Suppose you have an alternating renewal process: A new machine begins operating at time 0 and breaks down after a random amount of time whose...

Suppose you have an alternating renewal process: A new machine begins

operating at time 0 and breaks down after a random amount of time whose

distribution is F1. It takes a random amount of time whose distribution is

F2 to replace the broken machine with a new machine with characteristics

identical to the original. Successive periods are independent and alternate

between having distributions F1 and F2.

Dene fx(t) to be the probability that at time t there is an operating machine

and that it stays operative for at least x more time units. In what follows,

x is always xed.

(i) Write the renewal equation satised by fx(t).

(ii) Solve the equation.

(iii) Compute limt!1 fx(t).