Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition

This paper studies a single server queue in heavy traffic, with general inter-arrival and service time distributions, where arrival and service rates vary discontinuously as a function of the (diffusively scaled) queue length. It is proved that the weak limit is given by the unique-in-law solution to a stochastic differential equation in $[0,\infty)$ with discontinuous drift and diffusion coefficients. The main tool is a semimartingale decomposition for point processes introduced in \cite{dal-miy}, which is distinct from the Doob-Meyer decomposition of a counting process. Whereas the use of this tool is demonstrated here for a particular model, we believe it may be useful for investigating the scaling limits of queueing models very broadly.

💡 Research Summary

The paper investigates a single‑server GI/G/1 queue in heavy traffic where the arrival and service rates are piecewise constant functions of the (diffusively scaled) queue length. Unlike most heavy‑traffic literature that assumes smooth dependence of the rates on the state, the authors allow discontinuities at finitely many thresholds. The main contribution is to prove that, after centering and √n‑scaling, the queue‑length process converges weakly to a reflected diffusion on the non‑negative half‑line whose drift and diffusion coefficients are themselves discontinuous, taking different constant values on each interval defined by the thresholds.

The analytical engine behind the proof is the Daley‑Miyazawa semimartingale decomposition for general point processes, introduced in Daley and Miyazawa (2008). This decomposition writes a counting process as the sum of a deterministic linear term, a residual‑time correction, and a martingale component. Crucially, the finite‑variation part is not predictable but its first‑order term coincides with the law‑of‑large‑numbers limit of the point process, making it directly amenable to scaling‑limit analysis. In contrast, the classic Doob‑Meyer decomposition relies on a predictable intensity, which is tractable only for Poisson‑type inputs.

The model is built as follows. For each n, the state space is partitioned by thresholds ℓⁿ₀=0<ℓⁿ₁<…<ℓⁿ_{K−1}, with ℓⁿ_i≈√n·ℓ_i. When the queue length Xⁿ(t) lies in the i‑th interval Sⁿ_i, arrivals occur at rate λⁿ_i and services at rate μⁿ_i. Arrival and service epochs are generated by two independent renewal processes A and S with unit‑mean inter‑arrival times Z_A(j) and Z_S(j) and finite variances σ_A², σ_S². The cumulative time spent in each interval up to t is denoted Hⁿ_i(t); the integrated arrival and service intensities are Uⁿ(t)=∑ λⁿ_i Hⁿ_i(t) and Vⁿ(t)=∑ μⁿ_i Hⁿ_i(t). The counting processes are then Aⁿ(t)=A(Uⁿ(t)) and Dⁿ(t)=S(Vⁿ(t)), and the queue dynamics satisfy Xⁿ(t)=Xⁿ(0)+Aⁿ(t)−Dⁿ(t).

Heavy‑traffic scaling assumes λⁿ_i=nλ_i+√n \hat λ_i+o(√n) and μⁿ_i=nμ_i+√n \hat μ_i+o(√n) with λ_i=μ_i (critical load) for all i. Define b_i= \hat λ_i−\hat μ_i and σ_i=√(λ_iσ_A²+μ_iσ_S²). The drift and diffusion functions are piecewise constant: b(x)=∑{i=1}^K b_i 1{S_i}(x), σ(x)=∑{i=0}^K σ_i 1{S_i}(x). The limiting reflected SDE is X(t)=x₀+∫₀^t b(X(s)) ds + ∫₀^t σ(X(s)) dW(s) + L(t), ∫₀^t X(s) dL(s)=0, where L is the minimal non‑decreasing regulator ensuring X(t)≥0 (the Skorokhod map Γ).

The proof proceeds in several steps. First, the Daley‑Miyazawa decomposition is applied to A and S, yielding A(t)=t+R_A(t)−Z_A(0)+M_A(t), S(t)=t+R_S(t)−Z_S(0)+M_S(t), with martingales M_A, M_S and residual‑time processes R_A, R_S. Substituting the time‑changed arguments Uⁿ, Vⁿ gives analogous decompositions for Aⁿ and Dⁿ. The finite‑variation parts involve the integrals of λⁿ_i and μⁿ_i against the occupation times Hⁿ_i; after scaling they converge to the deterministic drift term ∫ b(X(s)) ds. The martingale parts, after √n‑scaling, satisfy a functional central limit theorem and converge jointly to a Brownian motion with variance σ²(X). Lemma 3.1 shows that simultaneous jumps of Aⁿ and Dⁿ are negligible, which guarantees that the cross‑variation of the two martingales vanishes, simplifying the diffusion coefficient.

A delicate part of the argument is controlling the behavior near the threshold points where the coefficients jump. Because the thresholds are of order √n, the process spends vanishingly small time in a shrinking neighborhood of each discontinuity, and the authors prove that the contribution of this “boundary layer” to the limit is o(1). This uses the continuity of the inter‑arrival and service time distributions (no atoms) to ensure that arrivals and departures do not occur simultaneously with positive probability.

Finally, the Skorokhod map Γ is invoked to handle the reflection at zero. The map is continuous under the J₁ topology, so convergence of the pre‑limit processes (the centered queue length and the scaled idle time) implies convergence of the reflected pair (X, L). Tightness follows from standard martingale criteria and the boundedness of the coefficients on each interval.

Theorem 2.1 establishes existence and uniqueness in law of the reflected diffusion with discontinuous coefficients and proves the weak convergence (ˆXₙ, ˆIₙ)⇒(X, L). The paper concludes by discussing extensions: functional central limit theorems for more general renewal inputs, multi‑dimensional networks with piecewise constant intensity surfaces, and controlled queueing systems where the limiting SDE would inherit additional local‑time terms at the discontinuity surfaces. The authors argue that the Daley‑Miyazawa decomposition provides a versatile framework for such problems, especially when traditional intensity‑based methods are inapplicable.