A two-species driven-diffusive model of classi- cal particles is introduced on a lattice with periodic boundary condition. The model consists of a finite number of first class particles in the presence of a second class particle. While the first class particles can only hop for- ward, the second class particle is able to hop both forward and backward with specific rates. We have shown that the partition function of this model can be calculated exactly. The model undergoes a non-equilibrium phase transition when a condensation of the first class particles occurs behind the second class particle. The phase transition point and the spatial correlations between the first class particles are calculated exactly. On the other hand, we have shown that this model can be mapped onto a two-dimensional walk model. The random walker can only move on the first quarter of a two-dimensional plane and that it takes the paths which can start at any height and end at any height upper than the height of the starting point. The initial vertex (starting point) and the final vertex (end point) of each lattice path are weighted. The weight of the outset point depends on the height of that point while the weight of the end point depends on the height of both the outset point and the end point of each path. The partition function of this walk model is calculated using a transfer matrix method.
