## 4 Optimization Problem Modeling
### Optimization Constraints
#### 4.4.4 Flow Conservation
By constructing the time-expanded graph \(G_{n}=(V,E,W_{n})\), we capture the dynamic topology of the inter-satellite network over time. The vertices \(V\) include the observation satellite \(OS_{n}\), relay satellites \(RS_{k}\) at different time slots, virtual ground station nodes \(GS_{t}\) representing the ground station at time \(t\), and the super node serving as the common destination. Edges \(E\) represent possible transmissions between nodes at consecutive time slots, with weights \(\mathrm{Wn}\) indicating the transmission costs. We apply [PERSON]'s algorithm to this graph to find the shortest path from \(OS_{n}\) to the super node. This path denotes the optimal sequence of transmissions through relay satellites and time slots, minimizing the overall transmission cost to the ground station. Figure 2 illustrates a schematic of the link formed by one observation satellite \(OS_{n}\) and six relay satellites. Applying [PERSON]'s algorithm to the augmented graph \(G_{n}\) identifies the shortest path between \(OS_{n}\) and the virtual ground station nodes. Let \(\tilde{\pi}^{o}_{n,t}=\{\tilde{\tau}^{o}_{n,k,t^{\prime}},...,\tilde{\tau}^{o }_{n,k,t^{\prime}},...,\tilde{\gamma}^{o}_{n,k,t^{\prime}},1\leq t^{\prime}\leq t\}\) represent the data set \(D^{o}_{n}\) from \(OS_{n}\) to \(GS_{t}\). Due to multiple hops and the dynamic nature of the inter-satellite link, the original transmission strategy may become infeasible, requiring adjustments. The strategy set \(\tilde{\pi}^{o}_{n,t}\) is checked for conflicts. If the maximum conflict time \(t_{1}=\arg\max\{\tilde{\tau}^{\prime}_{n,k,t^{\prime},t}\}\) of the relay satellite transmission strategy exceeds \(t\), the strategy is invalidated; if \(t_{1}<t\), the strategy set \(\tilde{\pi}^{o}_{n,t}\) is retained. The strategy with the minimum weight is selected as the local transmission scheduling strategy set \(\tilde{\pi}^{o}_{n,t^{*}}\) for \(OS_{n}\). Figure 2: Diagram of link state. In the context of independently designing transmission scheduling strategies for observation satellites, instances may arise where two distinct data tasks commence transmission from a similar origin and are destined for the same endpoint concurrently. Such scenarios precipitate a transmission scheduling conflict. To address this issue, the following section introduces an algorithm capable of adjusting transmission strategies through prioritization. For illustrative purposes, consider the transmission strategies \(\zeta^{r}_{n,k,k^{\prime},t}\) and \(\zeta^{r}_{n^{\prime},k^{\prime},t}\) associated with the conflicting data \(D^{o}_{n}\) and \(D^{o}_{n}\), respectively. The algorithm's fundamental concept is developed as follows:  
1. Assess the Priority of Present Data Tasks:At the current time, evaluate the transmission energy expended by both conflicting data tasks, assuming that the task with greater accumulated transmission energy is accorded higher priority. Define \(E_{n,t^{\prime}-1}\) as the total transmission energy accumulated by the data task \(D^{o}_{n}\) preceding time \(t^{\prime}\). The expression for \(E_{n,t^{\prime}-1}\) is given by the following: \[E_{n,t^{\prime}-1}=\sum_{k=1,t=1}^{K,t^{\prime}-1}E^{o}_{n,k,t}+\sum_{ \begin{subarray} {c}k=1,t^{\prime}-1\\ k^{\prime}\
eq k,t-1\end{subarray} }^{K,k^{\prime}-1}E^{r}_{n,k,k^{\prime},t}\] (30) The accumulated energy \(E^{o}_{n,t^{\prime}-1}\) in Equation (30) quantifies the total energy expended for transmitting data task \(D^{o}_{n}\) up to time \(t^{\prime}-1\), considering both observation-to-relay and relay-to-relay transmissions. Specifically, for conflicting data tasks \(D^{o}_{n}\) and \(D^{o}_{n}\), if \(E^{o}_{n,t^{\prime}-1}>E_{n^{\prime},t^{\prime}-1}\), the data task \(D^{o}_{n}\) is granted higher priority (i.e., the one that has already consumed more energy up to time \(t^{\prime}\)).