Stability of Skorokhod problem is undecidable
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Gamarnik_The stability.pdf
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Author(s)
Gamarnik, David
Katz, Dmitriy
Alternative Title
The stability of the deterministic Skorokhod problem is undecidable
Date Issued
October 2014
Journal
Queueing Systems
Publisher
Springer-Verlag
Citation
Gamarnik, David, and Dmitriy Katz. “The Stability of the Deterministic Skorokhod Problem Is Undecidable.” Queueing Systems 79, no. 3–4 (October 19, 2014): 221–249.
Version
Original manuscript
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[subscript 0] such that stability of the Skorokhod problem associated with a fluid model of an RBM in dimension d ≥ d[subscript 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 ≥ d[subscript 0].
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
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DOI of Published Version
https://doi.org/10.1007/s11134-014-9424-8
