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This near dichotomy implies that environmental processes within the halos of the MW and M31 \emph{rapidly} remove gas and quench star formation in their satellites after infall.  We combine the observed quiescent fractions of satellites of the MW/M31 with the virial-infall times of satellites in the ELVIS suite of cosmological simulations of MW/M31-like halos to infer statistically the timescales over which satellite dwarf galaxies are environmentally quenched.  The quenching timescales at $\mstar<10^8\msun$ are short: $< 2 - 3 \gyr$, depending on whether environmental preprocessing in lower-mass groups is important.  We compare with the timescales for more massive satellites from the literature, which suggests that environmental quenching timescales are longest ($\approx 9.5 \gyr$) 9.5\gyr$)  for satellites at $\mstar \approx 10 ^ 9 \msun$ and decrease to replace_contentlt;5\gyr$ $<5\gyr$  at $\mstar>5\times 10^9\msun$.