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