Some theoretical comments regarding the run-length properties of the synthetic and runs-rules X¯ monitoring schemes – part 2: steady-state

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 $\overline{X}$ 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 $\overline{X}$ 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 $\overline{X}$ schemes in practice and in further academic research.

Keywords:

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

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.