# Default Detection Rate-Dependent Software Reliability Model with Imperfect Debugging
## 1 Introduction
From the perspective of adjusting R-squared and mean squared error indicators, the model has excellent performance. The imperfect troubleshooting model established in reference [19; 20] incorporates new fault generation and change points, making software reliability more able to describe  
Figure 1: Research Ideas and Functions of SRGM. the real situation in testing. From the perspective of test coverage, reference [23] established an imperfect troubleshooting software reliability model related to the test coverage function. In general, These studies are from different perspectives and contents, but lack integrated research on imperfect debugging in the software testing process, thus can not accurately establish the quantitative relationship between fault detection, repair and introduction in the software testing process. The purpose of the test process is to verify and confirm whether the software meets the requirements. It can help to find problems that do not meet the requirements during the testing process and inform the developer to modify them in time to improve the reliability, so as to meet the desired goals. Objectively speaking, due to the difference in testing environment and testing strategy adopted by testers, test indicators have differences in the software testing process, which is representatively reflected in FDR (fault detection rate). From the perspective of establishing a mathematical model, the difference between models has a close relation to FDR. FDR can macroscopically describe the fault detection ability in a testing environment, which makes it the major point in evaluating the performance of the SRGM. In addition to testing coverage and all faults in software, FDRF (FDR Function) \(b(t)\) is an important factor affecting SRGM. Therefore, the study needs to analyze the influence of FDRF \(b(t)\) on models or reliability. This paper first develops a unified fault detection (including repair) model that establishes a single differential equation used in FDR studies. Then, the paper establishes another fault detection model including fault detection, repair and introduction and builds a system of differential equations used in FDR study. Through research on the performance of these two reliability models supported by FDR, it is possible to distinguish different FDR performances. In addition, the research verifies the importance of establishing a reliability model with abundant test information to enhance reliability. The method of software reliability process analysis with imperfect debugging proposed in this paper focuses on the influence of FDR. It can describe the software testing process more accurately. It is of great significance to improve software reliability. In this paper, the structure is arranged as follows: Section 2 gives a general review of imperfect debugging and FDR; Section 3 gives the SRGM modeling process related to the fault detection rate under imperfect debugging, proposes two specific models of imperfect debugging frameworks, and obtains a unified cumulative fault detection function; and Section 4 gives the specific imperfect debugging models supported by five types of FDR. Section 5 verifies the validity and rationality of the proposed model through published failure data sets. Finally, the summary makes conclusions and points out the directions for future research of SRGM.