  We study stationary states in a diluted asymmetric (kinetic) Ising model. We
apply the recently introduced dynamic cavity method to compute magnetizations
of these stationary states. Depending on the update rule, different versions of
the dynamic cavity method apply. We here study synchronous updates and random
sequential updates, and compare local properties computed by the dynamic cavity
method to numerical simulations. Using both types of updates, the dynamic
cavity method is highly accurate at high enough temperatures. At low enough
temperatures, for sequential updates the dynamic cavity method tends to a fixed
point, but which does not agree with numerical simulations, while for parallel
updates, the dynamic cavity method may display cyclic behavior. When it
converges and is accurate, the dynamic cavity method offers a huge speed-up
compared to Monte Carlo, particularly for large systems.
