# AoI Analysis of Satellite-UAV Synergy Real-Time Remote Sensing System
## 3 Derivation for MGF of AoI\({}_{0}\)  
### Preliminaries on SHS  
Under the three queuing disciplines described in Section 2.4, the queues depicted in Figure 2 are lossy where packets might be preempted or discarded. It is challenging to derive the moments and MGF of AoI\({}_{0}(t)\) in a lossy system by using the graphical approach [8]. Therefore, we solve this problem by utilizing the SHS framework for AoI, which was initially customized by [26; 44] for researching AoI. For the sake of completeness, the main idea of the SHS framework for AoI is briefly presented. In the SHS framework for AoI, the state of a system is partitioned into a discrete component \(q(t)\in\mathcal{Q}=\{0,1,\ldots,w\}\) that evolves as a continuous-time Markov chain and a continuous component \(\mathbf{x}\ (t)=[x_{0}(t),\cdots,x_{n}(t)]\) that describes the evolution processes related to AoI. In the state transition diagram \((\mathcal{Q},\mathcal{L})\) of Markov chain \(q(t)\), each \(q\in\mathcal{Q}\) and each transition \(l\in\mathcal{L}\) are, respectively, represented as a vertex and a directed edge \(\big{(}q_{l},q^{\prime}_{l}\big{)}\). Furthermore, the transition rate is \(\lambda^{(l)}\delta_{q_{l},q(t)}\). It is ensured by the Kronecker delta function \(\delta_{q_{l},q(t)}\) that the transition \(l\) may only occur when \(q(t)=q_{l}\). For the state \(q\in\mathcal{Q}\), the sets \(\mathcal{L}_{q}=\{l\in\mathcal{L}:q_{l}=q\}\) and \(\mathcal{L}^{\prime}_{q}=\big{\{}l\in\mathcal{L}:q^{\prime}_{l}=q\big{\}}\), respectively, denote the collections of outgoing and incoming transitions. As a result of the occurrence of transition \(l\), the system jumps from state \(q_{l}\) to state \(q^{\prime}_{l}\) and resets the continuous component from \(\mathbf{x}\) to \(\mathbf{x}^{\prime}=\mathbf{x}\mathbf{A}_{l}\). \(\mathbf{A}_{l}\) is a \((n+1)\)-order binary reset map matrix that depends on the characteristics of the specific system. The evolution of continuous component \(\mathbf{x}(t)\) is based on the differential equation \(\dot{\mathbf{x}}(t)=\mathbf{1}\) when \(q(t)=q\). Unlike ordinarily, the SHS may contain self-transitions and multiple transitions. Specifically, self-transitions represent the cases where a reset occurs in the \(\mathbf{x}(t)\) but the \(q(t)\) is unchanged. For the state pair \(i,j\in\mathcal{Q}\), multiple transitions represent that the discrete state \(q(t)\) may jump from \(i\) to \(j\) through transitions with different reset map matrices. Then, some aspects for deriving MGF using the SHS approach are introduced. \(\pi_{q}(t)\) denotes the probability of being in state \(q\in\mathcal{Q}\) at time \(t\). Furthermore, \(\mathbf{v}_{q}(t)\) denotes the correlation vector of \(q(t)\) and \(\mathbf{x}(t)\), and \(\mathbf{v}_{q}^{\mathrm{g}}(t)\) denotes the correlation vector of \(q(t)\) and \(e^{\mathbf{x}(t)}\), where \(\mathrm{g}\in\mathcal{R}\). \(\hat{\mathbf{A}}_{l}\) is defined as a diagonal companion matrix where if \(i=j\) and \([\mathbf{A}_{l}]_{j}=\mathbf{0}^{\mathrm{T}}\), then \([\hat{\mathbf{A}}_{l}]_{i,j}=1\); otherwise, \([\hat{\mathbf{A}}_{l}]_{i,j}=0\).