Well, Markov chains as used by programmers are generally discreet-time. Continuous-time Markov chains are continuous. Indeed, in the original model he begins with what seems to be a continuous-time Markov chain. You are correct you get to solve a linear differential equation with constant coefficients to get the distribution at any one time. Another interesting way to consider this stuff is in terms of the "logarithms" and "exponentiation" of Markov matrices.

http://en.wikipedia.org/wiki/Markov_chain http://en.wikipedia.org/wiki/Continuous-time_Markov_chain