The Batch Markov Modulated Poisson Process (BMMPP) is a subclass of the versatile Batch Markovian Arrival process (BMAP) which has been proposed for the modeling of dependent events occurring in batches (as group arrivals, failures or risk events). This paper focuses on exploring the possibilities of the BMMPP for the modeling of real phenomena involving point processes with group arrivals. The first result in this sense is the characterization of the two-state BMMPP with maximum batch size equal to K, the BMMPP2(K), by a set of moments related to the inter-event time and batch size distributions. This characterization leads to a sequential fitting approach via a moments matching method. The performance of the novel fitting approach is illustrated on both simulated and a real teletraffic data set, and compared to that of the EM algorithm. In addition, as an extension of the inference approach, the queue length distributions at departures in the queueing system BMMPP/M/1 is also estimated.