We study two kinetically constrained models in a quenched random environment. The first model is a mixed threshold Fredrickson-Andersen model on Z2, 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 Z2. 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.