A Markov chain approximation is used to determine the run length performance of a multivariate statistical process control chart. The Markov chain approach is widely used in the analysis of univariate control charts we extend the advantages of this type of analysis to a multivariate exponentially weighted moving averages control chart. The analysis can be applied whenever the multivariate control statistic can be modeled as a Markov chain and the run length performance depends on the off-target mean only through the noncentrality parameter.