Clock synchronization in symmetric stochastic networks

Manita, Anatoly

doi:10.1007/s11134-013-9386-2

Cited by 17 publications

(12 citation statements)

References 19 publications

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“…Subsequently to [5,6], there has been a line of work generally focused on modeling synchronization between elements (particles) of a large system, with mean-field interaction of the elements (particles), cf. [9][10][11][12][13][14].…”

Section: Other Related Workmentioning

confidence: 99%

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Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Stolyar¹

2020

Preprint

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We use probabilistic methods to study properties of mean-field models, arising as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle "jumps forward" at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle's location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles' distribution, when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of -and attraction tomean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

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Section: Other Related Workmentioning

confidence: 99%

“…This naturally leads to the corresponding meanfield models, describing behavior of large-scale systems. Further references to this line of work can be found in [13].…”

Section: Other Related Workmentioning

confidence: 99%

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Stolyar¹

2020

Preprint

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“…The problem is to find different time scales (t = t N → ∞ as N → ∞) on which the synchronization system x(t N ) demonstrates completely different qualitative behaviour. The complete description of times scales was obtained for several models [40,[46][47][48]51]. For example, in [47] it was shown that the "BM N -model" passes three different phases before it reaches the final synchronization.…”

Section: Introductionmentioning

confidence: 99%

“…For example, in [47] it was shown that the "BM N -model" passes three different phases before it reaches the final synchronization. The model of clock synchronization in WSNs (see [51]) has 5 different consecutive phases of qualitative behaviour. As it was explained in [47] and [51,Sect.…”

Section: Introductionmentioning

confidence: 99%

“…The model of clock synchronization in WSNs (see [51]) has 5 different consecutive phases of qualitative behaviour. As it was explained in [47] and [51,Sect. 5], each phase in evolution of a stochastic synchronization system is a cumulative result of competition between two opposite tendencies: with the course of time the free dynamics increases the "desynchronization" in the system while the interaction tries to decrease it.…”

Section: Introductionmentioning

confidence: 99%

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Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Manita

2014

Preprint

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We propose stochastic N -component synchronization models (x 1 (t), . . . , x N (t)), x j ∈ R d , t ∈ R + , whose dynamics is described by Lévy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components d (N ) kj (t) = x k (t)−x j (t) have limits in distribution as t → ∞. We give conditions of existence of natural (intrinsic) space scales for large synchronized systems, i.e., we are looking for such sequences {b N } that distribution of d (N ) kj (∞)/b N converges to some limit as N → ∞. It appears that such sequence exists if the Lévy process enters a domain of attraction of some stable law. For Markovian synchronization models based on α-stable Lévy processes this results holds for any finite N in the precise form with b N = (N − 1) 1/α . For non-Markovian models similar results hold only in the asymptotic sense. The class of limiting laws includes the Linnik distributions. We also discuss generalizations of these theorems to the case of non-uniform matrix-based intrinsic scales. The central point of our proofs is a representation of characteristic functions of d (N ) kj (t) via probability distribution of a superposition of N independent renewal processes.

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Limit Distributions in Stochastic Networks with Message-Passing Synchronization

Manita

2017

Analytical and Computational Methods in Probability Theory

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Clock synchronization in symmetric stochastic networks

Cited by 17 publications

References 19 publications

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Limit Distributions in Stochastic Networks with Message-Passing Synchronization

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