Question: 11.8 A certain component of a machine has a lifetime distribution F1(x). When this component fails, there is a probability p that it is instantaneously

11.8 A certain component of a machine has a lifetime distribution F1(x). When this component fails, there is a probability p that it is instantaneously replaced (0 <

p < 1). If the component is not replaced, the machine operates for a random amount of time (with distribution F2(x)) and then shuts down. After the system shuts down, the machine is repaired (and the component is replaced), which takes another random time length with distribution F3(x). Assume that the distributions have expectations

μ1, μ2, and μ3 and they are all finite. We also assume that all the random variables in question are independent of each other. Find the asymptotic proportion of time when the machine is operational and the component in question is also working.

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