Access control is an effective way to protect the radio access part of Long-Term Evolution-Advanced (LTE-A) network from the overload caused by a huge number of Machine-Type Communication Devices (MTCDs). A class of access control mechanisms is the Access Class Barring (ACB), which regulates the Machine-to-Machine (M2M) traffic in accordance with the available Random Access (RA) resources. In this paper, we extend the single power level ACB scheme to a multiple power level method to increase the number of successfully transmitted requests in the case of overload. Our analysis is based on the capture effect in the third step of RA procedure of the LTE-A system in which one of the transmitted requests by 2 or more co-tagged MTCDs, MTCDs which use the same preamble in the first step, can be decoded by the evolved-Node B (eNB). We first formulate the power level selection as an optimization problem assuming the perfect capture model without considering MTCDs' energy budget. Then, to take into account MTCDs' energy consumption, the scenario is extended for the signal-to-interference ratio-based capture model. In addition, we investigate the advantages of the proposed multiple power level RA method on discriminating the access of MTCDs with different priorities. The numerical results show that using the optimal parameters, the RA throughput can be improved in comparison with the single power level system at the cost of slightly increasing MTCDs' energy consumption and the complexity of RA procedure. [ABSTRACT FROM AUTHOR]