Some theoretical comments regarding the run-length properties of the synthetic and runs-rules X¯ monitoring schemes – part 1: zero-state

Abstract

In this paper, we discuss the short-term (also known as zero-state mode) run-length theoretical properties of the four different types of synthetic and runs-rules $\overline{X}$ monitoring schemes that were empirically analyzed in another paper. That is, we provide and point out how each corresponding type of the 2-of-(H + 1) runs-rules and synthetic charts’ transition probabilities matrices (TPMs) differ from each other in zero-state, for any positive integer H. Next, using these general TPMs and the standard Markov chain formulae, we derive the general form of the average run-length (ARL) vectors and the corresponding zero-state ARL expressions for any shift value for each of the four different types of the synthetic and runs-rules $\overline{X}$ monitoring schemes. Finally, we provide expressions to calculate the overall run-length performance for each of the schemes. While there is lots of literature available on empirical analysis of zero-state synthetic and runs-rules charts, there is very little on the corresponding theoretical analysis. We believe this paper will, in some part, fill this gap and encourage more research in this area.

Keywords:

• Average run-length (ARL)
• overall performance
• runs-rules charts
• synthetic charts
• transition probability matrix (TPM)
• zero-state

Acknowledgements

We would like to convey our gratitude to the Editorial staff and the reviewer(s) for taking out their valuable time to consider our paper and giving us comments that led to an improved paper. Part of this work was supported by the SARChI Chair at the University of Pretoria. Sandile Shongwe’s research was supported in part by the National Research Foundation (NRF) and Department of Science and Technology’s Innovation Doctoral scholarship (SFH14081591713; grant number: 95208) as well as Department of Statistics’ STATOMET and Marien Graham’s research was supported in part by NRF’s Thuthuka program (TTK14061168807, grant number: 94102).