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Problem 3: (Continious Markov Chains) [20 points] An aviation company has 3 jet plane for their operations. Each plane fails once in two weeks on average with exponentially distributed time between failures. The company has 1 serviceteams and each team needs 2 day on average to x a plane, with exponentially distributed repair time. The company needs at least 2 planes for proper functioning. (a) (5 points) Draw the transition diagram and indicate the instantaneous transition rates. Let the state number i denotes the number of operating planes. 1132 = 3/14 411= 2/14 410 = 1/14 1123 =1/2 112 =1/2 1101 = 1/2 (b) (5 points) Write down the rates vi and the transition probabilities Pu\" and Pi,,-_1 for each state i. (c) (5 points) Given that there are 3 working planes what is the probability that no failures will happen within the next day? (d) (5 points) Find the percentage of time that the company will not have enough planes (2 or more) for normal operation