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In cd-sIn  the Local Group, nearly all of the dwarf galaxies ($\mstar<10^9\msun$) that are satellites within $300\kpc$ (the virial radius) of the Milky Way (MW) and Andromeda (M31) have quiescent star formation and little-to-no cold gas. This contrasts strongly with more isolated dwarf galaxies, which are almost all actively star-forming and gas-rich.  This near dichotomy implies that environmental processes within the halos of the MW and M31 \emph{rapidly} remove gas and quench star formation in satellite dwarfs after infall.  We combine the observed quiescent fractions for 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 determine the typical timescales over which environmental processes quench satellite dwarf galaxies after infall.