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- Title
The stability of the deterministic Skorokhod problem is undecidable.
- Authors
Gamarnik, David; Katz, Dmitriy
- 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 $$d\ge d_0$$ .
- Subjects
FLUIDS; BROWNIAN motion; COMPUTABILITY logic; COMPUTER network traffic; QUEUEING networks; QUEUING theory; INTERMOLECULAR forces; CHEMICAL stability
- Publication
Queueing Systems, 2015, Vol 79, Issue 3/4, p221
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-014-9424-8