Perturbation Analysis for Denumerable Markov Chains with Application to Queueing Models

Eitan Altman (1) , Rudesindo Núñez-Queija (2) , Konstantin Avrachenkov (1)

1 MISTRAL - Modeling of Computer Systems and Telecommunication Networks : Research and Software Development
2 CWI - Centrum voor Wiskunde en Informatica

We study the parametric perturbation of Markov chains with denumerable state space. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, which has several ergodic classes, is perturbed in a way that allows rare transitions between the different ergodic classes of the unperturbed chain. In the previous works the singularly perturbed Markov chains were studied under restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. Our goal is to relax these by conditions that can be applied to queueing models (where the conditions mentioned above typically fail to hold). With the help of the $\nu$-geometric ergodicity approach, we are able to express explicitly the steady state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our tools to quasi birth and death processes and provide queueing examples.

Mots clés: DENUMERABLE MARKOV CHAINS; PERTURBATION ANALYSIS; GEOMETRIC ERGODICITY; QUASI BIRTH AND DEATH PROCESSES; QUEUEING MODELS

Domaines: Autre [cs.OH]

Eitan Altman, Rudesindo Núñez-Queija, Konstantin Avrachenkov. Perturbation Analysis for Denumerable Markov Chains with Application to Queueing Models. [Research Report] RR-4713, INRIA. 2003. ⟨inria-00071873⟩