We consider a discrete-time semi-Markov system, with a finite state space. The empirical and the exact maximum likelihood estimator for the semi-Markov kernel are given in the case of multiple parallel observations of the same process. Afterwards, we describe a reliability model described by a discrete-time semi-Markov process and we derive basic reliability measures, such as reliability, availability, failure rates and mean hitting times. Finally, we present a comparison between empirical and exact maximum likelihood estimators for these measures through a numerical application.