Exponential single server queues in an interactive random environment

Abstract

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.

Birth-Death Processes in Interactive Random Environments

“…In [6], another construction is provided for Markov processes in interactive random environments (pure Markov jump process) that allows simultaneous transitions for the Markov chain and environment states, for which a product form invariant measure is derived and applications to queueing and neural avalanches are discussed. We note the main differences in the construction of the joint Markov process in [14,15,23] from our paper: they allow simultaneous changes in the queueing and environment states, while our construction does not. Moreover, the environments in those papers are only a Markov jump process.…”

“…Birth-death processes are fundamental stochastic models in applied probability and queueing. Birth-death processes in random environments have been extensively studied [34,35,4,5] and used in various applications (see, e.g., queueing [14,15,10], inventory [23], population and biology [1] and epidemiology [25]). Most of the studies have been on models where the transitions of birth and death processes are affected by the environment (see, e.g., [34,35,4]).…”

“…In [15], the authors consider Jackson networks with interactive (Markov jump) random environments, where customers departing from the network may enforce the environment to jump immediately. In [23], single server queues with state dependent arrival and service rates which are also interactively affected by a Markov jump environment are studied, and both cases of an explicit product-form (separable) steady state distribution and of a non-separable steady-state distribution are considered. In [6], another construction is provided for Markov processes in interactive random environments (pure Markov jump process) that allows simultaneous transitions for the Markov chain and environment states, for which a product form invariant measure is derived and applications to queueing and neural avalanches are discussed.…”