# AoI Analysis of Satellite-UAV Synergy Real-Time Remote Sensing System
## 3 Derivation for MGF of AoI\({}_{0}\)
### Calculate The MGF of Ao\({}_{0}\) under LCFS-NP  
In this subsection, the SHS Markov chain in Figure 3 is built up, which models the discrete state under _LCFS-NP_. Based on the definitions and notations in Section 3.1, the MGF for the Ao\({}_{0}\) under _LCFS-NP_ is derived by using (13) and (14).  
As depicted in Figure 3, when GU \(k\) manages the arrived packets under _LCFS-NP_, the discrete component \(q\in\mathcal{Q}=\{0,1,2,3\}\). Each state in \(\mathcal{Q}\) represents a potential combination of the numbers of the status information in the \(queue_{1}\) and \(queue_{2}\) of GU. For example, a state \(q=(a_{1},\,a_{2})\) indicates that the \(queue_{1}\) at GU has \(a_{1}\) status information \(m_{1}\), and the number of \(m_{2}\) in \(queue_{2}\) is \(a_{2}\). Note that under the LCFS-NP queuing discipline, if there is already an \(m_{i}\) (\(i=1,2\)) stored in \(queue_{i}\), the newly arrived \(m_{i}\) cannot preempt the old one, and the GU discards the newly arrived \(m_{i}\). When \(q=(1,\,1)\), the GU is busy, which indicates that the GU is performing the fusion of \(m_{1}\) and \(m_{2}\) to generate the fused information \(m_{0}\). When either \(a_{1}\) or \(a_{2}\) is 0, the GU is idle. On the other hand, the continuous component \(\mathbf{x}(t)\) in the system is given by \(\mathbf{x}(t)=[x_{0}(t),x_{1}(t),x_{2}(t)]\). \(x_{0}(t)\) captures the continuous-time evolution of Ao\({}_{0}(t)\) in (9); \(x_{i}(t)\) (\(i\)=1,2) captures the continuous-time AoI evolution of \(m_{i}\) stored in the head position of \(queue_{i}\) and corresponds to \(\Delta^{1}_{m_{i}}(u_{m_{0}}(t))\) in Equation (9). As long as the discrete state \(q\) remains unchanged (i.e., no status information arrives and preempts the old one, or the fused information has been generated), we have \(\frac{\partial\mathbf{x}(t)}{\partial t}=1\). This means that the elements recording the age in the vector \(\mathbf{x}(t)\) grow linearly with time.  
The transitions are the directed edges \(l\in\{1,\cdots,6\}\). For transitions \(l\), Table 1 presents the transition pair \(q_{l}\to q^{\prime}_{l}\), the transition rate \(\lambda^{(l)}\), the reset map matrix \(\mathbf{A}_{l}\), and  
Figure 3: The discrete states and transitions under _LCFS-NP_ modeled by SHS Markov chain. The new arriving status information is not allowed to preempt the packet in service or 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.  
the transition reset mapping \(\mathbf{x}^{\prime}=\mathbf{x}\mathbf{A}_{l}\). The directed edge \(l\) indicates that transitions from state \(q_{l}\) to \(q^{\prime}_{l}\) occur at rate \(\lambda^{(l)}\). Additionally, in this paper, \(\mathbf{m}_{0}=[\begin{array}{cc}0&0&0\end{array}]^{\mathsf{T}}\), \(\mathbf{m}_{1}=[\begin{array}{cc}1&0&0\end{array}]^{\mathsf{T}}\), \(\mathbf{m}_{2}=[\begin{array}{cc}0&1&0\end{array}]^{\mathsf{T}}\), \(\mathbf{m}_{3}=[\begin{array}{cc}0&0&1\end{array}]^{\mathsf{T}}\). The description of \(l\) in Table 1 is as follows.
* \(l=1\) or 2: In state 0, transition \(l=1\) occurs when a new \(m_{1}\) arrives at GU \(k\). \(\mathbf{A}_{l}\) governs that \(x^{\prime}_{0}=x_{0}\) because no new \(m_{0}\) is generated at the end of state 0, \(x^{\prime}_{1}=0\) because the new \(m_{1}\) is fresh, and \(x^{\prime}_{2}=0\) because \(x_{2}\) is irrelevant in state 1. Similarly, the transition \(l=2\) occurs when a new \(m_{2}\) arrives. \(\mathbf{A}_{l}\) governs that \(x^{\prime}_{0}=x_{0}\), \(x^{\prime}_{2}=0\), and \(x^{\prime}_{1}=0\);
* \(  
According to (15), the mean of AoI\({}_{0}\) under _LCFS-NP_ can be derived as  
\[\text{AAoI}_{NP}=\{\mathbf{E}[MGF_{NP}(\bar{s})]|_{S=0}\}^{(1)}. \tag{19}\]