## 3 System Model and Working Mechanism
### Traffic Transmission Model between LEO Satellites and HAP Drones  
Suppose that there are \(I\) HAP drones at an altitude of \(h_{1}\) and \(J\) LEO satellites at an altitude of \(h_{2}\) in the space-air-ground integrated network. The LEO satellite set is denoted as \(\mathcal{J}=\{1,2,\cdots,I\}\), and the HAP drone set is denoted as \(\mathcal{I}=\{1,2,\cdots,I\}\). Each HAP drone is equipped with an omnidirectional antenna, and each LEO satellite is equipped with \(L\) steerable beams. The time interval is divided into \(M\) time slots with a length of \(M_{0}\), and the time slot set is denoted as \(\mathcal{M}=\{1,2,\cdots,M\}\). When \(M_{0}\) is sufficiently small, the matching between LEO satellites and HAP drones in each time slot can be treated as quasi-static. In each time slot, one LEO satellite beam can provide services for no more than one HAP drone, and one HAP drone can establish a connection with at most one LEO satellite. We define a LEO-drone matching matrix \(\mathbf{X}_{l\times l}[m]\) to describe the matching relationship between LEO satellites and HAP drones in the \(m\)-th time slot. If the \(i\)-th HAP drone is served by the \(j\)-th LEO satellite in the \(m\)-th time slot, there is \(x_{i,j}[m]=1\); otherwise, \(x_{i,j}[m]=0\). This work focuses on mobile users and IoT users in depopulated regions with almost no obstacles. Therefore, small-scale fading due to multi-path effects can be neglected. The channel gain from the \(i\)-th HAP drone to the \(j\)-th LEO satellite in the \(m\)-th time slot can be expressed as follows [27]:  
\[h_{i,j}[m]=\left(\frac{c}{4\pi f_{c}d_{i,j}[m]}\right)^{2}, \tag{1}\]  
where \(c\) and \(f_{c}\) represent the speed of light and the carrier frequency, respectively. \(d_{i,j}[m]\) represents the distance between the \(i\)-th HAP drone and the \(j\)-th LEO satellite in the \(m\)-th time slot. Based on this, the traffic transmission rate between the \(i\)-th HAP drone and the \(j\)-th LEO satellite can be expressed as follows:  
\[R_{i,j}[m]=W\log_{2}\!\left(1+\frac{P_{h}G_{i}G_{j}h_{i,j}[m]}{k_{B}T_{b}W} \right), \tag{2}\]  
where \(W\) is the bandwidth of LEO beams, \(P_{h}\) is the transmit power of HAP drones, and \(G_{i}\) and \(G_{j}\) represent the antenna gains of the transmitter of HAP drone and the receiver of LEO satellite, respectively [28]. \(k_{B}\) is Boltzmann's constant, and \(T_{b}\) is the system noise  
Figure 2: Traffic generation model for each HAP drone. temperature. When the channel gain between the HAP drone and the \(j\)-th LEO satellite exceeds a given threshold \(h_{0}\), it is considered that this HAP drone is within the visible range of the \(j\)-th LEO satellite. In the \(m\)-th time slot, the set of HAP drones within the visible range of the \(j\)-th LEO satellite can be expressed as follows:  
\[\mathcal{I}_{j}[m]=\big{\{}i|h_{i,j}[m]\geq h_{0}\big{\}}. \tag{3}\]  
As a result of the high-speed movement of LEO satellites, handover is required when the HAP drone moves outside the visible range of the LEO satellite. HAP drones are unable to send traffic to LEO satellites during the handover duration \(T_{h}\), which can be approximately expressed as follows:  
\[T_{h}=\kappa\times\frac{d_{i,j}[m]}{c}, \tag{4}\]  
where \(\kappa\) is the signaling that requires to be transmitted between the HAP drone and the LEO satellite during handover. Therefore, the available traffic transmission time in the \(m\)-th time slot can be represented as follows:  
\[T_{i,j}[m]= \begin{cases}T_{0}-\big{(}1-x_{i,j}[m-1]\big{)}\times\kappa\times \frac{d_{i,j}[m]}{c},&T_{i,j}^{remain}\geq T_{0}\\ T_{i,j}^{remain},&T_{i,j}^{remain}<T_{0}\end{cases} , \tag{5}\]  
where \(T_{i,j}^{remain}\) represents the remaining visible time between the \(j\)-th LEO satellite and the \(i\)-th HAP drone. In each time slot, a HAP drone can only choose one of the \(Q\) traffic types for transmission.