Shongwe, Sandile CharlesGraham, Marien Alet2018-09-172018Shongwe, S.C. & Graham, M.A. 2018, 'Some theoretical comments regarding the run-length properties of the synthetic and runs-rules monitoring schemes – Part 1 : Zero-state', Quality Technology and Quantitative Management, NYP.1684-370310.1080/16843703.2017.1389141http://hdl.handle.net/2263/66572In 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.en© 2018 International Chinese Association of Quantitative Management. This is an electronic version of an article published in Quality Technology and Quantitative Management, vol. x, no. y, pp. z-zz, 2018. doi : 10.1080/16843703.2017.1389141. Quality Technology and Quantitative Management is available online at: http://www.tandfonline.com/loi/ttqm20.Average run-length (ARL)Overall performanceRuns-rules chartsSynthetic chartsTransition probability matrix (TPM)Zero-stateMatrix algebraMarkov processesGraphic methodsPostprint Article