We study the stability of subcritical multi-class queueing networks with feedback allowed and a work-conserving head-of-the-line service discipline. Assuming that the fluid limit model associated to the queueing network satisfies a state space collapse condition, we show that the queueing network is stable provided that any solution of an associated linear Skorokhod problem is attracted to the origin in finite time. We also give sufficient conditions ensuring this attraction in terms of the reflection matrix of the Skorokhod problem, by using an adequate Lyapunov function. State space collapse establishes that the fluid limit of the queue process can be expressed in terms of the fluid limit of the workload process by means of a lifting matrix.
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