Optimal control problems on stratified domains: Application to single-station multiclass queueing systems with finite buffers and overflow costs

Open Access
Author:
Hong, Yunho
Graduate Program:
Industrial Engineering
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
April 27, 2007
Committee Members:
  • Soundar Rajan Tirupatikumara, Committee Chair
  • Alberto Bressan, Committee Chair
  • Susan H Xu, Committee Member
  • M Jeya Chandra, Committee Member
  • Tao Yao, Committee Member
Keywords:
  • Discontinuous Hamilton-Jacobi equation
  • Viscosity solution
  • Queueing theory
  • Optimal control theory
Abstract:
We study a class of optimal control problems on stratified domains (OCPSD) and apply the results to an optimal control problem for a single-station multiclass queueing system with finite buffers and overflow costs. In the optimal control problems on stratified domains, we assume that the state space $mathbb{R}^N$ admits a stratification as a disjoint union of finitely many embedded submanifolds ${mathcal M}_i$. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions, which guarantee the existence of an optimal solution, and study sufficient conditions for optimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobi equation with discontinuous coefficients, describing the value function. Our results are motivated by various applications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to a bounded domain, in the presence of an additional overflow cost at the boundary. The multiclass queueing system is modeled to have stochastic fluid flows. We model the problem as an optimal control problem with a closed piecewise smooth reflecting boundary and reflecting cost. For this, we extend the results of OCPSD into the stochastic fluid model. Then, we characterize the value function of the queueing control problem by the unique viscosity solution for a set of Hamilton-Jacobi equations. Furthermore, we validate the Markov chain approximation method in our problem. We illustrate the results and provide a numerical solution for an example problem.