4.6. A machine can be in either of two states: "up" or "down." It is up at time zero and thereafter alternates between being up and down. The lengths X,, X,.... of successive up times are independent and identically distributed random variables with mean

a, and the lengths Y,, Y,, ... of successive down times are independent and identically distributed with mean /3.

(a) In the long run, what fraction of time is the machine up?

(b) If the machine earns income at a rate of $13 per unit time while up, what is the long run total rate of income earned by the machine?

(c) If each down time costs $7, regardless of how long the machine is down, what is the long run total down time cost per unit time?