A　traffic　flow　model　describing　the　formation　and　dynamics　of　traffic　jams　was　introduced　by　Berthelin　et　al.　[＇A　model　for　the　formation　and　evolution　of　traffic　jams,＇Arch.　Ration.　Mech.　Anal.　187,　185-220　(2008)],　which　consists　of　a　pressureless　gas　dynamics　system　under　a　maximal　constraint　on　the　density　and　can　be　derived　from　the　Aw-Rascle　model　under　the　constraint　condition　ρ　≤　ρ∗　by　letting　the　traffic　pressure　vanish.　In　this　paper,　we　give　up　this　constraint　condition　and　consider　the　following　form:　{　ρ　t　+　(ρ　u)　x　=　0,　(ρ　u　+　p　(ρ))　t　+　(ρ　u　2　+　u　p　(ρ))　x　=　0,in　which　p　(ρ)　=　-　1　ρ.　The　Riemann　problem　for　the　above　traffic　flow　model　is　constructively　solved.　The　delta　shock　wave　arises　in　the　Riemann　solutions,　although　the　system　is　strictly　hyperbolic,　its　first　eigenvalue　is　genuinely　nonlinear,　and　the　second　eigenvalue　is　linearly　degenerate.　Furthermore,　we　clarify　the　generalized　Rankine-Hugoniot　relations　and　δ-entropy　condition.　The　position,　strength,　and　propagation　speed　of　the　delta　shock　wave　are　obtained　from　the　generalized　Rankine-Hugoniot　conditions.　The　delta　shock　may　be　useful　for　the　description　of　the　serious　traffic　jam.　More　importantly,　it　is　proved　that　the　limits　of　the　Riemann　solutions　of　the　above　traffic　flow　model　are　exactly　those　of　the　pressureless　gas　dynamics　system　with　the　same　Riemann　initial　data　as　the　traffic　pressure　vanishes.　©　2023　Author(s).