Loss Tomography from Tree Topologies to General Topologies

Loss tomography has received considerable attention in recent years and a number of estimators based on maximum likelihood (ML) or Bayesian principles have been proposed. Almost all of the estimators are devoted to the tree topology despite the general topology is more common in practice. Among the proposed estimators, most of them use an iterative procedure to search for the maximum of a likelihood equation obtained from observations. Without the detail of the solution space, an iterative procedure can be computationally expensive and may even converge to a local maximum. To overcome those, the following three questions need to be addressed: 1) whether there is a closed form solution to estimate the loss rates of a link or a path for the tree topology; 2) whether there is a unique maximum likelihood estimate (MLE) for a link or a path of the general topology; and 3) if so, how to obtain the MLE by a method other than an iterative procedure. This paper is devoted to address the three questions and provide the results obtained recently that include a closed form solution to estimate the loss rates of a tree network, a direct expression of the MLE for the general topology, a divide-and-conquer strategy to decompose a general network into a number of independent trees, and the method to estimate the loss rates of the decomposed trees.