We propose a novel random multiple access (RMA) scheme with quality of service (QoS) guarantees for machine-to-machine (M2M) communications. We consider a slotted uncoordinated data transmission period during which machine type communication (MTC) devices transmit over the same radio channel. Based on the latency requirements, MTC devices are divided into groups of different sizes, and the transmission frame is divided into subframes of different lengths. In each subframe, each group is assigned an access probability based on which an MTC device decides to transmit replicas of its packet or remain silent. The base station (BS) employs successive interference cancellation (SIC) to recover all the superposed packets. We derive the closed form expressions for the average probability of device resolution for each group, and we use these expressions to design the access probabilities. The accuracy of the expressions is validated through Monte Carlo simulations. We show that the designed access probabilities can guarantee the QoS requirements with high reliability and high energy efficiency. Finally, we show that RMA can outperform standard coordinated access schemes as well as some of the recently proposed M2M access schemes for cellular networks.