# Energy-Efficient and QoS-Aware Computation Offloading in GEO/LEO Hybrid Satellite Networks
## 2 Materials and Methods
### System Model
#### 2.1.4 Problem Formulation  
According to the time model constructed by Section 2.1.2, the processing delay, \(D_{n,i}\) of \(T_{n,i}\), includes the satellite computation part, the mission data transmission part, and the propagation part. Specifically, \(D_{n,i}\) is determined by the maximum value of the sum of the transmission delay, \(D_{n,i}^{j,tra}\), of LEO satellite \(i\), the propagation delay, \(D_{n,i}^{j,prop}\), from LEO satellite \(i\) to GEO satellite \(j\), and the computation delay, \(D_{n,i}^{j,geo}\), of GEO satellite \(j\) and the remaining mission processing time, \(D_{n,i}^{leo}\), on LEO satellite \(i\), which is denoted as  
\[\begin{split} D_{n,i}&=\max\left(D_{n,i}^{j,tra}+D_{ n,i}^{j,prop}+D_{n,i}^{j,geo},D_{n,i}^{leo}\right)\\ &=\max\left(\frac{\lambda_{n,i}B_{n,i}}{R_{n,i}^{j}}+\frac{h_{n,i }^{j,geo}-h_{n,i}^{leo}}{R_{light}}+\frac{m_{n,i}\lambda_{n,i}B_{n,i}}{f_{n,i}^ {j,geo}},\frac{m_{n,i}(1-\lambda_{n,i})B_{n,i}}{f_{n,i}^{leo}}\right)\end{split} \tag{11}\]  
According to the energy consumption model constructed by Section 2.1.3, the total energy consumption, \(E_{n,i}\), includes the computational energy consumption of LEO satellites and GEO satellites and the transmission energy consumption, which is calculated by  
\[\begin{split} E_{n,i}&=E_{n,i}^{leo}+E_{n,i}^{j,geo }+E_{n,i}^{j,tra}\\ &=\kappa^{leo}(J_{n,i}^{leo})^{2}(1-\lambda_{n,i})B_{n,i}m_{n,i} +\kappa^{geo}(f_{n,i}^{j,geo})^{2}\lambda_{n,i}B_{n,i}m_{n,i}\\ &\quad+\frac{p_{i}\lambda_{n,i}B_{n,i}}{\omega_{n,i}^{j}log_{2}( 1+\frac{p_{i}\lambda_{i}}{\sum_{w\in\mathcal{L}}p_{n}h_{w,i}+N_{0}})}\end{split} \tag{12}\]  
As mentioned above, the objective of this paper is to minimize the total energy consumption of GEO/LEO hybrid satellite networks without violating the QoS constraints of on-board missions, and the final constrained optimization model can be formulated as follows:  
min \[\left(\sum_{i=1}^{|L|}\sum_{n=1}^{|N_{i}|}E_{n,i}\right)\] (13) s.t. \[\begin{split} 0\leq\lambda_{n,i}\leq 1,& \forall n\in\mathcal{N}_{i},\forall i\in\mathcal{L}\\ &\quad D_{n,i}\leq\tau_{n,i},&\forall n\in\mathcal{N} _{i},\forall i\in\mathcal{L}\\ 0\leq f_{n,i}^{j,geo}\leq f_{j}^{GEO},&\forall j\in \mathcal{H}\\ \sum_{i=1}^{|L|}\sum_{n=1}^{|N_{i}|}f_{n,i}^{j,geo}\leq f_{j}^{ GEO},&\forall j\in\mathcal{H}\\ \sum_{j=1}^{|H|}\lceil\lambda_{n,i}^{j}\rceil\leq 1,& \forall n\in\mathcal{N}_{i},\forall i\in\mathcal{L}\end{split}\] (14)  
where Equation (13) shows that the optimization objective of this paper is to minimize the total energy consumption; Equation (14) denotes the offloading decision variable constraint of any mission; Equation (15) denotes that the total processing delay of any mission cannotexceed its own QoS constraint; Equations (16) and (17) ensure that the computational resources allocated to a mission cannot exceed the processing capacity of GEO satellites; and Equation (18) denotes that any mission can only be offloaded to the GEO satellite covering it for processing, and \(\lceil\cdot\rceil\) denotes rounding up, since \(\sum_{j=1}^{|H|}\lceil\lambda_{n,i}^{j}\rceil\leq 1\) indicates that there will not be two different GEO satellites \(j\) and \(j^{\prime}\) such that both \(\lambda_{n,i}^{j}\) and \(\lambda_{n,i}^{j^{\prime}}\) are greater than 0.