## 3 System Model
### Problem Formulation
, \tag{17}\]  
Among them \(\mathbf{y_{a}}=\left(s_{p},b_{pj},u_{q},a_{qi^{\prime}}^{c}f_{a_{qi^{\prime}}^{ p}}^{p_{pj}},t_{a_{qi^{\prime}}^{c}}^{p_{pj}}\right)\). \(s_{p}\) is the optimal satellite. \(b_{pj}\) is the optimal beam. \(u_{q}\) is the served user. \(a_{qi}^{c}\) is the transmitted packet. \(f_{a_{qi}^{pj}}^{b_{pj}}\) is the frequency index of the optimal radio resources. \(t_{a_{qi}^{c}}^{b_{pj}}\) is the time index of the optimal radio resources. The firefly's brightness \(\beta\) can be written in the following equation:  
\[\beta=\beta_{0}e^{-\gamma*(SINR*e^{\Omega*\beta^{2}})*\sigma^{2}}, \tag{18}\]  
where \(\beta_{0}\) is the value of initial attraction, that is, the value of attraction when \(r=0\). \(\gamma\) is the given fixed light absorption coefficient. \(r\) is the distance between two fireflies. CSI and QoS are the weight of \(r\). The distance between two fireflies can be expressed as:  
\[r_{ij}=\left\|y_{i}-y_{j}\right\|=\sqrt{\sum_{k=1}^{K}\left(y_{i,k}-y_{j,k} \right)^{2}}, \tag{19}\]  
where \(r_{ij}\) represents the difference between two radio resources. \(x_{i}\) and \(x_{j}\) represent different radio resources. \(k\) represents dimension, and \(y_{i,k}\) represents the position of the \(i\)-th firefly in dimension \(k\). The position of the \(i\)-th firefly moved by the influence of \(j\)-th firefly can be defined as:  
\[y_{i}=round\left(y_{i}+\beta_{0}e^{-\gamma*(SINR*e^{\Omega*\beta^{2}})*\pi_{ij }^{2}}\left(y_{j}-y_{i}\right)\right)+\alpha(rand-0.5), \tag{20}\]  
where the second part in the above equation represents the influence of attraction. The third part represents the random parameter. The \(rand\) represents the random value between \([0,1]\). \(\alpha\) is a random parameter. Since the radio resource allocation in the multi-beam satellite communication system is a complex discretization problem, this paper adopts \(round\) to discretize the radio resources. The ratio of the bandwidth to SINR of the user's packet \(a_{qi}\) is determined as fitness function \(h\left(B_{a_{qi}^{p}}^{T},SINR_{a_{qi}}\right)\) to find the optimal resources index. The bandwidth occupied by the packets \(a_{qi}\) of user \(u_{q}\) within time \(T\) is  
\[B_{a_{qi}}^{T}=\sum_{s_{p}\in S}\sum_{b_{pj}\in BM}w_{pjq}B_{s_{p}b_{pj}a_{qi} }^{T}=\sum_{s_{p}\in S}\sum_{b_{pj}\in BM}w_{pjq}\times B\times\frac{f_{a_{qi} ^{pj}}^{b_{pj}}}{N}\times t_{a_{qi}^{pj}}^{b_{pj}}. \tag{21}\]  
The total bandwidth of the beam \(b_{pj}\) of the satellite \(s_{p}\) is \(B\), which is divided into \(N\) sub-channels. The bandwidth occupied by each subchannel is \(\frac{B}{N}\) in a symbol. \(f_{a_{qi}^{c}}^{b_{pj}}\) is the number of subchannels of the beam \(b_{pj}\) occupied by the packet \(a_{qi}^{c}\) in the frequency domain, which is different in different beams of different satellite occupied by the packet \(a_{qi}^{c}\). The bandwidth occupied by the packet \(a_{qi}^{c}\) of the beam \(b_{pj}\) of the satellite \(s_{p}\) is \(w_{pjq}\times B\times\frac{f_{a_{qi}^{c}}^{b_{pj}}}{N}\) in a symbol. \(t_{a_{qi}^{c}}^{b_{pj}}\) is the number of symbols of the beam \(b_{pj}\) occupied by the packet \(a_{qi}^{c}\) in the time domain, which is different in different beams of different satellite occupied by the packet \(a_{qi}^{c}\). The bandwidth occupied by the packet \(a_{qi}^{c}\) of the beam \(b_{pj}\) of the satellite \(s_{p}\) is \(w_{pjq}\times B\times\frac{f_{a_{qi}}^{p_{pj}}}{N}\times t_{a_{qi}^{p_{j}}}^{b_{pj}}\). In summary, The bandwidth of all beams of all satellites occupied by the packet \(a_{qi}^{c}\) is \(B_{a_{qi}^{c}}^{T}\) in \(T\), that is, in a TTI. In the case of channels with different SINR, the index of the optimal radio resources, \(f_{a_{qi}^{c}}^{b_{pj}}\) and \(t_{a_{qi}^{c}}^{b_{pj}}\) are different. The bandwidth occupied by the packet is also different.