Some theoretical comments regarding the run-length properties of the synthetic and runs-rules X¯ monitoring schemes – part 2: steady-state. Issue 2 (4th March 2019) - Shongwe, Sandile C.; Graham, Marien A.
Abstract: In this paper, the long term (also known as the steady-state mode) run-length theoretical properties of the four different types of synthetic and runs-rules X ¯ monitoring schemes, that were empirically analysed in part 1 of this research work. That is, using the Markov chain imbedding technique (thoroughly discussed in Part 1 of this work), the closed-form expressions of the steady-state initial probabilities and average run-length ( ARL ) vectors are derived; so that the corresponding steady-state ARL and overall performance expressions, of each of the four different types of the synthetic and runs-rules X ¯ monitoring schemes, may be formulated. Since there is very little literature on steady-state analysis of the synthetic and runs-rules charts, the closed-form expressions derived here will ease the understanding and implementation of the different types synthetic and runs-rules X ¯ schemes in practice and in further academic research.