# Tethered Balloon Cluster Deployments and Optimization for Emergency Communication Networks
## 3 System Model and Problem Formulation
### Channel Model
#### 3.3.2 Air-to-Ground Channel Model  
The channel between balloons and users has not only LoS links but also non-line-of-sight (NLoS) links. The propagation of wireless signals in complex environments is affected by several factors. Firstly, buildings or other reflective surfaces such as the sea have high reflectivity, which tends to cause multi-path effects in signal propagation. Secondly, atmospheric humidity, shadowing effects, and other environmental factors may seriously attenuate the strength and quality of signals. Therefore, NLoS path loss is usually much larger in complex environments than LoS path loss. Considering the above analyses, the air-to-ground (ATG) propagation channel model is a weighted propagation loss model that combines LoS probability and NLoS probability. This model is particularly suitable for calculating the path loss of the communication system between aerial platforms and users. The probabilities of LoS and NLoS links between balloon \(n\) and user \(m\) can be expressed as, respectively,  
\[\mathbb{P}(LoS,\theta_{n,m})=\frac{1}{1+a\,exp(-b(\theta_{n,m}-a))}, \tag{9}\]  
\[\mathbb{P}(NLoS,\theta_{n,m})=1-\mathbb{P}(LoS,\theta_{n,m}), \tag{10}\]  
where \(\theta_{n,m}\) represents the elevation angle at which base station \(n\) establishes a LoS link with user \(m\). \(\mathbb{P}(LoS,\theta_{n,m})\) is the probability that a direct LoS propagation path exists between the transmitter and receiver. Its expression depends mainly on the following factors: elevation angle dependence and environmental factors. When the flight altitude is higher and the elevation angle is higher, \(\mathbb{P}(LoS,\theta_{n,m})\) is usually higher. \(\mathbb{P}(LoS,\theta_{n,m})\) is usually lower when environmental factors such as urban density, building height, and vegetation cover play a stronger role. \(a\) and \(b\) are constant values that depend on the type of environment, e.g., urban, rural, suburban, etc. \(a\) reflects the density of buildings. A larger value of \(a\) indicates a higher density of buildings. \(b\) is related to the geometrical distribution of the buildings. Larger values of \(b\), \(\mathbb{P}(LoS,\theta_{n,m})\) vary more rapidly with elevation. The path loss of the LoS link and the NLoS link between balloon \(n\) and user \(m\) can be expressed as, respectively,  
\[\mathbb{L}_{LoS}^{n,m} =(\frac{4\pi d_{n,m}f_{c}}{c})^{2}\,\eta_{LoS}, \tag{11}\] \[\mathbb{L}_{NLoS}^{n,m} =(\frac{4\pi d_{n,m}f_{c}}{c})^{2}\,\eta_{NLoS}. \tag{12}\]  
FSPL usually assumes that signals propagate in a vacuum without any obstacles or interference. However, in real-world environments, signals are affected by various factorssuch as atmospheric absorption, scattering, diffraction, reflection, etc., which can lead to additional losses. \(\eta_{LoS}\) and \(\eta_{NLoS}\) are the additional loss terms for LoS and NLoS links propagating with respect to FSPL due to atmospheric absorption and obstacle occlusion, respectively [23]. The weighted average path loss between balloon \(n\) and user \(m\) is given by  
\[L_{n,m}=\mathbb{P}(LoS,\theta_{n,m})\,\mathbb{L}_{LoS}^{n,m}+\mathbb{P}(NLoS, \theta_{n,m})\,\mathbb{L}_{NLoS}^{n,m}. \tag{13}\]  
The Signal-to-Interference-plus-Noise Ratio (SINR) accurately reflects the reality of the presence of interference in a wireless communication network. The success of a communication link does not depend on the presence of other transmitting links but rather on whether the total interference generated by all the links in the area transmitting data is greater than a predetermined threshold. Define \(\gamma_{n,m}\) to denote the SINR of the receiving node \(m\) when the balloon \(n\) sends data to the user \(m\), and \(m\) is considered to be covered when \(\gamma_{n,m}\) exceeds a threshold denoted by \(\Lambda\). \[\gamma_{n,m}=\frac{P_{n,m}\cdot g_{n,m}}{\sum\limits_{i\
eq n}P_{i,m}\cdot g_{ i,m}+\eta B_{n}}, \tag{14}\]  
\[C_{n,m}=\left\{