Shewhart Schemes with Variable Sampling Intervals Revisited

The efficiency of a control scheme may be improved by relating the sampling interval to the observed values of the control statistics: for instance, using a longer sampling interval if the sample statistic is close to the target and a shorter one otherwise. The resulting scheme is said to have variable sampling intervals (VSIs). Numerical results on the advantages of using the VSI feature have been presented by Reynolds et al. (1988), for the [image omitted]-scheme; Shobe (1988), for count data control schemes; and Reynolds et al. (1990), for cumulative sum (CUSUM) schemes. Moreover, Reynolds and Arnold (1989) and Reynolds (1989) showed that if the sampling interval is to be chosen from a range of possible values then, under some conditions, one-sided and two-sided schemes are optimal (in some specific sense) in case they make use of the shortest and largest possible sampling intervals. Bearing these results in mind, this paper discusses several properties of the in-control and out-of-control TS of Shewhart schemes with VSIs that have a discrete geometric compound distribution, in opposition to the geometric-like time to signal (TS) of Shewhart schemes with fixed sampling intervals (FSIs). This paper also focus on analytical answers to the question, What is the impact of VSIs in the performance of a Shewhart control scheme? Thus, we provide a qualitative basis for a more effective comparison of Shewhart schemes with FSIs and VSIs than the comparisons based entirely on numerical results.