Estimating the Occurrence Rate for Alpha-Series Process in Rayleigh Distribution

Abstract

The geometric process is sometimes appropriate for reliability and scheduling problems. Some previous studies suggested a possible alternative process that is alpha-series as to the geometric process when it decreases with time, as the decreasing geometric process shows that the expected number of events at an arbitrary time does not exist. In contrast, the expected number of events of the alpha-series process (ASP) exists at an arbitrary time under some conditions. In this paper, we assumed that the first arrival followed the Rayleigh distribution (RD). The modified moment estimator was proposed to estimate the alpha-series process parameters in the Rayleigh distribution and compare it with the maximum likelihood estimators. A simulation was conducted to compare the two estimators. The real-data application of intervals between successive failures of the Mosul Dam power station in Nineveh governorate in Iraq is provided to illustrate the results. When the initial occurrence time distribution is indicated to be RD, an estimate of the occurrence rate of an ASP is investigated in this study. Estimators are generated using modified moment (MM), and maximum likelihood (ML) approaches. According to the simulation study's findings, the MM estimator outperforms the ML estimator. In all cases, ASP with RD provides better data than the renewal process (RP) in real data sets. A test statistic has been devised to determine if the data conforms to an ASP.