Global Stability of Two-station Queueing Networks

This paper summarizes results of Dai and Vande Vate 15, 14] characterizing explicitly, in terms of the mean service times and average arrival rates, the global pathwise stability region of two-station open multi-class queueing networks with very general arrival and service processes. The conditions for pathwise global stability arise from two intuitively appealing phenomena: virtual stations and push starts. These phenomena shed light on the sources of bottlenecks in complicated queueing networks like those that arise in wafer fabrication facilities. We show that a two-station open multi-class queueing network is globally pathwise stable if and only if the corresponding uid model is globally weakly stable. We further show that a two-station uid model is globally (strongly) stable if and only if the average service times are in the interior of the global weak stability region. As a consequence, under stronger distributional assumptions on the arrival and service processes, the queueing network is globally stable in a stronger sense when the mean service times are in the interior of the global path-wise stability region. Namely, the underlying state process of the queueing network is positive Harris recurrent.