Control charts with random interarrival times between successive samplings

• Published in: Applied stochastic models in business and industry 2019-05, Vol.35 (3), p.651-670
• Main Authors: Koutras, Markos V., Rakitzis, Athanasios C.
• Format: Article
• Language: English
• Subjects: Control charts; CUSUM control chart; Performance assessment; phase‐type distributions; Plotting; Quality control; random sampling times; Random variables; runs rules; Signal monitoring; Statistical methods; time between inspections
• ISSN: 1524-1904; 1526-4025
• DOI: 10.1002/asmb.2358

In the classical setup used in process monitoring, the times between the collection of successive plotted samples are considered as nonrandom. However, in several real‐life applications, it seems plausible to assume that the time needed to collect the necessary information for plotting a point in the control chart has a stochastic nature. Under this scenario, instead of focusing on the number of points plotted on the chart until an out‐of‐control signal is initiated, the appropriate statistic to look at is the total time until a signal is generated. If we denote by L the run length of a control chart and by Yt,t = 1,2,…, the times between successive plotted points, then the compound random variable SL=∑t=1LYt expresses the time to signal of a monitoring scheme, under a particular sampling policy. In this paper, we illustrate how SL can be exploited to study various charts that are suitable for monitoring Poisson observations. We provide some results for the exact distribution of SL that may facilitate the task of the performance assessment of a control chart with random plotting times; illustrations and several numerical comparisons that are useful for quality control experts who wish to practice them are presented, and finally, an illustrative example elucidating the implementation of the proposed model is also provided.