\label{sec:preprocessing_duration_mass}
We next examine in more detail the durations and host-halo masses that satellites at \(z = 0\) experienced during their group preprocessing, in order to understand better its potential importance on their evolution.
Figure \ref{fig:preprocessing_time_mass_v_mass} (top) shows the distribution of the (maximum) host-halo mass that satellites were in during their group preprocessing as a function of their current \({M_{\rm star}}\), or subhalo \({M_{\rm peak}}\). The typical preprocessing host halo had \({M_{\rm vir}}\sim 10 ^ {11} {~\mbox{M}_\odot}\), with 68% spread of \(10 ^ {10 - 12} {~\mbox{M}_\odot}\), largely independent of satellite mass, though with scatter to lower host-halo masses at lower \({M_{\rm star}}\). About half of satellites were preprocessed in a host halo that does not survive to \(z = 0\), but instead falls into the MW/M31 halo and merges/disrupts, and this accounts for most of the preprocessing hosts with \({M_{\rm vir}}> 10 ^ {11.5} {~\mbox{M}_\odot}\) in Figure \ref{fig:preprocessing_time_mass_v_mass} (top).
Some of these satellites were preprocessed by the other of the paired MW/M31 halos, that is, fell into one of the MW/M31 halos, orbited beyond its \({R_{\rm vir}}\), and then fell into the paired MW/M31 halo. However, these satellites account for \(< 2\%\) of all preprocessed satellites across our mass range, in rough agreement with \citet{Knebe2011}. Thus, we conclude that this is not a particularly important population.
Figure \ref{fig:preprocessing_time_mass_v_mass} (bottom) shows the distribution of the time that satellites spent in their preprocessing host halo. The typical preprocessing time was \(\sim 1.2 {~\mbox{Gyr}}\), with 68% spread of \(0.5 - 3.5 {~\mbox{Gyr}}\), with weak dependence on satellite mass. However, at \({M_{\rm star}}> 10 ^ 7 {~\mbox{M}_\odot}\), no satellites were preprocessed longer than \(1.8 {~\mbox{Gyr}}\), while at \({M_{\rm star}}< 10 ^ 7 {~\mbox{M}_\odot}\), the scatter increases significantly, with some satellites having experienced up to \(7 {~\mbox{Gyr}}\) of preprocessing. The small preprocessing durations at \({M_{\rm star}}> 10 ^ 7 {~\mbox{M}_\odot}\) likely arises because those satellites have \({M_{\rm peak}}\) that approaches that of their preprocessing host-halo in Figure \ref{fig:preprocessing_time_mass_v_mass} (top), corresponding to shorter dynamical friction lifetimes. Thus, massive satellites could not have been preprocessed too long without merging/disrupting within their preprocessing host.
Overall, most preprocessing occurred within groups of \({M_{\rm vir}}= 10 ^ {10 - 12} {~\mbox{M}_\odot}\), masses that feasibly could influence satellite dwarfs galaxies, though environmental effects at these masses remain poorly understood. Furthermore, the typical preprocessing duration was \(0.5 - 3.5 {~\mbox{Gyr}}\), comparable to typical timescales over which satellite dwarf galaxies are environmentally quenched (Wetzel et al., in prep.). Thus, we conclude that such group preprocessing before joining the MW/M31 halo is likely an important component in the evolution of satellite dwarf galaxies.