Abstract: In this paper we not only present an estimation of the average run length (ARL) for CUSUM (cumulative sum) control chart but also give a numerical comparison of CUSUM, GLR (Generalized Likelihood Ratio), GEWMA (Generalized EWMA) and RFCuscore (Reference-free Cus-core) control charts in detecting the mean shifts of stable processes. The simulation results show that CUSUM has the best performance among the four charts in detecting mean shifts of any size if the index of stability a of stable distribution satisfies 1＜α＜2.