This　work　is　a　continuation　Of　Our　previous　work　[Z.-Q.　Shao,　D.-X.　Koncy,　Y.-C.　Li,　Shock　reflection　for　general　quasilinear　hyperbolic　systems　of　conservation　laws,　Nonlinear　Anal.　66　(1)　(2007)　93-124].　In　the　present　paper,　we　study　the　global　structure　instability　of　the　self-similar　Solution　u　=　U(x/t)　containing　shocks,　contact　discontinuities,　and　at　least　one　rarefaction　wave　of　the　initial-boundary　Riemann　problem　for　general　n　x　n　quasilinear　hyperbolic　systems　of　conservation　laws　on　the　quarter　plane　{(t,　x)　vertical　bar　t　＞=　0,　x　＞=　0}.　We　prove　the　nonexistence　of　global　piecewise　C-1　solution　to　a　class　of　the　mixed　initial-boundary　value　problem　for　general　n　x　n　quasilinear　hyperbolic　systems　of　conservation　laws　on　the　quarter-plane　{(t,　x)　vertical　bar　t　＞=　0,　x　＞=　0}.　Our　result　indicates　that　the　kind　of　Riemann　solution　it　=　U(x/t)　mentioned　above　is　globally　structurally　unstable.　As　an　application　of　our　result,　we　consider　the　model　proposed　by　Aw　and　Rascle　[A.　Aw,　M.　Rascle,　Resurrection　of　＂second　order＂　models　of　traffic　flow,　SIAM　J.　Appl.　Math.　60　(2000)　916-938]　describing　traffic　flow　on　a,　road　network.　Following　the　work　of　Garavello　and　Piccoli　[M.　Garavello,　B.　Piccoli,　Traffic　flow　on　a　road　network　using　the　Aw-Rascle　model,　Comm.　Partial　Differential　Equations　31　(2)　(2006)　243-2751,　we　prove　the　nonexistence　of　global　piecewise　C-1　solution　containing　one　rarefaction　wave　of　the　initial-boundary　value　problem　for　this　model.　(c)　2006　Elsevier　Ltd.　All　rights　reserved.