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\section{Introduction}  Galaxies in denser environments are more likely to have suppressed (quiescent) star formation and little-to-no cold gas than galaxies of similar stellar mass, $\mstar$, in less dense environments.  The observed environmental effects within the Local Group (LG) on the satellite galaxies within the halos of the Milky Way (MW) and Andromeda (M31) are particularly strong \citep[e.g.,][]{Einasto1974,Mateo1998,  GrcevichPutman2009, McConnachie2012, Phillips2014, SlaterBell2014, Spekkens2014}, SlaterBell2014},  even compared to the already strong effects on (more massive) satellites within massive groups/clusters. Specifically, the dwarf galaxies 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), from irregular to spheroidal morphologies, from having most of their baryonic mass in significant  cold atomic gas to having little-to-no measured cold gas, and from actively star-forming to quiescent. This trend has just a few exceptions: 4 gas-rich, star-forming galaxies persist within the halos of the MW (the LMC and SMC) and M31 (LGS 3 and IC 10), and 4 - 5 quiescent, gas-poor galaxies reside well beyond $\rvir$ of either the MW or M31: Cetus \citep{Lewis2007}, Tucana \citep{Fraternali2009}, KKR 25 \citep{Makarov2012}, KKs 3 \citep{Karachentsev2015}, and possibly Andromeda XVIII, though Cetus and Tucana may have orbited within the MW halo \citep{Teyssier2012}.  This efficient satellite quenching is particularly striking because, other than KKR 25 and KKs 3, at $\mstar<10^9\msun$ all known galaxies that are sufficiently isolated ($>1500\kpc$ from a more massive galaxy) are star-forming \citep{Geha2012, Phillips2014}.  Thus, the MW and M31 halos show the strongest signal of environmental influence over their satellites of any known systems, and the LG is a compelling laboratory for studying environmental processes on galaxies.  Several such processes within a host halo can regulate the gas content, star formation, morphology, and eventual disruption of satellite galaxies, including gravitational tidal forces \citep[e.g.,][]{Dekel2003}, galaxy-galaxy tidal interactions \citep[e.g.,][]{FaroukiShapiro1981}, galaxy-galaxy mergers \citep[e.g.,][]{Deason2014a}, and ram-pressure stripping of extended gas \citep[e.g.,][]{Larson1980, McCarthy2008} or cold inter-stellar medium \citep[e.g.,][]{GunnGott1972, Tonnesen2009}, some of which may be assisted by stellar feedback within the satellite \citep[e.g.,][]{NicholsBlandHawthorn2011,BaheMcCarthy2015}. Tonnesen2009}.  The key astrophysical challenge is understanding the relative importance of these processes, including which (if any) dominate, and how they vary across both satellite and host masses.  One strong constraint for understanding the relative effects of environmental processes is comes from  determining the timescale over which environmental quenching occurs, as previous works have explored at higher masses \citep[e.g.,][]{Balogh2000, Wetzel2013, Hirschmann2014, Wheeler2014}. Some For the satellite dwarf galaxies in the LG, 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. 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 simulatilons 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 observable mass range of $\mstar=10^{3-11}\msun$.