# A Study on the Maximum Reliability of Multi-UAV Cooperation Relay Systems
## 2 System Model and Problem Formulation
### Modeling Line of Sight Probability  
UAV communication platforms are low-altitude platform (LAPs); that is, the flight area is within low altitude airspace. AtG communication occurs in accordance to two main propagation groups according to whether they adhere to struct LoS conditions, while the non-line-of-sight (NLoS) conditions correspond to receivers with no UAV platform LoS but still receiving coverage via strong reflections and diffractions [10; 11]. The AtG communication channel from the UAV platform to the GTs is mainly dominated by LoS links. Radio signals emitted by the UAV platform base station propagate in free space, producing the free space path loss and reaching the ground building, incurring shadowing and scattering, which introduces an additional loss in the AtG link related to excessive path loss denoted by \(\eta\)[11; 12], as shown in Figure 2. Accordingly, the resulting path loss of the AtG communication (expressed in dB) can be modeled as follows:  
\[PL_{\xi}=PL_{\text{FS}}+\eta_{\xi}, \tag{1}\]  
where \(PL_{\text{FS}}\) represents the path loss of the free space between the UAV and ground users (GUs), and \(\xi\) refers to the propagation groups the propagation groups, which is classified according to the strict ground receiver LoS and NLoS conditions, while \(\xi\in\{\text{LoS},\text{NLoS}\}\). The maximum allowable path loss threshold was set to \(PL_{\text{max}}\), which corresponds to a circular coverage with radius \(r_{\text{max}}\). It can be noticed that the path loss between the UAV and any ground receiver within this circular coverage is less than or equal to \(PL_{\text{max}}\). It is important to note that the value of \(PL_{\text{max}}\) depends on the sensitivity of the receiver and the target communication's quality of service. Assuming the use of isotropic transmitters and receiver antennas, the total path loss between the UAV platform and all ground receivers can be measured as the mean value of the path loss, as follows:  
\[PL=\sum_{\xi}PL_{\xi}P_{\xi}, \tag{2}\]  
where \(P_{\xi}\) represents the probability of the occurrence of a certain propagation group \(\xi\), while \(\sum P_{\xi}=1\), \(PL_{\xi}=20\text{log}\Big{(}\frac{4\pi fd}{c}\Big{)}+\eta_{\xi}\), \(d=\sqrt{h^{2}+r^{2}}\) is the link distance between the UAV LAP and GUs, \(h\) is the UAV LAP altitude, and \(r\) is the ground distance between the UAV platform and GUs. In an urban environment, the probability of the geometric LoS between the ground transmitter at altitude \(h_{\text{TX}}\) and the receiver at altitude \(h_{\text{RX}}\) depends on three statistical  
Figure 2: Radio propagation of UAV aerial base station platform. parameters: \(\alpha\), \(\beta\), and \(\gamma\), which are related to the urban environment [10]. The probability \(P_{\text{LoS},n}\) that the height of an obtrusive building is smaller than height \(h_{\text{LoS},n}\) is given by  
\[P_{\text{LoS},n}=1-\exp\left(-\frac{h_{\text{LoS},n}^{2}}{2\gamma^{2}}\right), \tag{3}\]  
where \(h_{\text{LoS},n}\) is the height of a building that would obstruct the LoS in meters, while \(h_{\text{LoS},n}=h_{\text{TX}}-\frac{\left(n+\frac{1}{2}\right)\left(h_{\text {TX}}-h_{\text{RX}}\right)}{N_{n}}\), \(N_{n}\) is the number of buildings crossed by the LoS, while \(N_{n}=\text{floor}\left(r\sqrt{\alpha\beta}\right)\). Parameter \(\alpha\) is the ratio of land area covered by buildings to the total land area (dimensionless). Parameter \(\beta\) is the mean number of buildings per unit area (buildings/km2), \(r=h/\tan(\theta)\), while \(\theta\) is the elevation angle. Parameter \(\gamma\) is a variable used to determine the height distribution of obtrusive buildings, which is defined by the Rayleigh probability density function (PDF) of the height \(h_{\text{LoS},n}\), namely \(f(x)=\frac{x}{\gamma^{2}}\exp\left(-\frac{x^{2}}{2\gamma^{2}}\right)\).