Quillen equivalences for stable categories

For an abelian category A we investigate when the stable categories GPro_j(A) and GIn_j(A) are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on A that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from A to the category of chain complexes in A.