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Several environmental processes within a host halo regulate the gas content, star formation, morphology, and eventual tidal disruption of satellite galaxies, including gravitational tidal forces \citep[e.g.,][]{Dekel2003}, galaxy-galaxy interactions \citep[e.g.,][]{FaroukiShapiro1981} and mergers \citep[e.g.,][]{Deason2014a}, tidal shocking and resonant interactions with the host \citep[e.g.,][]{Mayer2001,DOnghia2010}, ram-pressure stripping of extended gas around the satellite \citep[e.g.,][]{Larson1980,McCarthy2008} of the cold inter-stellar medium \citep[e.g.,][]{GunnGott1972,Tonnesen2009}, many of which can be assisted by feedback from stars and/or AGN within the satellites \citep[e.g.,][]{BaheMcCarthy2015}.  One strong constraint on the relative effects of these processes comes in determining the dependence of the timescale of environmental quenching on both the mass of the host and the satellites, as has been done in previous works at higher satellite mas scales \citep[e.g.,][]{Balogh2000,Wetzel2013,Wheeler2014}. \citep[e.g.,][]{Balogh2000,Wetzel2013,Hirschmann2014,Wheeler2014}.  While previous worked examined the relative efficiency of satellite quenching in the LG \citep{Phillips2014,SlaterBell2014}, none have constrained the timescales over which environment acts to remove gas and quenching star formation as a function of satellite mass.  In this letter, we combine the observed quiescent fractions of satellites in the LG with the typical virial-infall times of such satellites from simulations to infer the environmental quenching timescales of the current satellite galaxies in the LG.