# Service-Driven Dynamic Beam Hopping with Resource Allocation for LEO Satellites
## 2 System Model and Problem Formulation  
### System Model  
A forward link model of the LEO beam hopping satellite system, as depicted in Figure 1, is constructed in this paper. Assume that there are \(M\) beam positions within the coverage area of a single LEO satellite. The set of ground beam positions is defined as \(\mathcal{M}=\{m\mid m=1,2,\cdots,M\}\). The satellite generates \(K\) service beams through Time Division Multiple Access to serve users. The set of satellite service beams is defined as \(\mathcal{K}=\{k\mid k=1,2,\cdots,K\}\), where \(K\) represents the maximum number of service beams and \(K<M\). Suppose that each beam has \(N\) power level options. The set of transmit powers of each satellite service beam is defined as \(P=\{p_{n}|n=1,2,\cdots,N\}\). A beam hopping period is divided into \(T\) time slots, and the beam hopping period is defined as \(\mathcal{T}=\{t\mid t=1,2,\cdots,T\}\). Due to the uneven distribution of ground users and the time-varying nature of their service requirements, the traffic demands of each beam position are different. To meet the differentiated traffic demands of ground beam positions, the satellite must determine which beam positions are illuminated by service beams. Specifically, the satellite needs to know the beam hopping pattern in time slot \(t\). Assume that each service beam can only illuminate one beam position at any time slot of the beam hopping period. The beam hopping pattern in time slot \(t\) is defined as \(\mathbf{X}_{t}=[x_{t}^{1},x_{t}^{2},\cdots,x_{t}^{m},\cdots,x_{t}^{M}]^{T}\), \(x_{t}^{m}\in\{0,1\}\),where \(x_{t}^{m}\) indicates whether beam position \(m\) is illuminated by a service beam in time slot \(t\). Specifically speaking, \(x_{t}^{m}=1\) means beam position \(m\) is illuminated by the beam, and \(x_{t}^{m}=0\) means the beam does not illuminate beam position \(m\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Definitions** \\ \hline \(M\) & Number of beam positions \\ \(K\) & Number of beams \\ \(N\) & Number of beam power \\ \(T\) & Beam hopping period \\ \(X_{t}\) & The beam hopping pattern in time slot \(t\) \\ \(x_{t}^{m}\) & Indication of whether the beam position \(m\) is illuminated in time slot \(t\) \\ \(pd_{t}^{m}\) & Newly arrived data packets in beam position \(m\) in time slot \(t\) \\ \(T_{th}\) & The survival time of data packets in the queue \\ \(\Phi\) & The data packet buffer queues for all beam positions \\ \(\Phi_{t}^{m}\) & The data packet buffer queue of beam position \(m\) in time slot \(t\) \\ \(\lambda\) & The packet arrival rate of beam position \(m\) in time slot \(t\) \\ \(p_{in}\) & The generation probability of data packets in each time slot \\ \(D_{t}\) & The traffic demands of all beam positions in time slot \(t\) \\ \(D_{t}^{m}\) & The service demand of beam position \(m\) in time slot \(t\) \\ \(d_{t}^{m}\) & The data in beam position \(m\) in time slot \(t\) \\ \(h_{t}^{m}\) & The channel coefficient between the beam \(k\) and the beam position \(m\) \\ \(H\) & Channel coefficient matrix between beam and ground beam position \\ \(G_{T}\) & The antenna transmission gain of beams \\ \(G_{R}\) & The antenna reception gain of beam positions \\ \(PL_{km}\) & Transmission path loss \\ \(B_{tot}\) & Total satellite bandwidth \\ \(P_{tot}\) & Total satellite power \\ \(P_{b}\) & Maximum beam power \\ \(C_{t}^{m}\) & The channel capacity of the beam position \(m\) in time slot \(t\) \\ \(\alpha\) & The weight between user satisfaction and delay fairness \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System notations. Suppose that the satellite provides corresponding packet buffer queues for each beam position to store the arriving traffic. The newly arrived packet at beam position \(m\) in time slot \(t\) is defined as \(pa_{t}^{m}\). Due to the limited on-board storage space, to reduce memory usage, a queuing delay threshold \(T_{th}\) is set for each packet buffer queue.