On the Average Delay for Routing Subject to Independent Deflections

Consider a packet walking along a directed graph with each node having two edges directed out. The packet is headed towards one of N destinations, chosen according to a probability distribution p. At each step, the packet is forced to use a nonpreferred edge with some probability q, independently of past events. Using information theory and sequential analysis, it is shown that the mean number of steps required by the packet to reach the destination is, roughly, at least H(p)/(1 - h(q)), where h is the binary entropy function and H(p) is the entropy (base two) of p. This lower bound is shown to be asymptotically achievable in the case the packet always begins at a fixed node. Also considered is the maximum, over all pairs of nodes in a graph, of the mean transit time from one node to the other. The work is motivated by the search for graphs which work well in conjunction with deflection routing in communication networks.