Csiszár's forward β-cutoff rate (given a fixed β > O) for a discrete source is defined as the smallest number R 0 such that for every R > R 0, there exists a sequence of fixed-length codes of rate R with probability of error asymptotically vanishing as e -nβ(R-R0). For a discrete memoryless source (DMS), the forward β-cutoff rate is shown by Csiszár  to be equal to the source Rényi entropy. An analogous concept of reverse β-cutoff rate regarding the probability of correct decoding is also characterized by Csiszár in terms of the Rényi entropy. In this work, Csiszár's results are generalized by investigating the β-cutoff rates for the class of arbitrary discrete sources with memory. It is demonstrated that the limsup and liminf Rényi entropy rates provide the formulas for the forward and reverse β-cutoff rates, respectively. Consequently, new fixed-length source coding operational characterizations for the Rényi entropy rates are established.