I am looking for a stochastic process model with the following features.
It is a continuous time Markov process---modelling, if you like, the evolution of a population.
New arrivals are added to the population at, say, Poisson rate.
The existing population following a continuous time Markov process with finite state space, e.g. $\{state1, state2, exit\}$. The holding times in each non-$exit$ state may be Poisson. Once a member of the population hits $exit$, he leaves the population.
The Markov transition matrix for existing population is time-invariant. (Pointers to generalization welcome but I would assume relaxing this would make statistical estimation very tricky.)
From perusing the literature, it seems basic models from queueing theory has some similarity with, but not quite the above. For what I am looking for, a new arrival in the queue is Markov jumping between different stations until he hits the "terminal station" and exits the queue.
I would like to also statistically estimate such a model, so parametric assumptions are OK in lieu of generality.