On the Reliability of Fat-Trees

In this paper we examine the reliability properties of the ideal fat-tree, a general model used to capture both distance and bandwidth constraints of various classes of fat-tree networks. We use the notion (see NPSY94]) of a random graph of type?G obtained by selecting edges of a given undirected graph G independently and with probability p, thus representing a network in which links fail independently and with probability f = 1 ? p. In addition, we may allow vertices of the graph to fail independently with probability f. We show that: 1. At least half of the edge disjoint paths connecting any subsets of terminal nodes of a fat tree (whose fat-nodes do not have internal edges) are preserved with probability tending to 1 as the number of nodes in the tree tends to innnity, even in the case of constant vertex and edge failure probabilities f < 0:25. Thus, and by also showing the preservation of the minimum cut (and thus of the maximum ow), we prove that the ideal fat-tree remains, with high probability, \fat-enough", in the sense of the \?fat" deenition given in BB94]. 2. In the case of the fat-nodes being random regular graphs of degree d (and thus ex-panders) we prove that at least (1 ? f) times the initial number of edge disjoint paths survive despite the edge faults with high probability, even for degrees d as small as 256 1?f. 3. There is a critical probability p c such that for p > p c the pruned butterry P d =p of type?P d has a connected component of linear size which includes at least a constant fraction of its terminal nodes, with high probability, while for p < p c such a component does not exist almost certainly.