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The BAR approach for multiclass queueing networks with SBP service policies

Anton Braverman, J. G. Dai, Masakiyo Miyazawa

TL;DR

The basic adjoint relationship (BAR) approach is extended to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines and makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting.

Abstract

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the "interchange of limits" approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being "tight." These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem.

The BAR approach for multiclass queueing networks with SBP service policies

TL;DR

The basic adjoint relationship (BAR) approach is extended to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines and makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting.

Abstract

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the "interchange of limits" approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being "tight." These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem.
Paper Structure (35 sections, 39 theorems, 291 equations, 1 figure)

This paper contains 35 sections, 39 theorems, 291 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A two-station, five-class queueing network
  3. Network description
  4. Markov process and its stability
  5. Heavy-traffic limit theorem
  6. Exponential distributions
  7. Basic adjoint relationship (BAR).
  8. Taylor expansion and asymptotic BAR.
  9. State space collapse (SSC).
  10. General bounded distributions
  11. Basic adjoint relationship (BAR).
  12. Multiclass queueing networks
  13. Service discipline.
  14. Notation facilitating an SBP discipline.
  15. Traffic equations.
  16. ...and 20 more sections

Key Result

Lemma 2.1

When conditions (eq:2s5c1)-(eq:2s5cv0) are satisfied, the following are satisfied

Figures (1)

  • Figure 1: A two-station, five-class reentrant line

Theorems & Definitions (66)

  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['pro:2s']}.
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 56 more