# Computation Offloading and Resource Allocation Based on P-DQN in LEO Satellite Edge Networks
## 3 System Model and Problem Description
### Channel Model  
Due to the high mobility of LEO satellites, the relative position between satellites and ground terminals changes rapidly, and so do the free space loss, atmospheric fading, and many other factors involved in the satellite-terrestrial link [44]. As shown in Figure 1, the link from terminals to the LEO satellite is named as satellite uplink, and that from the LEO satellite to the cloud server is named as the satellite downlink. Note that when returning processing results to terminals, the transmission latency (and thus the link) is ignored due to the small-sized result. Thus, when offloading task \(R_{k,i}^{\dagger}\) to satellites, the data transmission only goes through the satellite uplink; when offloading it to the cloud server through the LEO satellite, the data transmission must go through the satellite uplink first and then through the satellite downlink.  
Figure 1: LEO satellite edge networks.  
As stated in Section 3.1, the channel state of all terminals in the same satellite coverage is almost identical, so the satellite uplink channel state \(g_{k}^{t,L}\) in region \(k\) at slot \(t\) can be defined as  
\[g_{k}^{t,L}=G_{s}^{E}\cdot G_{k}^{t,L}\cdot G_{r}^{L}, \tag{1}\]  
where \(G_{s}^{E}\) denotes the transmit antenna gain of terminals, \(G_{k}^{t,L}\) is the channel fading between region \(k\) and the associated satellite at slot \(t\), and \(G_{r}^{L}\) represents the receiving antenna gain of satellites. In particular, the channel fading between terminals and satellites generally includes free space path loss, atmospheric fading, and small-scale fading (obeying the Rician distribution) [45], i.e.,  
\[G_{k}^{t,L}=\left(\frac{c}{4\pi d_{k}^{t,L}f_{\epsilon}}\right)^{2}\cdot\Phi \left(\alpha^{t,L}\right)\cdot\psi, \tag{2}\]  
where \(c\) represents the light speed, \(d_{k}^{t,L}\) is the distance between region \(k\) and its access satellite at slot \(t\), \(f_{\epsilon}\) denotes the carrier frequency, \(\Phi\left(\alpha^{t,L}\right)\) specifies the atmospheric fading, and \(\psi\) is the Rician distributed small-scale fading. More precisely, atmospheric fading \(\Phi\left(\alpha^{t,L}\right)\) is expressed as  
\[\Phi\left(\alpha^{t,L}\right)=10^{\left(\frac{M}{10\sin^{t,L}}\right)}, \tag{3}\]  
where \(\sin\alpha^{t,L}=H/d_{k}^{t,L}\), \(H\) is the orbital altitude of LEO satellites, and \(\delta\) is the attenuation through rain and clouds, separately. The Rayleigh fading channel models (e.g., in [46; 47]) is not used throughout.  
Further, the channel state of satellite downlink between LEO satellites and the cloud server \(M\) for region \(k\) at slot \(t\) is represented as \(g_{k}^{t,M}\) as  
\[g_{k}^{t,M}=G_{s}^{L}\cdot G_{k}^{t,M}\cdot G_{r}^{M}, \tag{4}\]  
where \(G_{s}^{L}\) is the transmit antenna gain of LEO satellites, \(G_{k}^{t,M}\) is the channel fading between LEO satellites and the cloud server for region \(k\) at slot \(t\), and \(G_{r}^{M}\) is the receiving antenna gain of cloud server. Likewise, the channel fading of satellite downlink also includes free space path loss, atmospheric fading and Rician distributed small-scale fading, respectively, i.e.,  
\[G_{k}^{t,M}=\left(\frac{c}{4\pi d_{k}^{t,M}f_{l}}\right)^{2}\cdot\Phi\left( \alpha^{t,M}\right)\cdot\psi, \tag{5}\]  
where \(d_{k}^{t,M}\) is the distance between \(M\) and satellite \(l_{k}\), \(f_{l}\) is the carrier frequency of LEO satellites, and \(\Phi\left(\alpha^{t,M}\right)\) is the atmospheric fading, i.e.,  
\[\Phi\left(\alpha^{t,M}\right)=10^{\left(\frac{M}{10\sin^{t,M}}\right)}, \tag{6}\]  
with \(\sin\alpha^{t,M}=H/d_{k}^{t,M}\).