# An On-Orbit Task-Offloading Strategy Based on Satellite Edge Computing
## 3 Model Introduction and Problem Analysis
### Objective Function  
As analyzed in the task model, the subtask \(o_{i}\) is authorized to begin computation when all the subtasks in \(PRE_{i}\) have been completed. Therefore, the subtasks are not all offloaded in the first time slot. Considering that the subtask assignment may select a certain satellite node after several time slots, we use \(T_{wait}^{i}\) to signify the inter-slot duration of waiting before the transmission of this subtask. Assume that the source node initiating the scheduling is \(v_{1}^{1}\). When subtask \(o_{i}\) is assigned from source node \(v_{1}^{1}\) to node \(v_{k}^{n}\), it needs to wait for \(n-1\) time slots at the source node. \(T_{wait}^{i}\) can be expressed as  
\[T_{wait}^{i}=(n-1)\Delta t. \tag{5}\]  
In addition, the original data of subtask \(o_{i}\) is transferred from source node \(v_{1}^{1}\) to \(v_{k}^{n}\) in the following time  
\[T_{trans}^{i}=\frac{D_{i}}{B}hop_{i}, \tag{6}\]  
where \(B(Mb/s)\) denotes the transmission rate of the ISL, and \(hop_{i}\) denotes the minimum number of hops required for a subtask \(o_{i}\) to transmit to the destination node \(v_{k}^{n}\). Additionally, the energy consumption resulting from the transmission of the subtask \(o_{i}\) through the ISL is defined as  
\[E_{trans}^{i}=T_{trans}^{i}P_{trans}\, \tag{7}\]  
where \(P_{trans}\) is the transmission power of the ISL. When the data of subtask \(o_{i}\) are all transmitted to the destination node \(v_{k}^{n}\), the computation time for this subtask is  
\[T_{com}^{i}=\frac{D_{i}\zeta_{i}}{C_{k}}, \tag{8}\]  
where \(C_{k}(CPUcycle/s)\) is the on-orbit processing performance of satellite node \(v_{k}\), and the calculated energy consumption of \(o_{i}\) on satellite node \(v_{k}\) is defined as  
\[E_{com}^{i}=T_{com}^{i}P_{com}^{k}, \tag{9}\]  
where \(P_{com}^{k}\) is the computational power of satellite node \(v_{k}\). The processing result \(RD\) must be transferred to the node allocated to the succeeding task \(o_{j}\) once the subtask \(o_{i}\) is computed, and the transmission time is stated as  
\[T_{re}^{i}=\frac{RD_{o_{i}}}{B}hop_{i}^{n}, \tag{10}\]where \(hop_{i}^{nr}\) denotes the minimum number of hops required for \(RDo_{i}\) to transmit to the node allocated \(o_{j}\). The transmission energy consumption of \(RDo_{i}\) is defined as  
\[E_{re}^{i}=T_{re}^{i}P_{trans} \tag{11}\]  
Then, the final completion time of subtask \(o_{i}\) is  
\[T_{end}^{i}=T_{wait}^{i}+T_{trans}^{i}+T_{com}^{i}+T_{re}^{i}. \tag{12}\]  
and the final energy consumption of subtask \(o_{i}\) is  
\[E^{i}=E_{trans}^{i}+E_{com}^{i}+E_{re}^{i}. \tag{13}\]  
We have specified that the start of subtask \(o_{j}\)'s computation must occur after the completion of its preceding subtask set \(PRE_{j}\). That is, the original data transfer completion time of subtask \(o_{j}\) follows the constraint  
\[T_{wait}^{i}+T_{trans}^{j}\geq T_{end}^{PRE_{j}}. \tag{14}\]  
The depth-first algorithm can be used to determine the order in which subtasks are executed. We must perform the calculations in the logical order of the activities for dependent subtasks. The task's total completion time is then equal to the time it took to complete the last exit subtask \(o_{l}\), i.e.,  
\[T_{trans}^{i}=\frac{D_{i}}{B}hop_{i}, \tag{6}\]0  
To simplify the model, the satellite nodes assigned to the exit subtask \(o_{l}\) in this study can be thought of as directly sending the calculation results to the ground cloud center after completing the task computation; the accompanying feedback delay is ignored, then \(T_{re}^{l}=0\). The total energy consumption can be defined as  
\[T_{trans}^{i}=\frac{D_{i}}{B}hop_{i}, \tag{6}\]1  
Furthermore, we define \(\Omega\) as the set of satellite nodes with high service load, any satellite node \(v_{g}\in\Omega\) can only be utilized as an auxiliary node for subtask transmission, and no subtasks can be scheduled for computational processing.