## 5 Numerical Results
### Fixed \(L_{k}\) and \(P_{a}=1\)  
In this subsection, we take \(L_{k}=L\), \(\forall k\in\mathcal{K}\), and \(P_{a}=1\), such that \(K_{a}=K\). Although these simplifying assumptions usually do not hold in practice, they are useful in unveiling the full potential of the investigated methods and in evaluating the performance losses when not assuming them, which is carried out in the following subsections. The performance results presented in Figures 2 and 3 have been generated with 10,000 Monte-Carlo realizations. Each realization is a frame or a time step in the QL framework, in which each device can transmit only one pilot and the payload packet. The number of antennas is kept fixed at \(M=100\), and the number of devices varies from 25 to 800 in steps of 25. Figure 2 shows average latency _versus_\(K\) results. The proximity of the proposed mQmMTC results for both scenarios with and without interference with the ideal cQmMTC protocol is noteworthy. Also, both the mQmMTC and the ideal cQmMTC results are below the baseline for any value of \(K\). Also, they are below the independent QmMTC for \(K>400\), which corroborates the cQmMTC superiority presented in [30]. Similarly, Figure 3 reveals the average network throughput \(\approx K\) results where the mQmMTC is very close to the ideal cQmMTC performance for both the scenarios with and without interference while being consistently superior to the baseline results for any number of devices \(K\). Furthermore, compared with iQmMTC, the obtained performances are very similar in the region of \(K\leq\tau_{p}=400\) devices, while the performance obtained by the proposed mQmMTC approach becomes remarkably superior for a higher number of devices. For example, with \(K=600\) active devices, while baseline and iQmMTC achieve network throughputs of \(\approx\)0.34 and 0.36, respectively, our mQmMTC protocol achieves a network throughput of \(\approx\)0.49, an improvement of \(\approx\)44% and \(\approx\)36%, respectively. The performance results presented in Figures 4 and 5 are generated with 80,000 Monte-Carlo realizations. The number of devices is fixed at \(K=400\), and the number of available pilots is also fixed at \(\tau_{p}=400\). The number of BS antennas varies in the range \(M\in[1,\,100]\), both in the home cell as well as in the neighboring cells. Figure 4 depicts the average latency with an increasing number of BS antennas \(M\), while Figure 5 reveals the behavior of the average network throughput vs. \(M\). Both figures of merit for the proposed mQmMTC approach improve with the increasing number of antennas \(M\) since the reward computations in (17) benefit from the large number of BS antennas (favorable propagation effect). Besides, in both cases the presence of ICI slightly deteriorates the congestion level estimates in (16) and the reward computations, leading to a small performance degradation. Despite that, the scenarios with and without ICI perform similarly, achieving improved connectivity performances by benefiting from the many BS antennas. The results of the ideal cQmMTC are also included in the figures as a lower bound (avg. latency) and upper bound (avg. throughput), respectively. The average percentual degradations of the results of mQmMTC regarding the ideal cQmMTC are also shown in both figures for \(M=30\) and \(M=100\). One can see that the improvement caused by increasing \(M\) from \(M=10\) to \(M=100\) is not as significant as when increasing \(M\) from \(M=1\) to \(M=10\). Therefore, we can conclude that our proposed mQmMTC RA protocol can achieve improved connectivity performance even with a small number of BS antennas. Indeed, \(M\approx 30\) antennas at the BS are revealed to be sufficient to attain reliable congestion level estimation \(\widehat{\phi}_{k}\) when a maximum acceptable degradation level of 3.5% is considered. Figures 6 and 7 present, respectively, the curves of the latency and network throughput _versus_\(L\). The results shown in both figures are generated with 10,000 Monte-Carlo realizations.