## 3 System Model
### Problem Formulation
Therefore, the system capacity is:  
\[C=\sum_{u_{q}\in U}\sum_{a^{c}_{qi}\in A}\sum_{s_{p}\in S}\sum_{b_{pj}\in BM}B _{a^{c}_{qi^{\prime}}s_{p}b_{pj}}\text{log}\left(1+SINR_{a^{c}_{qi^{\prime}}s_ {p}b_{pj}}\right). \tag{9}\]  
According to the above parameter description, the mathematical model of radio resource allocation problem with the optimization goal of maximizing the system capacity can be defined as:  
\[max\ C=\sum_{u_{q}\in U}\sum_{a^{c}_{qi}\in A}\sum_{s_{p}\in S}\sum_{b_{pj}\in BM }B_{a^{c}_{qi^{\prime}}s_{p}b_{pj}}\text{log}\left(1+SINR_{a^{c}_{qi^{\prime}} s_{p}b_{pj}}\right), \tag{10}\]  
\[s.t.\left[f^{b_{pj}}_{a^{c}_{qi}},t^{b_{pj}}_{a^{c}_{qi}}\right]\cup\left[f^{ b_{pj}}_{a^{c}_{qi^{\prime}}},t^{b_{pj}}_{a^{c}_{qi^{\prime}}}\right]=\varnothing, \forall i\
eq i^{{}^{\prime}},b_{pj}\in BM,a^{c}_{qi^{\prime}},a^{c}_{qi^{ \prime}}\in A, \tag{11}\]  
\[\left[f^{b_{pj}}_{a^{c}_{qi^{\prime}}},t^{b_{pj}}_{a^{c}_{qi}}\right]\cup\left[ f^{b_{pj}}_{a^{c}_{qi}},t^{b_{pj}}_{a^{c}_{qi^{\prime}}}\right]\
eq\varnothing, \forall j\
eq j^{{}^{\prime}},b_{pj},b_{pj^{\prime}}\in BM,a^{c}_{qi^{\prime}},a^{c}_{qi^{\prime}}\in A, \tag{12}\]\[\left[f^{p_{pj}}_{a_{qi}^{c}}t^{b_{pj}}_{a_{qi}^{c}}\right]\in\left\{\left[0,0 \right],\left[f^{p_{pj}}_{a_{qi}^{c}},t^{b_{pj}}_{a_{qi}^{c}}\right]\right\},b_{ pj}\in BM_{i},a_{qi}^{c}\in A, \tag{13}\]  
\[\sum_{s_{p}\in S}\sum_{b_{pj}\in BM}w_{pji}u_{qp}P_{sp_{pj}}\geq 1, \tag{14}\]  
\[P_{sp_{pj}}\leq P_{max}, \tag{15}\]  
\[B_{a^{c}_{qi}s_{p}b_{pj}}=w_{pjq}\times B\times\frac{f^{b_{pj}}_{a^{c}_{qi}}}{N}, \tag{7}\]0  
Equation (11) introduces that radio resources of the beam \(b_{pj}\) of the satellite \(s_{p}\) occupied by different packets, \(a_{qi}^{c}\) and \(a_{qi}^{c}\), cannot be overlapped at the same time. Equation (12) indicates that different packets, \(a_{qi}^{c}\) and \(a_{qi}^{c}\), can occupy the radio resources of the beam, \(b_{pj}\) and \(b_{pj}\), at the same time, which belong to different satellites, \(s_{p}\) and \(s_{p}\). Equation (13) indicates that the gateway can allocate resources or not according to the judgment conditions in the process of resource allocation. Equation (14) means that the user \(u_{q}\) can connect multiple beams of multiple satellites simultaneously. There will always be a beam of a satellite connected to the user. The transmitted power is limited by the total power given by Equation (15). As shown in Figure 5, a cross-layer and cross-dimension radio resources optimization model based on the weighted discrete firefly algorithm is recommended by considering the mapping model between cross-layer and cross-dimension resources and fireflies and the influence of CSI and QoS as the weight of fireflies' attraction. In this paper, the position of each firefly represents the radio resource \(x=\left(f^{b_{pj}}_{a_{qi}^{c}},t^{b_{pj}}_{a_{qi}^{c}}\right)\) of the beam \(b_{pj}\) of the satellite \(s_{p}\) and the CSI of radio resources and the QoS of packets are expressed as the weight of the brightness \(\beta\) of the firefly. The ratio of bandwidth and SINR \(Fit\) is the adaptive value. The matrix \(Y^{x}_{A}\) is used to represent the result of each optimization, and the dimension of the matrix is \(S\times BM\times U\times A\times N\times T\). \(S\) is the total number of satellites in the system. \(BM\) is the total number of beams in the system. \(U\) is the total number of users in the system. \(A\) is the total number of packets in the system. \(N\) is the total number of sub-channels provided by each beam of each satellite. \(T\) represents the exchange time of one request process, that is, the total time of one resources allocation process. The set of the optimal radio resources selected each time is represented as \(\mathbf{y}\). The representation of the matrix \(Y^{x}_{A}\) is as follows:  
\[B_{a^{c}_{qi}s_{p}b_{pj}}=w_{pjq}\times B\times\frac{f^{b_{pj}}_{a^{c}_{qi}}}{N}, \tag{7}\]1 Wireless resources and fireflies mapping model. \[Y_{A}^{x}= \begin{bmatrix}\mathbf{y}_{1}\\ \mathbf{y}_{2}\\ \vdots\\ \mathbf{y}_{a}\end{bmatrix}