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), p. 250-265 / Harvested from Numdam
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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.

Publié le : 2009-01-01
DOI : https://doi.org/10.1214/08-AIHP168
Classification:  60K35,  60K25,  90B22 
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     title = {Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {45},
     year = {2009},
     pages = {250-265},
     doi = {10.1214/08-AIHP168},
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     language = {en},
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Ferrari, Pablo A.; Martin, James B. Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) pp. 250-265. doi : 10.1214/08-AIHP168. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_1_250_0/

[1] D. Aldous and P. Diaconis. Hammersley's interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 (1995) 199-213. | MR 1355056 | Zbl 0836.60107

[2] O. Angel. The stationary measure of a 2-type totally asymmetric exclusion process. J. Combin. Theory Ser. A 113 (2006) 625-635. | MR 2216456 | Zbl 1087.60067

[3] F. Baccelli and P. Brémaud. Elements of Queueing Theory, 2nd edition. Springer, Berlin, 2003. | MR 1957884 | Zbl 1021.60001

[4] E. Cator and P. Groeneboom. Hammersley's process with sources and sinks. Ann. Probab. 33 (2005) 879-903. | MR 2135307 | Zbl 1066.60011

[5] B. Derrida, S. A. Janowsky, J. L. Lebowitz and E. R. Speer. Exact solution of the totally asymmetric simple exclusion process: shock profiles. J. Statist. Phys. 73 (1993) 813-842. | MR 1251221 | Zbl 1102.60320

[6] E. Duchi and G. Schaeffer. A combinatorial approach to jumping particles. J. Combin. Theory Ser. A 110 (2005) 1-29. | MR 2128962 | Zbl 1059.05006

[7] M. Ekhaus and L. Gray. A strong law for the motion of interfaces in particle systems. Unpublished manuscript, 1993.

[8] P. A. Ferrari. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91 (1992) 81-101. | MR 1142763 | Zbl 0744.60117

[9] P. A. Ferrari, L. R. G. Fontes and Y. Kohayakawa. Invariant measures for a two-species asymmetric process. J. Statist. Phys. 76 (1994) 1153-1177. | MR 1298099 | Zbl 0841.60085

[10] P. A. Ferrari, C. Kipnis and E. Saada. Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19 (1991) 226-244. | MR 1085334 | Zbl 0725.60113

[11] P. A. Ferrari and J. B. Martin. Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab. 35 (2007) 807-832. | MR 2319708 | Zbl 1117.60089

[12] P. A. Ferrari and J. B. Martin. Multiclass processes, dual points and M/M/1 queues. Markov Process. Related Fields 12 (2006) 273-299. | MR 2249632 | Zbl 1155.60043

[13] P. L. Ferrari. Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Comm. Math. Phys. 252 (2004) 77-109. | MR 2103905 | Zbl 1124.82316

[14] J. E. Garcia. Processo de Hammersley. Ph.d. Thesis (Portuguese). University of São Paulo, 2000. Available at http://www.ime.usp.br/~pablo/students/jesus/garcia-tese.pdf.

[15] J. M. Hammersley. A few seedlings of research. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics 345-394. Univ. California Press, Berkeley, Calif, 1972. | MR 405665 | Zbl 0236.00018

[16] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab. 4 (1976) 339-356. | MR 418291 | Zbl 0339.60091

[17] T. M. Liggett. Interacting Particle Systems. Springer, New York, 1985. | MR 776231 | Zbl 0559.60078

[18] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999. | MR 1717346 | Zbl 0949.60006

[19] T. Mountford and B. Prabhakar. On the weak convergence of departures from an infinite series of /M/1 queues. Ann. Appl. Probab. 5 (1995) 121-127. | MR 1325044 | Zbl 0826.60083

[20] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process J. Stat. Phys. 108 (2002) 1071-1106. | MR 1933446 | Zbl 1025.82010

[21] T. Sepäläinen. A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab. 1 (1996) (approx.) 51 pp. (electronic). | MR 1386297 | Zbl 0891.60093

[22] E. R. Speer. The two species asymmetric simple exclusion process. In On Three Levels: Micro, Meso and Macroscopic Approaches in Physics 91-102. C. M. M. Fannes and A. Verbuere (Eds). Plenum, New York, 1994. | Zbl 0872.60084

[23] W. D. Wick. A dynamical phase transition in an infinite particle system. J. Statist. Phys. 38 (1985) 1015-1025. | MR 802566 | Zbl 0625.76080