Abstract: Exact expression are derived for the probability functions of run lengths of one-sided cumulative sum (CUSUM) schemes when observations are exponentially distributed. Average run lengths (ARLs), standard deviations of run lengths (SDRLs) and percentiles of run length distributions can then be obtained by recursively evaluating the probability functions. The run length distributions of CUSUM schemes are found to be highly skewed, and consequently conclusions based on ARL alone can be misleading. Knowledge of run length distributions would provide a comprehensive understanding of CUSUM schemes. A comparison of the performance of CUSUM schemes is presented, and general considerations in the design of CUSUM schemes are discussed.
Key words and phrases: Quality control, cumulative sum (CUSUM), average run length (ARL), percentiles, exponential distribution, Poisson distribution.