Question: 14. Consider an irreducible continuous time Markov chain whose state space is the nonnegative integers, having instantaneous transition ratesI mage and stationary probabilities , .

14. Consider an irreducible continuous time Markov chain whose state space is the nonnegative integers, having instantaneous transition ratesI mage and stationary probabilities , . Let T be a given set of states, and letI mage be the state at the moment of the nth transition into a state in T.

(a) Argue that Image is a Markov chain.

(b) At what rate does the continuous time Markov chain make transitions that go into state j.

(c) For Image, find the long run proportion of transitions of the Markov chain Image that are into state i.

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