# A Hybrid Genetic Algorithm for Ground Station Scheduling Problems
## 3 Ground Resource Scheduling Model
### Constraint Condition
#### 3.3.2 Ground Resource Constraints  
Considering the ground resource constraints, we focus on four types of equipment resources: antennas, SDCs, demodulators, and RDs. The involved constraints include those related to the resource occupation time of data transmission tasks and the capabilities of each resource. We describe these constraints in detail below. If a visible time window meets the user's data transmission requirements and is converted into task \(k\), then the effective execution time of task \(k\) must be encompassed within it. The specific expression is shown in Equation (4). \[ \begin{split} time_{WL_{j}}^{s}<time_{s}^{k}<time_{e}^{k}<time_{WL_{ j}}^{s},\\ \forall j\in WL.Len,\forall k\in[1,n]\end{split}  \tag{4}\]  
\(time_{WL_{j}}^{s}\) represents the start time of the visible time window \(j\) between the satellite and the ground station; \(time_{WL_{j}}^{s}\) represents the end time of the visible time window \(j\) between the satellite and the ground station; and \(WL.Len\) denotes the number of visible windows. All the parameters of \(DL_{i}^{k}\) assigned to task \(k\) must match the required demodulator parameters for task \(k\). The specific expression is shown in Equation (5). \[\left\{ \begin{split} T_{D}^{k}.demMode\in DL_{i}^{k}.demMode, \forall k\in[1,n]\\ T_{D}^{k}.codeRate\in DL_{i}^{k}.codeRate,\forall k\in[1,n]\\ T_{D}^{k}.decMode\in DL_{i}^{k}.decMode,\forall k\in[1,n]\\ i\in[1,DL.Len]\end{split} \right. \tag{5}\]  
_demMode_ indicates the demodulation method; _codeRate_ represents the code rate; _decode_ represents the decoding method; and \(DL.Len\) denotes the number of demodulator resources at the ground station. The parameters of \(AL_{i}^{k}\) assigned to task \(k\) must be able to match the required antenna parameters for task \(k\). The specific expression is shown in Equation (6). \[\left\{ \begin{split} T_{A}^{k}.polar\in AL_{i}^{k}.polar,\forall k \in[1,n]\\ T_{A}^{k}.bands\in AL_{i}^{k}.bands,\forall k\in[1,n]\\ i\in[1,AL.Len]\end{split} \right. \tag{6}\]  
where polar indicates the antenna polarization mode; bands represent the antenna operating frequency range; and \(AL.Len\) denotes the number of antenna resources at the ground station. At any given time, each antenna, SDC, and demodulator can only be allocated to one task for use. Additionally, due to the potential for data confusion and loss when multiple data transmission tasks share the same RD to perform their respective tasks, the use of a shared RD is not considered when there are sufficient resources available. The operating input frequency and output frequency of the SDC assigned to task \(k\) must support the input frequency and output frequency required for task \(k\). The specific expression is shown in Equation (7). \[\left\{ \begin{split} T_{SDC}^{k}.inputFreq&\leq SDCL _{i}^{k}.inputFreq,\forall k\in[1,n]\\ T_{SDC}^{k}.outFreq&\leq SDCL_{i}^{k}.outFreq, \forall k\in[1,n]\\ i\in[1,SDCL.Len]\end{split} \right. \tag{7}\]  
_inputFreq_ represents the input frequency; _outFreq_ represents the output frequency; and \(SDCL.Len\) denotes the number of SDCs at the ground station. The parameters of the \(RDL_{i}^{k}\) assigned to task \(k\) must match the required RD parameters for the task. The specific expression is shown in Equation (8). \[\left\{ \begin{split} T_{R}^{k}.codeRate\leq RDL_{i}^{k}.codeRate, \forall k\in[1,n]\\ T_{R}^{k}.inputMode\in RDL_{i}^{k}.inputMode,\forall k\in[1,n]\\ i\in[1,RDL.Len]\end{split} \right. \tag{8}\]  
_codeRate_ represents the code rate; _inputMode_ indicates the input method, with TCP/IP protocol being a common input mode for RD; and \(RDL.Len\) denotes the number of RDs at the ground station. The number of crossings through the IFSM occupied by successfully allocated tasks must not exceed \(N_{cm}\). The specific expression is shown in Equation (9).