CUSUM design for detection of event-rate increases for a Poisson process

Abstract

In quantifying the performance of a CUSUM chart for detecting upward shifts in event rate, it has been recommended that steady-state evaluation of performance measures such as ARL be used. In this article, the methodology for making such evaluations using the Markov-chain approach is presented, for the case of an exponential CUSUM. This is much more efficient than the alternative of simulation, which is still in use. It is also shown that if one is using steady-state ARL as a measure of detection performance, one can find better choices for the CUSUM parameter k than that provided by the SPRT-based formula. Two types of shift in event rate are considered, and corresponding tables of recommended choices of CUSUM parameters (k, h) are presented for ten levels of in-control ARL, and for nine sizes of shift. These tables can assist quality engineers in the design of CUSUMs for monitoring inter-event times in steady-state operation. It is also shown that these exponential CUSUM tables may be used to find values for the parameters of a geometric CUSUM or a Bernoulli CUSUM chart for monitoring a proportion, provided the in-control value of the proportion is no more than approximately 0.5%.