Q1-) Consider a manufacturing system with two machines. Suppose that when both ma- chines are available, one is in use and the other is on standby. The probability that a machine in use fails during a day is p. When it fails its repair may start only the next day if the single repair facility is available. It takes two days to repair a failed machine. We can use a Markov Chain model to describe the evolution of this system. Let Xn = (i, j), n ≥ 0 denote the states of the Markov chain, where i is the number of machines in working condition and j is the number of elapsed repair days of a machine at the repair facility at the beginning of the n'th day. The corresponding transition probability matrix is (2,0) (1,0) (1,1) (0,1) (2,0) [1-p P 0 0 (1,0) 0 0 1-p Р P= (1,1) 1-p 0 0 P (0,1) 0 1 0 0 For parts (a)-(c) do not assume a specific value for p, leave your answer in terms of p. (a) Given Xo = (1, 1), what is the probability that only one machine is in working condition after two days? (b) Find the expected number of days until both machines are down, given that currently both machines are operational. (c) Find the steady state probabilities. (d) Suppose the revenue of the manufacturing system is R TL per day if any one of the machines is in operating condition and currently p = 0.3. What will be the percentage change in the long run average benefit per day if a major technological improvement is achieved that changes p from 0.3 to 0.2?