  The time-dependent barrier passage of an anomalous system-reservoir coupling
non-equilibrium open environment is studied where the heat bath is modulated by
an external noise. The time-dependent barrier passing probability is obtained
analytically by solving the generalized Langevin equation. The escaping rate
and effective transmission coefficient are calculated by using of the reactive
flux method in the particular case of external $\delta$-correlated noise
modulated internal Ornstein-Uhlenbeck process. It is found that not all the
cases of external noise modulating is harmful to the rate process. Sometimes it
is even beneficial to the diffusion of the particle. There exists an optimal
strength of external noise modulation for the particle to obtain a biggest
probability to escape from the potential well to form a notable rate of
effective net flux.
