Control charts are often designed with constant control limits. In this paper,
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 would be dynamic, which are more general and capable
of accommodating more complex situations in practice as compared to the use of
constant control limit. We limit the discussion to the situation when the sample
sizes are varying over time, with primary focus on the cumulative sum (CUSUM)
type control charts. 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 the fast initial response (FIR) feature is investigated.