A digital analog converter (DAC) used in a certain application has two components, each with independent lifetimes that have an exponential distribution with rate ; both are initially working. While both or just one of the two components is working then the DAC is used, but when both finally break down the DAC is immediately replaced by a new independent DAC with both components initially working. This continues forever.

YOU MUST SHOW YOUR WORK.

(a) Let X= the length of time until a DAC is replaced. Compute E(X)

(b) Find the long-run proportion of time that a DAC is used while both components are working.

(c) Find the long-run proportion of time that a DAC is used while only one component is working.

(d) It costs the company $c per unit time that both components are working 2 and $ c > c per unit time that only one component is working (regardless of 1 2 which one). Compute the long-run cost rate of using the DACs.

(e) Suppose that the two components have different rates for the exponential, and . Re-answer (d) now.