The subject of this paper is the heavy traffic behavior of a general class of queueing networks with first-in-first-out (FIFO) service discipline. For special cases that require various assumptions on the network structure, several authors have proved heavy traffic limit theorems to justify the approximation of queueing networks by reflecting Brownian motions. Based on these theorems, some have conjectured that the Brownian approximation may in fact be valid for a more general class of queueing networks. In this paper, we prove that the Brownian approximation does not hold for such a general class of networks. Our findings suggest that it may be fruitful to consider a more general class of approximating processes.
"On the Convergence of Multiclass Queueing Networks in Heavy Traffic." Ann. Appl. Probab. 4 (1) 26 - 42, February, 1994. https://doi.org/10.1214/aoap/1177005199