# Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning
## 1 Introduction  
Over time, software has become increasingly important in various fields. In the past, software helped with simple tasks or small parts in each industry; however, it has now been developed into a form that can perform basic roles in all-inclusive roles. If the current software is more dependent than in the past and is broken owing to large and small issues, such as certain coding and system errors, it is expected to cause significant damage to the entire industry. To address this problem, studies on software reliability, which measure how effectively the software works, are constantly being conducted. Software reliability tests the ability of software to not fail for a specified period and indicates how long the software can be used without failure. Studies on software reliability have been conducted based on the software reliability model, assuming the Markov model and the non-homogeneous Poisson process (NHPP) [1]. However, since software failures follow a Poisson distribution, most related studies have been conducted based on the software reliability model, assuming NHPP. Previous studies on software reliability began when a software reliability model based on NHPP was developed by [PERSON] and [PERSON] [2]. It was assumed that software failures occur independently and do not affect each other. Subsequently, a study was conducted on an S-shaped curve model in which the number of cumulative software failures increased along the S curve [3; 4; 5]. In addition, a software reliability model study, assuming incomplete debugging in which the defects found in the test phase were not corrected or removed, was also conducted [6]. Byexpanding on this, a study on a generalized incomplete debugging defect detection rate model was conducted [7; 8; 9; 10; 11]. The defect detection rate of each proposed model is in the form of a function. In addition, since the operating environment is different for each software, it is difficult to compare them equally. Thus, the software reliability model has been studied, considering uncertain factors in the operating environment [12; 13]. As software develops over time, it becomes composed of complex and intertwined structures; therefore, studies on software reliability models are being conducted, assuming that software failures occur independently [14]. A software reliability model that assumes that previous errors will continuously affect software failures if they are not well fixed has been proposed, and a dependent software reliability model that assumes an uncertain operating environment has been studied [15; 16]. As research progressed, several software reliability models studied under independent and dependent assumptions developed numerous models suitable for special cases, and generalizing them has been difficult. To address this, studies on software reliability models using deep learning, which is a nonparametric method, have been performed. Among the nonparametric software reliability models using deep learning, a study on software reliability models using deep neural networks was conducted [17], and a software reliability model using deep learning that applies failure data generated through open-source software was also proposed [18]. In addition, since software failure is a sequential data characteristic, the software reliability model using the recurrent neural network (RNN) and the long short-term memory (LSTM) was studied using this characteristic [19; 20; 21]. The software reliability model developed by NHPP has the problem that it fits well only in special cases because it adds special mathematical assumptions. Therefore, in this study, to solve this problem, we aim at the software reliability model utilizing deep learning among machine learning methods, which is a form that relies on given failure data. Since it is a data-dependent model, it is possible to develop a generalized software reliability model that considers all failures that occur in software without relying on special cases of software failures.