Many software reliability growth models based upon a non-homogeneous Poisson process (NHPP) have been proposed to measure and asses the reliability of a software system quantitatively. Generally, the error detection rate and the fault content function during software testing is considered to be dependent on the elapsed time testing. In this paper we have proposed three software reliability growth models (SRGM's) incorporating the notion of error generation over the time as an extension of the delayed S-shaped software reliability growth model based on a non-homogeneous Poisson process (NHPP). The model parameters are estimated using the maximum likelihood method for interval domain data and three data sets are provided to illustrate the estimation technique. The proposed model is compared with the existing delayed S-shaped model based on error sum of squares, mean sum of squares, predictive ratio risk and Akaike's information criteria using three different data sets. We show that the proposed models perform satisfactory better than the existing models.