This paper studies the instability of multiclass queueing networks. We prove that if a fluid limit model of the queueing network is weakly unstable, then the queueing network is unstable in the sense that the total number of customers in the queueing network diverges to infinity with probability 1 as time $t \to \infty$. Our result provides a converse to a recent result of Dai which states that a queueing network is positive Harris recurrent if a corresponding fluid limit model is stable. Examples are provided to illustrate the usage of the result.
J. G. Dai. "A fluid limit model criterion for instability of multiclass queueing networks." Ann. Appl. Probab. 6 (3) 751 - 757, August 1996. https://doi.org/10.1214/aoap/1034968225