7.4.6 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 X1;X2; : : :

of successive up times are independent and identically distributed random variables with mean , and the lengths Y1;Y2; : : : of successive down times are independent and identically distributed with mean .

(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?