  We suggest a novel stochastic discrete growth model which describes the
drifted Edward-Wilkinson (EW) equation $\partial h /\partial t = \nu
\partial_x^2 h - v\partial_x h +\eta(x,t)$. From the stochastic model, the
anomalous behavior of the drifted EW equation with a defect is analyzed. To
physically understand the anomalous behavior the height-height correlation
functions $C(r)=< |h({x_0}+r)-h(x_0)|>$ and $G(r)=< |h({x_0}+r)-h(x_0)|^2>$ are
also investigated, where the defect is located at $x_0$. The height-height
correlation functions follow the power law $C(r)\sim r^{\alpha'}$ and $G(r)\sim
r^{\alpha''}$ with $\alpha'=\alpha''=1/4$ around a perfect defect at which no
growth process is allowed. $\alpha'=\alpha''=1/4$ is the same as the anomalous
roughness exponent $\alpha=1/4$. For the weak defect at which the growth
process is partially allowed, the normal EW behavior is recovered. We also
suggest a new type queuing process based on the asymmetry $C(r) \neq C(-r)$ of
the correlation function around the perfect defect.
