A Note on the Aging Properties of the Run Length of Markov-Type Control Charts

Abstract

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.

Keywords:

• Phase-type distributions
• Run length
• Statistical process control
• Stochastic ordering

Subject Classifications:

• 60E15
• 62P30

ACKNOWLEDGMENTS

The first author would like to express his sincere thanks to Professors J. George Shanthikumar and Rhonda Righter, for their invaluable help and mind-blowing suggestions during the preparation of the initial draft of this article. He also extends his thanks to: Professor Dr. Wolfgang Schmid for invitating him to present this article in the session on Statistical Process Control at the 3rd. International Workshop in Sequential Methodologies (IWSM2011, Stanford University) and Professor Dr. Sven Knoth for the productive discussions during this workshop.

The first author was partially supported by CEMAT–IST during the preparation of the final draft of this article, while visiting the Department of Industrial Engineering and Operations Research, University of California, Berkeley, and gratefully acknowledges all the support given by Professor Rhonda Righter during that and two other previous visits.

Notes

Recommended by N. Mukhopadhyay