Scaling behavior in disordered sandpile automata (original) (raw)

The sandpile scheduler: How self-organized criticality may lead to dynamic load-balancing

Cluster Computing, 2014

This paper studies a self-organized criticality model called sandpile for dynamically load-balancing tasks arriving in the form of Bag-of-Tasks in large-scale decentralized system. The sandpile is designed as a decentralized agent system characterizing a cellular automaton, which works in a critical state at the edge of chaos. Depending on the state of the cellular automaton, different responses may occur when a new task is assigned to a resource: it may change nothing or generate avalanches that reconfigure the state of the system. The abundance of such avalanches is in power-law relation with their sizes, a scale-invariant behavior that emerges without requiring tuning or control parameters. That means that large-catastrophic-avalanches are very rare but small ones occur very often. Such emergent pattern can be efficiently adapted for non-clairvoyant scheduling, where tasks are load balanced in computing resources trying to maximize the performance but without assuming any knowledge on the tasks features. The algorithm design is experimentally validated showing that the sandpile is able to find near-optimal schedules by reacting differently to different conditions of workloads and architectures.

Disorder, memory and avalanches in sandpiles

EPL (Europhysics Letters), 1994

We construct a cellular-automaton model of a sandpile with unquenched disorder. This models the behaviour of a real sandpile in which the structure is disordered and grain rearrangements cause the structure to change with time. We find that the avalanches retain a memory of the evolving disorder and do not exhibit Self-Organised Criticality. SOC is retrieved in the limit of no disorder. We construct a phase diagram, for the scaling properties, which is parametrised in terms of disorder and its rate of change and we provide a framework for the interpretation of recent theory and experiments.

Scale-free energy dissipation and dynamic phase transition in stochastic sandpiles

Physical Review E, 1999

We study numerically scaling properties of the distribution of cumulative energy dissipated in an avalanche and the dynamic phase transition in a stochastic directed cellular automaton [B. Tadić and D. Dhar, Phys. Rev. Lett. 79, 1519 (1997)] in d = 1 + 1 dimensions. In the critical steady state occurring for the probability of toppling p ≥ p ⋆ =0.70548, the dissipated energy distribution exhibits scaling behavior with new scaling exponents τE and DE for slope and cutoff energy, respectively, indicating that the sandpile surface is a fractal. In contrast to avalanche exponents, the energy exponents appear to be p-dependent in the region p ⋆ ≤ p < 1, however the product (τE − 1)DE remains universal. We estimate the roughness exponent of the transverse section of the pile as χ = 0.44 ± 0.04. Critical exponents characterizing the dynamic phase transition at p ⋆ are obtained by direct simulation and scaling analysis of the survival probability distribution and the average outflow current. The transition belongs to a new universality class with the critical exponents ν = γ = 1.22±0.02, β = 0.56±0.02 and ν ⊥ = 0.761±0.029, with apparent violation of hyperscaling. Generalized hyperscaling relation leads to β+β ′ = (d−1)ν ⊥ , where β ′ = 0.195±0.012 is the exponent governed by the ultimate survival probability.

Stable Dynamics of Sand Automata

In this paper, we study different notions of stability for sand automata, dynamical systems inspired by sandpile models and cellular automata. First, we study the topological stability properties of equicontinuity and ultimate periodicity, proving that they are equivalent. Then, we deal with nilpotency. The classical definition for cellular automata being meaningless in that setting, we define a more suitable one. Finally, we prove that this dynamical behavior is undecidable. Full Text at Springer, may require registration or fee

Sandpiles and self-organized criticality

Physica A: Statistical Mechanics and its Applications, 1992

This article provides an overview of some of the models of sandpiles that have been studied in literature. The question whether real sandpiles exhibit self-organized criticality is discussed briefly, and a tentative explanation of the observed failure of scaling in the experiments by the IBM group offered.