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The ability of a control chart to detect a shift in the process can be determined by the average run length (ARL), a widely used performance indicator for single sampling interval (SSI) control charts. When it comes to variable sampling interval (VSI) schemes, average time to signal (ATS) is utilized instead of ARL. Although analysis of ARL has been extensively studied for SSI schemes, calculation of the ATS for VSI control charts is still in its infancy due to several theoretical challenges. In this context, the paper derives closed-form expressions for time to signal distribution and the ATS for a spatial design of VSI multivariate Bayesian control chart referred to as the dual sampling scheme (DSS). In a DSS strategy, the process is being monitored initially with a longer sampling interval, which is changed to the shorter sampling interval at a designed switching point. Unlike previous approximate developments where the switching point is ignored, the paper derives exact expressions for computation of time to signal distribution and ATS within such a Bayesian framework. Derivations of the ATS are based on an intuitively pleasing definition of an artificial absorbing state and coded values of the posterior probability using Markov chain theory.

Time to signal distribution of multivariate bayesian control chart with dual sampling scheme