Problem 2 (10 Points)

Consider a production line where each manufactured item may be defective with probability $p \in (0,1)$. The following inspection plan is proposed with a view to detecting defective items without checking every single one.

It has 2 phases: In phase A, the probability of inspecting an article is $r\in(0,1)$. In phase B, all the articles are inspected. One switches from phase A to phase B as soon as a defective item is detected. One switches from phase B to phase A as soon as a sequence of N successive acceptable items has been found.

let $\{X_n\}_{n\ge 0}$ be the process taking values $E_0,\cdots,E_N$, where if $j\in [0,N-1]$, $E_j$ means that the inspection plan is in phase B with j successive good items observed, and $E_N$ means that the plan is in phase A.

Prove that $\{X_n\}_{n\ge 0}$ is an irreducible HMC, give its transition graph, and show that it is ergodic. Find its stationary distribution.

Find the long-run proportion of items inspected and give the efficiency of the inspection plan, which is by definition equal to the ratio of the long-run proportion of detected defective items over the proportion of all defective items.