We consider the problem of approximating a variable bit rate (VBR) source by a simple process such that the corresponding buffer-related performance measure are close approximations of the true performance measures. Assuming that a VBR source can be modeled by a discrete-time batch Markovian arrival process (D- BMAP), we propose an approach for approximating it by a ﲭatched Markov process of finite memory obtained by information-theoretic techniques. We confirm analytically that the approximating performance measures become increasingly accurate with the memory of the matched Markov process. When the parameters of the D-BMAP are unknown, we estimate instead the parameters of a suitable Markov approximation from samples of an observed cell stream. We show that the estimated Markov process, with a fixed memory, comes closer with increasing sample size to the D-BMAP, as do the corresponding performance measures, in accordance with the law of iterated logarithm. Numerical examples are presented to illustrate the effectiveness of the approach.