Discrete Event Dynamic Systems

, Volume 18, Issue 4, pp 499–536 | Cite as

Zero-Automatic Networks

  • Thu-Ha Dao-Thi
  • Jean Mairesse


We continue the study of zero-automatic queues first introduced in Dao-Thi and Mairesse (Adv Appl Probab 39(2):429–461, 2007). These queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The simple M/M/1 queue and Gelenbe’s G-queue with positive and negative customers are the two simplest 0-automatic queues. All stable 0-automatic queues have an explicit “multiplicative” stationary distribution and a Poisson departure process (Dao-Thi and Mairesse, Adv Appl Probab 39(2):429–461, 2007). In this paper, we introduce and study networks of 0-automatic queues. We consider two types of networks, with either a Jackson-like or a Kelly-like routing mechanism. In both cases, and under the stability condition, we prove that the stationary distribution of the buffer contents has a “product-form” and can be explicitly determined. Furthermore, the departure process out of the network is Poisson.


Queueing theory Jackson network Kelly network Product form Zero-automatic 


  1. Brémaud P (1999) Markov chains: Gibbs fields, Monte Carlo simulation, and queues. Texts in applied mathematics, vol 31. Springer, New YorkMATHGoogle Scholar
  2. Chao X, Miyazawa M, Pinedo M (1999) Queueing networks. Customers, signals, and product form solutions. WileyGoogle Scholar
  3. Dao-Thi T-H, Mairesse J (2007) Zero-automatic queues and product form. Adv Appl Probab 39(2):429–461MATHCrossRefMathSciNetGoogle Scholar
  4. Dao-Thi T-H, Mairesse J (2006) Zero-automatic networks. In: Proceedings of valuetools, Pisa, Italy. ACMGoogle Scholar
  5. Dynkin E, Malyutov M (1961) Random walk on groups with a finite number of generators. Sov Math Dokl 2:399–402MATHGoogle Scholar
  6. Fourneau J-M, Gelenbe E, Suros R (1996) G-networks with multiple classes of negative and positive customers. Theor Comput Sci 155(1):141–156MATHCrossRefMathSciNetGoogle Scholar
  7. Gelenbe E (1991) Product-form queueing networks with negative and positive customers. J Appl Probab 28(3):656–663MATHCrossRefMathSciNetGoogle Scholar
  8. Gelenbe E, Pujolle G (1998) Introduction to queueing networks, 2nd edn. Wiley, ChichesterGoogle Scholar
  9. Guivarc’h Y (1980) Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Astérisque 74:47–98MATHMathSciNetGoogle Scholar
  10. Kelly F (1979) Reversibility and stochastic networks. Wiley, New-YorkMATHGoogle Scholar
  11. Ledrappier F (2001) 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. American Mathematical Society, pp 117–152Google Scholar
  12. Mairesse J (2005) Random walks on groups and monoids with a Markovian harmonic measure. Electron J Probab 10:1417–1441MathSciNetGoogle Scholar
  13. Mairesse J, Mathéus F (2006) Random walks on free products of cyclic groups. To appear in J London Math Soc 2006. Available at arXiv:math.PR/0509211, 2005
  14. Neuts M (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Johns Hopkins University Press, Baltimore, MdMATHGoogle Scholar
  15. Sawyer S, Steger T (1987) The rate of escape for anisotropic random walks in a tree. Probab Theory Relat Fields 76(2):207–230MATHCrossRefMathSciNetGoogle Scholar
  16. Serfozo R (1999) Introduction to stochastic networks. Springer, BerlinMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.LIAFACNRS-Université Paris 7Paris Cedex 05France

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