We investigate finite and semi-infinite particle systems on a one-dimensional integer lattice, where each particle undergoes a continuous-time nearest-neighbour random walk with intrinsic jump rates. The system is governed by an exclusion interaction that prevents multiple particles from occupying the same site.

For finite particle systems, we establish that the jump rates uniquely determine a maximal partition into stable sub-systems. We further demonstrate that this partition can be computed efficiently through a linear-time algorithm based on elementary steps or by solving a finite nonlinear system.

For semi-infinite particle systems, we consider initial conditions that involve finite perturbations of a close-packed configuration. Under suitable conditions, we show that the particles collectively form a semi-infinite “stable cloud.” Specifically, inter-particle separations converge to a product-geometric stationary distribution, while each particle’s location follows a strong law of large numbers with a common characteristic speed.