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Despite much research on the throughput of relaying networks under idealized interference models, many practical wireless networks rely on physical-layer protocols that preclude the concurrent reception of multiple transmissions. In this work, we develop analytical frameworks for the uplink of a multi- source single-channel relay-aided wireless system where transmissions are scheduled to avoid collisions. We study amplify-and-forward and decode-and-forward strategies, in both time-sharing and network-coded variants, and provide mathematical models to investigate their achievable rate regions. Both general and optimal power allocations are considered. We also find the cut-set outer bounds for the rate regions. Moreover, we present a comparison between these methods with the simple time sharing scheme. Our numerical results reveal that optimizing power allocation favors the time sharing scheme significantly more than it does the relaying schemes, so that time sharing under some circumstances can provide higher maximum sum rates, even if the links to the relay have strong channel gains. The proposed analysis provides a means to quantitatively evaluate the efficacy of relaying under the collision model, leading to pragmatic design guidelines.