Computing absorbing times via fluid approximations

Nicolas Gast (1, 2) , Bruno Gaujal (1, 2)

1 POLARIS - Performance analysis and optimization of LARge Infrastructures and Systems
2 UGA [2016-2019] - Université Grenoble Alpes [2016-2019]

In this paper, we compute the absorbing time Tn of a n-dimensional discrete time Markov chain made of n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time Tn is well approximated by a deterministic time tn that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1/n. We provide an asymptotic expansion of tn that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of Tn, relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime and the coupling times of random walks in high dimensional spaces.

Mots clés: Markov chain; Fluid approximation; Extreme values; Coupon collector; Coupling time

Domaines: Réseaux et télécommunications [cs.NI]; Performance et fiabilité [cs.PF]; Mathématiques [math]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Nicolas Gast, Bruno Gaujal. Computing absorbing times via fluid approximations. 2016. ⟨hal-01337950⟩