Distributed WFQ scheduling converging to weighted max-min fairness
This paper studies the fairness properties of distributed WFQ scheduling in buffered crossbars. Using a fluid model, we prove that these networks converge to ideal weighted max-min fair allocation. We simulate the cell-based system, and observe how close real rates come to the theoretical WMM fair: with buffer sizes of 2-5 cells per crosspoint, the average rate discrepancy is below 1%; the worst-case discrepancy falls below 4% with 4- to 8-cell buffers. We also study the transient phenomena when some flows come and go. We find that, after a change in a flow's state, rate changes visit flows along non-circular paths, stabilizing them to their new fair rates; interestingly, some flows can affect all other active flows in the crossbar. Regarding the factors that influence the stabilization delay, although larger buffers yield more accurate convergence, they increase stabilization delay. Larger networks may also take more time to stabilize. Finally, convergence speeds up as the difference between the present rates and the WMM fair rates increases. Transient behavior simulations verify these results. Although we deal with data crossbar networks only, our analysis reveals complicated, "multi-hop" interactions among the flows in the crossbar, similar to those that can appear in arbitrary, point-to-point networks with distributed WFQ schedulers and backpressure mechanisms, such as the scheduling sub-system of a multi-stage switching fabric.