## 4 Analysis for Network Throughput and Average Age of Information
### Average Age of Information  
The AoI is an important metric in any wireless communications network as it defines the freshness of data intended for a typical receiving node from the receiver's perspective. Note that timeliness is necessary to obtain a reliable system since a packet that arrives late may fail to achieve its intending purpose. At any given time stamp \(t\) (that is, observation time), the AoI \(\triangle(t)\) is the difference between the current observation point \(t\) and the time at which the packet was generated \(G(t)\)[32], i.e.,  
\[\triangle\left(t\right)=t-G(t). \tag{26}\]  
The evolution of the AoI can be formalized as  
\[\triangle\left(t+1\right)=\begin{cases}\triangle(t)+1,&\text{transmission failure}\\ t+1-G(t),&\text{otherwise}.\end{cases} \tag{27}\]  
This metric captures the sum of packet waiting time in the queue, as well as the time spent in service and can be represented as shown in Figure 2. Given that \(X_{k}\) is the number of slots between the successful delivery of any two consecutive \(k\)-th and \((k+1)\)-th packets, while \(Y_{k}\) depicts the sum of AoI, \(X_{k}\) is given as \[X_{k}=\sum_{i=1}^{M}I_{i}, \tag{28}\]  
where \(M\) is a random variable that depicts the number of attempted transmissions between two successfully received packets. Similarly,  
\[Y_{k}=\sum_{k=t_{k}}^{t_{k+1}}\triangle(k). \tag{29}\]  
From [15; 33], we know that the analysis for the average AoI can be obtained for any period of \(W\) slots where there is an \(L\) number of successfully delivered (i.e., decoded) packets. This average AoI is given as  
\[\triangle\left(W\right)=\frac{1}{W}\sum_{k=1}^{W}\triangle(k)=\frac{1}{W}\sum _{k=1}^{L}Y_{k}=\frac{L}{W}\frac{1}{L}\sum_{k=1}^{L}Y_{k}. \tag{30}\]  
Taking \(\lim_{W\rightarrow\infty}\frac{L}{W}=\frac{1}{E[X]}\) with \(\lim_{L\rightarrow\infty}\frac{1}{L}\sum_{k=1}^{L}Y_{k}=E[Y]\),  
\[\triangle=\lim_{W\rightarrow\infty}\triangle(W)=\frac{E[Y]}{E[X]}.\]  
A careful observation of Figure 2 shows that, \(Y_{k}\) can be obtained as  
\[Y_{k}=\sum_{m=1}^{X_{k}}m=\frac{X_{k}^{2}}{2}+\frac{X_{k}}{2}.\]  
From (30),  
\[\triangle=\frac{E[X^{2}]}{2E[X]}+\frac{1}{2}. \tag{31}\]  
It is worth noting that the distribution of \(X_{k}\) in (31) as given in (28) depends on the arrival rate \(v_{T}\) (i.e., probability of transmission) and the probability of successful transmission \(\mu_{y}^{\dagger}\). Hence the distribution of \(X_{k}\) is a random process. It becomes immediately clear that the average AoI observed at any tagged receiver depends on the spatial distributions of users, thus the need to obtain the analysis for the conditional average AoI as given in the following Lemma.  
Figure 2: Evolution of AoI. \(I_{i}\) denotes the inter-arrival time slots, \(X_{k}\) is the number of slots between successful delivery of two consecutive packets and \(Y_{k}\) is the sum of AoI between \(t_{k}\) and \(t_{k+1}\).  
**Lemma 3**.: _The conditional average AoI obtained at a tagged receiver \(y_{k}^{\text{Tr}}\) given that \(\Phi=\Phi_{p}^{I}\cup\Phi_{s}^{I}\) can be obtained as_  
\[\bigtriangleup(y_{k}^{\text{Tr}}|\Phi=\Phi_{p}^{I}\cup\Phi_{s}^{I})=\frac{1}{v_ {\text{Tr}}u_{y_{k}^{\text{Tr}}}^{i_{\text{Tr}}}}. \tag{32}\]  
Proof.: The proof is straightforward from [15; 33] and is omitted for brevity. From (32), the average AoI observed by any test PR and SR for each mode can be obtained based on the analysis in Section 3.2.