# Minimization of Average Peak Age of Information for Timely Status Updates in Two-Hop IoT Networks
###### Abstract  
Timely status updates are essential for Internet of Things (IoT) services. The freshness of these updates can be quantified using Age of Information (AoI). The worst-case behavior of AoI is evaluated by peak AoI (PAoI), denoting the maximum AoI just before each successful update. To characterize the time-averaged evolution of the PAoI over a long time horizon, we adopt the average PAoI as a performance metric. In this paper, we consider a two-hop status update system in IoT monitoring networks, where sensors periodically transmit short status packets to a remote edge server via a sink node. The sink node encodes status packets received from multiple sensors into a single longer packet to enhance the transmission reliability of short-packet communications. Here, we analyze the average PAoI in this setup as a function of system parameters and minimize this function by jointly optimizing three key parameters: (i) the number of status packets for joint coding at the sink node, (ii) the blocklength of a status packet in the first hop, and (iii) the blocklength of a coded packet in the second hop. Through numerical studies, we demonstrate the effectiveness of the proposed optimization in reducing the average PAoI. a 2025 15 7042 14 May 2025 14 May 2025  
highlighting the unique aspects of our work. Practical use cases of such systems include factory automation and smart environmental monitoring in IoT networks. In IoT monitoring networks, status packets are typically short (e.g., tens of bytes) to meet stringent low-latency requirements [9]. However, short-packet communications are prone to decoding errors due to operation in the finite blocklength regime. This contrasts with the infinite blocklength regime, where error-free channel capacity can be achieved based on [PERSON]'s formula, but at the cost of large transmission delays. In [10], [PERSON] et al. established a theoretical framework for error probability in the finite blocklength regime, providing a tight approximation formula for the PER as a function of blocklength. In this paper, we use this approximation formula to capture the PERs in two-hop transmissions, i.e., from a sensor to the sink node (first hop) and from the sink node to the edge server (second hop). To enhance the reliability of short-packet communications, multiple short packets can be concatenated into a longer packet [11; 12]. In this paper, we adopt joint packet coding at the sink node to concatenate status packets, following the approach in [5]. This approach involves dividing the sensors into groups. The sink node then collects status packets from each group, jointly encodes them into a longer packet, and transmits the coded packet to the edge server. For this system, we first note that the number of sensors per group significantly affects the AoI performance. We refer to this number as the _group size_ throughout the paper. In addition to group size, blocklength is a key parameter that determines AoI performance in short-packet communications. In recent years, there have been growing efforts to analyze and minimize AoI in the finite blocklength regime through blocklength optimization for various systems, e.g., [13; 14; 15; 16; 17; 18]. These studies derive optimal (or suboptimal) blocklengths to minimize the average AoI and, in some cases, the average PAol. A detailed discussion and comparison with our work are provided in Section 2. Motivated by these observations and prior studies, we optimize the following three key parameters to minimize the average PAol in a two-hop status update system with joint packet coding:  
1. The group size for joint packet coding;
2. The blocklength of a status packet in the first hop;
3. The blocklength of a coded packet in the second hop. Figure 1: A two-hop status update system consisting of \(G\) groups of \(K\) sensors, a sink node, and an edge server. This paper is inspired by [5], where the status update system shown in Figure 1 was studied under short-packet communication conditions.