  The Erlangian approximation of Markovian fluid queues leads to the problem of
computing the matrix exponential of a subgenerator having a block-triangular,
block-Toeplitz structure. To this end, we propose some algorithms which exploit
the Toeplitz structure and the properties of generators. Such algorithms allow
to compute the exponential of very large matrices, which would otherwise be
untreatable with standard methods. We also prove interesting decay properties
of the exponential of a generator having a block-triangular, block-Toeplitz
structure.
