Markov models have been widely used to model arrival processes at switches in packet- and cell-switched networks. However, recent experimental evidence suggests that such processes exhibit a long-range dependence (LRD) property which is not captured by these models. Fractal models are attractive because they capture the LRD property while providing parsimonious modeling of processes. Multi-state Markov models can capture the LRD property to some extent. However, they do not follow this principle since every state added to such a model also adds two adjustable parameters and thus increases the complexity of fitting experimental data to these parameters. In this paper, we show that a fractal model can be accurately approximated over a finite range of time scales by parsimonious multi-stage Markov models where the transition rates form a geometric progression along the stages, and each stage models a different time scale.