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Suitable homogeneous markov chain

Question:

Ferry boats and customers arrive at a ferry station in accordance with two independent homogeneous Poisson processes with intensitiesµ and λ , respectively. If there are k customers at the ferry station, when a boat arrives, then it departs with min(k, n) passengers (n is the capacity of each boat) If k > n

then the remaining k - n customers wait for the next boat. The sojourn times of the boats at the station are assumed to be negligibly small. Model the situation by a suitable homogeneous Markov chain {X(t), t ≥ 0} and draw he transition graph.

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## Q : Applying the optional stopping theorem

whichever occurs first. By using suitable martingales and applying the optional stopping theorem, determine