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.

We consider a system consisting of two not necessarily identicalexponential servers having a common Poisson arrival process. Uponarrival, customers inspect the first queue and join it if it isshorter than some threshold n. Otherwise, they join the secondqueue. This model was dealt with, among others, by Altman et al. [Stochastic Models 20 (2004) 149–172].We first derive an explicitexpression for the Laplace-Stieltjes transform of the distributionunderlying the arrival (renewal) process to the second queue. Second,we observe that given that the second serveris busy, the two queue lengths are independent.Third, we develop two computational schemes for thestationary distribution of the two-dimensional Markov process underlying thismodel, one with a complexity of$O(n \log\delta^{-1})$, the other with a complexity of $O(\log n\log^2\delta^{-1})$, where δ is the tolerance criterion.

The steady-state analysis of a quasi-birth-death process is possible by matrix geometric procedures in which the root to a quadratic matrix equation is found. A recent method that can be used for analyzing quasi-birth–death processes involves expanding the state space and using a linear matrix equation instead of the quadratic form. One of the difficulties of using the linear matrix equation approach regards the boundary conditions and obtaining the norming equation. In this paper, we present a method for calculating the boundary values and use the operator-machine interference problem as a vehicle to compare the two approaches for solving quasi-birth-death processes.