A new item will fail on its ith day of use with probability pi ,

∞

i=1 pi = 1.

An item that fails during a period is replaced by a new one at the beginning of the next period. Let An denote the age of the item in use at the beginning of period n. That is, An = i if the item in use is beginning its ith day. The random variables An can be interpreted as the age at time n of a renewal process whose interarrival times have mass function {pi, i ≥ 1}, with An = 1 signifying that a renewal occurs at time n.

(a) Argue that {An,n ≥ 1} is a Markov chain and give its transition probabilities.

(b) Suppose A0 = 1. If ∞pi j=i pj decreases in i, show that An stochastically increases in n.