EXPLICIT SOLUTIONS FOR CONTINUOUS-TIME QBD PROCESSES BY USING RELATIONS BETWEEN MATRIX GEOMETRIC ANALYSIS AND THE PROBABILITY GENERATING FUNCTIONS METHOD

Abstract

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 equationA0+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 matrixH(z). We show that: (a)H(z) and$\vec{b}(z)$can be explicitly expressed in terms of the tripleA0,A1, andA2; (b) when each matrix of the triple is lower (or upper) triangular, then (i)Rcan be explicitly expressed in terms of roots of$\det [H(z)]$; and (ii) the stability condition is readily extracted.