Quasi-Birth–Death (QBD) Markov chains on binomial-like trees and its application to multilevel feedback queues

A matrix analytic paradigm, termed Quasi-Birth–Death Markov chains on binomial-like trees, is introduced and a quadratically converging algorithm to assess its steady state is presented. In a bivariate Markov chain {(Xt, Nt), t≥0}, the values of the variable Xt are nodes of a binomial-like tree of order d, where the ith child has i children of its own. We demonstrate that it suffices to solve d quadratic matrix equations to yield the steady state vector, the form of which is matrix geometric. We apply this framework to analyze the multilevel feedback scheduling discipline, which forms an essential part in contemporary operating systems.