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

Masahiro Kobayashi, Masakiyo Miyazawa+1 more|arXiv (Cornell University)|Jan 6, 2026

Advanced Queuing Theory Analysis0 citations

TL;DR

The paper derives the stationary distribution in closed form for a history-dependent two-level M/M/1 queue, then establishes its diffusion limit under heavy traffic and provides approximation formulas and numerical validation.

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.

Motivation & Objective

• Motivate state- and history-dependent control of service capacity in queueing systems.
• Construct a two-level M/M/1 queue where arrival and service rates depend on current level and queue-length history.
• Derive the exact stationary distribution under a stability condition.
• Derive the diffusion limit of the stationary distribution in heavy traffic for each background state.
• Develop and assess approximation formulas for the stationary distribution using the diffusion limit.

Proposed method

• Model the system as a continuous-time Markov chain on the state space S = S1,1 ∪ S2,1 ∪ S1,2 ∪ S2,2 with levels and a background state B(t) ∈ {1,2}.
• Define level-dependent arrival and service rates λi, μi and level-crossing background transitions at thresholds ℓd and ℓu.
• Derive the stationary distribution π by solving balance equations (2.2)–(2.7) and express π in closed form (Theorem 2.1).
• Introduce the MGFs ψ, ψi,j to characterize the stationary distribution components (Equations 2.11–2.14).
• Formulate a heavy-traffic diffusion limit by analyzing a sequence of systems with scaling n, under Assumptions (a)–(d).
• Obtain the limiting density f = f1,1 + f2,1 + f1,2 + f2,2 (Theorem 3.1).

Figure 1: Sample path of $L(t)$

Experimental results

Research questions

• RQ1 What is the stationary distribution of a history-dependent two-level M/M/1 queue with level and history dependence?
• RQ2 What does the diffusion limit of the stationary distribution look like under heavy traffic for each background state?
• RQ3 How do the history-dependent controls influence the shape of the diffusion limit density components?
• RQ4 How accurate are the diffusion-based approximations for the stationary distribution in numerical experiments?

Key findings

• The stationary distribution exists under the stability condition ρ2 < 1 and has a closed-form expression (Theorem 2.1).
• The diffusion limit of the scaled stationary distribution exists and is given by a density f = f1,1 + f2,1 + f1,2 + f2,2 (Theorem 3.1).
• Some density components are exponential or uniform (f1,1 and f2,2 for b1 ≠ 0 or b1 = 0 respectively), while f2,1 and f1,2 exhibit nonstandard forms due to history dependence (Remark 3.1).
• The limit density is piecewise-defined over intervals tied to the thresholds ṽℓd and ṽℓu, reflecting level structure and history effects.
• The normalization constant C0 is explicitly given and used to express all density components (Equations 3.8, 3.4–3.7).
• Numerical examples show the diffusion approximation is accurate for the studied model.

Figure 2: Transition diagram of $(L(t),B(t))$