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We introduce a collision model for a random access and random spreading direct-sequence code-division multiple access (DS-CDMA) network: A collision is said to occur when no power control scheme can guarantee the signal-to-interference-plus-noise ratio (SINR) for all concurrent transmissions. Such a collision model represents an example system where the physical layer is equipped with advanced reception capability, as opposed to the conventional collision model where more than one concurrent transmission leads to a collision. We analyze the medium access control (MAC) layer throughput of random access networks based on a saturated slotted ALOHA model. Throughput results are presented first for a general reception model and then for the proposed collision model with random spreading and power control.