# Energy-Efficient and QoS-Aware Computation Offloading in GEO/LEO Hybrid Satellite Networks
## 2 Materials and Methods
### System Model
#### 2.1.2 Time Model  
As shown in Figure 1, the GEO/LEO hybrid satellite networks can be denoted as \(HSN=\{\mathcal{L}\cup\mathcal{H},\mathcal{E}\}\), where \(\mathcal{L}=\{1,2,\ldots,L\}\) denotes the set of LEO satellites, \(\mathcal{H}=\{1,2,\ldots,H\}\) denotes the set of GEO satellites, and \(\mathcal{E}\) denotes the set of inter-satellite links. In addition, we use \(\mathcal{N}_{i}=\{1,2,\ldots,N_{i}\}\) to describe the set of missions on LEO satellite \(i\). \(|L|,|H|\), and \(|N_{i}|\) denote the number of LEO satellites, GEO satellites, and missions on LEO satellite \(i\), respectively. The mission on LEO satellite \(i\) can be represented by a triplet \(T_{n,i}\triangleq(B_{n,i},\tau_{n,i},m_{n,i})\), where \(B_{n,i}\) denotes the data size of the mission, \(\tau_{n,i}\) denotes the QoS constraint of the mission, and \(m_{n,i}\) denotes the processing density (in CPU \(cycles/bit\)).  
In this paper, we are concerned with partial offloading and define \(\lambda_{n,i}\) as the ratio of \(T_{n,i}\) data volume offloaded to the GEO satellite (\(0\leq\lambda_{n,i}\leq 1\)). Thus, for \(T_{n,i}\), there is \(\lambda_{n,i}B_{n,i}\) amount of data that will be offloaded to the GEO satellite, while the amount of data that will be processed by the LEO satellite is \((1-\lambda_{n,i})B_{n,i}\).  
**Computation delay:** we define \(f_{n,i}^{leo}\) as the processing capacity provided to mission \(T_{n,i}\) by LEO satellite \(i\), then the LEO satellite processing time of mission \(T_{n,i}\) can be expressed as  
\[D_{n,i}^{leo}=\frac{m_{n,i}(1-\lambda_{n,i})B_{n,i}}{f_{n,i}^{leo}} \tag{1}\]  
where \(0\leq f_{n,i}^{leo}\leq f_{i}^{LEO}\) and \(f_{i}^{LEO}\) denotes the maximum processing capacity of LEO satellite \(i\).  
The GEO satellite \(j\) processing time of mission \(T_{n,i}\) can be calculated by  
\[D_{n,i}^{j,geo}=\frac{m_{n,i}\lambda_{n,i}B_{n,i}}{f_{n,i}^{j,geo}} \tag{2}\]  
where \(f_{n,i}^{j,geo}\) (\(0\leq f_{n,i}^{j,geo}\leq f_{j}^{GEO}\)) denotes the processing capacity provided to mission \(T_{n,i}\) by GEO satellite \(j\) and \(f_{j}^{GEO}\) denotes the maximum processing capacity of GEO satellite \(j\).  
**Transmission delay:** usually, the data size of the results is much smaller than the original data size, so we ignore the time of receiving the result of offloading missions [25].  
Figure 1: Computation offloading framework for GEO/LEO hybrid satellite networks.  
According to the Shannon-Hartley theorem, the maximum uplink transmission rate, \(R_{n,i}^{j}\), for transmitting mission \(T_{n,i}\) from LEO satellite \(i\) to GEO satellite \(j\) can be obtained by  
\[R_{n,i}^{j}=\omega_{n,i}^{j}log_{2}(1+\frac{p_{i}h_{i,j}}{\sum_{u\in\mathcal{L}} p_{u}h_{u,j}+N_{0}}) \tag{3}\]  
where \(\omega_{n,i}^{j}\) denotes the radio channel bandwidth for transmitting mission \(T_{n,i}\) from LEO satellite \(i\) to GEO satellite \(j\), \(p_{i}\) denotes the transmit power of LEO satellite \(i\), and \(h_{i,j}\) denotes the channel gain between LEO satellite \(i\) and GEO satellite \(j\). The interference term in the denominator \(\sum_{u\in\mathcal{L}}p_{u}h_{u,j}\) denotes the total interference signal power from other LEO satellites and \(N_{0}\) denotes the Gaussian noise power.  
Then, the uplink transmission delay of mission \(T_{n,i}\) on LEO satellite \(i\) is calculated by  
\[D_{n,i}^{j,trn}=\frac{\lambda_{n,i}B_{n,i}}{R_{n,i}^{j}} \tag{4}\]  
**Propagation delay:** we define \(R_{light}\) as the speed of light in a vacuum, \(h_{n,i}^{leo}\) as the altitude of LEO satellite \(i\), and \(h_{n,i}^{j,geo}\) as the altitude of GEO satellite \(j\). Then, the propagation delay from LEO satellite \(i\) to GEO satellite \(j\) can be expressed as  
\[D_{n,i}^{j,prop}=\frac{h_{n,i}^{j,geo}-h_{n,i}^{leo}}{R_{light}} \tag{5}\]