Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4202

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

Pablo A. Ferrari·James B. Martin
View original source ↗Read source document ↗Export metadata ↓

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 λand the (attempted) services with rate ρ>λ. 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.

Explore related subjects

math.PRmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pablo A. Ferrari, James B. Martin. 2007-07-28. Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues. https://arxiv.org/abs/0707.4202

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions

For the empirical measure of the first $n$ points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in $W_1$, both in expectation and almost surely along all sample sizes. The constant is $c_*=κ_1/(2\sqrtπ)\in(0,1/4]$, where $κ_1$ is the linear-growth Gaussian variational constant of Armegioiu--Goldman--Grotto--Trevisan. This answers the two-dimensional bounded-Lipschitz convergence question posed by Yukich in 1989. We give the complete endpoint argument, square--torus comparison, empirical reduction, and full-sequence probability argument.

math.PR↗
arXiv2026

Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime

Mansour, Rastegar and Roitershtein studied the roots of ballistic deposition on a cyclic strip of $K$ sites, namely the sites that receive a particle at height one, and the number $D_{i,K}$ of pairs of consecutive roots at distance $i+1$. They computed the first two moments of $D_{i,K}$ for $i\le7$. They conjectured that for every $i$ the mean and the variance are exactly linear in $K$, that $D_{i,K}$ satisfies a central limit theorem, and that the standardized vector $(D_{i,K})_{i\ge1}$ converges to a Gaussian process. We prove these conjectures. The key observation is that the roots are exactly the valleys of a uniformly random circular arrangement, so that $D_{i,K}$ counts occurrences of a family of consecutive patterns. We determine sizes from which the moments are exactly linear, which cannot be lowered in any case we checked, and give closed forms for the mean and variance constants and for the limiting covariance. We also prove a Wasserstein bound of order $(\mathbb{E} D_{i,K})^{-1/2}$ that is uniform in $i$, and a local limit theorem. Finally we describe the limit completely when $i$ grows with $K$: it is degenerate, Poisson or Gaussian according as $\mathbb{E} D_{i,K}$ tends to zero, to a positive constant, or to infinity.

math.PR↗
arXiv2026

Enhanced Dissipation under Variable Growth: Stochastic $p(x)$-Laplace Equations with Transport Noise

We study dissipation enhancement for stochastic $p(x)$-Laplace equations with divergence-free transport noise on $\mathbb T^d$, $d\ge2$. Under log-Hölder continuity and degenerate subcritical growth, we establish global variational well-posedness and an exact energy identity. For every prescribed positive coefficient, suitable transport noise yields averaged algebraic decay uniformly on bounded zero-mean $L^2$ balls. The noise is independent of the individual initial datum. This uniform enhancement holds despite the absence of homogeneous scaling and the two flux growth regimes associated with the variable exponent. We also identify $p^+$ as the optimal exponent for uniform modular coercivity near zero and obtain a strict averaged comparison with the unperturbed evolution at any prescribed time $T\ge1$ on initial-data shells with bounded gradient modular. For each prescribed decay rate and finite moment order, deterministic stepwise Fourier profiles with fixed intensity and increasing frequency range give exponential decay of the $L^2$ norm in mean square and almost surely. The absolute random prefactors have uniformly bounded moments of the prescribed order throughout the initial-data ball.

math.PR↗