[Paper Review] Stability of Skorokhod problem is undecidable

David Gamarnik, Dmitriy Katz|arXiv (Cornell University)|Jul 10, 2010

Advanced Data Processing Techniques20 references3 citations

TL;DR

This paper proves that determining the stability of the Skorokhod problem—specifically, whether trajectories of a fluid model are attracted to the origin—is an undecidable problem when the initial state is part of the input. The authors establish this by reducing the problem to the halting problem of a Turing machine, demonstrating that no algorithm can universally decide stability for general dimensions, thereby revealing a fundamental theoretical limitation in queueing theory and stochastic processes.

ABSTRACT

Skorokhod problem arises in studying Reflected Brownian Motion (RBM) on an non-negative orthant, specifically in the context of queueing networks in the heavy traffic regime. One of the key problems is identifying conditions for stability of a Skorokhod problem, defined as the property that trajectories are attracted to the origin. The stability conditions 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 that stability of Skorokhod problem 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 general dimensions.

Motivation & Objective

• To investigate the fundamental theoretical limitations in determining stability conditions for the Skorokhod problem in general dimensions.
• To establish that stability of a fluid model of a reflected Brownian motion is undecidable when the initial state is part of the input.
• To extend prior results on undecidability in queueing networks and constrained stochastic processes to the Skorokhod problem framework.
• To demonstrate that no constructive algorithm can determine whether a given Skorokhod problem is stable, even in principle.
• To conjecture that global stability (all paths attracted to origin) remains undecidable even when the initial state is not part of the input.

Proposed method

• Reduction of the Skorokhod problem's stability decision to the halting problem of a Turing machine.
• Construction of a fluid model dynamics that emulate the behavior of a Turing machine, using a system of linear differential equations with reflection constraints.
• Design of a reflection matrix and initial state such that the fluid path reaches the origin if and only if the corresponding Turing machine halts.
• Use of variable activation rules based on the state of counters and auxiliary variables to simulate Turing machine transitions.
• Formal proof that active variables (pushing processes) increase at unit rate only when the system is in a state corresponding to a valid Turing transition.
• Establishment of invariants and bounds on state variables to ensure the dynamics correctly simulate the Turing machine's evolution over time intervals.

Experimental results

Research questions

• RQ1 Can an algorithm determine whether a given Skorokhod problem with a specified initial state is stable?
• RQ2 Is the stability of a fluid model of a reflected Brownian motion undecidable when the initial state is part of the input?
• RQ3 Does the undecidability of stability persist when considering global stability (all paths attracted to origin) rather than path-specific stability?
• RQ4 Can the dynamics of a Skorokhod problem simulate a universal Turing machine, thereby encoding undecidable problems?
• RQ5 What are the fundamental theoretical limitations in characterizing stability for general-dimensional queueing networks modeled via fluid approximations?

Key findings

• Stability of the Skorokhod problem is undecidable when the initial state is part of the input, meaning no algorithm can determine whether a given instance is stable.
• The proof constructs a fluid model whose trajectory reaches the origin if and only if a corresponding Turing machine halts, thereby reducing the stability problem to the halting problem.
• The system of equations and reflection constraints is designed so that only specific variables (representing Turing machine states) become active in a way that mimics the machine's computation steps.
• The dynamics over time intervals (5t+3, 5t+5) preserve the state of counters and simulate Turing transitions, ensuring correctness of the emulation.
• The final state at time 5T+1 is zero if and only if the Turing machine halts at step T, confirming the correctness of the reduction.
• The authors conjecture that global stability—where all fluid paths are attracted to the origin—is also undecidable, extending the result beyond path-specific stability.