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 λ. 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 p^k(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 ρ<1, where ρ=defλp 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⩽dp/(1-ρ), thus showing that an average delay of Θ(d) is attainable for any fixed ρ<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 queueing network