# Exploiting the Capture Effect to Enhance RACH Performance in Cellular-Based M2M Communications
## 5 Simulation Results
### Performance of the Msg3 PR Scheme
#### 5.1.2 The RA Throughput  
Figure 6 illustrates the RA throughput when the \(PRS\) is changed with \(V=20\) and \(N_{max}=8\). In addition, we compare the performance of the conventional case, which does not consider the capture effect of Msg3 (i.e., \(E[S_{u},M_{v}=0|N]\)) [4; 5; 9; 14]. The RA throughput is clearly improved using the Msg3 PR scheme; particularly when heavy congestion occurs (i.e., \(N=3V=60\)), the RA throughput for \(PRS=1\) dB, 2 dB, 3 dB, and 4 dB is improved by approximately \(48\%,80\%,137\%\), and \(105\%\), respectively, as compared to the conventional case. This is due to the fact that the capture effect may occur among the devices that suffer from collisions that reduce the RA throughput, and the Msg3 PR can increase the capture probability. Note that the increase in \(PRS\) does not necessarily lead to an increase in RA throughput, as previously mentioned. In particular, the RA throughput in the case of 4 dB is lower than that in the case of 3 \(dB\) because the RA success probability of 4 dB-case is lower, as seen in Figure 5. Therefore, it is important to choose a proper \(PRS\) value. For example, choosing \(PRS=3\) dB would be preferred in terms of the RA throughput.  
Figure 7 illustrates the effect of changes in \(N_{max}\) on the RA throughput when \(PRS=2\) dB and \(V=20\). Comparing the RA throughputs in the cases when \(N_{max}=~{}4\) and \(N_{max}=~{}8\), a throughput difference is observed when the number of accesses is large because, if the congestion level is not severe (\(N\leq 25\)), retransmissions are not likely to occur more than four times. Hence, there is no throughput difference between these two cases. However, if the congestion level becomes severe, the trial number of most devices will reach to \(N_{max}\) due to severe collisions. Therefore, in this case, the probability of capture increases \(N_{max}\) is larger. However, this observation does not apply to the \(N_{max}=~{}12\) case; the performance difference between the cases with \(N_{max}=~{}8\) and \(12\) is negligible. This is due to the following reasons: (1) most devices (except the devices near the eNB) could not increase their ramping number enough due to the ramping number limitation, \(K_{max}\), and (2) the devices near the eNB are often captured before the ramping number is significantly increased. As a result, if \(N_{max}\) becomes sufficiently large, the overall throughput is no longer affected.  
In Figure 8, we also show the RA throughput varying with the size of the preamble set, \(V\), where \(PRS=2\) dB and \(N_{max}=8\). As analyzed in other studies [5; 6; 8; 12] (or, the conventional case in shown Figure 6), the RA throughput is maximized when the number of accesses is the same as the preamble size, i.e., \(n=V\). However, if the Msg3 PR is used, the number of throughput-maximizing arrivals increases significantly. This result further demonstrates that, because of the capture effect on the RA procedure, optimization is needed if diverse RACH congestion-control schemes, such as ACB or M2M-specific back-off, are used.  
Figure 6: Steady-state RA throughput in terms of power ramping steq (\(PRS\)).