Zero-Automatic Queues

  • Thu-Ha Dao-Thi
  • Jean Mairesse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3670)


We introduce and study a new model: 0-automatic queues. Roughly, 0-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all 0-automatic queues are quasi-reversible. When considering the two simplest and extremal cases of 0-automatic queues, we recover the simple M /M /1 queue, and Gelenbe’s G-queue with positive and negative customers.


M /M /1 queue G-queue quasi-reversibility product form 


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  1. 1.
    Cohen, J.W.: The single server queue, 2nd edn. North-Holland, Amsterdam (1982)MATHGoogle Scholar
  2. 2.
    Dao-Thi, T.-H., Mairesse, J.: Zero-automatic networks (in preparation)Google Scholar
  3. 3.
    Dao-Thi, T.-H., Mairesse, J.: Zero-automatic queues. LIAFA reseach report 2005-03, Univ. Paris 7 (2005)Google Scholar
  4. 4.
    Dynkin, E., Malyutov, M.: Random walk on groups with a finite number of generators. Sov. Math. Dokl. 2, 399–402 (1961)MATHGoogle Scholar
  5. 5.
    Fourneau, J.-M., Gelenbe, E., Suros, R.: G-networks with multiple classes of negative and positive customers. Theoret. Comput. Sci. 155(1), 141–156 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gelenbe, E.: Product-form queueing networks with negative and positive customers. J. Appl. Probab. 28(3) (1991)Google Scholar
  7. 7.
    Gelenbe, E., Pujolle, G.: Introduction to queueing networks, 2nd edn. John Wiley & Sons, Chichester (1998)Google Scholar
  8. 8.
    Guivarc’h, Y.: Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Astérisque 74, 47–98 (1980)MATHMathSciNetGoogle Scholar
  9. 9.
    Haring-Smith, R.: Groups and simple languages. Trans. Amer. Math. Soc. 279(1), 337–356 (1983)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ledrappier, F.: Some asymptotic properties of random walks on free groups. In: Taylor, J. (ed.) Topics in probability and Lie groups: boundary theory. CRM Proc. Lect. Notes, vol. 28, pp. 117–152. American Mathematical Society (2001)Google Scholar
  11. 11.
    Mairesse, J.: Random walks on groups and monoids with a Markovian harmonic measure. LIAFA research report 2004-05, Université Paris 7 (2004)Google Scholar
  12. 12.
    Mairesse, J., Mathéus, F.: Random walks on free products of cyclic groups and on Artin groups with two generators. LIAFA research report 2004-06, Université Paris 7 (2004)Google Scholar
  13. 13.
    Sawyer, S., Steger, T.: The rate of escape for anisotropic random walks in a tree. Probab. Theory Related Fields 76(2), 207–230 (1987)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Serfozo, R.: Introduction to Stochastic Networks. Springer, Berlin (1999)MATHGoogle Scholar
  15. 15.
    Stallings, J.: A remark about the description of free products of groups. Proc. Cambridge Philos. Soc. 62, 129–134 (1966)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Thu-Ha Dao-Thi
    • 1
  • Jean Mairesse
    • 1
  1. 1.LIAFA, CNRS-Université Paris 7France

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