Bayesian Monitoring of Waiting-Time Processes using Exponential Distribution

Abstract

The Bayesian control chart is an advanced statistical tool for monitoring process stability by incorporating prior knowledge with observed data. This study proposes a Bayesian control chart for monitoring waiting-time processes assuming an exponential distribution with a gamma conjugate prior. Posterior distributions of the process parameter are derived, and the posterior mean and standard deviation are used to construct dynamic control limits. A comprehensive simulation study is conducted by varying sample sizes and prior hyperparameters to evaluate the performance of the proposed chart. The results indicate that Bayesian control limits are generally narrower and more responsive to process shifts compared to classical control charts, leading to faster detection of out-of-control conditions. However, this increased sensitivity is associated with a higher false alarm rate, reflecting a trade-off between detection speed and in-control stability. The applicability of the proposed method is further demonstrated using real-life data, which supports the simulation findings. The study highlights the effectiveness of Bayesian approaches in process monitoring, particularly in situations with limited or uncertain data, while also emphasizing the importance of appropriate prior selection and control limit design. Future work may focus on developing probability-based or asymmetric control limits tailored for non-normal distributions.