steady state probabilities in decision making markov chain

Project Description:

“an important machine is known to never last more than four months. during its first month of operation, it fails 10% of the time. if the machine completes its first month, then it fails during its second month 20% of the time. if the machine completes its second month of operation, then it will fail during its third month 50% of the time. if the machine completes its third month, then it is sure to fail by the end of the fourth month. at the beginning of each month, we must decide whether or not to replace our machine with a new machine. it costs $500 to purchase a new machine, but if a machine fails during a month, we incur a cost of $1,000 (due to factory downtime) and must replace the machine (at the beginning of next month) with a new machine. three maintenance policies are under construction:
policy 1 plan to replace a machine at the beginning of its fourth month of operation.
policy 2 plan to replace a machine at the beginning of its third month of operation.
policy 3 plan to replace a machine at the beginning of its second month of operation.
which policy will give the lowest average monthly cost?

[note: you must provide a fully worked solution so that i can see how you derived the answer].ste
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