We introduce a class of one-dimensional lattice models in which a quantity, which may be thought of as an energy, is either transported from one site to a neighbouring one, or locally dissipated. Transport is controlled by a continuous bias parameter, q, which allows us to study symmetric as well as asymmetric cases. We derive sufficient conditions for the factorization of the N-body stationary distribution and give an explicit solution for the latter, before briefly discussing physically relevant situations.