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

Ferrari, Pablo A.; Martin, James B.

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 250-265.

Dans un processus de Hammersley nous considérons une infinité de particules sur la droite réelle; et il ne peut pas y avoir plus d'une particule sur chaque position. Une particule située en x et ayant pour plus proche voisine (sur sa droite) une particule située en y, saute avec un taux y-x à une position aléatoire choisie uniformément dans l'interval (x, y). Le couplage basique entre des trajectoires ayant des configurations initiales différentes induit un processus avec des particules de classes différentes. Nous donnons une construction explicite de la mesure invariante pour le processus ayant n classes de particules. Pour démontrer que la mesure est invariante nous introduisons un autre processus appelé «multi-ligne». La mesure invariante pour ce processus est un produit de plusieurs processus de Poisson. La définition du processus multi-ligne met en jeu les «points duaux» (de l'espace-temps), qui apparaient naturellement dans la construction graphique du processus renversé par rapport au temps.

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.

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