# Peak Age of Information Optimization in Cell-Free Massive Random Access Networks
## 1 Introduction  
With the rapid advancement of communication technologies, the Cell-Free Radio Access Network (CF-RAN) has emerged as a cutting-edge network architecture and a key technology for future 6G communications [1, 2]. CF-RAN comprises numerous geographically distributed single/multi-antenna Access Points (APs), which are connected to a Central Processing Unit (CPU) via high-speed, low-latency fronthaul links (e.g., optical fibers) to enable joint operations such as encoding, decoding, resource allocation, and interference coordination [3]. The collaborative operation of distributed APs effectively reduces interference and extends network coverage, making CF-RAN particularly suitable for future Internet of Things (IoT) communication scenarios requiring wide-area coverage, such as ultra-high-voltage grid monitoring and vehicular networks. Massive IoT applications predominantly adopt sporadic uplink transmission modes with small packets [4]. To address the signaling storm caused by grant-based access intraditional cellular networks, 3 GPP introduced the grant-free random access protocol in the 5G NR standard [5]. This protocol allows users to directly transmit data during random access without establishing air interface bearer links with the core network. However, this distributed decision-making complicates the coordination of user access behaviors. In massive access scenarios, random access channels are prone to congestion, leading to significant increases in data transmission delays. In emerging applications such as industrial IoT, information freshness is a critical metric [6]. Delays in information sampling or transmission cause the value of information to deteriorate rapidly. Packets containing \"stale\" information not only fail to create value but also exacerbate congestion, severely degrading application service quality and user experience [7]. Information freshness is quantified using the Age of Information (AoI), which reflects data timeliness [8]. The Peak AoI (PAoI) further captures the worst-case timeliness performance under dynamic loads [9]. Optimizing grant-free random access mechanisms in cellular networks to enhance information freshness has become a key research focus. Studies [10; 11] proposed dynamic transmission strategies based on AoI thresholds to mitigate channel contention by suppressing redundant transmissions. Studies [12; 13] further utilized deep reinforcement learning to optimize AoI by adjusting transmission probabilities and access control mechanisms. The aforementioned studies primarily focus on traditional single-cell cellular networks. Research on AoI optimization in CF-RAN scenarios is gradually gaining attention. For instance, [PERSON]. [14] developed a two-disk model to derive closed-form approximations of average AoI under grant-free random access, exploring and optimizing the impact of different variables. [PERSON]. [15] proposed a reinforcement learning framework to jointly optimize AoI and transmit power in uplink CF-RAN, designing state spaces, action spaces, and reward functions based on the soft actor-critic algorithm. [PERSON]. [16] introduced priority-aware frame structures and device grouping strategies for cell-free massive MIMO industrial IoT networks to satisfy differentiated AoI requirements. However, existing studies on CF-RAN AoI suffer from limitations, such as reliance on Poisson assumptions for user spatial distribution [14] or centralized control algorithms with high signaling overhead [15], which hinder their applicability in large-scale, distributed massive access scenarios. To address these challenges, this work extends the theoretical framework of [17] to grant-free CF-RAN massive random access networks. We construct a dynamic topology correlation model for multi-AP collaborative joint decoding scenarios. Based on this model, we derive expressions for average PAoI in single-AP and multi-AP networks.