## 5 Performance Evaluation  
In this section, a theoretical performance evaluation of the proposed novel frameless slotted-ALOHA protocol is presented based on the performance analysis in Section 4. Moreover, the computer simulation results using the Monte-Carlo technique are acquired in MATLAB, which verified the theoretical results. Performance and comparisons are obtained on the throughput along with the average number of backlogged users, average memory size, and the average successful transmission probability for the different values of \(M\). In addition, we compare the throughput of \(K=2\) in Reference [11] with that of \(K=3\) in this paper to verify the effectiveness of parameter setting by simulation. To prove the availability of the analysis results, we set the number of all active users in the system to 10, 20, 30 and 40, respectively. We regard the throughput performance as a function of the first transmission probability \(p_{f}\) based on its definition in Section 4.1. Then, the retransmission probability \(p_{r}\) is optimized for any given \(p_{f}\). It is found that the throughput achieves the best performance when \(p_{r}\) is 0.09. After this, a stochastic simulation is made according to the binomial distribution probability model and state transition matrix. The ranges of \(p_{f}\) and \(p_{r}\) are set from 0.01 to 1 and from 0.001 to 1, respectively. The Monte-Carlo simulation procedure can be seen in Figure 7.  
The basic steps of the Monte-Carlo simulation are as follows:  
Step 1: Generate a set of random numbers between zero and one using the random number method. The array length is the number of active users;  
Step 2: According to the binomial distribution model, the transition probability matrix is solved, and then a steady-state distribution is obtained;  
Step 3: Due to the occasionality of random simulation, we solved the average value after 100 repetitions. Figures 8 and 9 illustrate the trend of the system throughput with respect to the first transmission probability. According to Equations (12) and (15) in Section 4.1, we verified the consistency for the two definitions of throughput through numerical calculation, which is shown in Figure 8. To further verify the availability of the numerical results, Figure 9 demonstrates the comparison between the numerical results and the simulation results. We can see that under heavy load, i.e., \(p_{f}>0.1\), the throughput gradually decreases as \(M\) increases. Figure 8: Throughput vs. first transmission probability (Equation (12) vs. Equation (15)). Figure 7: Flow chart of the Monte-Carlo simulation. Figure 10 demonstrates the number of backlogged users as a function of the first transmission probability based on Equation (13). It can be seen that the minimum number of backlogged users is obtained at \(p_{f}=0.01\). This means that a higher probability of successful transmission can be achieved with a smaller \(M\). Thus, the solution to minimizing \(N_{a}\) is the opposite of maximizing \(T\) when \(M=10\). In effect, a trade-off needs to be made in these two solutions. The expected successful transmission probability of a transmitted packet is shown in Figure 11. The simulation results reveal good agreement with theoretical results. Reference [11] only derives an expression for the successful transmission probability of packets and does not represent the numerical simulation results. However, we not only present the theoretical analytical expression, i.e., Equation (20), but also show the simulation results for validation. We can see that a higher probability of successful transmission can be achieved with a smaller \(M\). Hence, the solution for maximizing \(P\) agrees with maximizing \(T\). Figure 10: Number of backlogged users vs. first transmission probability (Ana. vs. Sim.). Figure 9: Throughput vs. first transmission probability (Ana. vs. Sim.). Figure 12 shows the average buffer size of the proposed protocol on the basis of Equation (19).