  Queueing networks are gaining attraction for the performance analysis of
parallel computer systems. A Jackson network is a set of interconnected
servers, where the completion of a job at server i may result in the creation
of a new job for server j. We propose to extend Jackson networks by "branching"
and by "control" features. Both extensions are new and substantially expand the
modelling power of Jackson networks. On the other hand, the extensions raise
computational questions, particularly concerning the stability of the networks,
i.e, the ergodicity of the underlying Markov chain. We show for our extended
model that it is decidable in polynomial time if there exists a controller that
achieves stability. Moreover, if such a controller exists, one can efficiently
compute a static randomized controller which stabilizes the network in a very
strong sense; in particular, all moments of the queue sizes are finite.
