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One of the characteristic features of a stochastic process under resetting isthat the probability density converges to a nonequilibrium stationary state(NESS). In addition, the approach to the stationary state exhibits a dynamicalphase transition, which can be interpreted as a traveling front separatingspatial regions for which the probability density has relaxed to the NESS fromthose where it has not. One can establish the existence of the phase transitionby carrying out an asymptotic expansion of the exact solution. In this paper wedevelop an alternative, direct method for characterizing the approach to theNESS of a stochastic process with resetting that is based on the calculation ofthe so-called accumulation time. The latter is the analog of the mean firstpassage time of a search process, in which the survival probability density isreplaced by an accumulation fraction density. In the case of one-dimensionalBrownian motion with Poissonian resetting, we derive the asymptotic formula$|x-x_0|\approx \sqrt{4rD}T(x)$ for $|x-x_0|\gg\sqrt{D/r}$, where $T(x)$ is theaccumulation time at $x$, $r$ is the constant resetting rate, $D$ is thediffusivity and $x_0$ is the reset point. This is identical in form to thetraveling front condition for the dynamical phase transition. We also derive ananalogous result for diffusion in higher spatial dimensions and fornon-Poissonian resetting. We then consider the effects of delays such asrefractory periods and finite return times. In both cases we establish that theasymptotic behavior of $T(x)$ is independent of the delays. Finally, we extendthe analysis to a run-and-tumble particle with resetting. We thus establish theaccumulation time of a stochastic process with resetting as a useful quantityfor characterizing the approach to an NESS (if it exists) that is relativelystraightforward to calculate.

Accumulation time of stochastic processes with resetting