## 3 Materials and Methods
### Mathematical Problem Modeling  
Based on the CQF scheduling mechanism and the transmission mechanism of LEO satellite constellations mentioned above, this paper mathematically models the process of time-sensitive business transmission in Low Earth Orbit satellite constellations. Building upon the theory discussed earlier, the forwarding delay of a single satellite is controllable. Extending this to the initially partitioned parallel subnet, the transmission within the subnet transforms into a multi-objective optimization problem aimed at optimizing overall delay and energy consumption by controlling the flow of packet transmissions. This represents a classic Flexible Job Scheduling Problem (FJSP). In the context of the FJSP problem, where each workpiece can be processed on multiple machines for each operation, with processing times varying across machines, actual production can flexibly select resources based on the workload to enhance processing flexibility [31]. Therefore, it is evident that routing transmission and packet forwarding in Low Earth Orbit satellite constellations align with the elements and application scenarios of the FJSP problem, abstracting into an FJSP problem. To solve the FJSP problem abstracted from time-sensitive business transmission in Low Earth Orbit satellite networks, the mathematical symbols and definitions of the variables involved in the FJSP problem are presented in Table 2.  
Additionally, there are some constraint discriminants, as follows:  
\[x_{ijh}=\left\{ \begin{array}{ll}1,&if\text{ hop O}_{jh}\text{ choose satellite }i\\ 0,&\text{else}\end{array} \right. \tag{8}\]  
\[y_{ijhkl}=\left\{ \begin{array}{ll}1,&if\text{ O}_{jh}\text{ forwards before O}_{ikl}\\ 0,&\text{else}\end{array} \right. \tag{9}\]  
Typically, the general FJSP problem is subject to the following constraints:  
\[s_{jh}+x_{ijh}\times p_{ijh}\leq c_{jh} \tag{10}\]  
\[c_{jh}\leq s_{j(h+1)} \tag{11}\]  
\[s_{jh}+p_{ijh}\leq s_{kl}+L(1-y_{ijhkl}) \tag{12}\]  
\[c_{jh}\leq s_{j(h+1)}+L(1-y_{iklj(h+1)}) \tag{13}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(n\) & The total number of data packets \\ \(m\) & Total number of satellites \\ \(i,e\) & Satellite serial number \\ \(j,k\) & Packet serial number \\ \(h_{j}\) & The total number of hops of the \(j\) data packet \\ \(l\) & Hop count sequence number \\ \(p_{ijh}\) & Forwarding time of the \(h\) hop of the \(j\) data packet on satellite \(i\) \\ \(\text{O}_{jh}\) & The \(h\) hop of the \(j\) data packet \\ \(s_{jh}\) & Start time of the \(h\) hop forwarding of the \(j\) data packet \\ \(c_{jh}\) & Completion time of the \(h\) hop forwarding of the \(j\) data packet \\ \(m_{jh}\) & Number of optional satellites for the \(h\) hop of the \(j\) data packet \\ \(L\) & A positive number large enough \\ _makespan_ & Transmission completion time \\ \(TB\) & Total task burden \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Symbol definitions of FJSP. \[\sum_{i=1}^{m_{jh}}x_{ijh}=1 \tag{14}\]  
Equations (10) and (11) represent the constraints on the sequence of each hop for each data packet. Equations (12) and (13) indicate that at any given time, a single satellite's port can forward only one hop. Equation (14) signifies the machine constraint, meaning that at the same time, the same hop can only be forwarded by a single satellite port. In the process of solving the FJSP, the evaluation of the quality of a feasible solution requires measurement through an objective function. Common optimization objectives found in the relevant literature include minimizing _makespan_, minimizing total machine load, minimizing maximum machine load, and minimizing delay [32]. In this paper, the objective of solving the FJSP is to find a feasible solution that minimizes both the maximum transmission completion time and satellite buffer consumption simultaneously.