# Formation Control Algorithm of Multi-UAV-Based Network Infrastructure
## 3 Multi-UAV-Based Network Infrastructure Analysis
### Wireless Access Network  
This section addresses a generic form of the wireless access network model mathematically. Due to the resource limitation of the wireless medium and the randomized media access of the wireless devices, the network performance decreases when the number of the devices increase. As each UAV works as an AP, the UAVs might connect to the randomly spread users, which leads the large deviation of the number of connections between the UAVs. Some UAVs with relatively large number of the connections could suffer from the significant drop of the network performance, while some UAVs with relatively small number of the connections might waste the network resource. To elaborately distribute the network traffic to the UAVs, the mathematical model of the wireless access network and the numeric factors that affects the formation control should be designed. Also, there are valuable approaches to alleviate the risk of the packet collision such as Carrier Sense Multiple Access with Collision Detection (CSMA/CD) [28], variations of ALOHA protocol [29], and Orthogonal Frequency-Division Multiple Access (OFDMA) [30]. To give the generic view and enhance the flexibility to the model, we propose a mathematical expression of the aggregated throughput with a protocol-dependent parameter, which could be estimated by comparing the analytic solution and the simulated solution. First, we define a network capacity \(R\), which refers to the maximum data rate of a connection where only one mobile device is connected to a UAV, without any electrical interference and obstacle. Based on the \(R\), the ratio of the network throughput degradation is exponentially increased as the number of mobile devices in the same network (\(C(d_{j})\)) increases, due to the increased possibility of the collision. In a certain point of the time, the possibility of the collision when \(n\) devices are in the same medium is  
\[\frac{2^{n}-n-1}{2^{n}-1}. \tag{4}\]  
However, there are some previous studies allowing the concurrent transmissions by sub-channels [30] or code division [31], so the variable \(n\) could be divided into the approximated amount of the concurrent transmissions. Also, to reduce the collision probability, the mechanism that randomly backs off a number of time slots and detect the medium [28; 32] and its variations [33; 34; 35] were studied. Considering these improvements, the throughput degradation ratio, \(\mathcal{D}(n)\) can be expressed by augmenting Equation (4) to  
\[\mathcal{D}(n)=1-\frac{1}{\beta}\left(\frac{2^{\frac{n}{\alpha}}-\frac{n}{ \alpha}-1}{2^{\frac{n}{\alpha}}-1}\right) \tag{5}\]  
where \(\alpha\) and \(\beta\) are named to _concurrent transmission availability_ and _congestion relaxation degree_ of the protocol respectively, and \(n=|C(d_{j})|\). As \(\alpha\) or \(\beta\) increase, the degradation of the throughput decreases. Note that \(\alpha\) and \(\beta\) do not represents the physical property or performance of the protocols, for instance, \(\alpha=10\) does not suppose that an AP holds exactly 10 transmissions at maximum. Our generic model involves the common behavior of the wireless network and the parameters representing the evolution of the communication techniques. The verification of Equation (5) shows in Section 5.4. Since the objective of the paper is the improvement of the throughput of the infrastructure, we estimate \(\alpha\) and \(\beta\) of the WLAN protocol by the simulation and reflect to the formation algorithm. By the assumption stated in Section 3.1, the devices could require different amount of the network resource. When the multiple devices try to access the same wireless channel, the actual amount of the network resource provided by the UAV is proportionate to the required resource.