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Asymptotic Product-form Steady-state for Multiclass Queueing Networks: A Reentrant Line Case Study

Jim Dai, Dongyan Huo

Abstract

This paper serves as a companion to "Asymptotic Product-form Steady-state for Multiclass Queueing Networks with SBP Service Policies in Multi-scale Heavy Traffic." In this short paper, we illustrate the main results of the main paper through a two-station, five-class reentrant line under a specific static buffer priority policy, while avoiding heavy notations. For this example, we prove the asymptotic steady-state limit and uniform moment bound under general inter-arrival and service time distributions.

Asymptotic Product-form Steady-state for Multiclass Queueing Networks: A Reentrant Line Case Study

Abstract

This paper serves as a companion to "Asymptotic Product-form Steady-state for Multiclass Queueing Networks with SBP Service Policies in Multi-scale Heavy Traffic." In this short paper, we illustrate the main results of the main paper through a two-station, five-class reentrant line under a specific static buffer priority policy, while avoiding heavy notations. For this example, we prove the asymptotic steady-state limit and uniform moment bound under general inter-arrival and service time distributions.

Paper Structure

This paper contains 9 sections, 11 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Network and Service Discipline
  3. Markov Process and Stability
  4. Multi-scale Heavy-traffic and Product-form Limit
  5. Proof of Product-form Limit
  6. Exponential Distribution
  7. General Distribution
  8. Proof of Uniform Moment Bound
  9. Conclusion

Key Result

Proposition 3.1

Under Assumption assumption:2s5c-t-moment and assumption:multi-scale-2s5c, there exist a random vector $(Z_1^\ast, Z_4^\ast)\in\mathbb{R}_+^2$ such that where "$\Rightarrow$" denotes convergence in distribution. Moreover, $Z_1^\ast$ and $Z_4^\ast$ are independent and have the following marginal distributions with

Figures (1)

  • Figure 1: Two-Station Five-Class Re-entrant Line

Theorems & Definitions (14)

  • Proposition 3.1
  • Lemma 4.1
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Lemma 4.2: BravDaiMiya2023 Lemma 2.2
  • Lemma 4.3: BravDaiMiya2023 Lemma 2.3
  • Lemma 4.4: BravDaiMiya2023 Lemma 2.6
  • proof
  • ...and 4 more