## 3 System Model
### Problem Formulation  
According to the analysis of the multi-beam satellite communication system, the availability of multi-dimensional radio resources changes dynamically. At the same time, the influence of CSI and QoS should be considered when optimal resources are selected. Since the optimal resources in the multi-beam satellite communication system are discontinuous, packets need to be effectively segmented to adapt to the optimal resources. Therefore, we propose a multi-beam satellite cooperative transmission scheme based on cross-layer and cross-dimension resources optimization and irregular gradient adaptive packets segmentation. In this paper, in order to focus on the problem of resource allocation, we assume that satellites and beams have been connected to users. At the same time, considering the connection model between users and satellites mentioned above, the interference of the packet \(a_{qi}^{c}\) can be expressed as:  
\[I_{a_{qi}^{c}s_{p}b_{pj}}=\sum_{a_{qi^{\prime}}^{c}\in A,i^{\prime}\
eq i}u_{qp }P_{s_{p}b_{pj}}h_{a_{qi}^{c}s_{p}b_{pj}}+\sum_{a_{qi^{\prime}}^{c}\in A,i^{ \prime}\
eq q}u_{q^{\prime}p}P_{s_{p}b_{pj}}h_{a_{qi}^{c}s_{p}b_{pj}}, \tag{6}\]where the first part of Equation (6) is the interference to the packet \(a^{c}_{qi}\) caused by other packets \(a^{c}_{qi^{\prime}}\) of the user \(u_{q}\) which are transmitted by the same beam of the same satellite. The second part of Equation (6) is the interference to the packet \(a^{c}_{qi^{\prime}}\) caused by other packets \(a^{c}_{qi^{\prime}}\) of other users \(u_{q^{\prime}}\) which are transmitted by the same beam of the same satellite. \(P_{s_{p}b_{pj}}\) is the transmitting power of the beam \(b_{pj}\) of the satellites \(s_{p}\). In this paper, the transmitting power of each beam is set to the same, 60 W. \(h_{a^{c}_{qi^{\prime}}s_{p}b_{pj}}\) is the channel gain of the beam \(b_{pj}\) of the satellite \(s_{p}\) transmitted the packet \(a^{c}_{qi^{\prime}}\). According to the propagation model described above, different beams have different channel gains. The channel gain of each beam changes with changes in large scale fading, small scale fading, transmission delay, and doppler shift. The channel gains of different beams which are used to transmit packets are different. The packets are suffered from different interference. As the channel gain of the beams increases, the interference suffered by the packets increases. Let the binary variable \(w_{pjq}\) represent the connection between beams and users. When \(w_{pjq}=1\), it means that the user \(u_{q}\) connects to the beam \(b_{pj}\) of the satellite \(s_{p}\), otherwise, \(w_{pjq}=0\). The bandwidth occupied by segmentation packets \(a^{c}_{qi}\) is  
\[B_{a^{c}_{qi}s_{p}b_{pj}}=w_{pjq}\times B\times\frac{f^{b_{pj}}_{a^{c}_{qi}}}{N}, \tag{7}\]  
where \(B\) is the system bandwidth. \(N\) is the number of subchannels of the satellite \(s_{p}\) and \(f^{b_{pj}}_{a^{c}_{qi}}\) is the number of subchannels occupied by the packet \(a^{c}_{qi}\) in the frequency domain. At the receiver, we separately calculate the SINR of the beams that transmit different parts of the packet. The SINR of the beam \(b_{pj}\) of the satellite \(s_{p}\) that transmit the segmentation packet \(a^{c}_{qi}\)is  
\[SINR_{a^{c}_{qi}s_{p}b_{pj}}=\frac{w_{pjq}u_{qp}P_{s_{p}b_{pj}}h_{a^{c}_{qi}s_{ p}b_{pj}}}{I_{a^{c}_{qi^{\prime}}s_{p}b_{pj}}+N_{0}}, \tag{8}\]  
where \(N_{0}\) is Gaussian white noise. According to the propagation model described above, different beams of different satellites have different channel gains. The channel gain changes with the randomness of the channel. \(I_{a^{c}_{qi^{\prime}}s_{p}b_{pj}}\) is the interference suffered by the packet \(a^{c}_{qi}\) as mentioned by Equation (6). When the beam \(b_{pj}\) of the satellite \(s_{p}\) connect to the user \(u_{q}\) which generate the packet \(a^{c}_{qi^{\prime}}\), that is, \(w_{pjq}=1\) and \(u_{qp}=1\). \(w_{pjq}u_{qp}P_{s_{p}b_{pj}}h_{a^{c}_{qi^{\prime}}s_{p}b_{pj}}\) is the power to transmit the packet \(a^{c}_{qi}\).