This tutorial discusses the suitability of Markovian models to describe IP network traffic that exhibits peculiar scale invariance properties, such as self-similarity and long range dependence. Three Markov Modulated Poisson Processes (MMPP), and their associated parameter fitting procedures, are proposed to describe the packet arrival process by incorporating these peculiar behaviors in their mathematical structure and parameter inference procedures. Since an accurate modeling of certain types of IP traffic requires matching closely not only the packet arrival process but also the packet size distribution, we also discuss a discrete-time batch Markovian arrival process that jointly characterizes the packet arrival process and the packet size distribution. The accuracy of the fitting procedures is evaluated by comparing the long range dependence properties, the probability mass function at each time scale and the queuing behavior corresponding to measured and synthetic traces generated from the inferred models.