Some theoretical comments regarding the run-length properties of the synthetic and runs-rules 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 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 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.