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Journal of Statistical Physics

, Volume 140, Issue 5, pp 819–845 | Cite as

Non-Equilibrium Statistical Physics of Currents in Queuing Networks

  • Vladimir Y. Chernyak
  • Michael Chertkov
  • David A. Goldberg
  • Konstantin Turitsyn
Article

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.

Keywords

Statistics of non-equilibrium currents Open queueing networks Condensation phenomenon Birth-death processes 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Vladimir Y. Chernyak
    • 1
    • 2
  • Michael Chertkov
    • 1
    • 3
  • David A. Goldberg
    • 1
    • 4
  • Konstantin Turitsyn
    • 1
    • 5
  1. 1.Center for Nonlinear Studies and Theoretical DivisionLANLLos AlamosUSA
  2. 2.Department of ChemistryWayne State UniversityDetroitUSA
  3. 3.New Mexico ConsortiumLos AlamosUSA
  4. 4.Operations Research CenterMITCambridgeUSA
  5. 5.Landau Institute for Theoretical PhysicsMoscowRussia

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