Markov Chain approach to get control limits for a Shewhart Control Chart to monitor the mean of a Discrete Weibull distribution

Abstract

Typically, failure time is modeled using continuous distributions such as the Weibull or Gamma distributions. In many practical scenarios, data is recorded in terms of discrete counts, such as the number of days or cycles, therefore the Discrete Weibull distribution is employed to model such cases. In this paper, we propose the use of a Shewhart $\overline{X}$ control chart to monitor the mean of a Discrete Weibull process. While the distribution of the sum of Discrete Weibull random variables does not have a closed-form expression, it can be determined through a Markov Chain procedure, which enables the calculation of precise control limits. The Average Run Length ($ARL$) is the metric used to assess the performance of the control chart. Two numerical examples are provided to illustrate its practical application.