Markov chains can pop up everywhere. Not shown in this excellent intro is that the stationary distribution can be calculated as the eigenvector with an eigenvalue of 1.

I have used them in practice for load testing. Have thousands of bots randomly take actions using a transition table. The aggregate behaviour can predicted and converted into the frequency domain using the above trick and some other stuff (http://edinburghhacklab.com/2014/03/taming-randomized-load-t...)