Diffusion approximations for some multiclass queueing networks with FIFO service disciplines

The diffusion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. Ln addition to the usual assumptions for a heavy traffic limit theorem, a key condition that characterizes this class is that a J x J matrix G, known as the workload contents matrix, has a spectral radius less than unity, where J represents the number of sen ice stations. The (j, l)th component of matrix G can be interpreted as the long-run average amount of future work for station j that is embodied in a unit of immediate work at station l. This class includes, as a special case, the feedforward multiclass queueing network and the Rybko-Stolyar network under FIFO service discipline. A new approach is taken in establishing the diffusion limit theorem. The traditional approach is based on an oblique reflection mapping, but such a mapping is not well defined for the network under consideration. Our approach takes two steps: first establishing the C-tightness of the scaled queueing processes, and then completing the proof for the convergence of the scaled queueing processes by invoking the weak uniqueness for the limiting processes, which are semimartingale reflecting Brownian motions.