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A multiple packet receiver (MPR), which jointly performs timing acquisition of the incoming packets and decoding of the packet payloads, is analyzed. The payload decoding consists of iteratively performed parallel interference cancellation, followed by individual error control decoding. Timing acquisition is repeated at every stage of the interference cancellation in order to acquire packets that are otherwise covered by interference. The case in which packets are received at multiple power levels is considered, and it is proved that the system is capable to operate close to the multiple access channel capacity. An upper bound on the gap between the achievable spectral efficiency and the channel capacity is derived in terms of the preamble length (timing acquisition window).