Exponential single server queues in an interactive random environment

TL;DR

This paper studies exponential single server queues interacting with external environments, proving conditions for product-form steady states and analyzing ergodicity in complex coupled systems.

Contribution

It introduces a framework for analyzing bi-directionally interacting queue-environment systems, establishing conditions for steady state separability and ergodicity.

Findings

Proved product-form steady state distribution for a broad class of interactive queues.

Developed ergodicity and exponential ergodicity criteria using Lyapunov functions.

Provided bounds for throughput in non-separable systems based on related separable models.

Abstract

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.