Bounds on the deviation of discrete-time Markov chains from their mean-field model

We consider a generic mean-field scenario, in which a sequence of population models, described by discrete-time Markov chains (DTMCs), converges to a deterministic limit in discrete time. Under the assumption that the limit has a globally attracting equilibrium, the steady states of the sequence of...

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Bibliographic Details
Published in:Performance evaluation 2013-10, Vol.70 (10), p.736-749
Main Authors: Bortolussi, Luca, Hayden, Richard A.
Format: Article
Language:English
Subjects:
Convergence
Deviation
Dynamical systems
Error analysis
Markov chains
Markov population models
Mathematical models
Mean field approximation
Remote sensors
Steady state
Steady state error bounds for mean field
Steady state mean field approximation
Transient error bounds for mean field
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Summary:We consider a generic mean-field scenario, in which a sequence of population models, described by discrete-time Markov chains (DTMCs), converges to a deterministic limit in discrete time. Under the assumption that the limit has a globally attracting equilibrium, the steady states of the sequence of DTMC models converge to the point-mass distribution concentrated on this equilibrium. In this paper we provide explicit bounds in probability for the convergence of such steady states, combining the stochastic bounds on the local error with control-theoretic tools used in the stability analysis of perturbed dynamical systems to bound the global accumulation of error. We also adapt this method to compute bounds on the transient dynamics. The approach is illustrated by a wireless sensor network example.
ISSN:0166-5316
1872-745X
DOI:10.1016/j.peva.2013.08.012