Some theoretical comments regarding the run-length properties of the synthetic and runs-rules X¯ monitoring schemes – part 1: zero-state. Issue 2 (4th March 2019) - Shongwe, Sandile C.; Graham, Marien A.
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 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 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.