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

Abstract

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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.

How to cite

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Ferrari, 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&gt;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&gt;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 -

References

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  1. [1] D. Aldous and P. Diaconis. Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 (1995) 199–213. Zbl0836.60107MR1355056
  2. [2] O. Angel. The stationary measure of a 2-type totally asymmetric exclusion process. J. Combin. Theory Ser. A 113 (2006) 625–635. Zbl1087.60067MR2216456
  3. [3] F. Baccelli and P. Brémaud. Elements of Queueing Theory, 2nd edition. Springer, Berlin, 2003. Zbl1021.60001MR1957884
  4. [4] E. Cator and P. Groeneboom. Hammersley’s process with sources and sinks. Ann. Probab. 33 (2005) 879–903. Zbl1066.60011MR2135307
  5. [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. Zbl1102.60320MR1251221
  6. [6] E. Duchi and G. Schaeffer. A combinatorial approach to jumping particles. J. Combin. Theory Ser. A 110 (2005) 1–29. Zbl1059.05006MR2128962
  7. [7] M. Ekhaus and L. Gray. A strong law for the motion of interfaces in particle systems. Unpublished manuscript, 1993. 
  8. [8] P. A. Ferrari. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91 (1992) 81–101. Zbl0744.60117MR1142763
  9. [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. Zbl0841.60085MR1298099
  10. [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. Zbl0725.60113MR1085334
  11. [11] P. A. Ferrari and J. B. Martin. Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab. 35 (2007) 807–832. Zbl1117.60089MR2319708
  12. [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. Zbl1155.60043MR2249632
  13. [13] P. L. Ferrari. Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Comm. Math. Phys. 252 (2004) 77–109. Zbl1124.82316MR2103905
  14. [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. [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. Zbl0236.00018MR405665
  16. [16] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab. 4 (1976) 339–356. Zbl0339.60091MR418291
  17. [17] T. M. Liggett. Interacting Particle Systems. Springer, New York, 1985. Zbl0559.60078MR776231
  18. [18] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999. Zbl0949.60006MR1717346
  19. [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. Zbl0826.60083MR1325044
  20. [20] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process J. Stat. Phys. 108 (2002) 1071–1106. Zbl1025.82010MR1933446
  21. [21] T. Sepäläinen. A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab. 1 (1996) (approx.) 51 pp. (electronic). Zbl0891.60093MR1386297
  22. [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. Zbl0872.60084
  23. [23] W. D. Wick. A dynamical phase transition in an infinite particle system. J. Statist. Phys. 38 (1985) 1015–1025. Zbl0625.76080MR802566

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