# Stationary Distributions and Convergence for M/M/1 Queues in Interactive Random Environment

## TL;DR

This paper analyzes a Markovian single-server queue in an interactive random environment, deriving explicit stationary distributions and convergence rates for two types of environments, enhancing understanding of their long-term behavior.

## Contribution

It provides explicit stationary distributions and convergence rate estimates for M/M/1 queues in interactive random environments, including jump and jump-diffusion types.

## Key findings

- Stationary distribution is explicitly derived as a weighted geometric distribution.
- Exponential convergence rates to stationarity are explicitly estimated.
- Results apply to both pure jump and reflected jump-diffusion environments.

## Abstract

A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depends on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump-diffusion. In both cases, the joint dynamics is constructed so that the stationary distribution can be explicitly found in a simple form (weighted geometric). We also derive an explicit estimate for exponential rate of convergence to the stationary distribution via coupling.