In a totally asymmetric simple $K$-exclusion process, particles take nearest-neighbor steps to the right on the lattice Z, under the constraint that each site contain at most $K$ particles. We prove that such processes satisfy hydrodynamic limits under Euler scaling, and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function. Our technique requires no knowledge of the invariant measures of the process, which is essential because the equilibria of asymmetric $K$-exclusion are unknown. But we cannot calculate the flux function precisely. The proof proceeds via a coupling with a growth model on the two-dimensional lattice. In addition to the basic $K$-exclusion with constant exponential jump rates, we treat the sitedisordered case where each site has its own jump rate, randomly chosen but frozen for all time. The hydrodynamic limit under site disorder is new even for the simple exclusion process (the case $K = 1$). Our proof makes no use of the Markov property, so at the end of the paper we indicate how to treat the case with arbitrary waiting times.
"Existence of Hydrodynamics for the Totally Asymmetric Simple K-Exclusion Process." Ann. Probab. 27 (1) 361 - 415, January 1999. https://doi.org/10.1214/aop/1022677266