We present an analysis of galaxies in groups and clusters at $0.8<z<1.2$, from the GCLASS and GEEC2 spectroscopic surveys. We compute a conversion fraction $f_{rm convert}$ that represents the fraction of galaxies that were prematurely quenched by their environment. For massive galaxies, $M_{rm star}>10^{10.3}M_odot$, we find $f_{rm convert}sim 0.4$ in the groups and $sim 0.6$ in the clusters, similar to comparable measurements at $z=0$. This means the time between first accretion into a more massive halo and final star formation quenching is $t_psim 2$ Gyr. This is substantially longer than the estimated time required for a galaxys star formation rate to become zero once it starts to decline, suggesting there is a long delay time during which little differential evolution occurs. In contrast with local observations we find evidence that this delay timescale may depend on stellar mass, with $t_p$ approaching $t_{rm Hubble}$ for $M_{rm star}sim 10^{9.5}M_odot$. The result suggests that the delay time must not only be much shorter than it is today, but may also depend on stellar mass in a way that is not consistent with a simple evolution in proportion to the dynamical time. Instead, we find the data are well-matched by a model in which the decline in star formation is due to overconsumption, the exhaustion of a gas reservoir through star formation and expulsion via modest outflows in the absence of cosmological accretion. Dynamical gas removal processes, which are likely dominant in quenching newly accreted satellites today, may play only a secondary role at $z=1$.
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