# Joint Trajectory and Communication Design for Buffer-Aided Multi-UAV Relaying Networks
## 2 System Model and Problem Formulation  
### System Model  
As illustrated in Figure 1, a buffer aided multi-UAV relaying network is considered. The model consists of \(K\geq 2\) UAVs, a source node \(S\) and a destination node \(D\). Suppose \(S\) and \(D\) are stationary ground nodes and all UAVs are equipped with a buffer \(\mathbb{Q}\). For the convenience of analysis, the data rate instead of the packet rate is considered, and the data in the buffer follow the \"first-in-first-out\" rule. It is supposed that all UAVs operate in the time-division half-duplex mode and that the communication link between \(S\) and \(D\) is unavailable due to the shadow fading and small-scale fading. In addition, it is assumed that the considered network always selects two different UAVs to communicate with \(S\) and \(D\) according to the known CSI, respectively. There is always a UAV storing the data from \(S\), and \(D\) will be served by a UAV if and only if the UAV's buffer is not empty. For interference avoidance, UAV \(j\) stores the data from \(S\), and UAV \(k\) forwards the data to \(D\) over orthogonal time slots, so we do not consider the interference between UAVs. Without loss of generality, a three-dimensional (3D) Cartesian coordinate system is considered. Hence, the locations of \(S\) and \(D\) can be denoted as \(\left[w_{m}^{T},0\right]^{T},m\in\{S,D\}\), where \(w_{m}=\left[x_{m},y_{m}\right]^{T}\), \(m\in\{S,D\}\). In the given limited flight period of the UAVs, these UAVs are assigned the same communication bandwidth \(B\), and it is assumed that all UAVs fly at a fixed altitude \(H\), which corresponds to the minimum safe altitude to avoid collision with ground obstacles without the need to adjust the flight height of UAVs frequently. The given finite flight period \(T\) is assumed to be dispersed into \(N\) equal quasi-static time slots, and \(d_{t}\) represents the length of a time slot, i.e., \(T=Nd_{t}\). The duration \(d_{t}\) is designed to be sufficiently small to approximate the positions of UAVs accurately in each time slot. In the time slot \(n\), the location of UAV \(k\) is denoted by \(\left\{q_{k}\left[n\right]^{T},H\right\}\) where \(q_{k}\left[n\right]=\left[x_{k}\left[n\right],y_{k}\left[n\right]\right]^{T}\) is the horizontal trajectory of UAV \(k\). The maximum flight speed of the UAVs is defined as \(V_{max}\). Thus, during one time slot, the maximum flying distance of the UAV can be expressed as \(D=V_{\text{max}}d_{t}\). In order to make the UAV trajectory feasible, the distance between the initial and final location of the UAV during the flight period is assumed to satisfy the following constraint:  
\[\left\|q_{k}\left[1\right]-q_{k}\left[N\right]\right\|^{2}\leq D^{2},\forall k \tag{1}\]  
Figure 1: System model. Besides, in any time slot, all UAVs should be subjected to the maximum speed and collision avoidance constraints:  
\[\left\|q_{k}\left[n\right]-q_{k}\left[n+1\right]\right\|^{2}\leq D^{2}, \forall k,n=1,\ldots,N-1 \tag{2}\] \[\left\|q_{k}\left[n\right]-q_{j}\left[n\right]\right\|^{2}\geq V_ {\max}^{2},\forall n,k,j,k\
eq j \tag{3}\]  
The distance from UAV \(k\) to the ground nodes in time slot \(n\) is given by:  
\[d_{k,m}\left[n\right]=\sqrt{H^{2}+\left\|q_{k}\left[n\right]-w_{m}\right\|^{2} },m\in\left\{S,D\right\},\forall k,n \tag{4}\]  
In this paper, we consider the low altitude UAVs, so too high ground obstacles and UAV flight elevation angles are the key factors affecting whether air-to-ground (A2G) links are LoS links. Under practical conditions, the A2G channels of low altitude UAVs will be affected by large scale fading and multi-path fading [21]. However, A2G links are still dominated by LoS links. Therefore, it is not completely assumed that A2G channels are LoS channels or Rayleigh channels. In the Rician fading channel, there are both LoS signals (main signals) and non-line-of-sight (NLoS) signals. Therefore, in this paper, we assume that A2G communication channels obey Rician fading [18; 19].