# Study of Coded ALOHA with Multi-User Detection under Heavy-Tailed and Correlated Arrivals
## 1 Introduction  
Since the introduction of the ALOHA protocol [1, 2], many random access schemes have been proposed. Uncoordinated multiple access is necessary in communications systems where coordinated resource allocation is not possible or is too costly. Classical uncoordinated techniques, such as the ALOHA systems and carrier-sense-based techniques [3, 4], such as carrier-sense multiple access with collision avoidance (CSMA-CA), require a retransmission policy to provide reliable communications, which introduces the need for a separate feedback channel and incurs in a possibly large delay. Recently, a considerable interest for finding new solutions to provide reliable, low-latency communication in dynamical systems has emerged, motivated by the requirements of 5G communications, e.g., in visible light communications [5], in CDMA systems [6], or in LoRa-based communications [7]. An account of modern random access protocols is given in [8], and the role of multiple-access in non-orthogonal communications with many users is discussed in [9, 10]. One promising uncoordinated multiple access technique is coded-slotted ALOHA (CSA), also known as irregular repetition ALOHA (IRSA), which has been extensively studied in the literature in the asymptotic regime (where the packet replicas are spread over a large interval) and in the finite-length regime [11, 12, 13]. In CSA, transmissions are organized into fixed-length frames, each consisting of \(n\) slots, and users arrange the transmission of several copies of a packet in randomly chosen slots within the frame. This approach is known as frame-synchronous CSA (FS-CSA) [14]. Alternatively, the transmission of the packet copies can be performed immediately after the packet arrival, randomly within thenext \(n\) slots, avoiding the need for synchronization among users at the frame level. This approach is known as frameless or frame-asynchronous CSA [15; 16] (FA-CSA), where the time span for the transmission of the packet copies forms implicitly the local frame for each individual user. Extensions of CSA, and in particular of FA-CSA, to asynchronous, non-slotted random access have also been proposed and analyzed [17; 18]. In either version, CSA creates time diversity for the transmissions and makes it possible to sustain high throughput and low energy-consumption compared to slotted ALOHA, making this approach of combining random transmission with encoding at the packet level a good candidate for next generation random access with many simultaneous active users. CSA has recently been adopted in commercial satellite communication systems, such as in DVB-RCS2 [19], and is currently under study as the basis for next-generation multiple-access technology in internet-of-things (IoT) applications [20], and for massive machine-type communications (mMTC) in 5G networks under the so-called unsourced model where all users use the same codebook and the receiver aims to return an unordered list of messages. In these settings, CSA attains much better throughput and energy efficiency than random coding and pure ALOHA [21]. The majority of works on CSA assume a collision channel where decoding is possible only if a slot contains a single packet. As shown in [22; 23; 24; 25], for instance, there exists a connection among different types of systems and codes in graphs. Indeed, a CSA system can be described by a bipartite graph and successive interference cancellation (SIC) can be performed over the graph similarly to decoding of graph-based codes [26; 27; 28]. Therefore, the performance of CSA resembles that of graph-based codes, i.e., the packet loss rate curve displays generally a waterfall region for medium-to-high system loads and it shows and error floor for low loads. In the aforementioned protocols, all the replicas are transmitted using the same transmit power. Thus, the received power diversity caused by natural fading is not significant enough and the capture effect is not fully used to enhance the throughput of these protocols.