## 1 Introduction  
With the latest technological developments, the software industry is at the center of the fourth industrial revolution. The fourth industrial revolution relies on new and innovative information and communication technologies, cyber-physical systems, network communications, simulation, big data analysis, and cloud computing. Software systems play a vital role in controlling key machines in large industries such as aviation, medical, defense, and energy. In today's complex and rapidly changing environment, where software applications must be developed quickly and easily, software must be focused on rapidly changing information technology. Software systems improve solutions for immediate problems in a variety of industries and continue to offer customers convenience. The systems should be easy-to-use, error-free, and create a product that gives value to its users. To create good software, functions must be implemented that exactly match user requirements, and guarantee reliability, functionality, and performance. Software reliability, defined as the probability of failure-free operation under certain conditions and for a specific time, is one of the significant attributes of the software system development life cycle. Many software reliability models (SRMs) have been proposed to measure, predict, and ensure reliability. Moreover, the growth of reliability,and the trade-off between cost expenditure and optimal release both depend on the accuracy of the established SRM. The basic goal is to produce high-quality software at low cost in software engineering. However, because of the complexity of software systems, software development can be time-consuming and expensive. Therefore, the main focus of software companies is on improving the reliability and stability of a software system. This has prompted research into software reliability engineering and many software reliability growth models have been proposed in recent decades. SRMs are used to estimate and predict the reliability, number of remaining faults, failure intensity, total development cost, etc., of software. Discovering the reliability confidence intervals is done in the field of software reliability because it can aid the decision of software releases and control the related expenditures for software testing [1]. The purpose of many nonhomogeneous Poisson process (NHPP) software reliability models is to obtain an explicit formula for the mean value function m(t), which is applied to the software testing data and used to make predictions of software failures and reliability in the field environments [2]. The Goel-Okumoto (GO) [3] model is one of the most representative studies of SRMs. The GO model defines the mean value function m(t) and the intensity function \(\lambda\)(t) using an exponential distribution and estimates the reliability during mission time t by estimating the number of failures in the course of removing the remaining defects in the software. [PERSON] et al. [4] and [PERSON] [5] developed a model of the software fault detection process to evaluate software reliability based on test results during software development. Since then, many models using environmental changes have been developed; [PERSON] and [PERSON] [6] discussed a generalized model that captures the uncertainty of the environment and its effects upon the software failure rate. Recently, [PERSON] et al. [7] conducted software reliability modeling with uncertainty of testing environments. In addition, [PERSON] and [PERSON] [2,8] proposed NHPP SRMs considering fault removal efficiency and error generation, and the uncertainty of operating environments with imperfect debugging and testing coverage. [PERSON] et al. [9, 10, 11] studied NHPP SRMs with various fault detection rates considering the uncertainty of operating environments. Achieving the unique intended purpose of the software is more important than anything else. If a software failure occurs, such as when the software runs well and suddenly stops, it causes a huge loss to the enterprise and society.