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In this paper, we consider asynchronous spread spectrum multiple access (SSMA) communication systems with spreading sequences of Markov chains. We show that, given phase shifts and time delays, the empirical mean of the multiple access interference (MAI) for such systems satisfies the conditional large deviations principle (LDP) with the convex, good rate function. By refinement of the large deviations analysis, we also obtain exact asymptotics of bit-error probabilities in such systems. Comparing theoretical expressions of bit-error probabilities with experimental results, we confirm that for too small numbers of users compared to the lengths of spreading sequences, the central limit asymptotics become invalid, but the large deviations asymptotics turn out to be relevant.