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In the Local Group, nearly all of the dwarf galaxies ($\mstar\lesssim10^9\msun$) 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 comparatively isolated dwarf galaxies, which are almost all actively star-forming and gas-rich.
This near dichotomy implies a \emph{rapid} transformation after falling into the halo of the MW/M31.
We combine the observed quiescent fractions for satellites of the MW/M31 with the
virial-infall infall times of satellites from the ELVIS suite of cosmological simulations to determine the typical timescales over which environmental processes remove gas and quench star formation in satellite dwarf galaxies.
The quenching timescales at $\mstar<10^8\msun$ are short, $\lesssim2\gyr$, and decrease with decreasing satellite mass.
%and are comparable to a virial-radius crossing time.
These timescales can increase $1-2\gyr$ if environmental preprocessing in lower-mass groups prior to MW/M31 infall is important.