# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 1 Background  
The Weibull distribution is a popular continuous probability distribution that is commonly utilized to model the lifetimes or failure times of objects or systems. It was initially introduced by [PERSON] in 1951 [1] and has since found applications in various fields. For instance, in reliability engineering, [PERSON] et al. [2] employed the Weibull distribution to model the Hertzian fracture of Pyrex glass. In another study, [PERSON] and [PERSON] [3] used the Weibull distribution to analyze adhesive wear in metals. Similarly, [PERSON] and [PERSON] [4] applied the distribution to carbon fibers and small composite specimens. [PERSON] [5] investigated the failure of coating using the Weibull distribution, while [PERSON] et al. [6] focused on its use for brittle material. Additionally, [PERSON] et al. [7]employed the distribution to study compressive failure in high-performance polymers and [PERSON] et al. [8] analyzed concrete components using the Weibull distribution. The distribution mentioned above has been widely applied as a flexible modeling tool in various fields, addressing a diverse range of issues. It has been successfully utilized in disciplines such as quality control, weather forecasting, industrial engineering, electric systems engineering, communications systems engineering, hydrology, and more. For instance, [PERSON] and [PERSON] [9] utilized it for volcanic eruptions, while [PERSON] [10] and [PERSON] et al. [11] applied it to regional flood frequency analysis. [PERSON] [12] used it to describe the dynamics of foliage biomass on Scots pine. In the field of Economics, [PERSON] and [PERSON] [13] employed it to analyze unemployment duration data. In the context of wireless communications, the Weibull distribution is very flexible. [PERSON] and [PERSON] [14] analyzed the performance of multi-hop relaying systems over Weibull fading channels in terms of bit error rate and outage probability. A similar analysis of the bit error rate and outage probability over multi-hop Weibull fading channels was also conducted by [PERSON] et al. [15]. The Weibull distribution is defined by two parameters: the shape parameter (\(\
u\)) and the scale parameter (\(\lambda\)). The shape parameter determines the shape of the distribution curve, while the scale parameter determines the characteristic magnitude or scale of the failure times. Depending on the value of the shape parameter \(\
u\), the Weibull distribution can exhibit different shapes. When \(\
u>1\), the distribution is positively skewed, indicating a decreasing failure rate over time. This shape is commonly known as the bathtub curve and is often observed in reliability analysis. In this curve, failures are more likely to occur either early on, due to manufacturing defects or initial wear, or in the later stages, due to aging or wear-out effects. When \(\
u=1\), the distribution simplifies to the exponential distribution, which is the only continuous distribution with a constant hazard function on the positive axis. For \(\
u<1\), the distribution is negatively skewed, indicating an increasing failure rate over time ([PERSON] [16]). The versatility and importance of this distribution can be seen in its close relationship with several well-known distributions in statistics. It includes other distributions as special cases, such as the exponential distribution (when the shape parameter \(\
u\) is equal to 1) and the Rayleigh distribution (when \(\
u\) is equal to 2). Furthermore, if \(T\) follows a Weibull distribution, then \(Y=\lambda(1-\
u\log(T/\
u))\) follows an extreme-value distribution. By applying a simple log transformation, the Weibull distribution can also be converted into the Gumbel distribution. Additionally, it acts as a limit distribution for the Burr distribution, establishing a significant connection between these distributions ([PERSON] and [PERSON] [17]). These relationships further emphasize the importance and wide-ranging applicability of the Weibull distribution in statistical modeling and analysis.