Real-life decisions often need to take into account uncertainties about many future events. There are various models for processes that evolve over time in a probabilistic manner. Such processes are called stochastic processes. In this paper, we use both discrete and continuous time Markov chains to model an aircraft maintenance facility and explore the relationships of these models with standard queuing theory approaches. Our first step applies discrete time Markov chains based on Binomial and Poisson distributions to our model. Next, we extend the models to continuous time Markov chains associated with exponential distributions, and also relate the new model to queuing theory. Finally, we compare our discrete models to the continuous ones and generalize cases for which a discrete Markov chain can be effectively implemented into the standard queuing approach to help analyze the transient probability states. In the end, we analyze strengths and weaknesses of our methods as well as propose future work that could be accomplished. Several ideas are explored using Mathematica.