In　the　present　paper　the　author　investigates　the　generalized　non-　linear　initial-boundary　Riemann　problem　with　small　BV　data　for　general　n　×　n　quasilinear　hyperbolic　systems　of　conservation　laws　with　nonlinear　boundary　conditions　in　a　half　space　{(t,　x)|t　≥　0,　x　≥　0},　where　the　Riemann　solution　only　contains　shocks　and　contact　discontinuities.　Combining　the　techniques　employed　by　Li-Kong　with　the　modified　Glimm＇s　functional,　the　author　obtains　the　almost　global　existence　and　lifespan　of　classical　discontinuous　solutions　to　a　class　of　the　generalized　nonlinear　initial-boundary　Riemann　problem,　which　can　be　regarded　as　a　small　BV　perturbation　of　the　corresponding　nonlinear　initial-boundary　Riemann　problem.　This　result　is　also　applied　to　the　system　of　traffic　flow　on　a　road　network　using　the　Aw-Rascle　model.