\label{sec:infall_time_v_mass}
We first examine how the virial-infall times of satellites at \(z = 0\) depend on their mass. Figure \ref{fig:infall.time_v_mass} shows \({t_{\rm first\,infall}^{\rm since}}\) (top) and \({t_{\rm MW/M31\,infall}^{\rm since}}\) (bottom), or \({z_{\rm first\,infall}}\) and \({z_{\rm MW/M31\,infall}}\) on right axes, as a function of satellite \({M_{\rm star}}\), or subhalo \({M_{\rm peak}}\) on the top axes.
For both virial-infall metrics, lower-mass satellites fell in systematically earlier, though significant scatter persists at all masses. Such a trend with \({M_{\rm peak}}\) (and thus \({M_{\rm star}}\)) is a natural result of hierarchical structure formation, for two reasons. First, halos of a given \({M_{\rm peak}}\) are more common at later cosmic times, and \({M_{\rm peak}}\), by its definition, typically remains unchanged for satellites after infall. Thus, higher-mass satellites are more likely to have formed, and subsequently fallen in, at later time. Second, satellites with higher \({M_{\rm peak}}\) have shorter dynamical friction lifetimes (for fixed host-halo mass) before they tidally disrupt or merge with the host halo \citep{BoylanKolchin2008,Jiang2008,WetzelWhite2010}.
Our lowest-mass (ultra-faint) satellites first fell into any host halo typically \(\sim 10 {~\mbox{Gyr}}\) ago at \(z \sim 1.7\), and they first fell into the MW/M31 halo \(\sim 7.5 {~\mbox{Gyr}}\) ago at \(z \sim 1\). By contrast, our highest-mass satellites (corresponding to the LMC, SMC, NGC 205, M32) have \({t_{\rm first\,infall}^{\rm since}}\sim 6.5 {~\mbox{Gyr}}\) (\({z_{\rm first\,infall}}\sim 0.8\)) and \({t_{\rm MW/M31\,infall}^{\rm since}}\sim 5 {~\mbox{Gyr}}\) (\({z_{\rm MW/M31\,infall}}\sim 0.5\)). Thus, satellites at \(z = 0\) first fell into any host halo \(1.5 - 2.5 {~\mbox{Gyr}}\) before falling into the MW/M31 halo, a generic outcome of hierarchical structure formation, as we will examine further in Section \ref{sec:preprocessing_duration_mass}.
Overall, satellite dwarf galaxies at \(z = 0\) typically have evolved as satellites in a host halo for over half of their entire history, so the host-halo environment typically has had significant time to affect their evolution.