CUSUM procedures with probability control limits for monitoring processes with variable sample sizes

Control charts are usually designed with constant control limits. In this article, we consider the design of control charts with probability control limits aimed at controlling the conditional false alarm rate at the desired value at each time step. The resulting control limits are dynamic and thus are more general and capable of accommodating more complex situations in practice as compared with the use of a constant control limit. We consider the situation when the sample sizes are varying over time, with a primary focus on the CUmulative SUM (CUSUM)-type control charts. Unlike other methods, no assumptions about future sample sizes are required with our approach. An integral equation approach is developed to facilitate the design and analysis of the CUSUM control chart with probability control limits. The relationship between the CUSUM charts using probability control limits and the CUSUM charts with a fast initial response feature is investigated.