# Peak Age of Information Optimization in Cell-Free Massive Random Access Networks
## 2 System Model  
Consider a cell-free massive random access network serving massive Machine-Type Communication (mMTC) in IoT scenarios. Let \(\mathcal{M}=\{1,2,\ldots M\}\) denote the set of APs, each equipped with a single omnidirectional antenna. All APs collaboratively serve \(n\) Machine-Type Devices (MTDs), as illustrated in Figure 1. Unlike traditional cellular networks, MTDs in CF-RAN are not associated with a single AP but are dynamically served by neighboring AP clusters. Based on this architecture, MTDs are logically grouped according to the AP clusters that collaboratively serve them. Let \(\mathcal{G}_{A}\) denote the group of MTDs that can communicate with all APs in set \(A\), and the number of MTDs contained in this group is \(|\mathcal{G}_{A}|=n^{4}\). For example, the green dots in Figure 2 represent group \(\mathcal{G}_{\{1,2\}}\), which includes all MTDs served collaboratively by AP 1 and AP 2, and \(n^{\{1,2\}}=4\). Similarly, in Figure 2, there also exist groups \(\mathcal{G}_{\{1\}}\), \(\mathcal{G}_{\{2\}}\), \(\mathcal{G}_{\{2,3\}}\), and \(\mathcal{G}_{\{3\}}\). Let \(G\) denote the total number of logical groups.  
The CF-RAN employs the 3 GPP grant-free random access mechanism, specifically the Small Data Transmission (SDT) protocol defined in the 5G NR standard [5], to serve all MTDs. The duration of a complete SDT signaling and data exchange process is defined as one time slot. All MTDs transmit synchronously with fixed packet lengths, occupying one time slot per transmission. A packet is assumed to traverse an ideal noiseless channel and is successfully received only if no other packets are transmitted to the same AP within the same time slot. APs are connected to a CPU via error-free fronthaul links. The CPU performs deduplication to retain the most reliable packet and discard duplicates. Thus,  
Figure 1: Network scenario.  
Figure 2: Illustration of logical group division (taking a scenario with 3 APs as an example).  
CF-RAN employs joint packet decoding: a packet is successfully decoded if received by at least one AP. After decoding, APs broadcast ACK/NACK signals.  
Each MTD is equipped with a buffer queue of unit packet size. A new packet is generated at the beginning of each time slot with probability \(\lambda\in(0.1]\), where packet generation time is negligible. For an MTD belonging to logical group \(\mathcal{G}_{A}\), when its buffer is non-empty, it accesses the channel and transmits with probability \(q^{A}\). This group-specific access probability \(q^{A}\) corresponds to the \"access class barring\" factor in 3 GPP standards and serves as the optimization variable for minimizing PAoI. A First-Come-First-Served (FCFS) queue discipline is adopted. If the buffer is full, newly arrived packets are discarded.  
In CF-RAN dense deployments, AP coverage areas inevitably overlap, meaning data transmissions within a group are influenced not only by intra-group dynamics but also by inter-group interference. Based on this model, we analyze the successful transmission probability and the impact of inter-group interference on PAoI for any group \(\mathcal{G}_{A}\), as well as design optimization algorithms to minimize the network-wide average PAoI.