###### Abstract
[PERSON] and [PERSON] [18] proposed a cost model considering the time required to remove errors in a software system, thereby reducing the risk cost owing to software failure. In addition, [PERSON] and [PERSON] [19] proposed a cost model using the warranty cost to remove errors detected in a software system and reduce the risk cost owing to software failure. [PERSON] and [PERSON] [20] presented a software gain model in a random field environment, considering the time required to remove faults during in-house testing, the cost of removing faults during beta testing, and the risk cost owing to software failure. Moreover, they considered the benefits of reliable software execution during beta testing and field operation. [PERSON] and [PERSON] [21] presented a generalized cost model subject to random field environments, considering the cost of removing failures during testing and warranty periods and the penalty cost owing to system failure. [PERSON] et al. [15] presented a software cost model that considers the difference between testing and operating environments on the basis of an NHPP SRM. [PERSON] et al. [22] proposed a cost-reliability SRGM evaluation and selection algorithm to study the current software cost model and optimal release, which do not fully consider actual defects in the testing phase. [PERSON] et al. [23] proposed a framework for obtaining the optimal release and warranty times of software to minimize the overall cost under reliability constraints. [PERSON] and [PERSON] [24] proposed an SRM that considers defect removal efficiency for modern scenarios, including automated testing and debugging tools. The optimal release that satisfies reliability require ments was subsequently determined by considering the warranty, risk, and error-removal costs. [PERSON] et al. (2015) analyzed a software cost model based on an exponential SRGM. Probability distributions play important roles in various research fields, including statistics, physics, and engineering, where the Rayleigh distribution is frequently used ([PERSON], 1956). Named after the British physicist [PERSON], this distribution has wide applications in diverse fields, such as wireless communications, radar systems, and atmospheric science. The Rayleigh distribution is a continuous probability distribution used to model the magnitude of a positive random variable. In particular, it is a combination of the normal distributions of several random variables. The Rayleigh distribution has a probability density function of \(\mathrm{f}(t)=2a^{2}te^{-(at)^{2}}\) and a cumulative distribution function of \(\mathrm{F}(t)=1-e^{-(at)^{2}}\), where \(a\) is a scale parameter and \(t\geq 0\). The Rayleigh distribution has several important properties that make it useful for analyzing and modeling random phenomena. For example, it can only be applied to nonnegative random variables. Specifically, it is unimodal and has a single peak. Unlike many other probability distributions, the Rayleigh distribution has no defined mean because it does not have a finite upper bound. In addition, various degrees of skewness are exhibited. The Rayleigh distribution is used in reliability engineering to model the lifetime or failure time of a specific component or system. Fitting observed data to a Rayleigh distribution allows engineers to estimate product reliability and failure rates, which can be used to design maintenance strategies and optimize system performance. In summary, the Rayleigh distribution is an important probability distribution with wide-ranging applications in various fields. The ability to model positive-magnitude phenomena, along with its mathematical properties, is essential for statistical analysis, modeling, and simulation. Understanding the Rayleigh distribution provides researchers, engineers, and scientists with valuable insights into the behavior of random variables, allowing informed decision-making in their respective fields.