We determine the asymptotical satisfiability probability of a random at-most-k-Horn formula, via a probabilistic analysis of a simple version, called PUR, of positive unit resolution. We show that for $k=k(n)\rightarrow \infty$ the problem can be ``reduced'' to the case k(n)=n, that was solved in cs.DS/9912001. On the other hand, in the case k= a constant the behavior of PUR is modeled by a simple queuing chain, leading to a closed-form solution when $k=2$. Our analysis predicts an ``easy-hard-easy'' pattern in this latter case. Under a rescaled parameter, the graphs of satisfaction probability corresponding to finite values of k converge to the one for the uniform case, a ``dimension-dependent behavior'' similar to the one found experimentally by Kirkpatrick and Selman (Science'94) for k-SAT. The phenomenon is qualitatively explained by a threshold property for the number of iterations of PUR makes on random satisfiable Horn formulas.