The Geometry of Big Queues
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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 wordsqueueing theory Jackson networks large deviations large deviation principle optimal trajectories Hamilton equations dual Markov processes fluid dynamics
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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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