Runs tests for group control charts

When a process consists of several identical streams that are not highly correlated, an alternative to using separate control charts for each stream is to use a group control chart. Rather than plotting sample means from each stream at any time point, one could plot only the largest and/or smallest sample mean from among all the streams. Using the theory of stochastic processes and majorization together with numerical methods, the properties of a test that signals if r consecutive extreme values come from the same stream are examined. Both one and two-sided cases are considered. Average run lengths (ARL's), the least favorable configuration of the stream (population) means, and sample sizes necessary to have specified in-control and out-of-control ARL's are obtained. A test that signals if r - 1 out of r consecutive extreme values come from the same stream is also considered.