A note on the ageing character of the run length of Markov-type quality control schemes

Journal of Applied Probability, 41(4) (2004), 1243-1247
http://dx.doi.org/10.1239/jap/1101840572

In this letter, we refer to Li and Shaked (1997) concerning a discrete-time Markov chain $\{ X_n, n \geq 0\}$, with state space $\mathbb{N}_+ = \{0, 1, 2, \dots \}$ and initial state $k$. They showed that the first-passage time of $\{ X_n, n \geq 0\}$ to surpass a given threshold $x$ or for the maximal increment of this process to exceed a fixed critical value $y$, denoted $T_k(x,y)$, has increasing failure rate as long as:
(a) the transition matrix ${\bf P}$ is stochastically monotone convex and
(b) the matrix with the left partial sums of ${\bf P}$ is totally positive of order 2.
We show that if assumption (a) is replaced by an assumption of spatial homogeneity of ${\bf P}$, then $T_k(x,y)$ still has increasing failure rate when $k=0$.
This result is of special interest in statistical process control since this sort of first passage time naturally arises when dealing with Markov-type quality-control schemes such as the combined upper one-sided CUSUM-Shewhart, and stochastically monotone convex transition matrices are fairly unusual in a
quality-control setting.