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Non-Equilibrium Statistical Physics of Currents in Queuing Networks

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Abstract

We consider a stable open queuing network as a steady non-equilibrium system of interacting particles. The network is completely specified by its underlying graphical structure, type of interaction at each node, and the Markovian transition rates between nodes. For such systems, we ask the question “What is the most likely way for large currents to accumulate over time in a network ?”, where time is large compared to the system correlation time scale. We identify two interesting regimes. In the first regime, in which the accumulation of currents over time exceeds the expected value by a small to moderate amount (moderate large deviation), we find that the large-deviation distribution of currents is universal (independent of the interaction details), and there is no long-time and averaged over time accumulation of particles (condensation) at any nodes. In the second regime, in which the accumulation of currents over time exceeds the expected value by a large amount (severe large deviation), we find that the large-deviation current distribution is sensitive to interaction details, and there is a long-time accumulation of particles (condensation) at some nodes. The transition between the two regimes can be described as a dynamical second order phase transition. We illustrate these ideas using the simple, yet non-trivial, example of a single node with feedback.

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Authors and Affiliations

  1. Center for Nonlinear Studies and Theoretical Division, LANL, Los Alamos, NM, 87545, USA

    Vladimir Y. Chernyak, Michael Chertkov, David A. Goldberg & Konstantin Turitsyn

  2. Department of Chemistry, Wayne State University, 5101 Cass Ave, Detroit, MI, 48202, USA

    Vladimir Y. Chernyak

  3. New Mexico Consortium, Los Alamos, NM, 87544, USA

    Michael Chertkov

  4. Operations Research Center, MIT, Cambridge, MA, 02139, USA

    David A. Goldberg

  5. Landau Institute for Theoretical Physics, Moscow, 119334, Russia

    Konstantin Turitsyn

Authors
  1. Vladimir Y. Chernyak
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  2. Michael Chertkov
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  3. David A. Goldberg
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  4. Konstantin Turitsyn
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Correspondence to Michael Chertkov.

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Chernyak, V.Y., Chertkov, M., Goldberg, D.A. et al. Non-Equilibrium Statistical Physics of Currents in Queuing Networks. J Stat Phys 140, 819–845 (2010). https://doi.org/10.1007/s10955-010-0018-5

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