  Network tomography aims at estimating source-destination traffic rates from
link traffic measurements. This inverse problem was formulated by Vardi in 1996
for Poisson traffic over networks operating under deterministic as well as
random routing regimes. In this paper we expand Vardi's second-order moment
matching rate estimation approach to higher-order cumulant matching with the
goal of increasing the column rank of the mapping and consequently improving
the rate estimation accuracy. We develop a systematic set of linear cumulant
matching equations and express them compactly in terms of the Khatri-Rao
product. Both least squares estimation and iterative minimum I-divergence
estimation are considered. We develop an upper bound on the mean squared error
(MSE) in least squares rate estimation from empirical cumulants. We demonstrate
for the NSFnet that supplementing Vardi's approach with third-order empirical
cumulant reduces its averaged normalized MSE relative to the theoretical
minimum of the second-order moment matching approach by about 12%-18%. This
minimum MSE is obtained when Vardi's second-order moment matching approach is
based on the theoretical rather than the empirical moments.
