A machine has three components labeled 1, 2, and 3, whose times between failure are exponentially distributed with mean 1/1, 1/2, and 1/3, respectively. The

A machine has three components labeled 1, 2, and 3, whose times between failure are exponentially distributed with mean 1/λ1, 1/λ2, and 1/λ3, respectively. The machine needs all three components to work, thus when a component fails the machine is shut down until the component is repaired and the machine is brought up again. When repaired, a component is considered to be as good as new. The time to repair component 1 when it fails is exponentially distributed with mean 1/μ1. The time to repair component 2 when it fails is constant at 1/μ2, and the time to repair component 3 when it fails is a third-order Erlang random variable with parameter μ3.

a. What fraction of time is the machine working?

b. What fraction of time is component 2 being repaired?

c. What fraction of time is component 3 idle but has not failed?

d. Given that Bob arrived when component 1 was being repaired, what is the expected time until the machine is operational again?