It is now known that the usual traffic condition (the nominal load being less than 1 at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and first-buffer-first-served preemptive resume discipline in a reentrant line are positive Harris recurrent under the usual traffic condition.
"On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models." Ann. Appl. Probab. 5 (1) 49 - 77, February, 1995. https://doi.org/10.1214/aoap/1177004828