5.2. A component of a computer has an active life, measured in discrete units, that is a random variable T, where Pr{T = k} = a,, for k = 1, 2.... .

Suppose one starts with a fresh component, and each component is replaced by a new component upon failure. Let X,, be the age of the component in service at time n. Then {X_n} is a success runs Markov chain.

(a) Specify the probabilities p; and q,.

(b) A "planned replacement" policy calls for replacing the component upon its failure or upon its reaching age N, whichever occurs first.

Specify the success runs probabilities p; and q; under the planned replacement policy.