# Study of Coded ALOHA with Multi-User Detection under Heavy-Tailed and Correlated Arrivals
## 2 System Model  
### Coded-Slotted ALOHA  
In our system, packets arrive randomly to a group of nodes which share a slotted channel. We denote by \(\lambda_{a}\) the mean arrival rate (packets per slot), and do not assume that arrivals in different slots are statistically independent. Without loss of generality, we consider that all the packets have a transmission time equal to the duration of the time slot, and that the nodes attempt transmission independently at the beginning of a slot without sensing the channel previously. Contrary to ordinary ALOHA, for every packet to be transmitted, the sender generates \(\ell\) copies or replicas according to a predefined probability mass function that will be specified below in Section 2.2, and each of these replicas is transmitted in a random slot. Thus, the replicas constitute a form of repetition encoding as seen from each transmitting node. We introduce at this point some useful terminology: a node which picks a value \(\ell\) as its repetition schedule for a packet is a degree-\(\ell\) node, and a time slot in which \(r\) packets are transmitted simultaneously will be termed a degree-\(r\) slot. A key assumption in this paper is the \(k\)-MUD or \(k\)-MPR (multi-user detection, or multi-packet reception, equivalently) capability achievable with the use of SIC (successive interference cancellation): we suppose that, all the packets in a degree-\(i\) slot, for any \(1\leq i\leq k\) can be successfully decoded at the intended receiver. This implies also that the interference due to a correctly decoded packet can be completely removed from any other slot containing a replica of this packet, thus reducing the effective interference that would remain at that slot and enabling other decoding decisions. Note that, regardless of its successful or unsuccessful decoding, every packet is transmitted \(\ell\) times, where \(\ell\) is a random number. In this paper, we focus on two basic performance metrics for this system  
* The packet loss ratio (PLR). It is the probability that a packet is not decoded correctly. Observe that this event is the non-existence of resolvable patterns of decoding in any of the slots containing some replica of the packet. * The delay of a decoded packet. This is defined as the number of slots between the user's arrival and the slot where the packet is resolved and correctly decoded. As for the operation of the system, we distinguish and analyze two variants of coded-slotted ALOHA, depending on the use of a coordinated frame for grouping the transmissions. 1. _Frame synchronous CSA_ (FS-CSA). In this scheme, time is divided in frames, each consisting of \(n\) slots. Any new packet from a transmitter must wait until the beginning of the next frame for attempting the transmission of its \(\ell\) replicas, which are randomly and uniformly distributed over the frame time. A node with packets being scheduled for transmission in a frame is called an active node, thus the number \(M\) of active nodes in the frame is a random variable. For the receiver, in FS-CSA decoding is performed slot-by-slot by applying the following strategy: first, in a given slot all the interference caused by packets that have been successfully decoded in the previous slots of the frame is removed. If, after this step, the result is a degree-\(k\) (or lower) slot, then all the packets in it are decoded and the receiver proceeds to look for other slots in the frame from which the interference due to those newly decoded packets can be further removed. The decoding process finishes when the receiver cannot find any slots in the frame to which \(k\)-MUD can be applied to. Thus, notice that, at the end of a frame, all the packets from the active users will have been either decoded successfully or dropped at the receiver. 2. _Frame asynchronous CSA_ (FA-CSA). In this scheme, a new packet is transmitted immediately after its arrival, in the next slot, and its remaining replicas are transmitted at random times within the subsequent \(n-1\) slots, uniformly drawn.