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The Stability of Two-Station Multitype Fluid Networks

School of Industrial and Systems Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0205
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0205
dai{at}isye.gatech.edu
john.vandevate{at}isye.gatech.edu

This paper studies the fluid models of two-station multiclass queueing networks with deterministic routing. A fluid model is globally stable if the fluid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is defined by the nominal workload conditions and the "virtual workload conditions," and we introduce two intuitively appealing phenomena—virtual stations and push starts—that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a fluid solution that cycles to infinity, showing that the fluid network is unstable. When all the workload conditions are satisfied, we solve a network flow problem to find the coefficients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero, proving that the fluid level eventually reaches zero under any nonidling dispatch policy. Under certain assumptions on the inter arrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding fluid network is stable. Thus, the workload conditions are sufficient to ensure the global stability of two-station multiclass queueing networks with deterministic routing.

Subject classifications: Queueing networks, fluid models, stability, piecewise linear Lyapunov functions, network flows.
History: Received June 1997; revision received May 1998; revision received January 1999; accepted January 1999.

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