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Computational Method for Singular Control of Reflected Brownian Motion with Applications to Queueing Networks


핵심 개념
This paper presents a computational method for solving singular stochastic control problems motivated by queueing theory applications. The method approximates the original singular control problem by a drift control problem, which can be solved efficiently using a recently developed simulation-based approach.
요약
The paper considers two classes of singular stochastic control problems with a d-dimensional state process W evolving as a reflected Brownian motion. In the first formulation, the system is subject to exogenous reflection at the boundary, while in the second formulation, the reflection is endogenous. The authors propose approximating the original singular control problem by a drift control problem, where the control is restricted to have a specific form. This approximation can be solved efficiently using a recently developed computational method. The authors conjecture that the solution obtained from the drift control problem is nearly optimal for the original singular control problem as the upper bound on the drift rates becomes large. The paper presents several examples to demonstrate the viability of the proposed approach: A one-dimensional singular control problem with known analytical solution, which is used to compare the approximation accuracy for different upper bound values. A multi-dimensional singular control problem with a decomposable structure, which can be solved efficiently using the proposed method. Two queueing network examples - a tandem queue network and a criss-cross network - where the singular control approximation is used to derive near-optimal policies for the original discrete-flow queueing models. The authors show that the policies derived from the singular control approximation perform very close to the optimal policies obtained by solving the original Markov decision process formulations.
통계
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인용문
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by Baris Ata,J.... 위치 arxiv.org 04-03-2024

https://arxiv.org/pdf/2312.11823.pdf
Singular Control of (Reflected) Brownian Motion

심층적인 질문

How can the proposed computational method be extended to handle singular control problems with more general state space constraints beyond the orthant considered in this paper

The proposed computational method can be extended to handle singular control problems with more general state space constraints beyond the orthant by incorporating additional constraints and transformations into the drift control approximation. One approach is to use a change of variables to map the original state space to a more general space, allowing for a wider range of state constraints. This transformation should preserve the essential characteristics of the problem while enabling the application of the drift control method. Additionally, the method can be adapted to handle different types of state space constraints, such as polyhedral cones or bounded regions, by adjusting the formulation of the drift control problem and the associated cost functions to accommodate these constraints.

What are the theoretical guarantees on the near-optimality of the solutions obtained from the drift control approximation as the upper bound on the drift rates becomes large

The near-optimality of the solutions obtained from the drift control approximation as the upper bound on the drift rates becomes large can be theoretically guaranteed under certain conditions. As the upper bound parameter increases, the solutions obtained from the drift control approximation approach the optimal solutions of the original singular control problem. This convergence is supported by the structure of the singular control problem and the properties of the drift control method. The theoretical guarantees on near-optimality can be established through mathematical analysis, including convergence proofs and asymptotic behavior studies, demonstrating that the solutions obtained from the drift control approximation provide increasingly accurate approximations to the optimal solutions as the upper bound on the drift rates grows.

Can the insights gained from the singular control approximation be used to develop efficient heuristic policies for queueing network control problems that do not rely on heavy traffic assumptions

The insights gained from the singular control approximation can be leveraged to develop efficient heuristic policies for queueing network control problems that do not rely on heavy traffic assumptions. By analyzing the characteristics of the optimal policies derived from the singular control approximation, heuristic strategies can be formulated based on the key principles and patterns observed in the solutions. These heuristic policies can prioritize certain actions or decisions based on the system's state and the cost structure, aiming to approximate the behavior of the optimal policies derived from the singular control approximation. Additionally, machine learning techniques and optimization algorithms can be employed to learn from the singular control solutions and develop adaptive heuristic policies that perform well in practice.
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