<p>Detecting the change point between different distributions in queueing theory presents a significant challenge, especially when working with observed data. Tra-ditionally, both parametric and non-parametric methods have been employed to address this issue in Markov single-server finite queueing models. While these methods can successfully identify the change point, they often come with the draw-back of high variance in the estimates. This article introduces Bayesian approaches to improve precision in change point detection by utilizing squared error and pre-cautionary loss functions. The study is based on a dataset that records the number of customers present in the system immediately before the arrival of the<i> n</i>-th cus-tomer, which is analyzed through an embedded Markov chain framework. The findings demonstrate that Bayesian methods achieve comparable error rates to traditional approaches but with a significant reduction in variance, particularly when dealing with smaller sample sizes. To showcase the practical benefits of this Bayesian methodology, a comprehensive numerical example is provided, illustrat-ing how these approaches can enhance the accuracy and reliability of change point detection in queueing models.</p>