Run and Frequency Quota Rules in Process Monitoring and Acceptance Sampling

Motivated by certain models arising in the field of statistical process control and acceptance sampling, we study the waiting-time distribution for the first occurrence of a pattern belonging to a specific family of patterns in a bivariate sequence of trinomial trials. Such a setup can be applied in cases where the final quality of the output of an industrial process is determined by two correlated variables that are used to classify the two measured characteristics as “quite close to target quality,” “of satisfactory quality,” and “far from target quality.” A typical case where this happens is whenever registering the exact value of the characteristic of interest is expensive or difficult (e.g., when the observations come in the form of grouped data, due to the use of m-step gauges). The exact distribution of the waiting time is derived by using a Markov chain-embedding technique, which is then used in order to determine the most appropriate design for the quality-control problems discussed.