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We consider the serve-the-longest-queue discipline for a multiclass queuewith buffers of equal size, operating under (i) the conventional and (ii) theHalfin-Whitt heavy traffic regimes, and show that while the queue lengthprocess' scaling limits are fully determined by the first and second order datain case (i), they depend on finer properties in case (ii). The proof of thelatter relies on the construction of a {\it deterministic} arrival pattern.

A note on non-existence of diffusion limits for serve-the-longest-queue when the buffers are equal in size