Diffusion Approximation for an Input-queued Switch Operating under a Maximum Weight Matching Policy

Abstract

For N ≥ 2, we consider an N × N input-queued switch operating under a maximum weight matching policy. We establish a diffusion approximation for a (2N − 1)-dimensional workload process associated with this switch when all input ports and output ports are heavily loaded. The diffusion process is a semimartingale reflecting Brownian motion living in a polyhedral cone with N 2 boundary faces, each of which has an associated constant direction of reflection. Our proof builds on our own prior work [13] on an invariance principle for semimartingale reflecting Brownian motions in piecewise smooth domains and on a multiplicative state space collapse result for switched networks established by Shah and Wischik in [19].

Optimal Heavy-Traffic Queue Length Scaling in an Incompletely Saturated Switch

“…The mean and variance of the arrival process for this single server queue are then (1 − γ i ) and σ (ǫ) ij 2 respectively because of the independence of the arrival processes across the queues in the matrix. Then, using the Kingman bound for single server queue in Lemma 1, we get (7). Similarly lower bounding the total queue length for output port (column) j, i q (ǫ) ij (t) by a single server queue, we get (8).…”

“…State collapse in the case where multiple ports are saturated has been established in [2,16,17] and using the state-space collapse result in [16], a diffusion limit was established in [7]. However, properties of the diffusion limit (such as its steady-state distribution or mean queue lengths) were difficult to establish.…”

“…Another major contribution of the paper is that we express this function in a simple form, so that it can be generalized for use in other problems that exhibit state space collapse into other regions. The case where each port has a different rate of saturationis of theoretical importance since it corresponds to the situation where the drift vector is not the identity matrix in the diffusion limit [7]. In fact, the diffusion limit is no longer symmetric in the components of the limiting stochastic process (i.e., the diffusion limit is not symmetric across the ports), but here we show that the technique in [9] works even in this asymmetric situation to produce an exact formula for a certain linear combination of the queue lengths.…”