In this paper, we introduce the MMPP(1n) process, a model that may be used in Integrated Services Digital Networks to represent the superposition of several overflow streams belonging to different classes of traffic. Given an overflow system in a multiclass environment, we show that this Markov Modulated model may be used to match simultaneously the two first moments of each individual overflow process. In addition, it takes into account the presence of congestion and non-congestion periods for the system as well as the dependence among the flows of different classes. Our study is illustrated with numerical examples that prove the accuracy of our approach.