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

2019/02/11 by Ya. I. Belopolskaya, Guodong Pang, Belopolskaya, Yana +5
Business, Management and Accounting · Decision Sciences · Mathematics · #60K25 #68M20 #90B22 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1902.03941

openalex publication_date 2019/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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