  We develop randomized modifications of Markov chains and apply these
modifications to the routing chains of customers in Jacksonian stochastic
networks. The aim of our investigations is to find new rerouting schemes for
non standard Jackson networks which hitherto resist computing explicitly the
stationary distribution.
  The non standard properties we can handle by suitable algorithms encompass
several modifications of Jackson networks known in the literature, especially
breakdown and repair of nodes with access modification for customers to down
nodes, finite buffers with control of buffer overflow. The rerouting schemes
available in the literature for these situations are special cases of our
rerouting schemes, which can deal also with partial degrading of service
capacities and even with speed up of service.
  In any case we require our algorithms to react on such general changes in the
network with the aim to maintain the utilization of the nodes. To hold this
invariant under change of service speeds (intensities) our algorithms not only
adapt the routing probabilities but decrease automatically the overall arrival
rate to the network if necessary.
  Our main application is for stochastic networks in a random environment. The
impact of the environment on the network is by changing service speeds (by
upgrading and/or degrading, breakdown, repair) and we implement the
randomization algorithms to react to the changes of the environment. On the
other side, customers departing from the network may enforce the environment to
jump immediately. So our environment is not Markov for its own.
  The main result is to compute explicitly the joint stationary distribution of
the queue lengths vector and the environment which is of product form:
Environment and queue lengths vector, and the queue lengths over the network
are decomposable.
