## 3 Access Class Barring Configuration (ACBC) Scheme
### Methodology  
In this study, we aim to compare the tracking capabilities of the RLS adaptive algorithm to those of the LMS and simple fixed FIR filters in a transient environment: the traffic model 2 defined by the 3 GPP [2]. The latter represents a highly synchronized mMTC scenario, in which the UE arrivals follow a Beta \((3,4)\) distribution over 10 s. Several studies have confirmed the traffic model 2 leads to severe congestion if no access control scheme is implemented [2; 3; 4; 5; 12; 13]. Results presented in the following section were obtained by simulation. In particular, a C-based simulator is used, which closely replicates the arrival process of the UEs, the ACB scheme, and the RAP as described in the specifications [1; 22]. In each simulation, the adaptive algorithm is initialized and the filter weights are stabilized (if applicable). Then, \(n\) = 30,000 UE arrivals are scheduled within the distribution period \(t_{\text{dist}}\) which begins at \(i=0\). The \(j\)th SIB2 is broadcast at the \((jt_{\text{si}}+i_{r})\)th RAO, where \(i_{r}\) is a discrete random time shift, selected from a uniform distribution between 0 and \(t_{\text{si}}-1\). A simulation run is concluded when every UE has terminated the RA procedure. The number of simulation runs is set to the smallest number that ensures the cumulative results obtained up to the last simulation differ from those obtained up to the previous simulation by less than 0.01 percent. The first step in our analyses is to use a non-synchronized mMTC scenario, the traffic model 1, to identify an adequate configuration of the RLS algorithm. The traffic model 1 can be seen as a stationary scenario of finite duration with a low access intensity, in which UE arrivals follow a uniform distribution over 60 s. As such, the traffic model 1 (stationary scenario) is used to \"train\" the RLS algorithm to provide an adequate response in a transient scenario. For this, we set the UEs to ignore the ACB scheme and assess two main characteristics: the rate of convergence of the output of the filter and the relative amplitude of its variations with respect to the mean value. It is important to emphasize that similar trial and error approaches to set the configuration of adaptive algorithms are the most common in the practice, especially in transient scenarios [29]. The main characteristics of both traffic models are listed in Table 1.  
In our experiments, we assume the most typical configuration of the RACH, as suggested by the 3 GPP [2]. The number of available preambles is set to \(r=54\); this is the most typical value for this parameter. Furthermore, it allows us to evaluate the RAP in a scenario in which the number of successful accesses is greatly affected by \(r\), but also by the number of available uplink grants \(n_{\text{ug}}\). In addition, we assume the shortest possible period for the SIB transmissions, 80 ms. These values are listed in Table 2, along with other values selected for the most relevant configuration parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Traffic Model 1** & **Traffic Model 2** \\ \hline Number of UEs \(n\) & 30,000 & 30,000 \\ Distribution period \(t_{\text{dist}}\) (s) & 60 & \(10\) \\ Distribution over \(t_{\text{dist}}\) & Uniform & Beta\((3,4)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the different traffic models defined by the 3 GPP for the performance evaluation of the RAP [2]. To assess the benefits of the RLS algorithm, we test several values for parameter \(\delta\) and \(\lambda\), whereas other configuration parameters remain fixed. For example, \(M=32\) is a widely used filter length that led to satisfactory results in our preliminary tests with the RLS algorithm. Furthermore, it is the filter length that maximizes the benefits of the LMS algorithm [19]. Therefore, it is selected throughout the remainder of the paper for every filter.