Let's consider a supermarket with three cash registers. The operation of the cash registers is checked daily, and if a register is found to be faulty, it is sent to the workshop for repairs. The probability of a working cash register breaking down is 1/5, and the probability of a cash register undergoing repair becoming operational is 3/5. The processes of breakdown and repair of cash registers are independent of each other. 1. (25 pts) Model this problem using a discrete time Markov chain. Present the transition probability matrix. 2. (15 pts) What is the long-run proportion of days when none of the cash registers are operational?

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