Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector
$\vec{G}(z)$
of unknown generating functions satisfying
$H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$
where the row vector
$\vec{b}(z)$
contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and
$\vec{b}(z)$
can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of
$\det [H(z)]$
; and (ii) the stability condition is readily extracted.
Sharp bounds by probability-generating functions and variable drift
