A discrete time queueing model for the performance of an ATM system is analyzed using matrix analytic methods. Time is segmented into slots with each slot equal to the transmission time of one ATM cell. The ATM system is modeled as a single server queue with Markovian arrivals and service time equal to one slot. The arrival process includes as a special case the superposition of on-off sources, possibly heterogeneous. The queueing model is of the "M/G/1 type". By exploiting the structure of the "M/G/1 type" Markov chain, the complexity of the solution to the problem is reduced to only the inversion of a 2×2 matrix irrespective of the size of the Markov chain. This simplification allows us to investigate Constant Bit Rate (CBR) traffic performance (or quality-of-service) issues through a hybrid analysis/simulation approach. Specifically, we analyze the impact of on-off background traffic on the probability of consecutive cell losses, cell delay variation, and traffic shaper or playback buffer underflow and overflow probabilities of CBR traffic sources.