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In the Local Group, nearly all of the
satellite dwarf galaxies ($\mstar < 10 ^ 9 \msun$) that are
satellites within $300 \kpc$ (approximately the virial radius) of the Milky Way (MW) and Andromeda (M31) have quiescent star formation and little-to-no cold gas.
This is in strong contrast to more isolated dwarf galaxies at larger distances, 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} removed gas and quenched star formation in such 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 ($\approx 9.5 \gyr$) for satellites at $\mstar \approx 10 ^ 9 \msun$ and decrease to
$< replace_contentlt; 5 \gyr$ at $\mstar > 5 \times 10 ^ 9 \msun$.