On bifibrations of model categories

In this article, we develop a notion of Quillen bifibration whose purpose is to combine the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration p:E→B, we describe when a family of model structures on the fibers EA and on the basis category B combines into a model structure on the total category E, such that the functor p preserves cofibrations, fibrations and weak equivalences. Using this Grothendieck construction for model structures, we revisit the traditional definition of Reedy model structures, and possible generalizations, and exhibit their bifibrational nature.