An electronic device contains a number of interchangeable parts that operate independently. Each component is subject to random failure which renders it completely inoperable. All components are inspected at the end of every week. A component which is found to have failed is replaced with an identical new component. Based on historical data, a new component has 0.2 probability of failing in the first week. Components that are 1 week old have a 0.375 probability of failing in the second week. Components that are 2 weeks old have a 0.8 probability of failing in the third week. None of the components last more than 4 weeks. Let the process represented by the state variable Xt be in state

• 0 if a component is less than one week old in week t
• 1 if a component is one week old in week t
• 2 if a component is two weeks old in week t
• 3 if a component is three weeks old in week t

Provide solutions to the following questions:

a. What is the transition matrix P for this process?
b. Determine E(3). What is the value e(3) and what does it mean in the context of the problem?
c. If a device contains a component that is two weeks old, what is the probability that the next time it has a component that is two weeks old will be 5 weeks from now?
d. What is the mean recurrence time for components being new? Find your answer by taking the reciprocal of the steady state probability.