  We model satellite quenching at $z \sim 1$ by combining $14$ massive
($10^{13.8} < M_{\mathrm{halo}}/\mathrm{M}_{\odot} < 10^{15}$) clusters at $0.8
< z < 1.3$ from the GOGREEN and GCLASS surveys with accretion histories of $56$
redshift-matched analogs from the IllustrisTNG simulation. Our fiducial model,
which is parameterized by the satellite quenching timescale ($\tau_{\rm
quench}$), accounts for quenching in our simulated satellite population both at
the time of infall by using the observed coeval field quenched fraction and
after infall by tuning $\tau_{\rm quench}$ to reproduce the observed satellite
quenched fraction versus stellar mass trend. This model successfully reproduces
the observed satellite quenched fraction as a function of stellar mass (by
construction), projected cluster-centric radius, and redshift and is consistent
with the observed field and cluster stellar mass functions at $z \sim 1$. We
find that the satellite quenching timescale is mass dependent, in conflict with
some previous studies at low and intermediate redshift. Over the stellar mass
range probed ($M_{\star}> 10^{10}~\mathrm{M}_{\odot}$), we find that the
satellite quenching timescale decreases with increasing satellite stellar mass
from $\sim1.6~{\rm Gyr}$ at $10^{10}~\mathrm{M}_{\odot}$ to $\sim 0.6 - 1~{\rm
Gyr}$ at $10^{11}~\mathrm{M}_{\odot}$ and is roughly consistent with the total
cold gas (H{\scriptsize I}+H$_{2}$) depletion timescales at intermediate $z$,
suggesting that starvation may be the dominant driver of environmental
quenching at $z < 2$. Finally, while environmental mechanisms are relatively
efficient at quenching massive satellites, we find that the majority
($\sim65-80\%$) of ultra-massive satellites ($M_{\star} >
10^{11}~\mathrm{M}_{\odot}$) are quenched prior to infall.
