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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.  The quenching timescales at $\mstar<10^8\msun$ are short, $< 2 - 3 \gyr$, $<2-3\gyr$,  depending somewhat on whether environmental preprocessing in lower-mass groups is important, and are comparable to a virial-radius crossing time. We compare with timescales for more massive satellites from the literature, which implies that environmental quenching timescale increases rapidly with satellite mass to $\approx9.5\gyr$ at $\mstar\approx 10^9\msun$, but it then rapidly decreases with mass to $<5\gyr$ at $\mstar>5\times 10^9\msun$.  Thus, satellites with masses similar to the Magellanic Clouds exhibit the longest environmental quenching timescales.