Stability of a three-station fluid network

Authors:
J. G. Dai;J. J. Hasenbein;J. H. Vande Vate
Affiliations:
School of Industrial and Systems Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA E-mail: dai@isye.gatech.edu;Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, University of Texas at Austin, Austin, TX 78712-1063, USA E-mail: jhas@mail.utexas.edu;School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA E-mail: jvandeva@isye.gatech.edu
Venue:
Queueing Systems: Theory and Applications
Year:
1999

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Abstract

This paper studies the stability of a three-station fluid network. We show that, unlike the two-station networks in Dai and Vande Vate [18], the global stability region of our three-station network is not the intersection of its stability regions under static buffer priority disciplines. Thus, the “worst” or extremal disciplines are not static buffer priority disciplines. We also prove that the global stability region of our three-station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three-station network. We also show that the LP proposed by Bertsimas et al. [1] does not characterize either the global stability region or even the monotone global stability region of our three-station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen and Zhang [11] does not characterize the stability region of our three-station network under a static buffer priority discipline.