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The economic average cost Brownian control problem

Part of: Markov processes Operations research and management science Computer system organization

Published online by Cambridge University Press:  22 July 2019

Melda Ormeci Matoglu ,
John H. Vande Vate  and
Haiyue Yu
Melda Ormeci Matoglu*
Affiliation:
University of New Hampshire
John H. Vande Vate*
Affiliation:
Georgia Institute of Technology
Haiyue Yu
Affiliation:
University of New Hampshire Georgia Institute of Technology
*
*Postal address: Peter T. Paul College of Business and Economics, University of New Hampshire, 10 Garrison Avenue, Durham, NH 03824, USA. Email address: [email protected]
**Postal address: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Drive, NW, Atlanta, GA 30332, USA.
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Abstract

In this paper we introduce and solve a generalization of the classic average cost Brownian control problem in which a system manager dynamically controls the drift rate of a diffusion process X. At each instant, the system manager chooses the drift rate from a pair {u, v} of available rates and can invoke instantaneous controls either to keep X from falling or to keep it from rising. The objective is to minimize the long-run average cost consisting of holding or delay costs, processing costs, costs for invoking instantaneous controls, and fixed costs for changing the drift rate. We provide necessary and sufficient conditions on the cost parameters to ensure the problem admits a finite optimal solution. When it does, a simple control band policy specifying economic buffer sizes (α, Ω) and up to two switching points is optimal. The controller should invoke instantaneous controls to keep X in the interval (α, Ω). A policy with no switching points relies on a single drift rate exclusively. When there is no cost to change the drift rate, a policy with a single switching point s indicates that the controller should change to the slower drift rate when X exceeds s and use the faster drift rate otherwise. When there is a cost to change the drift rate, a policy with two switching points s < S indicates that the controller should maintain the faster drift rate until X exceeds S and maintain the slower drift rate until X falls below s.

Keywords

Brownian control problemoptimal rate controloptimal buffer sizesingular control

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 90B05: Inventory, storage, reservoirs 90B22: Queues and service
Secondary: 68M20: Performance evaluation; queueing; scheduling
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 300 - 337
Copyright
© Applied Probability Trust 2019 

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