The efficiency of greedy routing in hypercubes and butterflies

We analyze the following problem: Each node of the d-dimensional hypercube independently generates packets according to a Poisson process with rate A. Each of the packets is to be sent to a randomly chosen destination; each of the nodes at Hamming distance k from a packet's origin is assigned an a priori probability pk (1 _p)d- k. Packets are routed under a simple greedy scheme: each of them is forced to cross the hypercube dimensions required in increasing index-order, with possible queueing at the hypercube nodes. Assuming unit packet length and no other communications taking place, we show that this scheme is stable (in steady-state) if p < 1, where pde Ap is the load factor of the network; this is seen to be the broadest possible range for stability. Furthermore, we prove that the average delay T per packet satisfies T < AdE p , thus showing that an average delay of O(d) is attainable for any fixed p < 1. We also establish similar results in the context of the butterfly network. Our analysis is based on a stochastic comparison with a product-form network.