Problem $1(40$ points$)$

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 $\frac{1}{5},$ and the probability of a cash register undergoing repair becoming operational is $\frac{3}{5}.$ The processes of breakdown and repair of cash registers are independent of each other.

$(25$ pts$)$ Model this problem using a discrete time Markov chain. Present the transition probability matrix.

$(15$ pts$)$ What is the long$-$run proportion of davs when none of the cash registers are operational?