failed to work, due to some accident. In order to solve the communication problems of ground users, a UAV-BS is hovered over the target area to provide wireless service. The hovers in a circular trajectory with a radius of \(r\) and the period are \(T\). Let \(u\) represent the set of users that UAV-BS needs to serve. Let \((x_{i},y_{i})\) denote the horizontal location of the users \(i\); and the UAV locate at \((x_{o},y_{o},h)\). The air-to-ground (A2G) channel model proposed by the author's name [4] is adopted. The path losses of Line-of-Sight (LoS) \(L_{LoS}\) and non-Line-of-Sight (NLoS) \(L_{NLoS}\) are respectively proposed in Reference [4]:  
\[L_{LoS}\ =\ 20\log(\frac{4\pi fd_{i}}{c})+\eta_{LoS}, \tag{1}\]  
\[L_{NLoS}\ =\ 20\log(\frac{4\pi fd_{i}}{c})+\eta_{NLoS}, \tag{2}\]  
where \(f\) and \(c\) are respectively the carrier frequency and the speed of light. The distance between the UAV-BS and user \(i\) is denoted by \(d_{i}\ =\ \sqrt{h^{2}+r_{i}^{2}}\), where \(r_{i}\ =\ \sqrt{\left(x_{i}-x_{o}\right)^{2}+\left(y_{i}-y_{o}\right)^{2}}\). Furthermore, \(\eta_{LoS}\) and \(\ \eta_{NLoS}\) are respectively additional loss for Line-of-Sight (LoS) and non-Line-of-Sight (NLoS). The value of \(\eta_{LoS}\) and \(\eta_{NLoS}\) depend on the environment, and are given by Reference [4] in different environments. In addition, the Line-of-Sight (LoS) probability as follows:  
\[P_{LoS}\ =\ \frac{1}{1+\alpha\exp(-\beta(\frac{180}{\pi}\theta_{i}-\alpha))}, \tag{3}\]  
where \(\alpha\) and \(\beta\) are environmental constants, and \(\theta_{i}\ =\ \arctan(\frac{h}{r_{i}})\) is the elevation angle of the user \(i\) in the set \(u\). In addition, the probability of non-Line-of-Sight (NLoS) is \(P_{NLoS}\ =\ 1-P_{LoS}\). Let \(A\ =\ \eta_{LoS}-\eta_{NLoS}\), \(B\ =\ 20\log(\frac{4\pi f}{c})+\eta_{NLoS}\), and the probabilistic mean path loss as follows:  
\[L\ =\ P_{LoS}L_{LoS}+P_{NLoS}L_{NLoS}, \tag{4}\]  
\[L(h,r_{i})\ =\ \frac{A}{1+\alpha\exp(-\beta(\frac{180}{\pi}\arctan(\frac{h}{r _{i}})-\alpha))}+10\log(h^{2}+r_{i}^{2})+B. \tag{5}\]  
Equation (5) can be further rewritten as follows:  
\[L(\theta_{i},r_{i})\ =\ \frac{A}{1+\alpha\exp(-\beta(\frac{180}{\pi}\theta_{i}- \alpha))}+20\log(\frac{r_{i}}{\cos\theta_{i}})+B. \tag{6}\]  
For a given transmission power \(p_{t}\), the received power \(p_{r}\) at the user \(i\) is: \(p_{r}\ =\ p_{t}-L(h,r_{i})\). It is assumed that the user \(i\) is covered if the received power \(p_{r}\) exceeds a certain threshold \(p_{\min}\). If the link experiences a path loss less than a certain threshold \(L_{th}\), the user \(i\) is covered by the UAV-BS. According to References [17,18], for any QoS requirement \(L_{th}\), the optimal elevation angle \(\theta^{*}\) that maximizes the coverage radius is a constant and only depends on the environment. In addition, \(\theta^{*}\ =\ 20.34^{{}^{\circ}},42.44^{{}^{\circ}},54.62^{{}^{\circ}},75.52^{{}^{ \circ}}\) can be obtained for the suburban, urban, dense urban, and high-rise urban environments by solving Equation (6). The maximum coverage radius can also be evaluated by solving Equation (6). Finally, the optimal height \(h^{*}\) of the UAV-BS can be evaluated. Next, a scenario is discussed such that the UAV-BS at a height of \(h^{*}\) hovers in a circular trajectory with a radius of \(r\) and the period is \(T\). In addition, the instantaneous location of the UAV-BS is assumed to be \((x,y)\). The system model is shown in Figure 1.  
Therefore, the trajectory of the UAV-BS is \(x^{2}+y^{2}\ =\ r^{2}\). It can be written as a parametric equation, as follows:  
\[ \begin{array}{l}x\ =\ r\cos\theta\\ y\ =\ r\sin\theta\end{array} , \tag{7}\]  
Because of the circular trajectory of the UAV-BS, the parameter \(\theta\) is given by: \(\theta(t)\ =\ \frac{2\pi}{T}t\), and (7) can be further written as:  
\[ \begin{array}{l}x(t)\ =\ r\cos(\frac{2\pi}{T}t)\\ y(t)\ =\ r\cos(\frac{2\pi}{T}t)\end{array} .