The Murthy (1957) estimator has been extensively studied by various authors. The design based variance of the estimator has been studied by the various survey statisticians. The modifications of the Murthy (1957) estimator has been proposed by Shahbaz (2004) alongside the design based study. The model based study of the estimator has been presented in this monograph. We have also studied the stability of the variance estimator of the modified Murthy (1957) estimator under the linear super-population model. It has been observed that the anticipated variance of the modified Murthy (1957) estimator achieves the Godambe-Joshi (1965) lower bound and the variance estimator is stable.