In this article, we propose a stochastic model called the Gompertz software reliability model based on the familiar non-homogeneous Poisson process. It is shown that the proposed model can be derived from the well-known statistical theory of extreme-value and has the quite similar asymptotic property to the classical Gompertz curve. In a numerical example with the software failure data observed in a real software development project, we apply the Gompertz software reliability model to assess the software reliability and to predict the number of initial fault contents. We empirically conclude that our new model may function better than the existing models and is attractive in terms of goodness-of-fit test based on information criteria and mean squared error.