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Orthogonal frequency-division multiple access (OFDMA) systems are considered promising candidates for implementing next-generation wireless communication systems. They provide multiple channels that can be accessed via random access schemes. However, traditional random access schemes could result in an excessive amount of access delay. To address this issue, we develop a fast retrial scheme that is based on slotted Aloha and exploits the structure of OFDMA. A salient feature of this scheme is that when collisions occur instead of retrials occuring randomly in time, they occur randomly in frequency, i.e., the scheme randomly selects the subchannels for retrial. To further achieve fast access, retrials are designed to follow the 1-persistent type, i.e., no exponential backoff. To achieve the maximum throughput, we limit the maximum number of allowed retrials according to the load condition. We also consider the issue of designing for an appropriate reuse factor for random access channels in order to overcome the intercell interference problem in OFDMA multicell environments. Our finding is that full sharing, i.e., a reuse factor of one, performs best for given random access channels. Through analysis and simulation, we confirm that our fast retrial algorithm has the advantage of high throughput and low access delay, and the full sharing policy for random access channels shows high throughput as well as low collision.