  The steady states of three families of one-dimensional non-equilibrium models
with open boundaries, first proposed in [22], are studied using a matrix
product formalism. It is shown that their associated quadratic algebras have
two-dimensional representations, provided that the transition rates lie on
specific manifolds of parameters . Exact expressions for the correlation
functions of each model have also been obtained. We have also studied the
steady state properties of one of these models, first introduced in [23], with
more details. By introducing a canonical ensemble we calculate the canonical
partition function of this model exactly. Using the Yang-Lee theory of phase
transitions we spot a second-order phase transition from a power-law to a
jammed phase. The density profile of particles in each phase has also been
studied. A simple generalization of this model in which both the left and the
right boundaries are open has also been introduced. It is shown that double
shock structures may evolve in the system under certain conditions.
