Barrier-crossing processes are often described using Markovian models, where reaction rates are quantified by mean first-passage times (MFPTs), barrier-escape times, or state-correlation functions. However, for systems exhibiting non-Markovian dynamics, where memory effects play a significant role, these metrics can differ substantially, complicating the interpretation of reaction kinetics from experimental or simulated time-series data. Here, we investigate the numerical evaluation of MFPTs and escape times in non-Markovian systems. For the MFPTs, we focus on two distinct computational approaches: one that considers only first arrivals between states, equivalent to the waiting or dwell time, and another that includes all first passages for a barrier-crossing event. Using extensive simulations of a one-dimensional generalized Langevin equation, we show that these methods yield equivalent results in the Markovian limit but diverge significantly under strong non-Markovian conditions. We derive a relationship between their respective passage-time distributions, which we validate against simulation data. Analysis of these distributions reveals that mean reaction times can be dominated either by rapid state recrossing or by slow escape dynamics. As such, our results highlight the importance of examining full kinetic distributions, rather than just mean values, to accurately characterize barrier-crossing dynamics in systems with memory.
