Two high-altitude polar neutron monitors DOMC and DOMB (Dome C, Concordia station, Antarctic plateau, 3233 m a.s.l.) received a major electronics upgrade in 2019. While a typical standard neutron monitor data acquisition (DAQ) system only registers the number of pulses from a cosmic-ray particle detector, the new system digitizes all pulses with 2 MHz sampling rate and stores this information in raw data files. This feature makes it possible to conduct a pulse height-length analysis of the neutron monitor data on a routine basis. In this study, we have analysed several months of the cosmic-ray data recorded with the new DAQ system during 2019-2020 (more than 10 million pulses). We identified several pulse branches corresponding to different processes: (a) secondary particles from individual cosmic-ray cascades, (b) noise, (c) double pulses originated from particles of the same local cascade, (d) high multiple pulses likely related to atmospheric muons, (e) double pulses potentially caused by contamination by neutrons scattered in the neighbouring instrument. We also studied the waiting time distributions of pulses and have shown that two peaks can be clearly distinguished: (1) at about 1 millisecond, which is related to the intra-cascade particles, and (2) at 30-1000 milliseconds related to different uncorrelated cosmic-ray cascades. Our conclusions are supported by theoretical estimates of the waiting times in different scenarios.