Analysis of average run length for CUSUM procedure with negative exponential data

The Average Run Length (ARL) is a performance measure that is frequently used in control charts. Cumulative Sum (CUSUM) is a popular procedure in quality control as it is a sensitive detector of small shifts in values of distribution parameters. In this paper, we use an integral equation approach to derive explicit formulas for the ARL (the first passage times) for CUSUM when observations are negative exponential distributed. Simulations are carried out to compare the performance of the explicit formulas with that of numerical approximations. The computational time for the explicit formulas is found to be approximately 10 seconds, which is much less than the computational time required for numerical approximations