  We consider Markov processes, which describe e.g. queueing network processes,
in a random environment which influences the network by determining random
breakdown of nodes, and the necessity of repair thereafter. Starting from an
explicit steady state distribution of product form available in the literature,
we notice that this steady state distribution does not provide information
about the correlation structure in time and space (over nodes). We study this
correlation structure via one step correlations for the queueing-environment
process. Although formulas for absolute values of these correlations are
complicated, the differences of correlations of related networks are simple and
have a nice structure. We therefore compare two networks in a random
environment having the same invariant distribution, and focus on the time
behaviour of the processes when in such a network the environment changes or
the rules for traveling are perturbed. Evaluating the comparison formulas we
compare spectral gaps and asymptotic variances of related processes.
