In　this　paper,　we　consider　a　double-ended　queue　with　First-Come-First-Match　discipline　(also　known　as　matched　queues)　under　customers＇　flexible　and　impatient　behaviors.　Such　a　system　can　be　expressed　as　a　level-dependent　quasi-birth-and-death　(QBD)　process　with　infinitely　many　phases.　The　stability　condition　of　the　queueing　system　is　given　by　using　the　mean　drift　technique.　To　deal　with　the　level-dependent　QBD　process,　we　apply　the　RG-factorizations　to　obtain　stationary　probability　vectors.　Based　on　this,　the　queue　size　distributions　and　the　average　stationary　queue　lengths　are　given.　Furthermore,　we　provide　an　effective　method　to　discuss　the　sojourn　time　of　any　arriving　customer　and　to　compute　the　average　sojourn　time　by　using　the　technique　of　the　first　passage　times　and　the　phase-type　(PH)　distributions.　Finally,　some　numerical　examples　are　employed　to　illustrate　how　the　performance　measures　are　influenced　by　key　system　parameters.