## 3 Multichannel MAC Protocol with Directional Antenna for FANET
### Blind Rendezvous Algorithm with Directional Antenna
In a rendezvous period, there are \(M_{T}M_{R}N-\frac{M_{T}M_{R}N}{\min(M_{T},M_{R})}\) cases that cannot be searched. In order to scan all cases, it takes \(M_{T}M_{R}N\mathrm{T}-\max(M_{T},M_{R})N\mathrm{T}\) to scan for the system composed of this pair of UAVs. Only under the constraints \((M_{T}=1,M_{R}=N)\) and \((M_{T}=N,M_{R}=1)\) can the minimum delay of full rendezvous diversity, i.e., \(M_{T}M_{R}N\mathrm{T}\), be achieved. Consequently, the least WCDMR without coordination or information is \(N^{2}\mathrm{T}\) for clock synchronization discovery. In the scenario described in Section 2.3, if \((M_{T}\!\!=2,M_{R}\!\!=3)\), the antenna scan sequences for UAVs \(a\) and \(b\) are shown in Figure 8. UAV \(a\) switches to sector 1, but UAV \(b\) cannot switch to sector 4 at the same slot from Figure 8.  
If \((M_{T}\!\!=1,M_{R}\!\!=6)\), the antenna scan sequences for UAV \(a\) and \(b\) are shown in Figure 9. In slot 9, UAV \(a\) switches to sector 1, and UAV \(b\) can switch to sector 4 at the same slot. In the scenario of Section 2.3, there are 36 cases of the scenario, and it takes at least 36 slots to achieve full rendezvous of the main lobe. In the 36 slots, if \((M_{T}\!\!=1,M_{R}\!\!=6)\), the full rendezvous diversity is achieved. For achieving a minimum worst-case delay of blind rendezvous, Theorem 2 can be employed for clock synchronization discovery but not for clock asynchronization discovery. There are several blind rendezvous cases in which the time duration is less than T, so blind rendezvous with directional antennas cannot be achieved in \(N^{2}\mathrm{T}\). In Figure 10, according to Theorem 2, the time duration of the cases (1,4) and (3,6) equals T and can be searched for clock synchronization discovery. In Figure 11, according to Theorem 2, the time duration of the case (1,4) is more than T, enabling the neighbor discovery of this case for clock asynchronization discovery. However, the time duration of the case (3,6) is less than T, making it impossible to accomplish neighbor discovery for clock asynchronization discovery. From the above analysis, full rendezvous diversity cannot be attained for clock asynchronization discovery according to Theorem 2. Figure 8: Antenna scan sequences for UAV \(a\) and \(b\) if \((M_{T}\!\!=2,M_{R}\!\!=3)\). Figure 9: Antenna scan sequences for UAV \(a\) and \(b\) if \((M_{T}\!\!=1,M_{R}\!\!=6)\). **Theorem 3**. : _The minimum delay of the full rendezvous algorithm is achieved only when the constraints \((M_{T}=1,M_{R}=N+1)\) and \((M_{T}=N+1,M_{R}{=}1)\) are satisfied for clock asynchronization discovery. Additionally, the least-worst-case delay (supremum) of blind rendezvous without clock synchronization is \(N(N+1)\mathrm{T}\)._  
Proof of Theorem 3.: With the constraints \((M_{T}=1,M_{R}=N+1)\) and \((M_{T}=N+1,\)\(M_{R}{=}1)\), the time duration of all cases is more than T; hence, the full rendezvous diversity can be achieved for clock asynchronization discovery. Under the constraints \((M_{T}=1,\)\(M_{R}=N+1)\) and \((M_{T}=N+1,M_{R}{=}1)\), the delay of blind rendezvous is \(N(N+1)\mathrm{T}\). Based on the above analysis, we can conclude that under the constraints \((M_{T}=1,\)\(M_{R}=N)\) and \((M_{T}=N,M_{R}{=}1)\), the full rendezvous diversity cannot be achieved, and hence, we can get that the minimum \(M_{T}M_{R}\) is \(N+1\) for asynchronous neighbor discovery. Under the constraints \((M_{T}\
eq 1,M_{R}\
eq N+1)\), \((M_{T}\
eq N+1,M_{R}\
eq 1)\), and \(M_{T}M_{R}=N+1\), we can get and \(M_{R}\geq 2\). In \(N(N+1)\mathrm{T}\), there are at least \(N^{2}-\max(M_{T},M_{R})N\) cases that cannot be searched. Therefore, under the constraints \((M_{T}=1,M_{R}=N+1)\) and \((M_{T}=N+1,M_{R}{=}1)\), the least delay of full rendezvous can be achieved for clock asynchronization discovery and is \(N(N+1)\mathrm{T}\). Under the constraints \((M_{T}=1,M_{R}=N+1)\) and \((M_{T}=N+1,M_{R}{=}1)\), the least WCDMR without coordination or information is \(N(N+1)\mathrm{T}\) for clock asynchronization discovery.