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This paper proposes a new algorithm for computing the stationary distributionvector in continuous-time upper block-Hessenberg Markov chains. To this end, weconsider the last-block-column-linearly-augmented (LBCL-augmented) truncationof the (infinitesimal) generator of the upper block-Hessenberg Markov chain.The LBCL-augmented truncation is a linearly-augmented truncation such that theaugmentation distribution has its probability mass only on the last blockcolumn. We first derive an upper bound for the total variation distance betweenthe respective stationary distribution vectors of the original generator andits LBCL-augmented truncation. Based on the upper bound, we then establish aseries of linear fractional programming (LFP) problems to obtain augmentationdistribution vectors such that the bound converges to zero. Using the optimalsolutions of the LFP problems, we construct a matrix-infinite-product (MIP)form of the original (i.e., not approximate) stationary distribution vector anddevelop a sequential update algorithm for computing the MIP form. Finally, wedemonstrate the applicability of our algorithm to BMAP/M/$\infty$ queues andM/M/$s$ retrial queues.

A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains