# Joint Trajectory and Communication Design for Buffer-Aided Multi-UAV Relaying Networks
## 2 System Model and Problem Formulation
### System Model
Specifically, the channels are dominated by the LoS path with random phases, and the amplitudes of communication channels within one time slot are considered to be constant, because the horizontal location variation is virtually negligible as compared to the altitude of the UAVs during a sufficiently small time slot [18]. Then, the Rician fading channel model is given by:  
\[ \begin{split}& h_{k,m}\left[n\right]=h_{k,m}\left[n\right]^{LoS}+h_{ k,m}\left[n\right]^{NLoS}\\ &=\sqrt{\frac{\beta}{\left(G+1\right)d_{k,m}\left[n\right]^{2}}} \left(\sqrt{G}e^{j\frac{2\pi}{\lambda}d_{k,m}\left[n\right]}+g_{k,m}\left[n \right]\right)\\ &=\sqrt{\frac{G}{G+1}}\,\bar{h}_{k,m}\left[n\right]e^{i\theta_{k, m}\left[n\right]}+\sqrt{\frac{1}{G+1}}\,\bar{h}_{k,m}\left[n\right]g_{k,m}\left[n \right]\\ &\approx\sqrt{\frac{G}{G+1}}\sqrt{\frac{\beta}{d_{k,m}\left[n \right]^{2}}}e^{i\theta_{k,m}\left[n\right]}+\sqrt{\frac{1}{G+1}}\sqrt{\frac{ \beta}{d_{k,m}\left[n\right]^{2}}}g_{k,m}\left[n\right]\\ & m\in\left\{S,D\right\},\forall k,n\end{split}  \tag{5}\]  
where \(\beta\) is defined as the channel power gain at the reference distance \(d_{0}=1\) m and \(G\geq 0\) denotes the Rician factor. \(\bar{h}_{k,m}\left[n\right]=\sqrt{\frac{\beta}{d_{m,k}\left[n\right]^{2}}}\) represents the free space path loss of the channel. \(\theta_{m,k}\left[n\right]=\frac{2\pi}{\lambda}d_{m,k}\left[n\right]\) denotes the phase of the LoS channel, where \(\lambda=\frac{c}{f_{c}}\) is the wavelength. In \(\lambda=\frac{c}{f_{c}}\), \(c\) is defined as the speed of light and \(f_{c}\) is the carrier frequency. For simplicity, we let \(\theta_{m,k}\left[n\right]\) be modeled as a uniformly distributed random variable in \(\left[0,2\pi\right)\)[18]. Moreover, \(g_{m,k}\left[n\right]\) is a coefficient of the Rayleigh fading channel, subject to a complex Gaussian distribution with zero mean and unit variance. A binary variable \(\left\{a_{k,m}\left[n\right],m\in\left\{S,D\right\},\forall n,k\right\}\) is defined to indicate whether UAV \(k\) is to communicate with ground nodes \(S\) or \(D\) in time slot \(n\), i.e., \(a_{k,m}\left[n\right]=1\) means communicating, and \(a_{k,m}\left[n\right]=0\) means no communicating. It is also assumed that \(S\) and \(D\) communicate with different UAVs and at most one UAV communicates with \(S\) and \(D\) in the same time slot; thus the relay selection constraints can be obtained as:  
\[\sum_{k=1}^{K}a_{k,m}\left[n\right]\leq 1,\forall n,m\in\left\{S,D\right\} \tag{6}\] \[a_{k,S}\left[n\right]+a_{k,D}\left[n\right]\leq 1,\forall k,n\] (7) \[a_{k,m}\left[n\right]\in\left\{0,1\right\},m\in\left\{S,D\right\}, \forall k,n \tag{8}\]Besides, \(p_{S}\) is a constant of the transmit power of \(S\), and \(p_{k}\left[n\right]\) denotes the transmit power of UAV \(k\) in time slot \(n\), which satisfies both the average power constraint and peak power constraint, expressed by \(P_{mean}\) and \(P_{peak}\), respectively. Thus, the transmit power satisfies the following constraints:  
\[\frac{1}{N}\sum_{n=1}^{N}p_{k}\left[n\right]\leq P_{mean},\forall k,n \tag{9}\] \[0\leq p_{k}\left[n\right]\leq P_{peak},\forall k,n \tag{10}\]  
According to the above discussion and analysis, the maximum achievable rate from \(S\) to UAV \(j\) in time slot \(n\) can be written as:  
\[R_{jS}\left[n\right]=a_{j,S}\left[n\right]\log_{2}\left(1+\frac{p_{S}\left|h_{ j,S}\left[n\right]\right|^{2}}{\sigma^{2}}\right) \tag{11}\]  
where \(\sigma^{2}\) is defined as the additive white Gaussian noise (AWGN) at the receiver. Accordingly, these data will be temporarily stored in the buffer of UAV \(j\).