## 3 Problem Definition and Formulation
### Constraints
#### 3.2.7 Coupling Constraints Between Resource Allocation and Path Selection  
To ensure logical consistency between satellite node resource allocation and path selection strategies, this paper introduces the following coupling constraint:  
\[t_{ij}\leq\sum_{k=1}^{K_{j}}x_{jk}\cdot R_{i}\cdot\mathbb{I}(i\in P_{jk}),\quad \forall i\in\{1,...,n\},\forall j\in\{1,...,m\} \tag{9}\]  
where \(\mathbb{I}(i\in P_{jk})\) denotes the indicator function that equals 1 if satellite node \(i\) belongs to path \(P_{jk}\) of task \(j\), and 0 otherwise. \(R_{i}\) represents the total available resource capacity of satellite node \(i\). This constraint ensures system validity through dual mechanisms: First, resource allocation to node \(i\) is permitted only when \(i\) lies on an active path for task \(j\), ensuring path-resource topological alignment. Meanwhile, the \(R_{i}\) term enforces node capacity limits, while linearly coupling the resource allocation scale with path decisions via \(x_{jk}\), preventing over-subscription and maintaining operational stability under dynamic conditions.  
To facilitate a comprehensive understanding of the system model, we summarize the key variables and parameters in Tables 1 and 2. This table provides detailed definitions and units for each variable, ensuring clarity and consistency throughout the analysis.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Variable Name** & **Description** \\ \hline
**A** & Network topology matrix, representing the connection characteristics between nodes. \\
**T** & Task allocation matrix, representing the resource allocation for task \(j\) at satellite node \(i\). \\ \(t_{ij}\) & The amount of resources allocated to task \(j\) at satellite node \(i\). \\ \(t^{\prime}_{ij}\) & Adjusted resource allocation for task \(j\) at satellite node \(i\). \\ \(d_{j}\) & Transmission latency of task \(j\). \\ \(P_{j}\) & Transmission path for task \(j\). \\ \(s_{j}\) & Source node of task \(j\). \\ \(t_{j}\) & Destination node of task \(j\). \\ \(R_{i}\) & Maximum resource capacity of satellite node \(i\). \\
**X** & Set of all possible resource allocation, task scheduling, and path selection combinations. \\ \(\mathcal{U}\) & Overall network utility function. \\ \(L\) & Number of user categories (i.e., satellites) in the network \\ \(M\) & Number of ground stations \\ \(R_{l,m}\) & Actual throughput of the \(l\)-th category and \(m\)-th user \\ \(R_{\min,l,m}\) & Minimum acceptable throughput for the \(l\)-th category and \(m\)-th user \\ \(U_{l,m}\) & Utility of the \(l\)-th category and \(m\)-th user \\ \(U_{\min,l,m}\) & Minimum utility for the \(l\)-th category and \(m\)-th user \\ \hline \hline \end{tabular}
\end{table}
Table 1: Variables defined in scenario.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters Name** & **Description** \\ \hline \(C_{i}\) & Computational capacity of satellite node \(i\). \\ \(B_{i}\) & Bandwidth of satellite node \(i\). \\ \(Q_{i}\) & Storage capacity of satellite node \(i\). \\ \(\mathbf{R}_{i}\) & Resource vector of satellite node \(i\), including \(C_{i}\), \(B_{i}\), and \(Q_{i}\). \\ \(a^{B}_{ij}\) & Bandwidth between node \(i\) and node \(j\). \\ \(a^{z}_{ij}\) & Latency between node \(i\) and node \(j\). \\ \(D_{j}\) & Total resource requirement for task \(j\). \\ \(D_{\max}\) & Maximum acceptable latency for task \(j\). \\ \(C_{\max}\) & Maximum allowable change in computational complexity. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters List.