## 3 System Model and Problem Formulation
### Problem Formulation  
The UAVs in the network adopt the Time-Division Multiple Access (TDMA) scheme to transmit packets. As shown in Figure 1, UAV-4 and UAV-2 have overlapping coverage areas, and similarly, UAV-15 and UAV-16 also have overlapping coverage areas. When UAV-4 and UAV-2 individually transmit packets at the same slot, UAV-4 causes interference with UAV-2. In contrast, when UAV-16 and UAV-1 transmit packets in the same slot, co-channel interference does not happen. Because UAV-16 is far from UAV-1, the signal of UAV-20 cannot interfere with the UAVs to which UAV-1 transmits.  
Assume that we allocate \(K_{n}\) slots for UAV-\(n\), where \(1\leq K_{n}\leq N\). Consider \(T_{n}\) to be the corresponding slots selected by UAV-\(n\), where \(T_{n}=\{t_{1},t_{2},\ldots,t_{K_{n}}\},t_{i}\in\mathcal{T},1\leq i\leq K_{n}\).  
For an arbitrary individual UAV-\(n\) at an arbitrary time slot \(t\), \(t\in T_{n}\), we denote \(R_{n}(t)\) as the throughput at time slot \(t\) of UAV-\(n\), where \(R_{n}(t)\) is given in (8).  
\[R_{n}=\left\{\begin{aligned} & 0,\quad\exists i\in\mathcal{N},i\
eq n,d_{in}\leq d_{C}+d_{I},t\in T_{i}\\ & R_{0},\qquad\qquad\qquad\qquad\qquad\qquad\qquad else \end{aligned}\right. \tag{8}\]  
\(R_{0}\) is the fixed transmitting rate of each UAV, and \(T_{i}\) is the corresponding slots selected by UAV-\(i\). We note that the time slot length is normalized to 1. For an arbitrary individual UAV-\(n\), the total throughput can be calculated as  
\[Th_{n}=\sum_{t\in T_{n}}R_{n}(t) \tag{9}\]  
Then, the throughput of the whole multi-UAV network is  
\[Th=\sum_{n=1}^{N}\sum_{t\in T_{n}}R_{n}(t) \tag{10}\]  
An arbitrary individual UAV prefers to select more transmission time slots so that it can achieve higher throughput. But the higher number of transmission time slots will bring more collision at the same slot, which may cause a reduction in the throughput. From a multi-UAV network viewpoint, a higher number of selected time slots of each UAV yieldsmore transmission opportunities. Thus, our aim is to maximize the aggregate throughput of the network, as  
\[P1:\max\sum_{n=1}^{N}\sum_{t\in T_{n}}R_{n}(t) \tag{11}\]  
From a time slot viewpoint, we want more UAVs to transmit at the same slot, and we do not want them to interfere with each other. Denote \(N_{t}\) as the set of UAVs which select time slot \(t\) to transmit. For an arbitrary individual time slot \(t\), the throughput of the whole network is  
\[R(t)=\sum_{n\in N_{t}}R_{n}(t) \tag{12}\]  
Then, the throughput of the whole multi-UAV network can also be written as  
\[Th=\sum_{t=1}^{N}\sum_{n\in N_{t}}R_{n}(t) \tag{13}\]  
Obviously, the large no-collision UAV set of time slot \(t\) induces high network throughput. The problem of maximizing the aggregate throughput of the network transforms to find a group of the largest no-collision UAV set at time slot \(t\).  
\[P2:\max\sum_{t=1}^{N}\sum_{n\in N_{t}}R_{n}(t) \tag{14}\] \[\text{s. t.}\qquad\forall i,j\in N_{t},d_{ij}>d_{C}+d_{I} \tag{15}\]  
P1 and P2 are equivalent. Problem 1 finds the maximum value of the sum of the throughput of all nodes, and problem 2 finds the maximum value of the sum of the rates on all time slots from the perspective of time slots.