I have a pretty good handle on Hidden Markov Models, but I can't think of any examples of a Markov process with a directly observable state. Any examples?
They are all over the place, albeit you sometimes need to make simplifying assumptions. You could model workplace participation as a Markov model. People can be in training, working, unemployed, on disability leave, maternity leave, not seeking work, retired. Retirement would be an absorbing state in this model. Individuals transition between the states, and you might want to know the overall breakdown in the society. Or you might want to model the transition matrix for different groups (say those with post-secondary training and those without).
You could model this as continuous time, or discretize. Say, consider the changes if any every three months, or whatever.
Another model could be Telecom subscription in a market with a fixed number of Telecoms. People enter the market, churn between Telecoms, and then possibly exit the system (maybe by dying, moving away, or switching technologies altogether).