Question: In Example, is the continuous-time Markov chain $ X(t): t ≥ 0 % a birth and death process?
Example: In Ponza, Italy, a man is stationed at a specific port and can be hired to give sightseeing tours with his boat. If the man is free, it takes an interested tourist a time period, exponentially distributed with mean 1/µ1, to negotiate the price and the type of tour. Suppose that the probability is α that a tourist does not reach an agreement and leaves. For those who decide to take a tour, the duration of the tour is exponentially distributed with mean 1/µ2. Suppose that tourists arrive at this businessman's station according to a Poisson process with parameter λ and request service only if he is free. They leave the station otherwise. If the negotiation times, the duration of the tours, and the arrival times of the tourists at the station are independent random variables, find the proportion of time the businessman is free.