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Prоblem 3 Lunа аnd Nаоmi run a pet grоoming shop. Pets arrive according to a Poisson process with rate . Each arriving pet is a dog with probability and a cat with probability , independently of previous arrivals. Naomi takes care of dogs only, and Luna takes care of cats only. Naomi's service time is exponentially distributed with rate dogs per hour. Luna's service time is exponentially distributed with rate cats per hour. Upon arrival of a dog, if Naomi is available, she takes care of it; otherwise, the dog leaves and does not return. Similarly, upon arrival of a cat, if Luna is available, she takes care of it; otherwise, the cat leaves and does not return. At the end of service, pets leave the shop and do not return. (a) Write down the rate matrix for a continuous-time Markov chain with state space , where means there are no pets in the shop, means there is one cat only, means there is one dog only, and means there is one cat and one dog in the shop. Is this a birth-and-death chain?(b) Let denote the state of the Markov chain at time . Find