# AoI Analysis of Satellite-UAV Synergy Real-Time Remote Sensing System
## 3 Derivation for MGF of AoI\({}_{0}\)
### Calculate the MGF of AoI\({}_{0}\) under LCFS-PS  
In this subsection, the SHS Markov chain in Figure 4 is built up, which models the discrete state under _LCFS-PS_. Then, the MGF for AoI\({}_{0}\) under _LCFS-PS_ is derived.  
As depicted in Figure 4, when GU \(k\) manages the arrived packets under _LCFS-PS_, the state \(q\in\mathcal{Q}=\{0,1,2,3\}\). The meaning of the discrete component \(q\) and the continuous component \(\mathbf{x}(t)\) under LCFS-PS is similar to that under LCFS-NP, as described in Section 3.2. Note that unlike Section 3.2, under the LCFS-PS, newly arrived \(m_{i}\) can preempt the old one already stored in the \(queue_{i}\) at GU, and GU discards the old one. The transitions are labeled \(l\in\{1,\ldots,10\}\). The detailed description of the transitions \(l\) in Table 2 is as follows.  
* \(l\in\{1,2,3,4,5,6\}\): The descriptions are same as the ones of the transitions \(l\in\{1,2,3,4,5,6\}\) in Table 1;
* \(l=7\) or 8: In state 1, transition \(l=7\) occurs when a new \(m_{1}\) arrives at GU \(k\). The new \(m_{1}\) preempts the old one stored in \(queue_{1}\). \(\mathbf{A}_{l}\) governs that \(x_{0}^{\prime}=x_{0}\), because no new \(m_{0}\) is generated when the transition occurs, \(x_{1}^{\prime}=0\) because the new \(m_{1}\) is fresh, and \(x_{2}^{\prime}=0\) is due to the irrelevance of \(x_{2}\) to state 1. Similarly, In state 2, the transition occurs when a new \(m_{2}\) arrives, and the new \(m_{2}\) preempts the old one stored in \(queue_{2}\). \(\mathbf{A}_{l}\) governs that \(x_{0}^{\prime}=x_{0}\), \(x_{2}^{\prime}=0\), and \(x_{1}^{\prime}=0\);
* \(l=9\) or 10: In state 3, transition \(l=9\) occurs when a new \(m_{2}\) arrives at GU \(k\). The new \(m_{2}\) preempts the old one stored in \(queue_{2}\). \(\mathbf{A}_{l}\) governs that \(x_{0}^{\prime}=x_{0}\), because no new \(m_{0}\) is generated when the transition occurs, \(x_{2}^{\prime}=0\) because the new \(m_{2}\) is fresh, \(x_{1}^{\prime}=x_{1}\) because \(x_{1}\) is irrelevant to \(m_{2}\). Similarly, transition \(l=10\) occurs when a new \(m_{1}\) arrives. \(\mathbf{A}_{l}\) governs that \(x_{0}^{\prime}=x_{0}\), \(x_{1}^{\prime}=0\), and \(x_{2}^{\prime}=x_{2}\).  
Figure 4: The discrete state and transitions under _LCFS-PS_ modeled by SHS Markov chain. The new arriving status information is allowed to preempt the packet in service and the packet in waiting. When \(a_{1}=a_{2}=1\), the GU is busy, which indicates that the GU is performing the fusion of \(m_{1}\) and \(m_{2}\). Otherwise, GU is idle.  
Similar to Section 3.2, with some algebraic operations by using Equations (12)-(14), the MGF of AoI\({}_{0}\) under _LCFS-PS_ is given by  
\[MGF_{PS}(s)=\frac{\lambda_{1}\lambda_{2}\mu\beta_{5}(\lambda_{2}\gamma_{2}\gamma_ {4}+{\lambda_{1}}^{3}\gamma_{5}+{\lambda_{1}}^{2}\gamma_{6}+\lambda_{1}\gamma_{ 7})}{\beta_{1}\beta_{2}\beta_{3}\gamma_{3}^{3}\gamma_{1}\gamma_{2}\alpha}, \tag{20}\]  
where \(\gamma_{1}=s-\lambda_{1}-2\mu\), \(\gamma_{2}=s-\lambda_{2}-2\mu\), \(\gamma_{3}=-s+\lambda_{1}+\lambda_{2}\), \(\gamma_{4}=s\lambda_{2}+2s\mu-4\lambda_{2}\mu\), \(\gamma_{5}=-\bar{s}+2\lambda_{2}+4\mu\), \(\gamma_{6}=\bar{s}^{2}-5s\lambda_{2}+4{\lambda_{2}}^{2}-8\bar{s}\mu+12\lambda_ {2}\mu+8\mu^{2}\), \(\gamma_{7}=2\bar{s}^{2}\lambda_{2}-5s{\lambda_{2}}^{2}+2{\lambda_{2}}^{3}+2 \bar{s}^{2}\mu-12s\lambda_{2}\mu+12{\lambda_{2}}^{2}\mu-4s\mu^{2}+16\lambda_{ 2}\mu^{2}\).  
According to Equation (15), the mean of AoI\({}_{0}\) under the _LCFS-PS_ queuing discipline is derived as AAoI\({}_{PS}=\{\mathbf{E}[MGF_{PS}(s)]|s_{=0}\}^{(1)}\).