TY - JOUR

T1 - Non-equilibrium statistical physics of currents in queuing networks

AU - Chernyak, Vladimir Y.

AU - Chertkov, Michael

AU - Goldberg, David A.

AU - Turitsyn, Konstantin

N1 - Funding Information:
Acknowledgements We are thankful to David Gamarnik for consulting us on many issues related to Queuing Theory, and Sergey Foss, Bill Massey and Alexander Rybko for enlightening conversations. This material is based upon work supported by the National Science Foundation under CHE-0808910 (VC) and CCF-0829945 (MC via NMC). The work at LANL was carried out under the auspices of the National Nuclear Security Administration of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396. KT acknowledges support of an Oppenheimer Fellowship at LANL, and DAG work on the project was a part of his summer internship (GRA program) at LANL.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Birth-death processes

KW - Condensation phenomenon

KW - Open queueing networks

KW - Statistics of non-equilibrium currents

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U2 - 10.1007/s10955-010-0018-5

DO - 10.1007/s10955-010-0018-5

M3 - Article

AN - SCOPUS:77955553109

SN - 0022-4715

VL - 140

SP - 819

EP - 845

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5

ER -