## 2 Random Access Procedure (RAP) and Access Class Barring (ACB) Scheme  
This section describes the random access (RA) as defined in the 3 GPP standards. It comprises the access control and the contention-based RAP [1; 20]. Throughout this study, the ACB, as defined in the 3 GPP standards [1], is studied and we assume no other access control scheme is implemented. Figure 1 provides a brief description of the RA in 3 GPP networks with the ACB scheme. The first step for a UE to switch from idle to connected mode is to acquire the network configuration parameters. Once these have been obtained, the UE is subject to the ACB scheme and, finally, it performs the RAP. The network operates in a time-slotted channel in which the minimum unit for scheduling is the subframe. The configuration parameters are broadcast by the cellular base station (BS) through the System Information Blocks (SIBs). The SIB1 includes, among other parameters, the periodicity of other SIBs in the parameter _si-Periodicity_\(\in\{80,160,\ldots,5120\}\) ms [1]. The SIB2 includes some basic parameters such as the periodicity of the time/frequency resources in which preamble transmissions are allowed \(t_{\text{rao}}\) (known as random access opportunities, RAOs) and the barring parameters [1; 4; 5; 21]. The \(j\)th SIB2 transmission includes a barring rate \(p_{\text{acb}}(j)\) and a mean barring time \(t_{\text{acb}}(j)\)[1]. Only special categories of high-priority UEs may not be subject to the ACB scheme. The rest of the  
Figure 1: Random access (RA) in LTE-A: ACB scheme and RAP. UEs subject to the ACB scheme must perform a barring check before initiating the RAP (i.e., before the transmission of their first preamble) as described in Algorithm 1[1; 22]. ```
1:repeat
2: Acquire the mean barring time \(t_{\text{acb}}(j)\) and barring rate \(p_{\text{acb}}(j)\) broadcast in the \(j\)th SIB2. 3: Generate \(\mathcal{U}\left[0,1\right)\equiv\) a random number with uniform distribution between 0 and 1. 4:if\(\mathcal{U}\left[0,1\right)\leq p_{\text{acb}}(j)\)then
5: Initiate the RAP. 6:else
7: Generate a new \(\mathcal{U}\left[0,1\right)\). 8: Select the barring time as \[t_{\text{barred}}=\left(0.7+0.6\,\mathcal{U}\left[0,1\right)\,\right)t_{ \text{acb}}(j).\] (1)
9: Wait for \(t_{\text{barred}}\). 10:endif
11:until the RAP is initiated. ```  
**Algorithm 1** Access class barring (ACB) scheme. UEs that succeed in a barring check are no longer subject to the ACB scheme [23] and proceed to perform the RAP as follows. **Preamble:** Each UE randomly selects one out of the \(r\) available preambles and sends it towards the BS in a RAO through the random access channel (RACH). Multiple UEs can access the BS in the same RAO using different preambles due to their orthogonality. As a result, BS decodes the preambles transmitted without (wireless channel) errors by exactly one UE. In this study, we assume the BS cannot decode preambles transmitted by multiple UEs simultaneously; we refer to these as collided preambles. This assumption goes in line with the 3 GPP recommendations for the performance analysis of the RACH [2] and with most of the literature [3; 14; 15; 24; 25; 26; 27]. The interested reader may refer to [12] for details on the two main assumptions related to collisions during the RAP and their impact on performance. Next, the UE waits for a predefined time window, known as the RA response (RAR) window, to receive a RAR message from the BS. **RAR:** The BS computes an identifier for each successfully decoded preamble and sends the RAR message through the physical downlink control channel (PDCCH). It includes, among other data, uplink grants for the transmission of the next message, Msg3. Up to one RAR message can be sent per subframe, but it may contain several uplink grants; each of which is associated with a successfully decoded preamble. The PDCCH resources are limited, so a maximum of \(n_{\text{ug}}\) uplink grants can be sent within a RAR window.