A Stochastic Differential Equation-Based Software Reliability Growth Model with Inverse-Weibull Intensity for Open-Source Software

Abstract

In this paper, we propose a continuous-state software reliability growth model (SRGM) for open-source software in which the fault-detection rate is governed by an Itô-type stochastic differential equation (SDE) with an intensity function driven by the Inverse-Weibull distribution (IW), allowing heavy-tailed early-failure behaviour typical of community-driven projects. We derive closed-form expressions for the expected number of detected faults and analytically evaluate the instantaneous and cumulative Mean Time Between Failures (MTBF) based on the nominal intensity function. Model parameters are estimated via the Method of Maximum Likelihood (MLE), utilising numerical optimisation to maximise the complex log-likelihood function derived from the SDE scheme. Using 22 weeks of Apache bug-count data, we compare the new model with the classical Goel–Okumoto and Delayed S-shaped SRGMs. The proposed model yields the lowest MSE, SSE, and RMSE and the highest coefficient of determination ($R^{2}$), while providing a robust probabilistic framework for reliability assessment.