THE EFFICIENCY OF GREEDY ROUTING IN HYPERCUBES AND BUTTERFLIES

We analyze the following problem: Each node of the d-dimensional hyper cube independently generates packets according to a Poisson process wi th rate lambda. 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 a re routed under a simple greedy scheme: each of them is forced to cros s the hypercube dimensions required in increasing index-order, with po ssible queueing at the hypercube nodes. Assuming unit packet length an d no other communications taking place, we show that this scheme is st able (in steady-state) if rho < 1, where rho=(def)lambda p 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 pac ket satisfies T less than or equal to dp/1-rho, thus showing that an a verage delay of Theta(d) is attainable for any fixed rho < 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 queu eing network.