## 3 Access Class Barring Configuration (ACBC) Scheme
### Methodology
In addition, we select an intuitive value for the mean barring rate \(t_{\text{acb}}=1\) s among those available in the SIB2; as it will be seen throughout the following section, this value leads to a sufficiently high \(P_{s}\geq 0.95\) for all the selected filtering methods under the traffic model 2. Certainly, an optimal value for this parameter could be selected, but it is out of the scope of this paper to optimize the performance of each of the studied filtering methods under the selected scenario. Furthermore, selecting a value higher than \(t_{\text{acb}}=1\) only causes a slight increase in the access delay when compared to the results presented in this paper. The interested reader may refer to [19], where our ACBC scheme with the LMS algorithm was optimized, and apply an analogous procedure for the RLS algorithm. The configuration parameters for the filtering methods studied in this paper are listed in Table 3.  
In particular, we are set to find configurations of the RLS and fixed FIR filters that successfully reduce the variations of \(u(j)\) with the fastest possible convergence toward \(\mathbb{E}\left[u(j)\right]\). Under the traffic model 1, \(n=\) 30,000 UE accesses are uniformly distributed within 60 s. Since RAOs occur once every \(t_{\text{rao}}=5\) subframes and the subframe duration is 1 ms, there are \(i_{\text{dist}}=\) 12,000 RAOs in the distribution period. Let \(A\) be the random variable that defines the number of UE arrivals at an arbitrary RAO within the distribution period. Hence, \(a(i)\), shown in Figure 2, is the outcome of a single experiment for random variable \(A\) at the \(i\)th RAO. Please observe \(A\sim\text{B}\left(n,1/i_{\text{dist}}\right)\), hence, we have \(\mathbb{E}\left[A\right]=n/i_{\text{dist}}=2.5\). Since the intensity of access attempts under the traffic model 1 is considerably low when compared to the RAP capacity, the number of successful accesses per RAO \(s(i)\) highly resembles that of UE arrivals (i.e., \(s(i)\approx a(i)\), for all \(i\)). As a consequence, \(\mathbb{E}\left[u(j)\right]\) can be approximated by substituting \(\mathbf{s}(i)\) with \(\mathbf{a}\), a column vector of length \(t_{\text{si}}\) whose entries are \(\mathbb{E}\left[A\right]\), in (5). \[\mathbb{E}\left[u(j)\right]\approx 1-\mathbf{h}^{\mathsf{T}}\mathbf{a}=1-\frac{ \mathbb{E}\left[A\right]}{c\left(r,n_{\text{ug}}\right)} \tag{19}\]  
which gives \(\mathbb{E}\left[u(j)\right]\approx 5/6\) for \(r=54\). This value has been confirmed by simulation and by an analytical model [12]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Setting** \\ \hline Available preambles & \(r=54\) \\ Subframe length & 1 ms \\ RAO periodicity & \(t_{\text{rao}}=5\) subframes \\ RAR window size & 5 subframes \\ Available uplink grants per RAR window & \(n_{\text{ug}}=15\) \\ Period of SIB transmissions & 80 ms \\ SIB2 periodicity in RAOs & \(t_{\text{si}}=16\) \\ Maximum number of preamble transmissions & 10 \\ Backoff indicator & 20 ms \\ Detection probability for the \(k\)th preamble transmission & \(P_{d}(k)=1-1/e^{k}\) \\ Maximum number of \(M\)sg3 and \(M\)sg4 transmissions & 5 \\ Detection probability for \(M\)sg3 and \(M\)sg4 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for the selected network configuration. It has been observed in the literature that parameter \(\lambda\) plays a similar role in the RLS algorithm as the complement of parameter \(\mu\) in the LMS algorithm [28]. In our previous studies we observed \(\mu=1/800\) is an adequate value for the adaptation step size in our ACBC scheme. Hence, results presented in Figure 5 confirm the relation between \(\lambda\) and \(\mu\), as an adequate response is obtained with \(\lambda=0.999\approx 1-1/800=1-\mu\). The latter value of \(\lambda\) is selected throughout the remainder of the paper, in combination of \(\delta=0.001\). As it can be seen, \(p_{\text{acb}}(j)\) with this combination converges toward \(\mathbb{E}\left[u(j)\right]\) around \(j=M\).