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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. [ABSTRACT FROM AUTHOR]

This article considers a batch service polling system. We first study the case in which the server visits the queues cyclically, considering three different service regimes: gated, exhaustive, and globally gated. We subsequently analyze the case (the so-called “Israeli Queue”) in which the server first visits the queue with the “oldest” customer. In both cases, queue lengths and waiting times are the main performance measures under consideration. [ABSTRACT FROM AUTHOR]