## 3 Problem Description and Modeling  
### Scenario  
The SEC scenario is shown in Figure 1, in which delay-sensitive tasks generated by observation satellites are further uploaded to SEC satellites for processing. There are SEC satellites \(S=\{1,\,2,\,\ldots,\,m\}\) and \(n\) observation satellites \(D=\{1,\,2,\,\ldots,\,n\}\) in total. Each SEC satellite \(s\in S\) is equipped with a server whose computing capacity and buffer size are \(C_{s}\) and \(M_{s}\), respectively. Due to the limited energy supply in orbit, the maximum energy consumption for each SEC satellite is limited to \(H_{s}\). Every SEC satellite also connects with four neighboring satellites via LISL, so that network topology can be represented as an \(m\,\times\,m\) matrix \(G\), where \(G[k,l]=1\) if an LISL exists between satellites \(k\) and \(l\), and otherwise, \(G[k,l]=0\). Some useful notations are listed in Table 1.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(m\) & Total number of SEC satellites. \\ \(n\) & Total number of observation satellites. \\ \(S\) & Set of SEC satellites. \\ \(D\) & Set of observation satellites. \\ \(C_{s}\) & Computing capacity of SEC satellite \(s\). \\ \(M_{s}\) & Buffer space of SEC satellite \(s\). \\ \(H_{s}\) & Energy consumption limitation on SEC satellite \(s\). \\ \(G\) & Network topology matrix. \\ \(T=\{t_{1},\,t_{2},\,\ldots,\,t_{n}\}\) & Set of tasks \(t\). \\ \(\Theta=\{\theta_{1},\,\theta_{2},\,\ldots,\,\theta_{m}\}\) & Task offloading solution. \\ \(F(\Theta)\) & Objective value of solution \(\Theta\). \\ \(\mathcal{R}(\theta_{s}\,\odot\,t)\) & Removal impact, indicating the variation of \(F(\theta_{s})\) after removing \(t\) from \(\theta_{s}\). \\ \(\mathcal{I}(\theta_{s}\,\odot\,t)\) & Inclusion impact, indicating the minimum variation of \(F(\theta_{s})\) after inserting \(t\) into \(\theta_{s}\). \\ \(\mathcal{R}_{s}=[R_{s1},\,R_{s2},\,\ldots,\,R_{sn}]^{\mathrm{T}}\), & Consensus vectors of satellite \(s\), where \(R_{s}\) implies the removal impacts of tasks, \(W_{s}\) represents the considered offloaded satellites of tasks, and \(\mathcal{Q}_{s}\) indicates the latest timestamp of satellites. \\ \(\overline{\
u}_{s}\) & Pending removal tasks in satellite \(s\). \\ \([\mu_{u},\,\mu_{b}]\) & Influence horizon. \\ \(\overline{U}_{s}=[U_{s1},\,U_{s2},\,\ldots,\,U_{s(n+q)}]^{\mathrm{T}}\) & Task classification vector of satellite \(s\). \\ \(\Phi_{s}\) & Candidate insertion positions of sequence \(\theta_{s}\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations.  
Figure 1: An example of the SEC scenario with delay-sensitive observation-data-processing tasks.  
Each observation satellite \(i\in D\) generates a delay-sensitive task \(t_{i}\), and all tasks \(T=\{t_{1},\)\(t_{2},\)\(\ldots,\)\(t_{i},\)\(\ldots,\)\(t_{n}\}\) are indivisible. Every task \(t_{i}\in T\) can be characterized by a tuple \(<\)\(\lambda_{i}\), \(\mu_{i}\), \(\sigma_{i}\)\(>\), where \(\lambda_{i}\) represents the input data size of \(t_{i}\), \(\mu_{i}\) denotes the workload of \(t_{i}\), and \(\sigma_{i}\) refers to the deadline.  
According to [23], observation satellite \(i\in D\) can access a specific SEC satellite \(a\in S\) covering \(i\) for data uploading. Initially, data from task \(t_{i}\in T\) are uploaded from observation satellite \(i\) to access satellite \(a\). Then, task \(t_{i}\) can be executed directly on \(a\) or offloaded to another satellite in \(S\) via LISLs. Let \(s\in S\) be the offloaded satellite. Since tasks can be executed only after the required data are received, all offloaded tasks in \(s\) must not exceed the available buffer size (represented as \(MA_{s}\)), and these tasks are processed sequentially. After \(t_{i}\) is completed, its data in \(s\) will be released. Then, the whole process is continued until all generated delay-sensitive tasks are finished. Due to a typically small size of outputs, the results feedback process is ignored in this paper.