This paper studies the uid models of two-station multiclass queueing networks with deterministic routing. A uid model is globally stable if the uid 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 dened 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 uid solution that cycles to innity, showing that the uid network is unstable. When all of the workload conditions are satised, we solve a network ow problem to nd the coecients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero proving that the uid level eventually reaches zero under any non-idling dispatch policy. Under certain assumptions on the interarrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding uid network is stable. Thus, the workload conditions are sucient to ensure the global stability of two-station multiclass queueing networks with deterministic routing. To appear in Operations Research

This paper studies the stability of a three-station fluid network. We 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. The linear program proposed by Bertsimas, Gamarnik and Tsitsiklis [1] does not characterize either the global stability region or even the monotone global stability region of our three-station network.