## 2 System Model
### Problem Formulation  
Within a single BH cycle, the \(N_{\text{sat}}\) LEO satellites can schedule up to \(MNL\) resource units and each user is allowed to occupy at most, one such unit. The overall optimization objective is to maximize the utilization of these multi-dimensional resources while jointly optimizing the allocation of frequency and power for each satellite's beams and associated users, so as to maximize the total system throughput. However, due to the strong coupling across the spatio-temporal-frequency-power dimensions, direct optimization of the global objective incurs prohibitively high computational complexity. To address this, we decompose the original problem into two sub-objectives, enabling a stepwise optimization approach. This hierarchical strategy allows for the design of low-complexity algorithms capable of achieving near-optimal solutions with significantly reduced computational overhead.  
The first optimization objective aims to maximize the number of scheduled system resources, which can be formulated as the following objective function, P1:P1: max \(\sum_{t=1}^{M}\sum_{b=1}^{N}\sum_{f=1}^{L}\alpha_{t,b,f}^{i}\) s.t. \[\sum_{i}\alpha_{t,b,f}^{i}\leq 1,\forall\text{triplet}\{t,b,f\}\] (6b) \[\sum_{t}\sum_{b}\sum_{f}\alpha_{t,b,f}^{i}\leq 1,\forall i\] (6c) \[\alpha_{t,b,f}^{i}\in\{0,1\}.\] (6d)  
In objective function P1, constraint (6b) ensures that each spatio-temporal-frequency resource unit can be assigned to at most, one user within a single BH cycle. Constraint (6c) guarantees that each user can be allocated no more than one such resource unit per BH cycle. Constraint (6d) specifies that \(\alpha_{t,b,f}^{i}\) is a binary allocation variable.  
The second optimization objective aims to allocate satellite beam power, intra-beam user power, and user frequency resources in order to maximize the total throughput of users served by the satellite system. This objective can be formulated as the following optimization problem, P2:  
\[\text{P2: max}\sum_{t=1}^{M}\sum_{b=1}^{N}\sum_{i=1}^{U_{t,b}}R_{t,b,f_{i}}^{i} \tag{7a}\] \[\text{s.t.}\sum_{i=1}^{U_{t,b}}P_{i}^{i}\leq P_{t}^{b},\forall t,b\] (7b) \[P_{t}^{i}\geq 0,\forall i,t\] (7c) \[P_{t}^{b}\leq P_{\text{max}},\forall t,b\] (7d) \[\sum_{b=1}^{N}P_{t}^{b}\leq P_{s},\forall t\] (7e) \[\sum_{i=1}^{U_{t,b}}f_{t,b}^{i}=B_{s},\forall t,b\] (7f) \[f_{t,b}^{i}\geq 0,\forall i,t\] (7g) \[i\
otin D_{\text{GEO}}. \tag{7h}\]  
\(U_{t,b}\) denotes the number of users served by the \(b\)-th beam during the \(t\)-th time slot and \(\left\{u_{t,b^{\prime}}^{1},u_{t,b^{\prime}}^{2},\ldots,u_{t,b}^{U_{t,b}}\right\}\) represents the corresponding user set.  
In objective function P2, the associated constraints are interpreted as follows:  
* Constraint (7b) specifies that the total power allocated to users under each beam must not exceed the power assigned to that beam.
* Constraint (7c) specifies that the power allocated to each user must be non-negative.
* Constraint (7d) specifies that the power allocated to each beam must not exceed the predefined beam-level power limit.
* Constraint (7e) specifies that the total power allocated across all beams must not exceed the satellite's onboard power budget.
* Constraint (7f) specifies that the total bandwidth allocated to users under each beam must equal the total bandwidth available to the satellite.
* Constraint (7g) specifies that the bandwidth allocated to each user must be non-negative.
* Constraint (7h) specifies that scheduled LEO users must be located outside the GEO protection zones in order to eliminate potential interference to GEO system operations.  
Based on the two objective functions defined above, we perform dimension-wise optimization for spatio-temporal-frequency-power resource allocation in the LEO system.