# 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-PW  
In this subsection, the SHS Markov chain in Figure 5 is built up, which models the discrete state under _LCFS-PW_. Then, the MGF for AoI\({}_{0}\) is derived. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(I\) & \(\mathbf{q}_{I}\rightarrow\mathbf{q}_{I}^{\prime}\) & \(\mathbf{\lambda}^{(I)}\) & \(\mathbf{\mathrm{x}}\mathbf{\mathrm{A}}_{I}\) & \(\mathbf{\mathrm{A}}_{I}\) \\ \hline
1 & \(0\to 1\) & \(\lambda_{1}\) & \([\,x_{0}\ \ 0\ \ 0\ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
2 & \(0\to 2\) & \(\lambda_{2}\) & \([\,x_{0}\ \ 0\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
3 & \(1\to 3\) & \(\lambda_{2}\) & \([\,x_{0}\ x_{1}\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{2}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
4 & \(2\to 3\) & \(\lambda_{1}\) & \([\,x_{0}\ \ 0\ x_{2}\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{3}\,]\) \\ \hline
5 & \(3\to 0\) & \(\mu\) & \([\,x_{1}\ \ 0\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{2}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
6 & \(3\to 0\) & \(\mu\) & \([\,x_{2}\ \ 0\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{3}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
7 & \(1\to 1\) & \(\lambda_{1}\) & \([\,x_{0}\ \ 0\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
8 & \(2\to 2\) & \(\lambda_{2}\) & \([\,x_{0}\ \ 0\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
9 & \(3\to 3\) & \(\lambda_{2}\) & \([\,x_{0}\ x_{1}\ \ 0\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{2}\ \mathbf{\mathrm{m}}_{0}\,]\) \\ \hline
10 & \(3\to 3\) & \(\lambda_{1}\) & \([\,x_{0}\ \ 0\ x_{2}\ \ ]\) & \([\,\mathbf{\mathrm{m}}_{1}\ \mathbf{\mathrm{m}}_{0}\ \mathbf{\mathrm{m}}_{3}\,]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transitions in Figure 4.  
Figure 5: The discrete state and transitions under _LCFS-PW_ modeled by SHS Markov chain. The new arriving status information is only allowed to preempt the packet in waiting. When both \(a_{1}\geq 1\) and \(a_{2}\geq 1\) are true, the GU is busy. When either \(a_{1}\) or \(a_{2}\) is \(0\), the GU is idle. As depicted in Figure 5, when GU \(k\) manages the arrived packets under _LCFS-PW_, the state \(q\in\mathcal{Q}=\{0,1,\cdots,6\}\). The \(a_{1}\) and \(a_{2}\) in the state \(q=(a_{1},\,a_{2})\) denote the number of status information stored in \(queue_{1}\) and \(queue_{2}\) of GU, respectively. When both \(a_{1}\geq 1\) and \(a_{2}\geq 1\) are true, the GU is busy, indicating that the GU is performing the fusion of \(m_{1}\) and \(m_{2}\) to generate \(m_{0}\). Moreover, when GU is busy, the status information arranged and stored in the second position in \(queue_{1}\) and \(queue_{2}\) is in waiting. When either \(a_{1}\) or \(a_{2}\) is 0, the GU is idle, and the status information in \(queue_{1}\) or \(queue_{2}\) is in waiting. Under the LCFS-PW, the newly arrived \(m_{i}\) is only able to preempt the old ones that have been stored in \(queue_{i}\) and in waiting. Once a \(m_{i}\) has been preempted, it is discarded by the GU. On the other hand, the continuous component \(\mathbf{x}(t)\) in the system captures the continuous time processes, which are represented by the vector \(\mathbf{x}(t)=[x_{0}(t),x_{1}(t),x_{2}(t),x_{3}(t),x_{4}(t)]\). Moreover, \(x_{0}(t)\) captures the evolution of AoI\({}_{0}(t)\) in Equation (9); \(x_{i}(t)\) (\(i\)=1,2) captures the evolution of AoI of \(m_{\tilde{t}}\) stored in the head position of \(queue_{\tilde{t}}\); and \(x_{\tilde{t}+2}(t)\) captures the evolution of AoI of \(m_{\tilde{t}}\) stored in the second position of \(queue_{\tilde{t}}\). Similar to the description in Section 3.2, the elements recording the age in \(\mathbf{x}(t)\) grow linearly with time as long as the discrete state remains unchanged.