Attractiveness of Brownian queues in tandem

  • Eric A. Cator
  • Sergio I. LópezEmail author
  • Leandro P. R. Pimentel


Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.


Brownian queue Tandem queues Last-passage percolation Exclusion process 

Mathematics Subject Classification

60K25 60K35 



The authors would like to thank an anonymous referee for her helpful comments that greatly improved the presentation and clarity of this work.


  1. 1.
    Baryshnikov, Y.: GUEs and queues. Probab. Theory Relat. Fields 119, 256–274 (2001)CrossRefGoogle Scholar
  2. 2.
    Cator, E.A., Groeneboom, P.: Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34, 1273–1295 (2006)CrossRefGoogle Scholar
  3. 3.
    Cator, E.A., Pimentel, L.P.R.: Busemman functions and equilibrium measures in last-passage percolation models. Prob. Theory Relat. Fields. 154, 89–125 (2012)CrossRefGoogle Scholar
  4. 4.
    Ferrari, P.A.: Shocks in the Burgers equation and the asymmetric simple exclusion process. In: Goles, E., Martínez, S. (eds.) Statistical Physics, Automata Networks and Dynamical Systems. Mathematics and its Applications, vol. 75, pp. 25–64. Springer, Dordrecht (1992)Google Scholar
  5. 5.
    Ferrari, P.L., Spohn, H., Weiss, T.: Scaling limit for Brownian with one-sided collisions. Ann. Appl. Probab. 25, 1349–1382 (2015)CrossRefGoogle Scholar
  6. 6.
    Ferrari, P.L., Spohn, H., Weiss, T.: Brownian motions with one-sided collisions: the stationary case. Eletron. J. Probab. 69, 1–41 (2015)Google Scholar
  7. 7.
    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model. Probab. Theory Relat. Fields 169, 223–255 (2015)CrossRefGoogle Scholar
  8. 8.
    Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles and Busemann functions for the corner growth model. Probab. Theory Relat. Fields 169, 177–222 (2015)CrossRefGoogle Scholar
  9. 9.
    Glynn, P.W., Whitt, W.: Departures from many queues in series. Ann. Appl. Probab. 1, 546–572 (1991)CrossRefGoogle Scholar
  10. 10.
    Gravner, W.J., Tracy, C.A., Widom, H.: Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102(5–6), 1085–1132 (2001)CrossRefGoogle Scholar
  11. 11.
    Hambly, B.M., Martin, J.B., O’Connell, N.: Concentration results for a Brownian directed percolation problem. Stoch. Process. Appl. 102, 207–220 (2002)CrossRefGoogle Scholar
  12. 12.
    Harrison, M., Williams, R.: On the quasireversibility of a multiclass Brownian service station. Ann. Probab. 18, 1249–1268 (1990)CrossRefGoogle Scholar
  13. 13.
    Ichiba, T., Karatzas, I.: On collisions of Brownian particles. Ann. Appl. Probab 20, 951–977 (2012)CrossRefGoogle Scholar
  14. 14.
    Karatzas, I., Pal, S., Shkolnikov, M.: Systems of Brownian particles with asymmetric collisions. Ann. Inst. H. Poincaré Probab. Stat. 52(1), 323–354 (2016)CrossRefGoogle Scholar
  15. 15.
    López, S.I.: Convergence of tandem Brownian queues. J. Appl. Probab. 53(2), 585–592 (2016)CrossRefGoogle Scholar
  16. 16.
    López, S.I., Pimentel, L.P.R.: On the location of the maximum of a process: lévy, Gaussian and multidimensional cases. Stochastics 90(8), 1221–1237 (2018)CrossRefGoogle Scholar
  17. 17.
    Loynes, R.M.: The stability of a queue with non-independent interarrival and service times. Proc. Camb. Philos. Soc. 58, 497–520 (1962)CrossRefGoogle Scholar
  18. 18.
    Mairesse, J., Prabhakar, B.: The existence of fixed points for the \(\cdot /GI/1\) queue. Ann. Probab. 31, 2216–2236 (2003)CrossRefGoogle Scholar
  19. 19.
    Martin, J.B.: Last passage percolation with general weight distribution. Markov Proc. Relat. Fields. 12, 273–299 (2006)Google Scholar
  20. 20.
    Mountford, T., Prabhakar, B.: On the weak convergence of departures from an infinite series of \(\cdot /M/ 1\) queues. Ann. Appl. Probab. 5(1), 121–127 (1995)CrossRefGoogle Scholar
  21. 21.
    O’Connell, N., Yor, M.: Brownian analogues of Burke’s theorem. Stoch. Process. Appl. 2, 285–304 (2001)CrossRefGoogle Scholar
  22. 22.
    Pal, S., Pitman, J.: One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Prob. 18, 2179–2207 (2008)CrossRefGoogle Scholar
  23. 23.
    Prabhakar, B.: The attractiveness of the fixed points of a \(\cdot /GI/1\) queue. Ann. Probab 31, 2237–2269 (2003)CrossRefGoogle Scholar
  24. 24.
    Seppäläinen, T.: A scaling limit for queues in series. Ann. Appl. Probab. 7, 855–872 (1997)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Radboud UniversityNijmegenThe Netherlands
  2. 2.Facultad de CienciasUNAMMexico CityMexico
  3. 3.Instituto de MatemáticaUFRJRio de JaneiroBrasil

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