## 3 Access Class Barring Configuration (ACBC) Scheme
### Methodology
The horizontal line is located at \(\mathbb{E}\left[u(j)\right]\). As it can be seen, regardless of the selected filtering method, the 95 th percentile of access delay and the expected number of preamble transmissions are reduced by more than 26% and 12%, respectively, when compared to the optimal static ACB implementation. It can also be seen in Figure 7 that the different values of \(\rho\) for the LPF method have a minimal impact on performance. Specifically, the difference in each of the two KPIs shown in the figure for the different valued of \(\rho\) is less than 4%. On the other hand, there is a significant difference between the KPIs obtained with each of the adaptive algorithm configurations. In particular, the ALE leads to the longest access delay but to the lowest number of transmissions, whereas the opposite occurs with the PALE. The RLS leads to an access delay that is in between both LMS configurations, but to the highest number of preamble transmissions. This difference is caused by the \"pulling\" effect of the PALE and RLS algorithms, which is absent from the ALE. Figure 8 clearly illustrates the difference in the \(p_{\text{acb}}(j)\) calculated with the ALE, PALE, RLS, and LPF methods. The number of UE arrivals and successful accesses at each RAO with the RLS algorithm are also illustrated as a reference in Figure 8a. From this figure, it can be seen the number of UE accesses and successful accesses per RAO are highly variable. Furthermore, the number of successful accesses per RAO closely follows that of UE arrivals in the interval \(0\leq i\leq 343\). Then, for \(i\geq 343\), the number of successful accesses approaches \(n_{\text{ug}}\). This behavior is highly desirable, as it maximizes the utilization of resources and, in turn, reduces the access delay of UEs. Next, Figure 8b,c exhibit that the LMS algorithm with the ALE configuration operates as a low-pass filter with output \(p_{\text{acb}}(j)\). That is, the calculated \(p_{\text{acb}}(j)\) with the ALE configuration highly resembles that with the LPF method with both \(\rho\in\{0,2\}\). On the other hand, the PALE and the RLS \"pull\" the latter toward \(d(j)=1\). As a consequence, the \(p_{\text{acb}}(j)\) calculated with the PALE and RLS are higher than those with the ALE and LPF methods. However, it is important to observe that the \"pulling\" effect is greater for the PALE than for the RLS. For instance, \(p_{\text{acb}}(j)\) increases more rapidly for the PALE than for the RLS as the number of UE arrivals drops (i.e., after the 2000 th RAO, which corresponds to the 150 th SIB2 transmission). The reason for this effect is the rapid response of the RLS, which converges more rapidly toward \(\mathbb{E}\left[u(j)\right]\), calculated with the \(M\) most recent values of \(u(j)\). As a result, the benefits provided by the RLS are in between those provided by the ALE and the PALE. Figure 7: 95 th percentile of access delay and expected number of preamble transmissions obtained with our ACBC scheme and with each of the studied filtering methods; the KPIs obtained with the optimal configuration of an ACB scheme with fixed parameters (static ACB) are also included. The success probability for these is higher than 0.95. On the other hand, the calculated \(p_{\mathrm{acb}}(j)\) with both \(\rho\in\{0,2\}\) is closely similar in the period \(j\in[50,175]\), and the difference is observed when rapid changes in the number of successful accesses occur. Naturally, the rate of change of \(p_{\mathrm{acb}}(j)\) with respect of \(s(i)\) is more rapid for \(\rho=2\) than for \(\rho=0\). However, this difference is not evident in the 95 th percentile of access delay, illustrated in Figure 7.