# Service-Driven Dynamic Beam Hopping with Resource Allocation for LEO Satellites
## 2 System Model and Problem Formulation
### Problem Formulation  
The total packet delay \(t_{s,e}^{delay}\) is defined as \(t_{s}-t_{e}\), where \(t_{s}\) is the time slot when the packet is completely transmitted, and \(t_{e}\) is the time slot when the packet is generated. The queuing and transmission delays are calculated statistically over the interval from \(t_{e}\) to \(t_{s}\).  
To minimize the traffic delay and improve user satisfaction to the greatest extent, a dynamic beam hopping multi-objective optimization problem is established. The problem can be modeled as  
\[\begin{split}\min_{x_{t}^{m},p_{t}^{k}}&\alpha\cdot \sum_{m\in\psi}\sum_{t\in T}\sum_{p_{t}^{m}\in\Phi_{t}^{m}}t_{s,e}^{delay}-(1- \alpha)\cdot\sum_{m\in\psi}\sum_{t\in T}\frac{C_{t}^{m}}{D_{t}^{m}},\alpha\in[ 0,1]\\ \text{s.t.}& C1:\sum_{m\in\psi}x_{t}^{m}=K,x_{t}^{m }\in\{0,1\},\ \forall t\in T\\ & C2:0\leq p_{t}^{k}\leq p_{b},\ \forall k\in K,\ \forall t\in T\\ & C3:\sum_{k\in K}p_{t}^{k}\leq P_{tot},\ \forall t\in T\\ & C4:C_{t}^{m}\leq D_{t}^{m},\ \forall m\in\psi,\ \forall t\in T\\ & C5:t_{s,e}^{delay}\leq T_{th}\end{split} \tag{14}\]  
In this optimization problem, we take the beam hopping pattern \(x_{t}^{m}\) and the beam transmit power \(P_{t}^{k}\) of each time slot as optimization variables. \(\alpha\) represents the weight coefficient between delay and user satisfaction. Constraint \(C1\) indicates that the number of illuminated beam positions within any time slot is consistent with the maximum number of satellite service beams. \(C2\) restricts the range of the transmit power of a single beam. Similarly, \(C3\) constrains the upper limit of the sum of the transmit powers of all service beams within any time slot. Considering the problem of users' quality of service, \(C4\) specifies the upper limit of the actual data throughput of the illuminated beam positions. \(C5\) describes the maximum dwell time of data packets through the delay threshold. Due to the limitation of on-board memory size, data packets in the data packet buffer queue that exceed this threshold will be discarded.  
The optimization problem in Equation (14) is non-convex and non-linear. Due to the long-term accumulation characteristic of the objective function, the problem is reformulated as a sequential decision-making task to determine the power allocation and beam hopping pattern for each time slot. The traffic data volume of the packet buffer queue for each beam position in the next time slot depends on the traffic data volume of this beam position in the current time slot, the beam hopping pattern in the current time slot, and the newly arrived traffic data volume in the next time slot. Therefore, the traffic data volume of the next time slot can be expressed as  
\[D_{t+1}=D_{t}-\mathbf{X}_{t}C_{t}+ND_{t+1} \tag{15}\]  
where \(D_{t}\) represents the traffic data volume of the current time slot. \(C_{t}\) is the beam allocation capacity of the current time slot, and \(ND_{t+1}\) means the traffic data volume of the newly arrived packets in the next time slot. It can be seen from the above formula that the problem satisfies the characteristics of the MDP.