Head start values—that is, nonzero initial values for the summary statistics of Markov-type control schemes such as the well-known cumulative sum (CUSUM) schemes—are throughly recommended in the literature and used by quality control practitioners. The rationale is as follows: if the process is operating in control, the summary statistic of the control scheme is soon brought to zero, so that the expected effect of the head start is minimal; otherwise, the operator is alerted to the out-of-control situation much sooner, which may prevent start-up problems.

This article assesses the impact of the adoption of a head start on the run length, indisputably the most popular performance measure of control schemes. It also brings stochastically monotone matrices into focus in the statistical process control setting and investigates their influence on the survival function, the alarm rate function and the aging character of the run length of one-sided Markov-type control schemes.

The derived properties, which may be of great interest to quality control practitioners, are illustrated through examples.