An instrument has a component subject to random failure. If the instrument is operating correctly at a given point in time, then with probability 0.15 it will fail during the next 20 minute period. If the component fails, it can be replaced with a new one, an operation that takes 20 minutes.
The current distributor of new components does not guarantee that all components are in proper working order, it points out that 2% of the supplied components are defective; however, this can be discovered only after the faulty component has been installed. If the supplied component is defective, the instrument will have to undergo a new replacement operation. Consider that when a failure occurs, it always happens at the end of a 20 minute period.
a. Determine the transition matrix associated with this Markov process.
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b. Calculate the steady state probabilities c. Consider that each replacement component has a cost of $ 0.30 and that the opportunity cost in terms of profit lost during the time the instrument does not work is $ 10 per hour.
What is the average cost per 20 minute period?