  We consider an interacting particle system on the lattice involving pushing
and blocking interactions, called PushASEP, in the presence of a wall at the
origin. We show that the invariant measure of this system is equal in
distribution to a vector of point-to-line last passage percolation times in a
random geometrically distributed environment. The largest co-ordinates in both
of these vectors are equal in distribution to the all-time supremum of a
non-colliding random walk.
