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\section{Introduction}  Galaxies in denser environments are more likely to have suppressed (quiescent) star-formation rates (SFR) and little-to-no cold gas than galaxies of similar stellar mass, $\mstar$, in less dense environments.  The observed environmental effects in the Local Group (LG), specifically on the satellite dwarf galaxies within the halos of the Milky Way (MW) and Andromeda (M31) (M31),  are particularly strong \citep[e.g.,][]{LyndenBell1974,Mateo1998,McConnachie2012,Phillips2014,SlaterBell2014,Spekkens2014}, even compared to the effects on (more massive) satellites within massive groups/clusters. Specifically, the dwarf galaxies dwarfs  around the MW/M31 show a strikingly sharp and nearly complete transition in their properties within $\approx 300 \kpc$ (approximately the virial radius, $\rvir$, of the MW or M31), transitioning from (1) having irregular to elliptical/spheroidal morphologies, (2) having most of their baryonic mass in cold atomic/molecular gas to having little-to-no measured cold gas, and (3) being actively star-forming to quiescent \citep[e.g.,][and references therein]{McConnachie2012}. This trend has just a few exceptions: four gas-rich, star-forming galaxies persist within the halos of the MW (the LMC and SMC) and M31 (LGS 3 and IC 10), and 3 - 4 quiescent, gas-poor galaxies reside just beyond $\rvir$ of the MW (Cetus and Tucana) and M31 (KKR 25 and possibly Andromeda XVIII), though Cetus and Tucana likely orbited within the MW halo \citep{Teyssier2012}.  Thistrend of  efficient satellite quenching is particularly striking because, other than KKR 25, at $\mstar<10^9\msun$  all known galaxiesat $\mstar<10^9\msun$  that are isolated (not within $1500 \kpc$ (beyond $1500\kpc$  of a more massive galaxy) areactively  star-forming \citep{Geha2012}. Thus, the MW and M31 halos exert the strongest environmental influence on their galaxy populations satellites  of any observed known  systems, making so  the LG one of is perhaps  the most compelling laboratories to study laboratory for studying  environmental effects processes  on galaxy evolution. galaxies.  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}.  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 examine 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, considering LG.  Motivated by the results of \citet{Wetzel2015}, we also consider  the possible impact of group preprocessing. preprocessing on satellites before falling into the MW/M31 halo.