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Problems of Information Transmission

, Volume 55, Issue 2, pp 174–200 | Cite as

The Geometry of Big Queues

  • A. A. PuhalskiiEmail author
Communication Network Theory
  • 2 Downloads

Abstract

We use Hamilton equations to identify most likely scenarios of long queues being formed in ergodic Jackson networks. Since the associated Hamiltonians are discontinuous and piecewise Lipschitz, one has to invoke methods of nonsmooth analysis. Time reversal of the Hamilton equations yields fluid equations for the dual network. Accordingly, the optimal trajectories are time reversals of the fluid trajectories of the dual network. Those trajectories are shown to belong to domains that satisfy a certain condition of being “essential.” As an illustration, we consider a two-station Jackson network. In addition, we prove certain properties of substochastic matrices, which may be of interest in their own right.

Key words

queueing theory Jackson networks large deviations large deviation principle optimal trajectories Hamilton equations dual Markov processes fluid dynamics 

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Notes

Acknowledgement

The author is grateful to S.A. Pirogov and A.N. Rybko for helpful discussions and advice on improving the presentation.

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© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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