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 the are influenced by environment’s state movements environment depend on status (bidirectional interaction). is constructed in a way to encompass various models from recent Operations Research literature, where queue coupled an inventory or reliability issues. With Markovian joint queueing-environment process, we prove separability for large class such interactive systems; is, steady distribution product form explicitly given. processes decouple asymptotically state. For nonseparable systems, develop ergodicity criteria via Lyapunov functions. By examples explain principles bounding departure served customers (throughputs) systems throughputs related separable as upper lower bound.