The geometric approximation to the cusum run length distribution

It has long been known that at larger run lengths, the distribution of cusum run length, T, converges to the geometric form cλT. Currently, the only approximation to c is 1 and one approximation is available for λ which goes back to Page (1954). The usefulness of these approximations has never been tested. Two new approximations for c and three for λ are derived. All are compared to numerically-derived values c and λ for a small but diverse set of cusums whose increments are normally-distributed. The resulting approximations to cλT are finally compared to actual numerically-derived run length percentiles.