Luna runs a pet grooming place. The place has 1 service chair and 1 waiting chair. Customers arrive at rate of 5 per hour: with probability 0.2 every arriving customer is a cat and with probability 0.8 a dog, independent of the past. Luna's policy is that only customers of the same species would be in her grooming shop at the same time. Hence, if there is a dog in service and the waiting chair is empty, then any cat arriving the shop leaves immediately, however, an arriving dog will take the waiting chair; and similarly if there is a cat in service and the waiting chair is empty, then any arriving dog leaves immediately, however, a cat will take the waiting chair. If both chairs are taken, both new arriving cats and dogs would leave the shop without being served and not return. A haircut for a cat lasts a random exponential time with rate 2 per hour and for dogs 3 per hour, independent of past arrivals and service times. Consider a Markov chain on the states S := {0, d, c, dd, cc}, where x S refers to the customers in the shop: 0 means no customers, d means only one dog in the shop (in service), c means only one cat in the shop (in service), dd means two dogs in the shop, and cc means two cats in the shop. (a) Write down the generator matrix for (Xt). Can the chain be expressed as a birth-death chain (possibly with reordering the elements of S)? (b) Find the stationary distribution of this process. (c) Suppose that for a dog haircut Luna charges $10 and for cat's $30. In the long-run, what is the average (per hour) revenue of Luna? Make sure to justify your answer rigorously. (d) If the shop is currently empty, what is the expected time until the rst time that it is full?