## 3 Problem Definition and Formulation
### System Architecture Analysis  
STIN system comprises multiple heterogeneous satellite nodes, each with unique resource configurations. These nodes are interconnected via satellite intercommunication links, which form a dynamically changing network. Ground stations function as control centers for coordinating and scheduling the satellite nodes, while user terminals interact with the network by initiating task requests and receiving processing results. Due to the inherent characteristics of satellite networks, task scheduling and resource management must account for uncertainties such as dynamic topology, varying channel quality, and potential node failures.  
Each satellite node \(i\) is characterized by a resource vector as  
\[\mathbf{r}_{i}=(c_{i},s_{i},b_{i},p_{i}) \tag{1}\]  
where \(c_{i}\) denotes the computational capacity, measured in Flops; \(s_{i}\) represents the storage capacity, measured in GB; \(b_{i}\) indicates the bandwidth, measured in Gbps; and \(p_{i}\) signifies the processing capability coefficient, reflecting the node's parallel computing ability, typically within the range [0,1].  
These resources are influenced by network states, topology changes, and other factors, and they evolve dynamically as the network operates. Precise modeling of each satellite node's resources is crucial for effective task offloading and resource allocation.  
Tasks in the network are characterized by their computational, storage, and bandwidth requirements. Each task \(j\) is represented by the vector as  
\[\mathbf{t}_{j}=(t_{j}^{c},t_{j}^{s},t_{j}^{b}) \tag{2}\]  
where \(t_{j}^{c}\) denotes the computational requirement, measured in Flops; \(t_{j}^{s}\) represents the storage requirement, measured in GB; and \(t_{j}^{b}\) indicates the bandwidth requirement, measured in Gbps.  
To effectively schedule and offload tasks, a task assignment matrix \(\mathbf{X}\) is defined, where \(X_{ij}\in\{0,1\}\) indicates whether task \(j\) is offloaded to satellite node \(i\). The task assignment must satisfy  
\[\sum_{i=1}^{N}X_{ij}=1,\ \forall j \tag{3}\]  
ensuring that each task is assigned to exactly one satellite node for processing.  
During task offloading, data are transmitted and processed through satellite nodes. Let \(d_{ik}^{j}\) represent the amount of data transmitted from satellite node \(i\) to user terminal \(k\) for task \(j\), and \(\delta_{ik}^{j}\) denote the transmission latency. The data flow and transmission latency are modeled as  
\[\mathbf{D}_{t}=\left\{d_{ik}^{j},X_{ij}\right\},\ \forall i,j,k \tag{4}\]  
where \(d_{ik}^{j}\) are the data transmitted from node \(i\) to terminal \(k\) for task \(j\), and \(X_{ij}\) indicates the task assignment. The transmission latency \(\delta_{ik}^{j}\) is calculated based on the allocated bandwidth \(b_{i}\) and the transmitted data amount \(d_{ik}^{j}\) as follows:  
\[\delta_{ik}^{j}=\frac{d_{ik}^{j}}{b_{i}} \tag{5}\]  
Network congestion, route choices, and channel quality also affect transmission delays, which are necessary to account for these changing factors in practical models. The combined effect of resource allocation, task assignment, and data transmission dynamics defines the overall performance and efficiency of the STIN system. Accurate modeling ensures optimal use of resources, efficient task execution, and minimized system delays, while facing network uncertainties and changing operational conditions.  
To facilitate a comprehensive understanding of the system model, we summarize the key variables and parameters in Table 1. This table provides detailed definitions and units for each variable, ensuring clarity and consistency throughout the analysis.