Exponential single server queues with state dependent arrival and service
rates are considered 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 structure of the environment is constructed in a way to
encompass various models from the recent Operation Research literature, where a
queue is coupled e.g. 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 criteria via Lyapunov
functions. By examples we show principles for bounding throughputs of
non-separable systems by throughputs of two separable systems as upper and
lower bound