## 3 Access Class Barring Configuration (ACBC) Scheme
### Recursive Least-Squares (RLS) Algorithm  
The RLS is an adaptive algorithm that minimizes the total squared error between its output and the desired response. In particular, it is a recursive implementation of the method of least squares. The RLS can be derived from the least-mean square (LMS) algorithm, described in the next subsection, and is renowned for its fast rate of convergence. For instance, the rate of convergence of the RLS can be up to an order of magnitude faster than that of the LMS. In exchange, its numerical complexity is in the order of \(\mathcal{O}\left(M^{2}\right)\), which is higher than that of the LMS. Nevertheless, the mathematical formulation and implementation of the RLS is relatively simple ([PERSON], 2016). The internal structure of the RLS algorithm is illustrated in Figure 3. A buffer has been incorporated to clearly illustrate that the ratio of idle resources during the last \(M\) SIB2 intervals  
\[\mathbf{u}(j)=[u(j),u(j-1),\ldots,u(j-M+1)] \tag{8}\]  
is the input to the algorithm. That is, a single value of \(u(j)\) is the input to the buffer as indicated by the thin arrow, and the output of the buffer is a vector, as indicated by the thick arrows. The RLS is summarized in Algorithm 2, and consists of a filtering and an adaptive process. In the filtering process, the output of the FIR filter \(y(j)\) is computed from \(\mathbf{u}(j)\). In the adaptive process, the output \(y(j)\) is compared to the desired response \(d(j)\) to adapt the weights of the FIR filter  
\[\mathbf{w}(j)=[w_{0}(j),w_{1}(j),\ldots,w_{M-1}(j)]\,. \tag{9}\]  
The parameter \(\varepsilon(j)\) is the a priori error, which represents an estimate of the desired response \(d(j)\) based on the old least-squares estimate of \(\mathbf{w}\left(j-1\right)\). It is essential to emphasize that \(\varepsilon(j)\) is different to the _a posteriori_ estimation error \(e(j)\), calculated in the LMS algorithm (see next subsection). Regarding the configuration of the RLS in our ACBC scheme, the desired response is set to be \(d(j)=1\) for all \(j\). By doing so, the RLS algorithm is set to minimize the distance between the output \(y(j)\) and its maximum value, \(d(j)=1\), which in turn \"pulls\" the output toward the aforementioned maximum value. This increases the barring rate \(p_{\mathrm{acb}}(j)\) and, in turn, reduces the access delay of UEs. This effect will be described in detail on page 16. The RLS algorithm has two parameters that must be selected empirically, the regularization parameter \(\delta\) and the forgetting factor \(\lambda\). The latter determines the rate at which the effect of the previous inputs on the algorithm is minimised. There is no exact method to select \(\delta\) nor \(\lambda\). However, the following recommendations exist [28]. * Regularization parameter \(\delta\): Select a small positive value when the input (in our case \(u(j)\)) is relatively high with respect to its sudden variations. Select a large positive value otherwise. * Forgetting factor \(\lambda\): Select a positive value that is close to, but less than, 1. This ensures that past the values of the input \(u(j)\) are forgotten by the algorithm. On the other hand, \(\lambda=1\) corresponds to the method of least squares. The beginning of Section 4 focuses on finding adequate values for \(\delta\) and \(\lambda\). Figure 3: Block diagram of the RLS adaptive filter algorithm.