## 3 System Model Description
### Content Delivery Model  
In the satellite network system, user-requested content can be responded to directly at the user's terminal or be responded to by a ground station, a remote sensing satellite, depending on whether the content is actively pushed and where the content is cached. Due to the short queuing and processing delays, this article only considers propagation and transmission delays as content delivery delay.  
**Direct delivery mode \((m^{da}=1):\)** If the content of the request has been cached in the user terminal, the user terminal responds directly, and the delay \(d_{u_{i}f}^{da}(t)\) for user \(u_{i}\) to obtain content \(f\) is 0.  
**Access delivery mode \((m^{ac}=1):\)** If the requested content has been cached in access remote sensing satellite \(s0\), the content is directly transmitted to the user terminal. The delay \(d_{u_{i},f}^{ac}(t)\) for user \(u_{i}\) to obtain content \(f\) can be calculated as follows  
\[d_{u_{i},f}^{ac}(t)=\frac{dis(u_{i},s_{0})}{v_{a}}+\frac{C_{f}}{R_{s_{0},u_{i }}(t)}, \tag{6}\]  
where \(dis(u_{i},s_{0})\) represents the Euclidean distance from user \(u_{i}\) to access satellite \(s_{0}\), \(v_{a}\) is the propagation rate of the satellite-ground link, and \(R_{s_{0},u_{i}}(t)\) is the transmission rate between access satellite \(s_{0}\) and \(u_{i}\) at time slot \(t\).  
**Collaborative delivery mode \((m^{co}=1):\)** If the requested content is not cached in the access remote sensing satellite \(s_{0}\), but another remote sensing satellite \(s_{j}\) has cached the content, then the user's request will be transmitted to \(s_{j}\) through the communication satellite network, and the content will be distributed through inter-satellite links from \(s_{j}\). Due to the dynamically varying link rates, the content delivery delay dynamically varies across multiple transmission paths between \(s_{0}\) and \(s_{j}\). We use the Dijkstra algorithm to solve the shortest path \(\text{Path}_{\min_{0},s_{j}}^{t}\) with minimum content delivery delay.  
The propagation delay for user \(u_{i}\) to obtain content f is denoted by \(d\_{prop^{co}}(t)\), which can be calculated as follows  
\[d\_{prop^{co}}(t)=\frac{dis(u_{i},s_{0})}{v_{a}}+\frac{\sum_{i=0}^{\left| \text{Path}_{\min_{0},s_{j}}^{t}\right|-2}dis(s_{i},s_{i+1})}{v_{b}}, \tag{7}\]  
where \(\text{Path}_{\min_{0},s_{j}}^{t}\) denotes the content delivery path from access satellite \(s_{0}\) to satellite \(s_{j}\), \(\text{Path}_{\min_{0},s_{j}}^{t}=s_{0},s_{1},...,s_{\left|Path_{\min_{0},s_{j} }^{t}\right|-2},s_{j}\), and \(v_{b}\) is the inter-satellite link propagation rate. The content transmission delay of user \(u_{i}\) to obtain content \(f\) is denoted by \(d\_{trans^{co}}(t)\), which can be calculated as  
\[d\_{trans^{co}}(t)=\frac{C_{f}}{R_{s_{0},u_{i}}(t)}+\sum_{i=0}^{\left|\text{ Path}_{\min_{i},s_{0},s_{j}}^{t}\right|-2}\frac{C_{f}}{R_{s_{i},s_{i+1}}(t)}. \tag{8}\]  
where \(R_{s_{i},s_{i+1}}(t)\) is the transmission rate between satellite \(s_{i}\) and \(s_{j}\) at time slot \(t\). Then, the delay \(d_{u_{i},f}^{co}(t)\) for user \(u_{i}\) to obtain content \(f\) is  
\[d_{u_{i},f}^{co}(t)=d\_{prop^{co}}(t)+d\_{trans^{co}}(t). \tag{9}\]  
**Ground station delivery mode \((m^{ga}=1):\)** If the requested content is not cached by the remote sensing satellite network, the content will be distributed from the ground station through the inter-satellite links. Due to the distance between the ground station and the user, users can only obtain content from the ground station through satellite networks, resulting in significant delay when users obtain content from the ground station. This paper uses a large constant \(\varsigma\) to represent the delay \(d_{u,f}^{ga}(t)\) for users requesting content from ground stations.