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The stability of the deterministic Skorokhod problem is undecidable

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Abstract

The Skorokhod problem arises in studying reflected Brownian motion (RBM) and the associated fluid model on the non-negative orthant. This problem specifically arises in the context of queueing networks in the heavy traffic regime. One of the key problems is that of determining, for a given deterministic Skorokhod problem, whether for every initial condition all solutions of the problem staring from the initial condition are attracted to the origin. The conditions for this attraction property, called stability, are known in dimension up to three, but not for general dimensions. In this paper we explain the fundamental difficulties encountered in trying to establish stability conditions for general dimensions. We prove the existence of dimension \(d_0\) such that stability of the Skorokhod problem associated with a fluid model of an RBM in dimension \(d\ge d_0\) is an undecidable property, when the starting state is a part of the input. Namely, there does not exist an algorithm (a constructive procedure) for identifying stable Skorokhod problem in dimensions \(d\ge d_0\).

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Acknowledgments

With great pleasure we acknowledge several enlightening discussions with Maury Bramson. David Gamarnik: This work was supported by NSF Grant CMMI-0726733

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Authors and Affiliations

  1. Operations Research Center and Sloan School of Management, MIT, Cambridge, MA, 02139, USA

    David Gamarnik

  2. T.J. Watson Research Center, IBM, Yorktown Heights, NY, 10598, USA

    Dmitriy Katz

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  1. David Gamarnik
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  2. Dmitriy Katz
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Correspondence to David Gamarnik.

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Dedicated to the memory of A.V. Skorokhod.

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Gamarnik, D., Katz, D. The stability of the deterministic Skorokhod problem is undecidable. Queueing Syst 79, 221–249 (2015). https://doi.org/10.1007/s11134-014-9424-8

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  • DOI: https://doi.org/10.1007/s11134-014-9424-8

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