# Minimization of Average Peak Age of Information for Timely Status Updates in Two-Hop IoT Networks
## 6 Numerical Studies
### Comparison  
Last, we evaluate the effectiveness of the proposed JBG optimization in minimizing the average PAoI. To this end, we adopt the method in [5] as a reference benchmark and compare its performance with that of our proposed scheme. In [5], the group size \(K\) is optimized to minimize the average AoI and the bounded AoI, while the blocklengths \(L_{1}\) and \(L_{2}\) are determined from (1), under the constraint that the PERs \(\epsilon_{1}\) and \(\epsilon_{2}\) are fixed (e.g., \(\epsilon_{1}=\epsilon_{2}=0.5\times 10^{-3}\)). We apply the approach from [5] to minimize the average PAoI and present the results in Figure 7 for \(N\in\{10,20,30,40,50\}\). Our proposed scheme outperforms the benchmark in [5] in terms of average PAoI, achieving reduction rates of \(\{8.70,7.80,6.90\}\%\) on average for \(\rho\in\{1,3,5\}\) dB. This demonstrates that optimizing the blocklengths in addition to the group size further reduces the average PAoI, with the effect being more pronounced in lower SNR environments, though at the cost of additional computation required to solve (19) and (20).  
We note that PAoI is closely related to packet delay, which is also a critical performance metric in IoT networks. Under the generate-at-will model, the sojourn time of a packet corresponds to its delay, as there is no queueing delay. Consequently, the packet delay of sensor \((g,k)\) is given \(T_{(g,k)}\), as specified in (4). It follows from (9) that this delay is bounded above by the average PAoI of sensor \((g,k)\), implying that a lower average PAoI generally leads to reduced packet delays. Hence, in industrial-IoT monitoring scenarios--where an end-to-end latency budget on the order of 100 ms is common--even a modest decrease in the average PAoI can help reduce the share of packets that miss the latency deadline, delivering a tangible benefit. Importantly, the only additional cost introduced by the proposed JBG optimization is a single offline fixed-point computation; at run time, the decoder still performs one FEC pass, and the sink node maintains the same single-packet buffer as in [5]. Therefore, the proposed scheme increases the latency margin without changing the control cycle or protocol, and the performance gain remains evident while imposing virtually no extra memory overhead.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Number of** & **Average SNR** & **Optimal Group** & **Optimal Blocklengths** \\
**Sensors**\(N\) & \(\rho\) **(dB)** & **Size \(K^{*}\)** & \(\big{(}L_{1}^{*},L_{2}^{*}\big{)}\) \\ \hline \multirow{4}{*}{10} & 1 & 1 & \((214,214)\) \\  & 3 & 1 & \((156,156)\) \\  & 5 & 1 & \((118,118)\) \\ \hline \multirow{4}{*}{50} & 1 & 2 & \((214,407)\) \\  & 3 & 2 & \((157,298)\) \\  & 5 & 2 & \((119,227)\) \\ \hline \multirow{4}{*}{100} & 1 & 5 & \((214,963)\) \\  & 3 & 4 & \((156,573)\) \\  & 5 & 4 & \((119,437)\) \\ \hline \multirow{4}{*}{150} & 1 & 6 & \((214,1146)\) \\  & 3 & 6 & \((156,843)\) \\  & 5 & 5 & \((119,540)\) \\ \hline \multirow{4}{*}{200} & 1 & 8 & \((214,1508)\) \\  & 3 & 8 & \((156,1111)\) \\ \cline{1-1}  & 5 & 8 & \((118,848)\) \\ \hline \end{tabular}
\end{table}
Table 3: Impact of the number of sensors \(N\) on optimal group size \(K^{*}\) in (13) and optimal blocklengths \((L_{1}^{*},L_{2}^{*})\) in (14) that minimize the average PAoI for each average SNR \(\rho\in\{1,3,5\}\) dB.