Waldhausen's wS construction of K-theory assigns K-groups to an arbitrary small Waldhausen category, my main goal in reading this construction was to apply it to the case of exact categories with weak equivalencec being isomorphisms. Now my question is, what are the other examples of Waldhausen categories? Are their K-groups serious studied?
Also if I am not mistaken, the category of cofibrant objects in a model category satisfy the axioms of a Waldhausen category, but may not be a small category. Can we talk about the K-groups of some suitable small subcategory of such a category?