Diffusion limit for the stationary distribution of a history-dependent two-level M/M/1 queue

TL;DR

This paper studies a history-dependent two-level M/M/1 queue in which both the level and queue-length history influence arrival and service rates via a background state. It derives a closed-form stationary distribution for the joint process $(L,B)$ under $\rho_2<1$ and establishes a heavy-traffic diffusion limit for the scaled queue length $L^{(n)}/\sqrt{n}$, yielding a density decomposed into four components $f=f_{11}+f_{21}+f_{12}+f_{22}$ with some components being nonstandard due to history effects. The components $f_{11}$ and $f_{22}$ are exponential (or uniform when $b_1=0$), while $f_{21}$ and $f_{12}$ capture history-induced structure and are not exponential or uniform. Using the limiting density, the authors provide a diffusion-based approximation for the mean queue length and validate its accuracy through numerical experiments, highlighting practical applicability for design and control in congestion-aware systems.

Abstract

Recently, Atar and Miyazawa [2] introduced a multi-level GI/G/1 queue with a finite number of levels, where both the arrival and service rates depend on the level corresponding to the current queue length. For this model, they proved that the diffusion limit of its queue length process in heavy traffic is the level-dependent reflected Brownian motion of [6]. In a subsequent study, Kobayashi et al. [4] derived the corresponding diffusion limit of the stationary distribution. These studies are motivated by the control of service capacity depending on the queue length. We are interested in the more general case where this control may also depend on the history of the queue length. As the first step toward such a generalization, we specialize the multi-level GI/G/1 queue to a two-level M/M/1 queue. We then extend the dynamics of this model so that its arrival and service rates depend not only on the current queue length but also on the recent history of queue lengths. Under the stability condition for this model, we first compute its stationary distribution in closed form, then derive its diffusion limit in heavy traffic. Finally, using this diffusion limit, we derive approximation formulas for the stationary distribution and then numerically assess their accuracy.

Figures (3)

• Figure 1: Sample path of $L(t)$
• Figure 2: Transition diagram of $(L(t),B(t))$
• Figure 3: Graphs of changing $n$ and $b_{1}$

Theorems & Definitions (8)

• Theorem 2.1
• Remark 2.1
• Theorem 3.1
• Remark 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 5.1
• Corollary 5.1