In　this　paper,　we　develop　a　more　general　framework　of　block-structured　Markov　processes　in　the　queueing　study　of　blockchain　systems,　which　can　provide　analysis　both　for　the　stationary　performance　measures　and　for　the　sojourn　time　of　any　transaction　or　block.　In　addition,　an　original　aim　of　this　paper　is　to　generalize　the　two-stage　batch-service　queueing　model　studied　in　Li　et　al.　(Blockchain　queue　theory.　In:　International　conference　on　computational　social　networks.　Springer:　New　York;　2018.　p.　25–40)　both　“from　exponential　to　phase-type”　service　times　and　“from　Poisson　to　MAP”　transaction　arrivals.　Note　that　the　MAP　transaction　arrivals　and　the　two　stages　of　PH　service　times　make　our　blockchain　queue　more　suitable　to　various　practical　conditions　of　blockchain　systems　with　crucial　factors,　for　example,　the　mining　processes,　the　block　generations,　the　blockchain　building　and　so　forth.　For　such　a　more　general　blockchain　queueing　model,　we　focus　on　two　basic　research　aspects:　(1)　using　the　matrix-geometric　solution,　we　first　obtain　a　sufficient　stable　condition　of　the　blockchain　system.　Then,　we　provide　simple　expressions　for　the　average　stationary　number　of　transactions　in　the　queueing　waiting　room　and　the　average　stationary　number　of　transactions　in　the　block.　(2)　However,　on　comparing　with　Li　et　al.　(2018),　analysis　of　the　transaction–confirmation　time　becomes　very　difficult　and　challenging　due　to　the　complicated　blockchain　structure.　To　overcome　the　difficulties,　we　develop　a　computational　technique　of　the　first　passage　times　by　means　of　both　the　PH　distributions　of　infinite　sizes　and　the　RG　factorizations.　Finally,　we　hope　that　the　methodology　and　results　given　in　this　paper　will　open　a　new　avenue　to　queueing　analysis　of　more　general　blockchain　systems　in　practice　and　can　motivate　a　series　of　promising　future　research　on　development　of　blockchain　technologies.　©　2019,　The　Author(s).