## 3 System Modelling and Analysis
### Markov Model Analysis for the System  
This part makes a theoretical analysis of the novel frameless slotted-ALOHA system. Let \(N(0\leq N\leq M)\) represent the number of backlogged users at the beginning of each slot.  
Figure 4: Standard time feedback process for system.  
Figure 5: Active user states.  
Assume \(p_{f}\
eq p_{r}\), and let \(P_{r}(i,N)\) be the probability that \(i\) out of \(N\) backlogged users send packets in a slot. Then  
\[P_{r}(i,N)=C_{N}^{i}(1-p_{r})^{N-i}p_{r}^{i}. \tag{4}\]  
Let \(P_{f}(i,N)\) represent the probability that \(i\) out of \(M-N\) free users send packets in a slot, which can be expressed as  
\[P_{f}(i,N)=C_{M-N}^{i}(1-p_{f})^{M-N-i}p_{f}^{i}. \tag{5}\]  
As a result, according to the above four feedback cases: Idle (I), True Success (TS), Conflict (C) and Pseudo Success (PS), the corresponding probability of the four cases, based on Equations (4) and (5), can be expressed as follows:  
\[P_{I}(N)=P_{f}(0,N)P_{r}(0,N), \tag{6}\]  
\[P_{TS}(N)=P_{f}(1,N)P_{r}(0,N)+P_{f}(0,N)P_{r}(1,N), \tag{7}\]  
\[\begin{split} P_{PS}(N)=P_{f}(3,N)P_{r}(0,N)+P_{f}(0,N)P_{r}(3,N)+ P_{f}(1,N)P_{r}(2,N)\\ +P_{f}(2,N)P_{r}(1,N)+P_{f}(2,N)P_{r}(0,N)+P_{f}(0,N)P_{r}(2,N)+ P_{f}(1,N)P_{r}(1,N),\end{split} \tag{8}\]  
\[P_{C}(N)=1-P_{I}-P_{TS}-P_{PS}. \tag{9}\]  
According to the probabilities of the four feedback cases, i.e., Equations (6)-(9), the transition probability \(p_{N,N+n}\) from \(N\) state to \(N+n\) state is  
\[p_{N,N+n}=\begin{cases}0,&n\geq 4\\ P_{f}(3,N)[1-P_{r}(0,N)],&n=3\\ P_{f}(2,N)[1-P_{r}(0,N)-P_{r}(1,N)]+P_{f}(3,N)P_{r}(0,N),&n=2\\ P_{f}(1,N)[1-P_{r}(0,N)-P_{r}(1,N)-P_{r}(2,N)]+P_{f}(2,N)[P_{r}(0,N)+P_{r}(1,N )],&n=1\\ P_{f}(0,N)[1-P_{r}(1,N)-P_{r}(2,N)-P_{r}(3,N)]+P_{f}(1,N)[P_{r}(0,N)+P_{r}(1,N )+P_{r}(2,N)],&n=0\\ P_{f}(0,N)[P_{r}(1,N)+P_{r}(2,N)+P_{r}(3,N)].&n=-1\end{cases} \tag{10}\]  
Taking the number of backlogged users in each slot, \(N\), as the system state, the above state transition process constitutes a \(M+1\) states discrete-time Markov chain, as shown in Figure 6. It can be seen that the maximum value of the state increase transition is three, because this work is based on \(K=3\); that is, the number of backlogged users changes by no more than three after a slot. The state reduction transition can only be one, because the number of backlogged users can only be reduced by one with a successful retransmission.  
When the system reaches a steady state, the frequency from other states to a certain state is equal to the frequency from this state to other states; that is, traffic equilibrium. For any state \(N(0\leq N\leq M)\) in Figure 6, its steady-state probability is set as \(\pi_{N}\), which is the probability that there are \(N\) backlogged users in a slot. Moreover, the \(\pi_{N}\) satisfies \(\pi_{0}+\pi_{1}+\pi_{2}+\cdots+\pi_{M}=1\), then \((\pi_{0},\pi_{1},\cdots,\pi_{M})\) is a stable distribution of the system.  
Figure 6: State transition diagram of a novel frameless slotted-ALOHA system.  
the steady-state probability \(\pi_{N}(0\leq N\leq M)\) can be calculated using elimination by substitution. Thus, the stationary distribution \((\pi_{0},\pi_{1},\cdots,\pi_{M})\) of the system can be obtained, which is a preparation for the simulation of the various indicators in Section 5.