## 2 System Model and Problem Definition
### Network Model and Antenna Model  
We consider a FANET deployed over a large area. Each UAV in the FANET is equipped with two independent transceiver terminals with directional antennas. One transceiver is responsible for exchanging control packets and is denoted as TS1. The other transceiver transmits data packets and is denoted as TS2. Both TS1 and TS2 work in half duplex mode. To reduce costs, we consider using switched beam and phased array antennas [33] for the directional antennas on TS1 and TS2, respectively. The directional antenna has a main lobe angle \(\theta\).  
Switched beam antenna model: The angle of each sector is denoted by \(\theta_{S}\) (\(0<\theta_{S}\leq 2\pi\)) which is also the width of the main lobe and sidelobes, and the directional antenna pattern consists of \(N-1\) sidelobes and a main lobe as illustrated in Figure 1. The relationship between \(\theta_{S}\) and \(N\) is expressed as:  
\[\theta_{S}=\frac{2\pi}{N} \tag{1}\]  
The communication coverage is divided into \(N\) non-overlapping sectors, with sector indexes ranging from \(1\) to \(N\) in counterclockwise order. The main lobe needs to scan \(N\) times to cover all sectors at a minimum. As an example in Figure 1, the angle of the sector is \(\frac{\pi}{3}\), and therefore the main lobe of the directional antenna can scan all directions by 6 scans. The indexes of the sectors are \(\mathbf{u}=\{1,2,3,4,5,6\}\).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **Definition** & **Symbol** & **Definition** \\ \hline \multirow{2}{*}{T\({}_{\text{rs}}\)} & The worst-case discovery delay for a pair & \multirow{2}{*}{\(T_{\text{TS2-RTS}}\)} & \multirow{2}{*}{The delay of TS2-RTS} \\  & of UAVs & & \\ \hline \multirow{2}{*}{\(M_{T}\)T} & The time duration for the transmitter UAV & \multirow{2}{*}{\(T_{\text{TS2-CTS}}\)} & \multirow{2}{*}{The delay of TS2-CTS} \\  & The minimal slot duration to exchange & & \\ \multirow{2}{*}{\(M_{R}\)T} & discovery beacons between the pair of UAVs & & \\  & The length of the segment of \(0^{\prime}\)s in a specific & & \\  & control sequence & & \\ \hline \end{tabular}
\end{table}
Table 2: _Cont._  
Figure 1: Switched beam antenna model.  
Phased array antenna model: Compared to the \(N\) fixed directions of the switched beam antenna, the main lobe of the phased array antenna can point in any direction. We denote \(\theta_{P}\) as the main lobe angle of TS2. Note that \(\theta_{S}\) is usually bigger than \(\theta_{P}\) to accelerate neighbor discovery. When the main lobe rendezvous is achieved for TS2, the boresight of the transmitter and receiver overlap, which is shown in Figure 2. In Figure 2, A and B represent the transmitter and receiver, respectively. C and D are the intersection points of the two beam edges. Hence, \(\angle CAB\), \(\angle DAB\), \(\angle\)\(CBA\) and \(\angle DBA\) are \(\frac{\theta_{P}}{2}\).  
Let \(\varphi\) denote the direction of communication and \(\psi\) denote the direction of main lobe boresight, and we adopt the following antenna model [23; 34]:  
\[g(\varphi)=\left\{\begin{aligned} & G(\theta)\,,|\varphi-\psi|\, \leq\,\frac{\theta}{2}\\ & 0\,,\text{otherwise}\end{aligned}\right., \tag{2}\]