Perturbation analysis for denumerable Markov chains with application to queueing models

Abstract

We study the parametric perturbation of Markov chains with denumerable state spaces. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, with several ergodic classes, are perturbed such that (rare) transitions among the different ergodic classes of the unperturbed chain are allowed. Singularly perturbed Markov chains have been studied in the literature under more restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. We relax these conditions so that our results can be applied to queueing models (where the conditions mentioned above typically fail to hold). Assuming ν-geometric ergodicity, we are able to explicitly express the steady-state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our results to quasi-birth-anddeath processes and queueing models.

Stability of size-based scheduling disciplines in resource-sharing networks

“…To determine the sufficient condition for stability of Q 0 , we will consider the network in a limiting regime, obtained by scaling the dynamics of some classes with a common parameter and passing ↓ 0. This technique is usually referred to as analytic perturbation or nearly-complete decomposability and has successfully been applied to study steady-state performance as a function of , as ↓ 0, see for instance [3,12].…”

Opinion propagation in bounded medium-sized populations

“…For finite state spaces (in contrast to infinite ones, see e.g. [3,18]), this is fairly easy to establish.…”

“…If a matrix depends analytically on a parameter, then the corresponding eigenvalues and eigenvectors are also analytic in case of null-space perturbation [1]. Another possible path towards proving analyticity is via V -uniform ergodicity of the unperturbed Markov process with generator Q (0) (see a.o [3]), which is equivalent to the existence of a spectral gap (the distance between eigenvalue 0 of the generator matrix Q (0) and the eigenvalue that is its nearest neighbour). For finite Markov chains, there is a spectral gap as long as there is only one recurrent class for µ = 0.…”

Regular Perturbation of V-Geometrically Ergodic Markov Chains

“…In general this is not a Taylor expansion, except for instance when D ε has the form D ε = ǫD with D ∈ L(B 1 ). This special case is discussed in [AAN04]. In fact, for general perturbations, even in the case when (1) is fulfilled, obtaining Taylor expansions for π ε causes difficulties when the derivatives of the perturbed kernels P ε (x, ·) w.r.t.…”