The Stability of Two-Station Multitype Fluid Networks

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 de+ned 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 in+nity, showing that the uid network is unstable. When all the workload conditions are satis+ed, we solve a network ow problem to +nd the coe/cients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero, proving that the uid level eventually reaches zero under any nonidling 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 su/cient to ensure the global stability of two-station multiclass queueing networks with deterministic routing.