  A random walk model in a one dimensional disordered medium with an
oscillatory input current is presented as a generic model of boundary
perturbation methods to investigate properties of a transport process in a
disordered medium. It is rigorously shown that an admittance which is equal to
the Fourier-Laplace transform of the first-passage time distribution is
non-self-averaging when the disorder is strong. The low frequency behavior of
the disorder-averaged admittance, $<\chi > -1 \sim \omega^{\mu}$ where $\mu <
1$, does not coincide with the low frequency behavior of the admittance for any
sample, $\chi - 1 \sim \omega$. It implies that the Cole-Cole plot of $<\chi>$
appears at a different position from the Cole-Cole plots of $\chi$ of any
sample. These results are confirmed by Monte-Carlo simulations.
