Software reliability modelling is the mathematical technique used to evaluate the reliability of a software system. The non-homogeneous Poisson process is a prominent approach in this field. More than half of the models in this group are based on S-shaped functions, primarily the 2-parameter S-shaped function. This paper proposes a new model based on the 3-parameter S-shaped function, which is an expanded form of the 2-parameter S-shaped function obtained by adding a growth rate controller. Real data from industrial software development projects are used to verify the usability of the proposed model. The proposed model is shown to perform better than the existing models, especially with respect to the predictive performance. Furthermore, the rate of convergence of the proposed model is acceptable, with a rate of 76.47%.