Exact solution for average run length of CUSUM charts for MA(1) process

In this paper we apply Fredhom type integral equations method to derive explicit formula of the average run length (ARL) for a Cumulative Sum (CUSUM) chart, when observations are described by a first order moving average MA(1) process, with exponential white noise. We compare the computational time between our analytical explicit expressions for the ARL performance with the one obtained via Gauss-Legendre numerical scheme for integral equations. We found that those methods are in excellent agreement however, the computational time of the former takes approximately 1 second while the latter method consumes the computational time 11 minutes approximately.