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Bit flipping and time to recover

Journal article
Authors Anton Muratov
Sergei Zuyev
Published in Journal of Applied Probability
Volume 53
Issue 3
Pages 650-666
ISSN 0021-9002
Publication year 2016
Published at Department of Mathematical Sciences, Mathematical Statistics
Pages 650-666
Language en
Subject categories Mathematics

Abstract

We call 'bits' a sequence of devices indexed by positive integers, where every device can be in two states: 0 (idle) and 1 (active). Start from the 'ground state' of the system when all bits are in 0-state. In our first binary flipping (BF) model the evolution of the system behaves as follows. At each time step choose one bit from a given distribution P on the positive integers independently of anything else, then flip the state of this bit to the opposite state. In our second damaged bits (DB) model a 'damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both the BF and DB models to show recurrent or transient behaviour, depending on the properties of the distribution P. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a central limit theorem for the number of active bits for both models.

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