When a large number of machine-to-machine (M2M) terminals attempt to access the Long-Term Evolution (LTE)/LTE Advanced (LTE-A) cellular network using the physical random access channel (PRACH), congestion and overload may result, which can lead to serious degradation of performance for both M2M and human-to-human (H2H) terminals. The main cause for this is the inherent complexity of the four-way handshake used for random access, which is well suited for H2H terminal access but unsuitable for massive M2M access. In this paper, we describe an efficient scheme for concurrent M2M and H2H access on the PRACH, which separates the resources for M2M and H2H access at the level of preamble codes and avoids the use of the four-step handshake for M2M terminals by implementing a carrier sense multiple access with collision avoidance (CSMA/CA) overlay network using the designated preamble codes. We analyze the performance of the scheme for both H2H and M2M traffic and show the values of the most important design parameters that enable this scheme to support concurrent access by H2H and M2M terminals with little performance degradation.