  A classic network tomography problem is estimation of properties of the
distribution of route traffic volumes based on counts taken on the network
links. We consider inference for a general class of models for integer-valued
traffic. Model identifiability is examined. We investigate both maximum
likelihood and Bayesian methods of estimation. In practice, these must be
implemented using stochastic EM and MCMC approaches. This requires a
methodology for sampling latent route flows conditional on the observed link
counts. We show that existing algorithms for doing so can fail entirely,
because inflexibility in the choice of sampling directions can leave the
sampler trapped at a vertex of the convex polytope that describes the feasible
set of route flows. We prove that so long as the network's link-path incidence
matrix is totally unimodular, it is always possible to select a coordinate
system representation of the polytope for which sampling parallel to the axes
is adequate. This motivates a modified sampler in which the representation of
the polytope adapts to provide good mixing behavior. This methodology is
applied to three road traffic data sets. We conclude with a discussion of the
ramifications of the unimodularity requirements for the routing matrix.
