Exponential Single Server Queues in an Interactive Random Environment

We consider exponential single server queues with state-dependent arrival and service rates that 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 (bidirectional 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; that is, the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state. For nonseparable 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 nonseparable systems by throughputs of related separable systems as upper and lower bound.