# On Edge Pruning of Communication Networks under an Age-of-Information Framework
## 5 Average AoI Characterization  
In this section, we demonstrate the complete characterization of the total time taken by a network that follows the SP topology and how the AoI behaves in such topologies.  
After applying the reduction introduced in the previous section, our system becomes a system with a single source-destination link in which messages are delivered with the first-come-first-serve policy. Here, we consider the case in which messages are generated at the source node following to a Poisson process characterized with an average arrival rate of \(\mu\).  
Figure 3: Links deliver probabilistically under the exponential distribution with expected delivery times of \(1/\lambda_{1},\ldots,1/\lambda_{n}\) after the arrival of the message to the source node. (**a**) is parallel links reduction. (**b**) is series links reduction.  
Going back to Figure 2, considering the Poisson process specifically is due to the fact that the time intervals between the generation of the messages \(Y\) are independent exponentially distributed random variables. Those independent variables are characterized with an average of \(\frac{1}{\mu}\).  
In addition to that the delivery times of the messages our network supports at each time are independent exponentially distributed random variables as well characterized with an average of \(1/\lambda_{\text{total}}\) that is the result of applying both the parallel and series reductions.  
We now show the characterization of the average age of information in this system. We can directly infer from Figure 2 that the average AoI will be  
\[\Delta=\mathbb{E}[Q]/\,\mathbb{E}[Y] \tag{18}\]  
Since \(Q_{i}\) represents the area of a trapezoidal shape that is the result of the difference of two right-angle isosceles triangles with congruent sides \(T_{i}+Y_{i}\) and \(T_{i}\). We conclude that \(Q_{i}=0.5(T_{i}+Y_{i})^{2}-0.5T_{i}^{2}=Y_{i}T_{i}+0.5Y_{i}^{2}\).  
Hence,  
\[\Delta=\frac{\mathbb{E}[YT]+0.5\,\mathbb{E}[Y^{2}]}{\mathbb{E}[Y]} \tag{19}\]  
However, we may divide the system time \(T_{i}\) into a wait time \(W_{i}\) and delivery time \(D_{i}\), and since the delivery time \(D_{i}\) is independent from the interarrival time \(Y_{i}\), we get  
\[\mathbb{E}[T_{i}Y_{i}]=\mathbb{E}[(W_{i}+D_{i})Y_{i}]=\mathbb{E}[W_{i}Y_{i}] +\mathbb{E}[D_{i}]\,\mathbb{E}[Y_{i}] \tag{20}\]  
where \(\mathbb{E}[D_{i}]=1/\lambda_{\text{total}}\) in the case of a reduced network into a source-destination system with an exponentially distributed link, and \(\mathbb{E}[Y_{i}]=\frac{1}{\mu}\). However, the waiting time \(W_{i}=T_{i-1}-Y_{i}\). This means that the expected waiting time given a specific interarrival time \(Y_{i}=y\) will equal  
\[\mathbb{E}[(T_{i}-y|Y_{i}=y]=\mathbb{E}[T-y]=\int_{y}^{\infty}(t-y)f_{T}(t)dt \tag{21}\]  
In the special case when the probability density function (PDF) \(f_{T}(t)\) of such a system is given by [35]:  
\[f_{T}(t)=(\lambda_{\text{total}}-\mu)e^{(\mu-\lambda_{\text{total}})t} \tag{22}\]  
Therefore, this yields in having  
\[\mathbb{E}[W_{i}Y_{i}]=\int_{0}^{\infty}y\,\mathbb{E}[W_{i}|Y_{i}=y]f_{Y_{i}} (y)dy=\frac{\mu}{\lambda_{\text{total}}^{3}-\lambda_{\text{total}}^{2}\mu} \tag{23}\]  
Now we have everything set up to directly get the average AoI under in case that our system reduces into the simplest case, which is a source-destination system with a single exponentially distributed link:  
\[\Delta=\frac{1}{\lambda_{\text{total}}}+\frac{1}{\mu}+\frac{\mu}{\lambda_{ \text{total}}^{3}-\lambda_{\text{total}}^{2}\mu} \tag{24}\]