An open multiclass network is said to be {\em well behaved in heavy traffic}, or simply well behaved, if it satisfies a conventional heavy traffic limit theorem. In such a theorem the heavy traffic limit is derived from (or simply is) a regulated or reflected Brownian motion whose dimension equals the number of stations in the network. Recent examples show that not all multiclass networks are well behaved in this sense, and a top priority for heavy traffic researchers is to understand the exceptional set of ill behaved network models. In this paper we consider a large family of open multiclass queueing networks, associate with each network a deterministic fluid analog, and develop the following conjecture: A multiclass network is well behaved in heavy traffic if and only if its balanced fluid analog exhibits a certain stability property. This general conjecture resonates with other contemporary developments in queueing network theory, where fluid models play an increasingly important role. However, our view of fluid models differs from the conventional one in ways that seem to be important for heavy traffic theory.

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