## 3 Access Class Barring Configuration (ACBC) Scheme  
This section describes the structure of our ACBC scheme and the configuration of the selected filtering methods. It is important to emphasize our ACBC scheme, first introduced in [8], was designed to operate under the 3 GPP specifications, hence, it can be directly implemented in 4G and 5G networks. Figure 2 illustrates the block diagram that describes the operation of the RAP with our ACBC scheme. Two main blocks are clearly identified in the figure: the RA and the ACBC scheme. The RA, depicted in the upper part of Figure 2, operates as described in Section 2. Let \(a(i)\) be the number of UEs that attempt to switch from idle to connected mode for the first time at the \(i\)th RAO; these UEs are subject to the ACB scheme. Hereafter we refer to \(a(i)\) simply as the UE arrivals. Next, let \(f(i)\) be the number of UEs whose barring check at the \(i\)th RAO is successful. These UEs begin the RAP at the \(i\)th RAO; hence, \(f(i)\) is also the number of UEs whose first preamble is transmitted at the \(i\)th RAO. Finally, \(s\left(i\right)\) is the number of UEs that will receive an uplink grant at the \(i\)th RAR window. That is, as a response to a preamble transmitted at the \(i\)th RAO. Therefore, the discrete time index \(i\) stands for the epoch number. The epoch duration is one RAO since the RA is performed at each and every RAO. The lower part of Figure 2 depicts the structure of our ACBC scheme, where the BS calculates the barring parameters: mean barring time \(t_{\mathrm{acb}}(j)\) and barring rate \(p_{\mathrm{acb}}(j)\) that will be broadcast through the \(j\)th SIB2. The SIB2 is broadcast once every \(t_{\mathrm{si}}\) RAOs, hence, these parameters are adapted according to the perceived access intensity throughout this period. As such, the discrete time index \(j\) stands for the epoch number when the epoch duration is \(t_{\mathrm{si}}\) RAOs. Therefore, the ACBC scheme operates at a time scale that is \(t_{\mathrm{si}}\) times greater than that of the RA block. Specifically, the \(j\)th SIB2 is broadcast at the \(i=(jt_{\mathrm{si}})\)th RAO. Therefore, the barring parameters remain constant from the \((jt_{\mathrm{si}}+1)\)th until the \(((j+1)\,t_{\mathrm{si}})\)th RAO. Throughout this study we assume \(t_{\mathrm{acb}}(j)\) is selected beforehand and remains fixed throughout the operation of the network. Hence, the latter is simply denoted as \(t_{\mathrm{acb}}\). In the following, we describe the process to calculate \(p_{\mathrm{acb}}(j)\). The BS calculates the ratio of idle to available resources immediately before the \(j\)th SIB2 transmission. For this, let \(n(i)\) be the number of contending UEs (i.e., the total number of preamble transmissions) at the \(i\)th RAO. Please observe the number of successful preambles at an arbitrary RAO  
Figure 2: Block diagram of the RAP with our ACBC scheme. The random access is performed at each RAO, whereas the ACBC can only be performed once every \(t_{\mathrm{si}}\) RAOs. \(i\) is a random variable. From there, the theoretical PRACH (uplink) capacity can be defined as the maximum expected value of the latter random variable for a given number of available preambles \(r\) and for any \(n(i)\in\mathbb{R}_{\geq 0}\). The PRACH capacity is achieved when the number of contending UEs is  
\[n^{*}(i)=\left[\log\left(\frac{r}{r-1}\right)\right]^{-1} \tag{2}\]  
and is calculated by [PERSON] et al. (2015) as  
\[c\left(r\right)=n^{*}(i)\left(1-\frac{1}{r}\right)^{n^{*}(i)-1}. \tag{3}\]  
Next, let \(c\left(r,n_{\text{ug}}\right)\) be the theoretical capacity of the channels involved in the RAP: PRACH and PDCCH. The capacity of the PDCCH is exclusively determined by the number of available uplink grants per RAR window \(n_{\text{ug}}\)([PERSON], 2008). Hence, the capacity of the RAP is the minimum between the PRACH capacity and the PDCCH capacity,  
\[c\left(r,n_{\text{ug}}\right)=\min\left\{c\left(r\right),n_{\text{ug}}\right\}. \tag{4}\]  
We refer to \(c\left(r,n_{\text{ug}}\right)\) simply as the RAP capacity.