Tree Structured QBD Markov Chains and Tree‐Like QBD Processes

Tree Structured Qbd Markov Chains and Tree-like Qbd Processes

In this paper we show that an arbitrary tree structured QBD Markov chain can be embedded in a tree-like QBD process with a special structure. Moreover, we present an algebraic proof that applying the natural fixed point iteration (FPI) to the nonlinear matrix equation V = B + ∑d s=1 Us(I − V )Ds that solves the tree-like QBD process, is equivalent to the more complicated iterative algorithm pre...

Kronecker-Based Infinite Level-Dependent QBD Processes

Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its under...

Analysis of a QBD process that depends on background QBD processes

We define a class of Markov chains that are called recursive foreground-background quasi-birthand-death (RFBQBD) processes, and describe approximate (nearly exact) analyses of an RFBQBD process. An RFBQBD process consists of a foreground QBD process whose transitions depend on the level of a background QBD process, where the transitions of the background QBD process may depend on the level of a...

Stability and Performance of Stack Algorithms for Random Access Communication Modeled as a Tree Structured Qbd Markov Chain

QBD Markov chains on binomial-like trees and its application to multilevel feedback queues

Amatrix 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 i-th child has i children of its own. We demonstrate that it suffices to solve...