This article discusses situations in which one is interested in evaluating the run-length characteristics of a cumulative sum control scheme when the underlying data show the presence of serial correlation. In practical applications, situations of this type are common in problems associated with monitoring such characteristics of data as forecasting errors, measures of model adequacy, and variance components. The discussed problem is also relevant in situations in which data transformations are used to reduce the magnitude of serial correlation. The basic idea of analysis involves replacing the sequence of serially correlated observations by a sequence of independent and identically distributed observations for which the runlength characteristics of interest are roughly the same. Applications of the proposed method for several classes of processes arising in the area of statistical process control are discussed in detail, and it is shown that it leads to approximations that can be considered acceptable in many practical situations.