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For the satellite dwarf galaxies in the LG, some works have shown that the environmental quenching \emph{efficiency} is higher than than for more massive satellites in massive groups/clusters \citep{Phillips2014, SlaterBell2014}.  In this letter, we combine the observed quiescent fractions for satellites in the LG with the typical infall times of such satellites from cosmological simulations to infer the timescales over which environmental processes remove gas and quench star formation in the current satellite galaxies in the MW/M31 halos.  Motivated by the results of \citet{Wetzel2015}, we also consider the possible impact of group preprocessing on satellites before they fell into the MW/M31 halos.  We also compare with the above works on more massive satellites, allowing us to examine satellite quenching timescales across the observablemass  range of $\mstar=10^{3-11}\msun$.