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Diffusion approximation of the stationary distribution of a two-level single server queue

Masakiyo Miyazawa

TL;DR

This work analyzes a single-server queue whose arrival and service parameters switch at a threshold, forming a 2-level GI/G/1 model. By applying heavy-traffic scaling and a BAR-based framework with Palm distributions, the authors derive the diffusion-like limit of the scaled queue length, showing a truncated exponential density below the threshold and an exponential tail above it under mild parametric conditions. The result encompasses generally distributed inter-arrival times and workloads and recovers the exponential-case as a special instance; a complete BAR-based proof is provided and a process-limit interpretation is discussed. This diffusion-approximation insight advances understanding of state-dependent queues and supports energy-management considerations by informing threshold choices that balance energy consumption and waiting costs.

Abstract

We consider a single server queue which has a threshold to change its arrival process and service speed by its queue length, which is referred to as a two-level single server queue. This model is motivated by an energy saving problem for a single server queue whose arrival process and service speed are controlled. To get its performance in tractable form, we study the limit of the stationary distribution of the queue length in this two-level queue under scaling in heavy traffic. Except for a special case, this limit corresponds to its diffusion approximation. It is shown that this limiting distribution is truncated exponential (or uniform if the drift is null) below the threshold level and exponential above it under suitably chosen system parameters and generally distributed inter-arrival times and workloads brought by customers. This result is proved under a mild limitation on arrival parameters using the so called BAR approach studied in Braverman, Dai and Miyazawa (2017, 2023) and Miyazawa (2017, 2023). We also intuitively discuss about a diffusion process corresponding to the limit of the stationary distribution under scaling.

Diffusion approximation of the stationary distribution of a two-level single server queue

TL;DR

This work analyzes a single-server queue whose arrival and service parameters switch at a threshold, forming a 2-level GI/G/1 model. By applying heavy-traffic scaling and a BAR-based framework with Palm distributions, the authors derive the diffusion-like limit of the scaled queue length, showing a truncated exponential density below the threshold and an exponential tail above it under mild parametric conditions. The result encompasses generally distributed inter-arrival times and workloads and recovers the exponential-case as a special instance; a complete BAR-based proof is provided and a process-limit interpretation is discussed. This diffusion-approximation insight advances understanding of state-dependent queues and supports energy-management considerations by informing threshold choices that balance energy consumption and waiting costs.

Abstract

We consider a single server queue which has a threshold to change its arrival process and service speed by its queue length, which is referred to as a two-level single server queue. This model is motivated by an energy saving problem for a single server queue whose arrival process and service speed are controlled. To get its performance in tractable form, we study the limit of the stationary distribution of the queue length in this two-level queue under scaling in heavy traffic. Except for a special case, this limit corresponds to its diffusion approximation. It is shown that this limiting distribution is truncated exponential (or uniform if the drift is null) below the threshold level and exponential above it under suitably chosen system parameters and generally distributed inter-arrival times and workloads brought by customers. This result is proved under a mild limitation on arrival parameters using the so called BAR approach studied in Braverman, Dai and Miyazawa (2017, 2023) and Miyazawa (2017, 2023). We also intuitively discuss about a diffusion process corresponding to the limit of the stationary distribution under scaling.
Paper Structure (22 sections, 24 theorems, 178 equations, 2 figures)

This paper contains 22 sections, 24 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Framework for the 2-level $GI/G/1$ queue
  3. A sequence of the 2-level $GI/G/1$ queues
  4. Assumptions
  5. Main results
  6. BAR's for the 2-level $GI/G/1$ queue
  7. Palm distribution and BAR
  8. Asymptotic BAR for the pre-limit process $X^{(r)}(\cdot)$
  9. Expansion of the asymptotic BAR
  10. Proof of Theorem \ref{['mythr:2LQ-main']}
  11. Proof of Proposition \ref{['mypro:2LQ-main-2']}
  12. Proof of Proposition \ref{['mypro:2LQ-3f']}
  13. Concluding remarks
  14. Process limit
  15. Intuitive verification of (\ref{['eq:2LQ-Ai']})
  16. ...and 7 more sections

Key Result

Lemma 3.1

Assume the assumptions (mysec:sequence.b) and (mysec:sequence.e), then $\alpha^{(r)}_{e}$ is finite and positive, and $\alpha^{(r)}_{s} = \alpha^{(r)}_{e}$.

Figures (2)

  • Figure 3.1: Sample path of the queue length process of the $r$-th model
  • Figure A.1: Transition diagram

Theorems & Definitions (39)

  • Remark 3.1
  • Lemma 3.1
  • Theorem 3.1
  • Remark 3.2
  • Corollary 3.1
  • Proposition 3.1
  • Remark 3.3
  • Remark 4.1
  • Lemma 4.1
  • proof
  • ...and 29 more