A production process contains a machine that deteriorates rapidly in both quality and output under heavy usage, so that it is inspected at the end of each day. Immediately after inspection, the condition of the machine is noted and classified into one of four possible states:

The process can be modeled as a Markov chain with its (one-step) transition matrix P given by

(a) Find the steady-state probabilities.
(b) If the costs of being in states 0, 1, 2, 3, are 0, $1,000, $3,000, and $6,000, respectively, what is the long-run expected average cost per day?
(c) Find the expected recurrence time for state 0 (i.e., the expected length of time a machine can be used before it must be replaced).

State 0 2 Condition Good as new Operable-minimum deterioration Operable-major deterioration Inoperable and replaced by a good-as-new machine 3-16 18-20 2-16-8 120 -| 78 34 0 0 0 0 0 0 a-0123