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At $\mstar\gtrsim10^9\msun$, the long quenching timescales suggests quenching driven by gas depletion in the absence of cosmic accretion, caused by the stripping of extended gas around the satellite, after infall (``strangulation'').  This scenario also explains the decline of the quenching timescale with increasing $\mstar$, because higher-$\mstar$ (non-satellite) galaxies generally have lower $\mgas/\mstar$ \citep[in either cold atomic or molecular gas, e.g.,][Bradford et al., submitted]{Schiminovich2010, Huang2012, Boselli2014} and thus shorter gas depletion timescales in the absence of accretion.  Conversely, at $\mstar\sim10^9\msun$, galaxies have $\mgas/\mstar\approx1$, with gas depletion timescales comparable to a Hubble time.  Thus, satellite quenching timescales at $\mstar\gtrsim10^9\msun$ do not necessarily \emph{require} strong environmental processes beyond truncated gas accretion \citep[see also discussions in][]{Wetzel2013, Wheeler2014,Phillips2014,  McGee2014}. However, strangulation cannot explain the rollover in satellite quenching times at $\mstar\lesssim10^9\msun$, because the gas-rich dwarf galaxies of the LG also have $\mgas\gtrsim\mstar$ \citep{GrcevichPutman2009} and thus contain enough cold gas to fuel star formation for a Hubble time even absent accretion.  Thus, the rapid decline at lower $\mstar$ \emph{requires} an additional process(es) to remove gas from satellite dwarf galaxies after infall.