  We study semi-infinite particle systems on the one-dimensional integer
lattice, where each particle performs a continuous-time nearest-neighbour
random walk, with jump rates intrinsic to each particle, subject to an
exclusion interaction which suppresses jumps that would lead to more than one
particle occupying any site. Under appropriate hypotheses on the jump rates
(uniformly bounded rates is sufficient) and started from an initial condition
that is a finite perturbation of the close-packed configuration, we give
conditions under which the particles evolve as a single, semi-infinite "stable
cloud". More precisely, we show that inter-particle separations converge to a
product-geometric stationary distribution, and that the location of every
particle obeys a strong law of large numbers with the same characteristic
speed.
