Effects of white noise on the transient characteristics of a model nonequilibrium system is analyzed. Various types of characteristic time scales are studied deterministically and stochastically. Special attention is paid to the effect of mixed-order multiplicative noise on the stochastic nonlinear relaxation time. Numerical results show that the critical slowing-down, which was found to exist near the noise-induced instability, is not observed in the nonlinear relaxation time. We discuss these results and comment on the merit of the formulation of nonlinear relaxation time.