# Minimization of Average Peak Age of Information for Timely Status Updates in Two-Hop IoT Networks
## 3 System Model  
In this section, we provide a detailed description of the system model depicted in Figure 1. We consider a status update system comprising \(N\) homogeneous sensors, a sink node, and an edge server. The sensors are divided into \(G=\frac{N}{K}\) groups, where \(K\) represents the group size, i.e., the number of sensors per group. The \(k\)-th sensor in group \(g\) is indexed as \((g,k)\), where \((g,k)\in\{1,2,\ldots,G\}\times\{1,2,\ldots,K\}\). We consider an uplink scenario in which status packets are transmitted from each sensor to the edge server via the sink node in two hops. We assume that the sink node operates in half-duplex mode; thus, no first-hop transmissions occur during the second-hop transmission, and vice versa. The uplink transmissions follow a round-robin schedule on a group-by-group basis, as described below. Suppose that group \(g\) is scheduled for the first-hop transmissions. Then, each sensor in group \(g\), from sensor \((g,1)\) to sensor \((g,K)\), transmits its status packet to the sink node in order. Specifically, once sensor \((g,k)\) completes its transmission to the sink node, the next sensor \((g,k+1)\) immediately begins transmitting its status packet. (The analysis in this paper remains applicable to other TDMA-based policies for scheduling sensors in a group, provided that sensor \((g,k)\) is defined as the \(k\)-th scheduled sensor in group \(g\).) At the same time, the sink node continuously collects status packets from all sensors in group \(g\). The transmission time of each status packet to the sink node is given by \(T_{1}=\frac{L_{1}}{B}\), where \(L_{1}\) denotes the blocklength of each status packet and \(B\) represents the system bandwidth. Hence, the total duration of the first-hop transmissions in group \(g\) is given by \(K\cdot T_{1}\). Once the first-hop transmissions for group \(g\) are completed, i.e., the sink node has collected \(K\) status packets from all sensors in group \(g\), it jointly encodes these \(K\) status packets into a single coded packet and transmits it to the edge server in the second-hop transmission. The transmission time in the second hop is given by \(T_{2}=\frac{L_{2}}{B}\), where \(L_{2}\) denotes the blocklength of the coded packet. Once the second-hop transmission for group \(g\) is completed, i.e., the edge server has received the coded packet of group \(g\) from the sink node, the next group \(g+1\) (or group \(1\) if \(g=G\)) is scheduled for the first-hop transmissions to the sink node. These first-hop transmissions are followed by the second-hop transmission of the corresponding coded packet from the sink node to the edge server, and this process repeats for all groups. We now analyze the PER in short-packet communications based on the results of [10]. We assume that packets are transmitted over additive white Gaussian noise (AWGN) channels. Then, the PER in the \(i\)-th hop for \(i=1,2\) can be tightly approximated as follows:  
\[\epsilon_{i}\approx Q\left(\frac{\frac{1}{2}\log_{2}(1+\rho_{i})-\frac{S_{i}}{ L_{i}}}{\log_{2}(e)\sqrt{\frac{1}{2L_{i}}(1-(1+\rho_{i})^{-2})}}\right). \tag{1}\]Here, \(Q(x):=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{x^{2}}{2}}\,du\) is the Q-function, \(\rho_{i}\in\{\rho_{1},\rho_{2}\}\) is the received signal-to-noise ratio (SNR) at the sink node (\(i=1\)) or the edge server (\(i=2\)), and \(S_{i}\in\{S,K\cdot S\}\) is the number of information bits per status packet (\(i=1\)) or coded packet (\(i=2\)). A status packet sent by a sensor is successfully received and decoded at the edge server if and only if no packet error occurs in either the first-hop or second-hop transmission. Hence, the probability of successful decoding at the edge server is given by  
\[p_{s}=(1-\epsilon_{1})(1-\epsilon_{2}). \tag{2}\]  
Next, we describe the status update procedure at the edge server. Upon receiving a coded packet of a group (e.g., group \(g\)), the edge server decodes it to recover the status information of all sensors in group \(g\).