[Paper Review] Multiclass processes, dual points and M/M/1 queues

Pablo A. Ferrari, Jean Martin|ArXiv.org|Sep 20, 2005

Random Matrices and Applications22 references21 citations

TL;DR

This paper introduces a multiclass coupling framework for the TASEP and Hammersley-Aldous-Diaconis (HAD) processes using a multi-line process with independent Bernoulli product measures as invariant measures. The key contribution is a multiclass Burke's theorem showing that the stationary output of a tandem queue system with priority classes matches the input distribution, generalizing classical Burke’s theorem to multiclass queues.

ABSTRACT

We consider the discrete Hammersley-Aldous-Diaconis process (HAD) and the totally asymmetric simple exclusion process (TASEP) in Z. The basic coupling induces a multiclass process which is useful in discussing shock measures and other important properties of the processes. The invariant measures of the multiclass systems are the same for both processes, and can be constructed as the law of the output process of a system of multiclass queues in tandem; the arrival and service processes of the queueing system are a collection of independent Bernoulli product measures. The proof of invariance involves a new coupling between stationary versions of the processes called a multi-line process; this process has a collection of independent Bernoulli product measures as an invariant measure. Some of these results have appeared elsewhere and this paper is partly a review, with some proofs given only in outline. However we emphasize a new approach via dual points: when the graphical construction is used to construct a trajectory of the TASEP or HAD process as a function of a Poisson process in ZxR, the dual points are those which govern the time-reversal of the trajectory. Each line of the multi-line process is governed by the dual points of the line below. We also mention some other processes whose multiclass versions have the same invariant measures, and we note an extension of Burke's theorem to multiclass queues which follows from the results.

Motivation & Objective

• To establish invariant measures for multiclass versions of the TASEP and HAD processes using a novel coupling method.
• To demonstrate that the same invariant measure applies to both TASEP and HAD processes, despite their different dynamics.
• To introduce the concept of dual points in the graphical construction to enable time-reversal analysis and multi-line process construction.
• To extend Burke’s theorem to multiclass queueing systems by showing that the output distribution matches the input distribution under priority service.
• To unify the analysis of multiclass systems across different particle systems, including discrete-time and sequential variants.

Proposed method

• Use the basic coupling to define a multiclass process via a map R that assigns class labels based on the number of particles across n coupled configurations.
• Construct a multi-line process αt on X^n with independent Bernoulli product measures ν = ν^ρ¹ × ⋯ × ν^ρⁿ as invariant measure.
• Define the multiclass process ξt = Tαt, where T is a deterministic function mapping coupled configurations to class labels.
• Introduce dual points in the graphical construction of TASEP and HAD processes, which govern time-reversal and enable recursive construction of the multi-line process.
• Prove invariance of the measure μ on Y_n by showing that the multi-line process preserves ν and that Tαt evolves as the coupled multiclass process.
• Apply the framework to discrete-time and sequential versions of TASEP and HAD, showing invariance of μ under these dynamics.

Experimental results

Research questions

• RQ1 Can the same invariant measure be constructed for both TASEP and HAD processes in the multiclass setting?
• RQ2 How do dual points in the graphical construction facilitate the construction of the multi-line process and time-reversal analysis?
• RQ3 Does the multiclass invariant measure μ remain invariant under sequential or discrete-time updates of the TASEP and HAD processes?
• RQ4 Can the classical Burke’s theorem be generalized to multiclass priority queues in a discrete-time setting?
• RQ5 Why does the multiclass invariant measure fail to be invariant under the parallel TASEP or asymmetric simple exclusion process (ASEP) with p < 1?

Key findings

• The invariant measure μ for the multiclass TASEP and HAD processes is the same as the stationary output of a tandem system of multiclass M/M/1 queues with independent Bernoulli arrival and service processes.
• The multiclass process ξt is stationary under the measure μ, with first-class particle density ρ¹ and k-th class density ρ^k − ρ^{k−1} for k = 2, ..., n.
• The measure μ satisfies a multiclass Burke’s theorem: the m-class marginal of the n-class output distribution is equal in law to the m-class input distribution, implying invariance under truncation.
• The multi-line process with independent Bernoulli product measures ν is invariant, and the map Tαt generates the coupled multiclass process, proving invariance of μ.
• The framework extends to sequential TASEP (left-to-right or right-to-left updates), where μ remains invariant, but fails for the parallel TASEP due to loss of ordering in the coupling.
• For the ASEP with p < 1, the measure μ is not invariant, indicating a fundamental difference in dynamics compared to the TASEP and HAD processes.