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Heavy Traffic Diffusion Limit for a Closed Queueing Network with Single-Server and Infinite-Server Stations

Amir A. Alwan, Barış Ata

TL;DR

The paper derives a diffusion approximation for a closed queueing network with $J$ single-server and $K$ infinite-server stations in heavy traffic, using a continuous-mapping approach to establish a multidimensional reflected diffusion as the limit of diffusion-scaled queue lengths and idle times. The analysis scales the single-server rates with $n$ while keeping infinite-server rates fixed, and imposes a two-level routing structure, enabling a Skorokhod-problem description and a nonlinear regulator mapping for the limit. The model captures origin-destination dynamics and heterogeneous travel times, providing a tractable framework for diffusion-scale control in ride-hailing-like systems and beyond. This work generalizes prior diffusion limits by allowing multiple infinite-server nodes and two-level routing, laying theoretical groundwork for high-dimensional stochastic-control methods in closed networks.

Abstract

This paper studies the limiting behavior of a closed queueing network with multiple single-server and infinite-server stations. Under a heavy traffic asymptotic regime$\unicode{x2014}$where the number of jobs and single-server service rates grow large while infinite-server rates remain fixed$\unicode{x2014}$we prove a weak convergence result for the queue length and idleness process vector, providing an approximation for the original system.

Heavy Traffic Diffusion Limit for a Closed Queueing Network with Single-Server and Infinite-Server Stations

TL;DR

The paper derives a diffusion approximation for a closed queueing network with single-server and infinite-server stations in heavy traffic, using a continuous-mapping approach to establish a multidimensional reflected diffusion as the limit of diffusion-scaled queue lengths and idle times. The analysis scales the single-server rates with while keeping infinite-server rates fixed, and imposes a two-level routing structure, enabling a Skorokhod-problem description and a nonlinear regulator mapping for the limit. The model captures origin-destination dynamics and heterogeneous travel times, providing a tractable framework for diffusion-scale control in ride-hailing-like systems and beyond. This work generalizes prior diffusion limits by allowing multiple infinite-server nodes and two-level routing, laying theoretical groundwork for high-dimensional stochastic-control methods in closed networks.

Abstract

This paper studies the limiting behavior of a closed queueing network with multiple single-server and infinite-server stations. Under a heavy traffic asymptotic regimewhere the number of jobs and single-server service rates grow large while infinite-server rates remain fixedwe prove a weak convergence result for the queue length and idleness process vector, providing an approximation for the original system.

Paper Structure

This paper contains 17 sections, 12 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Queueing Network Model
  4. Model Primitives
  5. State Dynamics
  6. Heavy Traffic Assumption and Main Result
  7. Auxiliary Results
  8. Main Convergence Results
  9. Convergence of Fluid Scaled Processes
  10. Convergence of Diffusion Scaled Processes
  11. Proof of Lemma \ref{['Lemma:A.1']}
  12. Existence of an element in $D_K^T$ satisfying (\ref{['eq:A.1']}) for all $t\in [0,T]$
  13. Uniqueness of the element in $D_K^T$ satisfying (\ref{['eq:A.1']}) for all $t\in [0,T]$
  14. Extension to a unique element in $D^K$ satisfying (\ref{['eq:A.1']}) for all $t\in [0,\infty)$
  15. Proof of Proposition \ref{['Prop:4.1']}
  16. ...and 2 more sections

Key Result

Theorem 1

As $n\rightarrow\infty$, $(\hat{Q}^n,\hat{I}^n,\hat{V}^n)\Rightarrow (Q^*, I^*, V^*)$, where $(Q^*,I^*, V^*)$ is a $(2J+K)$-dimensional process with continuous sample paths in $\mathbb{R}^{2J}_+\times \mathbb{R}^K$ that satisfies the following equalities for all $j\in [J]$, $k\in [K]$, and $t\ge 0$:

Theorems & Definitions (23)

  • Theorem 1
  • Proposition 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 13 more