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One strong constraint for understanding the relative effects of environmental processes is determining the timescale over which environmental quenching occurs, as previous works have explored at higher masses \citep[e.g.,][]{Balogh2000, Wetzel2013, Hirschmann2014, Wheeler2014}.  Some works have shown that the environmental quenching \emph{efficiency} for the satellite dwarf galaxies in the LG is higher than than for more massive satellites in massive groups/clusters \citep{Phillips2014, SlaterBell2014}, but no works yet have constrained the quenching \emph{timescales} at these masses.  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 halo. halos.