Stability of Skorokhod problem is undecidable
Author(s)Gamarnik, David; Katz, Dmitriy
The stability of the deterministic Skorokhod problem is undecidable
MetadataShow full item record
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].
DepartmentMassachusetts Institute of Technology. Operations Research Center; Sloan School of Management
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.