Software Reliability Growth Model Combining Testing Effort Function and Burr-Type Fault Detection Rate

Software reliability growth models (SRGMs) often assume a linear relationship between the fault detection rate (FDR) and testing effort function (TEF), which fails to capture their dynamic and nonlinear characteristics. To address this limitation, this paper proposes a novel SRGM framework that employs Burr-III and Burr-XII distributions to characterize the FDR, integrated with S-shaped TEFs. To tackle the parameter estimation challenge for such complex models, we designed a hybrid GRU-HMM deep learning framework. Experiments on multiple real-world datasets demonstrate that the proposed models (particularly III-is and XII-is) significantly outperform traditional baseline models in both goodness-of-fit and prediction accuracy. Quantitatively, on the DS1 dataset, the III-is model reduced the MSE from 110.7 to 102.9 and improved the AIC from 108.3 to 91.7 compared to the best baseline. On the DS2 dataset, the XII-is model notably decreased the MSE from 64.2 to 48.9. These results not only validate the theoretical advantage of combining Burr distributions with S-shaped TEFs in modeling nonlinear, multi-phase testing dynamics but also provide a practical solution for high-precision reliability assessment and resource planning in complex software testing environments.