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

Pablo A. Ferrari; James B. Martin

Annales de l'I.H.P. Probabilités et statistiques (2009)

- Volume: 45, Issue: 1, page 250-265
- ISSN: 0246-0203

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topFerrari, Pablo A., and Martin, James B.. "Multiclass Hammersley–Aldous–Diaconis process and multiclass-customer queues." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 250-265. <http://eudml.org/doc/78019>.

@article{Ferrari2009,

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 λ and the (attempted) services with rate ρ>λ. Then put 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 reversed process. The coupled process is a transformation of the multi-line process and its invariant measure is the transformation described above of the product measure.},

author = {Ferrari, Pablo A., Martin, James B.},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {multi-class Hammersley–Aldous–Diaconis process; multiclass queuing system; invariant measures; multi-class Hammersley-Aldous-Diaconis process},

language = {eng},

number = {1},

pages = {250-265},

publisher = {Gauthier-Villars},

title = {Multiclass Hammersley–Aldous–Diaconis process and multiclass-customer queues},

url = {http://eudml.org/doc/78019},

volume = {45},

year = {2009},

}

TY - JOUR

AU - Ferrari, Pablo A.

AU - Martin, James B.

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

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2009

PB - Gauthier-Villars

VL - 45

IS - 1

SP - 250

EP - 265

AB - 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 λ and the (attempted) services with rate ρ>λ. Then put 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 reversed process. The coupled process is a transformation of the multi-line process and its invariant measure is the transformation described above of the product measure.

LA - eng

KW - multi-class Hammersley–Aldous–Diaconis process; multiclass queuing system; invariant measures; multi-class Hammersley-Aldous-Diaconis process

UR - http://eudml.org/doc/78019

ER -

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