## 2 System Model  
### System Setup  
This study investigates the resource scheduling problem within a single BH cycle, in line with the scheduling characteristics of BH systems. Based on the analysis in [24], the ground-track displacement of a LEO satellite within a single BH cycle is negligible compared to the beam coverage radius, and user positions are assumed to be static over this short duration. Thus, we adopt a snapshot-based model where satellite positions and user distribution remain unchanged within each BH cycle. The two satellite systems operate over a shared frequency spectrum but serve distinct service objectives: the LEO network is designed for dynamic, high-density user access, whereas the GEO network is tasked with providing stable and continuous coverage. In the simulation scenario, a representative ground service area is selected, which is served by a heterogeneous satellite network consisting of \(N_{\text{sat}}\) LEO satellites and one GEO satellite as shown in Figure 1.  
In the time domain, one BH cycle is divided into \(M\) discrete time slots. During each time slot, each satellite determines the positions of its activated beams. In the spatial domain, each LEO satellite is equipped with a multi-beam antenna array and can simultaneously illuminate up to \(N_{\text{beam}}\) beams, with each beam covering a circular area of radius \(R\). Within a single BH time slot, the total number of active beams across all \(N_{\text{sat}}\) satellites is constrained by \(N=N_{\text{sat}}\cdot N_{\text{beam}}\). To mitigate intra-system interference, the illuminated beams in each time slot must maintain a minimum inter-beam distance \(d_{\min}\geq 4R\) as suggested in [25]. To mitigate inter-system interference, a protection region \(D_{\text{GEO}}\) is established around the GEO users, within which the activation of LEO satellite beams is prohibited. In the remainder of this study, all LEO users are assumed to be located outside of the GEO protection region to ensure inter-system interference is avoided during resource scheduling [26]. In the frequency domain, each LEO satellite is allocated the entire system bandwidth \(B_{\text{s}}\), and each beam can serve up to \(L\) users, with the bandwidth equally distributed among them. The total transmit power of each satellite is limited to \(P_{\text{s}}\) and the sum of the power allocated to all beams must not exceed this limit. By partitioning the time, space, and frequency domains, a three-dimensional spatio-temporal-frequency resource grid is constructed. Within a single BH cycle, the \(N_{\text{sat}}\) LEO satellites can schedule up to \(MNL\) resource units, where each user is limited to occupying at most, one unit of this grid. To represent user-resource assignment, we define a binary variable \(a^{i}_{t,b,f}\in\{0,1\}\), which indicates whether the \(i\)-th user is assigned to the \(t\)-th time slot, the \(b\)-th beam, and the \(f\)-th sub-band. For a given user \(i\), the received signal power can be expressed as follows:  
\[S^{i}_{t,b,f}=P^{i}_{t,b,f}\cdot G^{i}_{b}\cdot PL^{i}_{b,f} \tag{1}\]  
where \(P^{i}_{t,b,f}\) denotes the transmit power allocated to the \(i\)-th user on the \(f\)-th sub-band by the \(b\)-th beam in the \(t\)-th time slot. \(G^{i}_{b}\) represents the beamforming gain from the \(b\)-th beam  
Figure 1: GEO–LEO scenario schematic diagram. to the \(i\)-th user, and \(PL_{b,f}^{i}\) denotes the path loss between the \(b\)-th beam and the \(i\)-th user. The receiver antenna at the user terminal is assumed to be omnidirectional, and thus its gain is omitted. The interference received by the \(i\)-th user is defined as follows:  
\[I_{t,b,f}^{i}=\sum_{b^{\prime}\
eq b}P_{t,b^{\prime},f}^{i}\cdot G_{b^{\prime} }^{i}\cdot PL_{b^{\prime},f}^{i} \tag{2}\]  
where \(b^{\prime}\
eq b\) represents the aggregated interference from non-serving beams associated with irrelevant links.