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Machine-to-machine (M2M) communications typically exhibit features such as a large number of devices, low data-rates, small-sized packets, and low or no mobility, while human-to-human (H2H) communications typically support a small number of users, high data-rates, large-sized packets, and high mobility. To support M2M communications in future cellular systems, one of the most challenging problems is to resolve a collision problem in random access because of access attempts from a large number of devices. For a large class of fixed-location M2M services such as smart metering and remote sensing, each machine device has fixed uplink timing alignment (TA) due to a fixed distance between the machine device and its eNodeB. We propose a novel random access scheme based on its fixed TA information for M2M communications at a large number of fixed-location machine devices in future orthogonal frequency division multiple access (OFDMA)-based cellular systems like Long Term Evolution (LTE) system. The proposed random access scheme yields significantly lower collision probability, shorter access delay, and higher energy-efficiency, compared with the conventional random access scheme.