# Formation Control Algorithm of Multi-UAV-Based Network Infrastructure
## 3 Multi-UAV-Based Network Infrastructure Analysis
### Assumptions and Notations  
We assume that a number of the terrestrial mobile devices are randomly spread at a flat area \(S\). \(M\) refers to the set of the mobile devices in \(S\), and \(m_{i}\) refers to the \(i\)th device where \(1<i\leq|M|\). At the same manner, \(D\) refers to the set of the UAVs composing the network infrastructure, and \(d_{j}\) refers to the \(j\)th device where \(1<j\leq|D|\). Location of each device is referred to \(p_{m_{i}}\), which is unknown on the side of the UAVs and the network infrastructure. Also, position of each UAV is referred to \(p_{d_{j}}\), and we assume that \(p_{d_{j}}(t)\) is periodically collected through the navigation devices (e.g., GPS) and shared by the UAVs' ad hoc channel. Each device could have different amount of the network resource requirement \(r_{m_{i}}\), depending on what application each device uses. We additionally assume that \(r_{m_{i}}\) could not exceed the maximum data rate of the data transmission \(r_{max}\). Each device is trying to access the Internet by scanning the nearby AP providers (UAVs) and connecting to the one of them.  
For mobile devices to connect the Internet, they scan the APs around the locations. We refer the maximum range of the AP connection provided by the UAVs to the \(R_{AP}\). We assume that the mobile devices select an AP (UAV) broadcasting the beacon signal with the best SINR, which can be expressed to  
\[\mathrm{AP}(m_{i})=\operatorname*{arg\,max}_{d_{j}\in s_{m_{i}}}SINR(m_{i},d_{ j}) \tag{1}\]  
where \(s_{m_{i}}\) refers to the set of the UAVs scanned by \(m_{i}\) and \(SINR(a,b)\) refers to the SINR of the \(a\) beacon signal from the device \(b\). The assumption is valid since the existing AP selection mechanisms commonly shape the decision metric according to the received signal strength or SINR [26; 27]. On the UAV's side, a set of connected mobile devices are referred to \(C(d_{j})\), which can be obtained by  
\[C(d_{j})=\{i|\mathrm{AP}(m_{i})=d_{j}\}. \tag{2}\]  
We assume that the channel used by AP network of the UAVs and the channel used by ad hoc network are completely isolated, which means that there is no channel interference between the AP and ad hoc network. The assumption also can be accomplished by using the different frequency band (e.g., 2.4 GHz for AP and 5.0 GHz for ad hoc). The assumption is not only for the simplicity of the modeling, but the expected performance of the network infrastructure. If APs and ad hoc use the same channel, the medium is shared by the nearby UAVs as well as the devices connected in the same AP. Furthermore, since an expected network resource requirement of the ad hoc connection is the sum of resource requirement of the AP connections, AP network performance might be significantly lowered by the contention with the ad hoc connections.  
Also, we assume that the ad hoc network routing protocol results in the optimized routing path while the UAVs are changing their locations. Since the network infrastructure specifies the GCS as a sync node, the UAVs spread directionally or omni-directionally around the GCS. We refer the number of connected UAVs with a certain UAV \(d_{j}\) in a single hop to \(N_{d_{j}}\), which can be calculated to  
\[N_{d_{j}}=|\{d_{k}|dist(d_{k},d_{j})<R_{adhoc}\}| \tag{3}\]  
where \(R_{adhoc}\) refers to the maximum range of the one-hop communication of ad hoc network.