  We study two kinetically constrained models in a quenched random environment.
The first model is a mixed threshold Fredrickson-Andersen model on
$\mathbb{Z}^{2}$, where the update threshold is either $1$ or $2$. The second
is a mixture of the Fredrickson-Andersen $1$-spin facilitated constraint and
the North-East constraint in $\mathbb{Z}^{2}$. We compare three time scales
related to these models - the bootstrap percolation time for emptying the
origin, the relaxation time of the kinetically constrained model, and the time
for emptying the origin of the kinetically constrained model - and understand
the effect of the random environment on each of them.
