Evaluation of the run-length probability distribution for CUSUM charts:: Assessing chart performance

This article provides a fast and accurate algorithm to compute the run-length probability distribution for cumulative sum charts to control process mean. This algorithm uses a fast and numerically stable recursive formula based on accurate Gaussian quadrature rules throughout the whole range of the computed run-length distribution and, therefore, improves the numerical efficiency and accuracy of existing methods. The algorithm may detect whether or not the geometric approximation is adequate and, when it is possible, it allows switching to the geometric recursion. The procedure may be applied not only to the normal distribution but also to nonsymmetric and long-tailed continuous distributions, some examples of which are provided. Methods to assess chart performance according to the run-length distribution, as well as some multivariate issues in statistical process control, are considered.