Lazy Chris has three identical lightbulbs in his living room that he keeps on all the time. Because of his laziness, Chris does not replace a lightbulb when it fails. (Maybe Chris does not even notice...

Lazy Chris has three identical lightbulbs in his living room that he keeps on all the time. Because of his laziness, Chris does not replace a lightbulb when it fails. (Maybe Chris does not even notice that the bulb has failed!) However, when all three bulbs have failed, Chris replaces them at the same time. The lifetime of each bulb is exponentially distributed with a mean of 1/λ, and the time to replace all three bulbs is exponentially distributed with a mean of 1/μ.

a. Give the state-transition-rate diagram of the process.

b. What is the probability that only one lightbulb is working?

c. What is the probability that all three lightbulbs are working?