Explicit solutions for continuous-Time qbd processes by using relations between matrix geometric analysis and the probability generating functions METHOD

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 of unknown generating functions satisfying where the row vector contains unknown boundary probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and 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; and (ii) the stability condition is readily extracted.