## 3 System Modelling and Analysis
### The Receiver for the Slotted-ALOHA System  
It is assumed that the channel coefficients of each slot are fixed and will change due to fading. Then, the superimposed signal obtained by the receiver in a given slot [11], can be expressed as  
\[y_{j}=\sum_{m=1}^{M}c_{m,j}h_{m,j}u_{m,i}+n_{j}, \tag{3}\]  
where \(n_{j}\) represents the additive white Gaussian noise (AWGN), \(n_{j}\sim(0,\sigma_{n}^{2})\). \(h_{m,j}\) is the channel coefficient between user \(m\) and the receiver in slot \(j\). \(u_{m,j}\) is the \(i-th\) packet of user \(m\). If packet \(u_{m,i}\) is transmitted in slot \(j\), then \(C_{m,j}=1\), otherwise \(C_{m,j}=0\). To ensure that conflicting packets can be successfully recovered, we assume that the system has reasonable channel coding, and that the receiver can perform ideal channel detection. According to the slot state of the frameless slotted-ALOHA system in Figure 2, at any given slot, all states of the system can be divided into four cases: Idle (I), True Success (TS), Conflict (C) and Pseudo Success (PS). To facilitate understanding, Figure 3 shows the feedback of the receiver based on the slot state given in Figure 2. In different cases, the receiver will give different responses. Figure 2: Slot state of the frameless slotted-ALOHA system. Case 1 (Idle, I): There is no active user in the slot, i.e., no packets need to be sent, so the channel is idle. The receiver does not need to send feedback. Case 2 (True success, TS): There is only one active user, i.e., a packet needs to be sent in this slot. The receiver sends an acknowledgement (ACK) to this user. Case 3 (Conflict, C): There are at least four active users, i.e., at least four packets are transmitted by a shared wireless channel in this slot. The receiver sends a negative acknowledgement (NACK) to all users because a destructive collision occurred. Case 4 (Pseudo success, PS): There are two or three active users, i.e., two or three packets need to be sent by one channel. Pseudo success can be divided into two situations, (I) and (II):  
(I) If there are two active users in a slot. The receiver sends feedback information to the senders in the following manners:  
1. If both users are in the same state, the receiver randomly selects a user to send an ACK and sends a NACK to the other user. 2. If two users are in different states, the receiver sends an ACK to the backlogged users and a NACK to the free users. 3. If there are three active users in a slot. The receiver sends responses to the senders in the following manners:  
1. If all three users are in the same state, the receiver randomly selects one of them to send an ACK and sends a NACK to the remaining two users. 2. If two of them are backlogged and the remaining one is free, the receiver randomly selects one of the backlogged users to send an ACK to and sends a NACK to the remaining two users. 3. If two of them are free and the remaining one is backlogged, the receiver will send an ACK to the backlogged user and send a NACK to the free users. Figure 4 shows the working principle of the above feedback mechanism based on each case in Figure 3 that represent its role distinctly. The time axis is divided into slots according to the fundamental principle shown in Figure 1. Recall that in the example in Equation (1), SIC technology is applied to decode all of the packets contained in the conflict after the receiver receives a replica of the conflicting packets. For example, in Figure 4, the second user sends a replica immediately after it receives the signaling NACK2. Then, other conflicting packets can be decoded by SIC. The transmission process of any packet is carried out in one slot. Figure 3: Feedback response of the receiver in four cases. All users in the system are divided into a backlogged state or a free state. The state of a user at the beginning of a slot is determined by the feedback it receives. If a sender receives an ACK or a NACK, the user is free or backlogged.