## 3 System Model and Problem Formulation
### Problem Formulation  
After allocating bandwidth and corresponding power to the cells, the channel capacity of cell \(i\) in the time slot \(t\) can be calculated according to [PERSON]'s formula as follows:  
\[C_{t}^{i}=\chi_{t}^{i}B_{t}^{i}\log(1+\frac{C}{N}_{t}) \tag{11}\]  
in which \(\chi_{t}^{i}\) is a binary variable used to indicate whether cell \(i\) is illuminated in the time slot \(t\), and \(\chi_{t}^{i}=1\) indicates illumination, otherwise \(\chi_{t}^{i}=0\). \(B_{t}^{i}\) is the allocated bandwidth, and \(\frac{C}{N}_{t}^{i}\) is the signal-to-noise ratio obtained by Equations (4), (5), (8) and (9). The traffic demand of cell \(i\) at time slot \(t\) is \(D_{t}^{i}\). The throughput of a cell is defined as the minimum of demand and capacity, i.e., \(Thro_{t}^{i}=\min\{C_{t}^{i},D_{t}^{i}\}\). While improving the system throughput, we hope that the delay of each service request is as small as possible to ensure the quality of service. As mentioned in Section 3.1, there are \(N\) queues on the satellite to store the request data for each cell, and the traffic stored in the queue is denoted by \(D_{t}=\{D_{t}^{i}|i\in\mathbb{N}\}\), in which \(D_{t}^{i}\) represents the stored traffic of cell \(i\) in time slot \(t\). Since the waiting time of each data packet in the queue is different, the demand can be further subdivided into \(D_{t}^{i}=\sum\limits_{l=1}^{T_{ttl}}\phi_{t,l}^{i}\), where \(\phi_{t,l}^{i}\) denotes the number of packets that have been waiting in queue \(i\) for \(l\) time slots. Packets that are not responded to within \(T_{ttl}\) time will be discarded and regarded as access failures. The average waiting delay of data packets in the queue of cell \(i\) in time slot \(t\) is  
\[\tau_{i}^{i}=\frac{\sum\limits_{l=1}^{T_{ttl}}l\cdot\phi_{t,l}^{i}}{\sum \limits_{l=1}^{T_{ttl}}\phi_{t,l}^{i}} \tag{12}\]  
Figure 5: Illustration diagram of the Constraint Map. In the entire beam hopping period, the number of data packets requested in each cell is denoted as \(n_{total}^{i}\), the number of packets that dropped due to timeout is \(n_{fail}^{i}\), and the access success rate can be calculated by the following equation:  
\[ \begin{split} SucRate=\frac{\sum\limits_{i=1}^{N}\left(n_{total}^{i }-n_{fail}^{i}\right)}{\sum\limits_{i=1}^{N}n_{total}^{i}}\end{split}  \tag{13}\]  
The average response delay of all data packets in the entire beam hopping period is  
\[ \begin{split}\tau_{average}=\frac{\sum\limits_{i=1}^{N}\sum \limits_{T_{all}}^{T_{all}}l\cdot\phi_{l}^{i}}{\sum\limits_{i=1}^{N}\sum \limits_{T_{all}}^{T_{all}}\phi_{l}^{i}}\end{split}  \tag{14}\]  
in which \(\phi_{l}^{i}\) denotes the number of packets in cell \(i\) that waited for \(l\) slots before being responded during the entire beam hopping period. While minimizing the delay, we want to ensure the fairness of service in each cell, which can be characterized by minimizing the variance of the average waiting delay of data packets between cells as follows:  
\[ \begin{split} Fair=\text{var}(\tau_{i})=\text{var}(\frac{\sum \limits_{l=1}^{T_{all}}l\cdot\phi_{l}^{i}}{\sum\limits_{l}^{T_{all}}\phi_{l}^{ i}})\end{split}  \tag{15}\]  
In summary, we can model the entire beam hopping assignment problem as an optimization model as follows:  
\[ \begin{split} P1:\max\sum\limits_{i=1}^{T}\sum\limits_{i=1}^{N} Thro_{t}^{i}\\ P2:\max SucRate\\ P3:\min\tau_{average}\\ P4:\min Fair\\ s.t.\\ C1:B_{t}^{i}\subseteq\{\beta_{1},\beta_{2},\cdots,\beta_{m}\}\\ C2:P_{t}^{i}\subseteq PHb_{t}^{Bin}\qquad\forall\beta_{m}\in B_{t}^{i} \\ C3:\sum\limits_{i=1}^{N}P_{t}^{i}<P_{tot}\quad\quad\forall i\in N\\ C4:\sum\limits_{i=1}^{N}\chi_{t}^{i}\leq K\,,\ \chi_{t}^{i}\in\{0,1\}\end{split}  \tag{16}\]  
In the above optimal problem, \(\chi_{t}^{i}\), \(B_{t}^{i}\) and \(P_{t}^{i}\) are the variables to be optimized, which are defined as the illumination pattern and bandwidth and power allocation of each beam at each time slot. \(P1\) and \(P2\) aim to achieve maximum system throughput and access success rate during the entire beam hopping period.