## 2 Problem Statement and Model Formulation
### Integer Linear Programming Model  
Considering that the transmission cost of each task is different, we define the decision variable of the integer linear programming model (denoted as M1) as \(x_{ij}^{d}\) which takes the value one when task \(d\) is transmitted using link \((i,j)\) and zero otherwise. It is worth noting that each task is defined with a priority attribute, and a task with higher priority is transmitted first. However, the objective function of the M1 model is to minimize the total transmission cost. Therefore, we consider the reciprocal of each task priority as the weight of each task in the objective function. The integer linear formulation of the multi-task ISRPTWC can be expressed as follows:  
\[Min\ \sum_{d\in D}\biggl{(}\frac{1}{Pri_{d}}\biggr{)}\sum_{i\in N}\sum_{j\in N}c _{ij}^{d}x_{ij}^{d} \tag{1}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline \(S\) & Satellite set, \(S=\left\{s_{1},\cdots,s_{|S|}\right\}\). \\ \(GATA\) & Gateway station set, \(GATA=\left\{g_{1},\cdots,g_{|GATA|}\right\}\). \\ \(T\) & Timeslot set, \(TS=\left\{t_{1S},\cdots,t_{|S|TS|}\right\}\). \\ \(N\) & A set of nodes consisting of different timeslot satellites and letter gateway stations for the current superframe and the next superframe, \(N=N_{1}\cup N_{2}\). \\ \(N_{1}\) & The set of nodes of the current superframe, \\  & \(N_{1}=\left(\left\{ts_{1},\cdots,t_{|TS|}\right\}\right\}\), \(\left\{s_{1},\cdots,s_{|S|P|},g_{|S|+1},\cdots,g_{|S|+|GATA|}\right\}\right)\). \\ \(E\) & The set of edges consisting of connection routes between satellites with different timeslots. \\ \(G\) & Direct Graph, \(G=(N,E)\). \\ \(D\) & Task Set, \(|D|=n\). \\ \(ST\) & Task upload nodes Set, \(ST=\left\{st_{1},\cdots,st_{n}\right\}\), \(ste\Bigl{(}\left\{ts_{1},\cdots,t_{|TS|}\right\}\Bigr{\{}s_{1},\cdots,s_{|S|} \Bigr{\}}\Bigr{)}\). \\ \(AR\) & Task downlink nodes Set, \\  & \(AR=\left\{ar_{1},\cdots,ar_{n}\right\}\), \(are\Bigl{(}\left\{ts_{1},\cdots,t_{|TS|}\right\}\Bigr{\{}s_{1},\cdots,s_{|GATA| }\Bigr{\}}\Bigr{)}\). \\ \(SA\) & Set of task upload and download nodes, \(\left\{ST,AR\right\}\). \\ \(0\) & Task virtual start node. \\ \(2n+1\) & Task virtual termination node. \\ \(d\) & Task Index. \\ \(Dur_{d}\) & Maximum transmission time for task \(d\),\(d\in D\). \\ \(Pri_{d}\) & Transmission priority of task \(d\), \(d\in D\). \\ \(Blo_{d}\) & Maximum blocking time for task \(d\), \(d\in D\). \\ \(Cap_{i}\) & Capacity of node \(i\). \\ \((i,j)\) & Index of link between adjacent nodes \(i\) and \(j\). \\ \(c_{ij}^{d}\) & Transmission cost of link \(i\) to \(j\). \\ \(t_{ij}^{d}\) & Transmission time of link \(i\) to \(j\). \\ \(Cap_{ij}\) & Transmission capacity of link \(i\) to \(j\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sets, indices, and parameters. \[\sum_{st_{ij}\in ST}\sum_{j\in N}x^{d}_{st_{ij}}=1,\;\forall d\in D \tag{2}\] \[\sum_{j\in N}x^{d}_{st_{ij}}-\sum_{j\in N}x^{d}_{st_{ij}}=0,\forall st _{i}\in ST,\forall ar_{i}\in AR\] (3) \[\sum_{j\in ST}x^{d}_{0j}=1,\;\forall d\in D\] (4) \[\sum_{j\in N}x^{d}_{ji}-\sum_{j\in N}x^{d}_{ij}=0,\;\forall d\in D,\forall i\in N\] (5) \[\sum_{i\in AR}x^{d}_{i,2n+1}=1,\;\forall d\in D\] (6) \[\sum_{i\in N}\sum_{j\in N}t^{d}_{ij}x^{d}_{ij}\leq Dur_{d},\; \forall d\in D\] (7) \[\sum_{d\in D}\sum_{i\in N\cup\{0\}}x^{d}_{ij}\leq Cap_{j},\forall i\in N\] (8) \[\sum_{d\in D}x^{d}_{ij}\leq Cap_{ij},\forall i,j\in N,i.satellite \
eq j.satellite\] (9) \[\sum_{k=0}^{Blo_{d}}x^{d}_{i+k*|S|,j+(k+1)*|S|}\leq Blo_{d},\; \forall i\in N_{1},\forall d\in D\] (10) \[\sum_{st_{ij}\in ST}\sum_{j\in N}x^{d}_{st_{ij}}=1,\;\forall d\in D \tag{2}\]0 (11) \[\sum_{st_{ij}\in ST}\sum_{j\in N}x^{d}_{st_{ij}}=1,\;\forall d\in D \tag{2}\]1  
The objective function (1) is to minimize the total transmission cost. Constraints (2) and (3) represent that each task has a unique upload node and a downlink node.