Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

TL;DR

This paper studies the multiclass Hammersley-Aldous-Diaconis process, linking it to multi-tiered queue models and introducing a multi-line process with product Poisson measures as invariant distributions.

Contribution

It provides a novel construction of invariant measures for multiclass particle systems using tandem queues and a new multi-line coupling process.

Findings

Invariant measures for multiclass processes are derived from tandem queue constructions.

A multi-line process with product Poisson invariant measure is introduced.

The approach generalizes to multiple classes using a sequence of queues.

Abstract

In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate \lambda and the (attempted) services with rate \rho>\lambda. Then put the first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n-1 queues in tandem with n-1 priority-types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the system. The coupled process is a transformation of the multi-line process and its invariant measure the transformation described above of the product measure.