Store-forward and its implications for proportional scheduling

The Proportional Scheduler was recently proposed as a scheduling algorithm for multi-hop switch networks. For these networks, the BackPressure scheduler is the classical benchmark. For networks with fixed routing, the Proportional Scheduler is maximum stable, myopic and, furthermore, will alleviate certain scaling issues found in BackPressure for large networks. Nonetheless, the equilibrium and delay properties of the Proportional Scheduler have not been fully characterized. In this article, we postulate on the equilibrium behaviour of the Proportional Scheduler through the analysis of an analogous rule called the Store-Forward allocation. It has been shown that Store-Forward allocates asymptotically according to the Proportional Scheduler. Further, for Store-Forward networks, numerous equilibrium quantities are explicitly calculable. For FIFO networks under Store-Forward, we calculate the policies stationary distribution and end-to-end route delay. We discuss network topologies where the stationary distribution is product-form, a phenomenon which we call product form resource pooling. We extend this product form notion to independent set scheduling on perfect graphs, where we show that non-neighbouring queues are statistically independent. Finally, we analyse the large deviations behaviour of the equilibrium distribution of Store-Forward networks in order to construct Lyapunov functions for FIFO switch networks.