Since the early 1970s tremendous growth has been seen in the research of software reliability growth modeling. In general, software reliability growth models (SRGMs) are applicable to the late stages of testing in software development and they can provide useful information about how to improve the reliability of software products. A number of SRGMs have been proposed in the literature to represent time-dependent fault identification / removal phenomenon; still new models are being proposed that could fit a greater number of reliability growth curves. Often, it is assumed that detected faults are immediately corrected when mathematical models are developed. This assumption may not be realistic in practice because the time to remove a detected fault depends on the complexity of the fault, the skill and experience of the personnel, the size of the debugging team, the technique, and so on. Thus, the detected fault need not be immediately removed, and it may lag the fault detection process by a delay effect factor. In this paper, we first review how different software reliability growth models have been developed, where fault detection process is dependent not only on the number of residual fault content but also on the testing time, and see how these models can be reinterpreted as the delayed fault detection model by using a delay effect factor. Based on the power function of the testing time concept, we propose four new SRGMs that assume the presence of two types of faults in the software: leading and dependent faults. Leading faults are those that can be removed upon a failure being observed. However, dependent faults are masked by leading faults and can only be removed after the corresponding leading fault has been removed with a debugging time lag. These models have been tested on real software error data to show its goodness of fit, predictive validity and applicability.