Employing the Rayleigh Distribution to Estimate the Reliability of a French Fries Cutting Machine and Using a Markov Chain to Estimate the Probability of Transitioning from One State to Another for the Short and Long Term

Abstract

In this study we used two methods to estimate the hazard of the cutting machine of French fries which are reliability analysis with Rayleigh distribution and Markov chine to estimate the probability transition from a state to another, the data that had been used were 70 observations of failure time which has been taken from Bazian manufacturer for producing French fries, after computing the reliability analysis we transform the real data above into the dummy variable based on the arithmetic mean which is calculated from the real failing time data (Mean = 46.53 min) as when the failing time below 46.53min takes zero otherwise takes one, and the transition frequency matrix has been calculated, the results presents that the Rayleigh distributions probability density function exhibits a decreasing trend over time, a substantial portion of failure times the system failure is most likely to occur between 10 and 80 minutes, marking a critical period. The probability density function significantly decreases after 120 minutes. The reliability curve shows a decline over time; the likelihood of the system remaining reliable beyond 40 minutes is about 0.6, while the probability drops to 0.2 for reliability beyond 80 minutes. Additionally, there is a 51.86% chance that a failure will occur within 46.53 minutes and be followed by another failure within the same timeframe. Similarly, there is a 51.86% chance that a failure occurring after 46.53 minutes will be followed by another failure occurring after the same duration value is 0.4814.

Keywords

• Probability theory, Reliability analysis, Rayleigh Distribution, and Markov chain.