Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Rami Atar, Masakiyo Miyazawa

TL;DR

The paper analyzes a multi-level GI/G/1 queue in heavy traffic with discontinuous, level-dependent rates and introduces the Daley-Miyazawa semimartingale decomposition as a core analytic tool. It proves that the diffusion limit is a unique-in-law reflecting diffusion with discontinuous drift and diffusion coefficients, obtained via a rigorous tightness and martingale-based argument. The main result demonstrates convergence of the scaled queue-length process to this limit and outlines a framework potentially applicable to a broad class of non-Markovian queueing models. This approach offers a principled method for diffusion approximations in systems with discontinuities and time-varying rates tied to the state of the queue.

Abstract

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.

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

TL;DR

The paper analyzes a multi-level GI/G/1 queue in heavy traffic with discontinuous, level-dependent rates and introduces the Daley-Miyazawa semimartingale decomposition as a core analytic tool. It proves that the diffusion limit is a unique-in-law reflecting diffusion with discontinuous drift and diffusion coefficients, obtained via a rigorous tightness and martingale-based argument. The main result demonstrates convergence of the scaled queue-length process to this limit and outlines a framework potentially applicable to a broad class of non-Markovian queueing models. This approach offers a principled method for diffusion approximations in systems with discontinuities and time-varying rates tied to the state of the queue.

Abstract

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 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.

Paper Structure

This paper contains 15 sections, 10 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Paper organization and notation
  4. Model and main results
  5. Sequence of queueing models
  6. Heavy traffic conditions and main theorem
  7. Daley-Miyazawa decompositions
  8. Proof of main result
  9. Preliminaries
  10. Key lemmas and proof of Theorem \ref{['th1']}
  11. Tools
  12. Proof of Lemma \ref{['lem01']}.
  13. Proof of Lemma \ref{['lem02']}.
  14. Proof of Lemma \ref{['lem03']}.
  15. Proof of Lemma \ref{['lem:e-mp']}

Key Result

Theorem 2.1

There exists a weak solution to 09, and it is unique in law. Moreover, denoting by $(X,L,W)$ a solution corresponding to $x_0=0$, and by $(\hat{X}^n,\hat{I}^n)$ the processes associated with the mutli-level queue as in hat-XI, one has $(\hat{X}^n,\hat{I}^n)\Rightarrow(X,L)$ in $\mathscr{D}({\mathbb

Theorems & Definitions (15)

  • Theorem 2.1
  • Remark 2.2: About our assumptions
  • Remark 2.3: About extensions
  • Lemma 3.1
  • Remark 3.2
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • ...and 5 more