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The problem of buffer overflow in variable-length-to-block and block-to-variable-length coding of fixed-rate finite-state homogeneous Markov sources for transmission through fixed:rate noiseless channels is investigated. Asymptotically optimal converging upper and lower bounds on the probability of overflow are derived. They decrease exponentially with the buffer size . The least rates that achieve exponents for both coding methods are obtained, as are the corresponding optimal word assignments. It is shown that for the class of state-calculable sources, variable-length-to-block and block-to-variable-length rates are equal.