Steady-state average run length(s): Methodology, formulas, and numerics

The average run length (ARL), with its various phenotypes, is the prevailing performance measure for evaluating control charts, or change-point detection schemes. Essentially, the ARL counts the number of observations until the corresponding procedure flags a change. To enable a fair comparison between competing designs, one frequently deploys the steady-state ARL. Differing from the older concept of the zero-state ARL (which assumes that the to-be-detected change occurs immediately at startup or never), the former measure postulates this change's appearance after reaching some steady state. Considering different notions (primarily conditional and cyclical ones) of the measure, we recapitulate its historical development; provide a critical discussion of its often-careless exploitation, including a few misconceptions; and derive some new mathematical characterizations that permit its easy calculation.

Automated summary

The average run length (ARL) is the prevailing performance measure for evaluating control charts, counting observations until a change is flagged. The steady-state ARL is used for fair comparison between competing designs, postulating the appearance of change after reaching a steady state, unlike the zero-state ARL assumption.