A change in a production process must be detected quickly so that a corrective action can be taken. Thus, it comes as no surprise that the run length (RL) is usually used to describe the performance of a quality control chart.

This popular performance measure has a phase-type distribution when dealing with Markov-type charts, namely, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts, as opposed to a geometric distribution, when standard Shewhart charts are in use.

In this article, we briefly discuss sufficient conditions on the associated probability transition matrix to deal with run lengths with aging properties such as new better than used in expectation, new better than used, and increasing hazard rate.

We also explore the implications of these aging properties of the run lengths, namely when we decide to confront the in control and out-of-control variances of the run lengths of matched in control Shewhart and Markov-type control charts.

CEMAT - Center for Computational and Stochastic Mathematics