  We consider exponential single server queues with state-dependent arrival and
service rates which evolve under influences of external environments. The
transitions of the queues are influenced by the environment's state and the
movements of the environment depend on the status of the queues (bi-directional
interaction). The environment is constructed in a way to encompass various
models from the recent Operations Research literature, where a queue is coupled
with an inventory or with reliability issues. With a Markovian joint
queueing-environment process we prove separability for a large class of such
interactive systems, i.e. the steady state distribution is of product form and
explicitly given. The queue and the environment processes decouple
asymptotically and in steady state. For non-separable systems we develop
ergodicity and exponential ergodicity criteria via Lyapunov functions. By
examples we explain principles for bounding departure rates of served customers
(throughputs) of non-separable systems by throughputs of related separable
systems as upper and lower bound.
