Approximations for the Random Minimal Spanning Tree with Application to Network Provisioning

This paper considers the problem of determining the mean and distribution of the length of a minimal spanning tree (MST) on an undirected graph whose arc lengths are independently distributed random variables. We obtain bounds and approximations for the MST length and show that our upper bound is much tighter than the naive bound obtained by computing the MST length of the deterministic graph with the respective means as arc lengths. We analyze the asymptotic properties of our approximations and establish conditions under which our bounds are asymptotically optimal. We apply these results to a network provisioning problem and show that the relative error induced by using our approximations tends to zero as the graph grows large.