# Software Reliability Growth Model with Dependent Failures and Uncertain Operating Environments
## 1 Introduction  
Since software is used in various industries, software reliability is an extremely important issue. For example, in the medical industry, software is used to provide medical services to underprivileged individuals. If a problem with software reliability occurs, incorrect medical information may be provided. Thus, the importance of software reliability cannot be overemphasized. The software used in various industries has many different functions and its structure is complex. In addition, the reliability (stability) of both software and hardware is extremely important. Software reliability issues can cause financial and human losses. Thus, many researchers have been studying software reliability for decades. To conduct software reliability studies, software reliability growth models (SRGM) should be understood. A SRGM is a tool used to estimate the quality and reliability of software products and provide information to developers and consumers. Consumers can refer to such information to select products for purchase, and developers can judge the reliability of a product and efficiently manage the development process. In most existing SRGMs, the mean value function is considered to follow a non-homogeneous Poisson process (NHPP). Each model has a unique mean value function that considers the failure intensity, detection rate, number of remaining failures, and various environments (e.g., development, testing, and operation). In particular, the form and parameters of the mean value function (m(t)) are significantly affected by the environments and assumptions made. Previous studies have considered various SRGMs when discussing software reliability. The model proposed by [PERSON] and [PERSON] [1] is the most fundamental approach used in research on SRGMs and has become the focal point of various studies. [PERSON] et al. [2] proposed a model using an S-shaped curve. There have also been studies on models thatreflect not only the shape of the function, but also various environments. [PERSON] et al. [3,4] introduced models that consider an error introduction rate that follows an exponential function and a generalized SRGM that reflects the testing coverage. [PERSON] et al. [5] proposed a model for considering the test coverage. [PERSON] [6] discussed a model that considers a logistic time-dependent fault-detection rate function. In addition, [PERSON] [7] introduced a V-tub-shaped rate function. [PERSON] et al. [8] presented an improved model for a generalized inflection S-shaped testing effort function. [PERSON] et al. [9] also developed a generalized inflection S-shaped SRGM. [PERSON] et al. [10] proposed a SRGM that assumes imperfect debugging by a two-step process that considers fault observation and fault removal. [PERSON] [11] presented a new Software Reliability Growth Model (SRGM) with the structure of a Logistic Growth Model in which the fault detection rate increases as the test department's skill improves as the test progresses. [PERSON] et al. [12] developed a SRGM with a bathtub-shaped fault detection rate function and proposed the conditional maximization algorithms to fit the models. There are also SRGM studies that introduce different assumptions from previous studies. For example, the controlled test environment and actual operating environment of the consumers can vary significantly. [PERSON] et al. [13,14] discussed models that consider the uncertainty of the operating environments using new parameters as random variables. [PERSON] et al. [15] discussed a method to evaluate reliability analysis considering various types of uncertainty. Most SRGMs assume that failures are independent. However, a failure is occasionally dependent. [PERSON] et al. [16] proposed a method for efficiently judging the software reliability using the model development, assuming a dependent failure and applying a sequential probability ratio test. [PERSON] et al. [17] developed a new model that assumes the presence of dependent defects.