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

We consider the serve-the-longest-queue discipline for a multiclass queue with buffers of equal size, operating under (i) the conventional and (ii) the Halfin-Whitt heavy traffic regimes, and show that while the queue length process’ scaling limits are fully determined by the first and second order data in case (i), they depend on finer properties in case (ii). The proof of the latter relies on the construction of a deterministic arrival pattern.