Stochastic models with boundaries and quadratic algebras

We derive a series of stochastic models on linear chains having their stationary probability distributions described by the matrix product Ansatz. We consider models with different species on a line with closed or open ends. The algebras we obtain and the type of representation depend on the boundary conditions. In the language of quantum chains we obtain the ground state of -state quantum chains with free boundary conditions or non-diagonal boundaries in one or both ends.