A Markov modulated growth collapse model

We consider a growth collapse model in a random environment where the input rates may depend on the state of an underlying irreducible Markov chain and at state change epochs there is a possible downward jump to a level which is a random fraction of the level just before the jump. The distributions of these jumps are allowed to depend on both the originating and target states. Under a very weak assumption we develop an explicit formula for the conditional moments (of all orders) of the time stationary distribution. We then consider special cases and show how to use this result to study a growth collapse process in which the times between collapses have a phase type distribution.