Metastability of Queuing Networks with Mobile Servers

  • F. Baccelli
  • A. Rybko
  • S. Shlosman
  • A. Vladimirov


We study symmetric queuing networks with moving servers and FIFO service discipline. The mean-field limit dynamics demonstrates unexpected behavior which we attribute to the metastability phenomenon. Large enough finite symmetric networks on regular graphs are proved to be transient for arbitrarily small inflow rates. However, the limiting non-linear Markov process possesses at least two stationary solutions. The proof of transience is based on martingale techniques.


Mean-field limit Non-linear Markov process Random walks on graphs Martingale 


  1. 1.
    Antunes, N., Fricker, C., Robert, P., Tibi, D.: Stochastic networks with multiple stable points. Ann. Probab. 36(1), 255–278 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baccelli, F., Błaszczyszyn, B.: Stochastic Geometry and Wireless Networks. NOW Publishers, Breda (2009)zbMATHGoogle Scholar
  3. 3.
    Baccelli, F., Rybko, A., Shlosman, S.: Queuing networks with varying topology—a mean-field approach. Inf. Transm. Probl. 52(2), 178–199 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Durrett, R.: Probability: Theory and Examples. Cambridge series on statistical and probabilistic mathematics. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    El Gamal, A., Mammen, J., Prabhakar, B., Shah, D.: Throughput-delay trade-off in wireless networks. In: Proceedings of IEEE INFOCOM, Hong Kong (2004)Google Scholar
  6. 6.
    Gibbons, R., Hunt, P., Kelly, F.: Bistability in Communication Networks, Disorder in Physical Systems, pp. 113–128. Oxford University Press, New York (1990)Google Scholar
  7. 7.
    Grossglauser, M., Tse, D.: Mobility increases the capacity of ad hoc wireless networks. IEEE Trans. Netw. 10(4), 477–486 (2002)CrossRefGoogle Scholar
  8. 8.
    Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Trans. Inf. Theory 46(2), 388–404 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rybko, A., Shlosman, S., Vladimirov, A.: Spontaneous resonances and the coherent states of the queuing networks. J. Stat. Phys. 134(1), 67–104 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  10. 10.
    Schonmann, R.H., Shlosman, S.: Wulff droplets and the metastable relaxation of the kinetic ising models. Commun. Math. Phys. 194, 389–462 (1998)MathSciNetCrossRefzbMATHADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • F. Baccelli
    • 4
  • A. Rybko
    • 3
  • S. Shlosman
    • 1
    • 2
    • 3
  • A. Vladimirov
    • 3
  1. 1.Skolkovo Institute of Science and TechnologyMoscowRussia
  2. 2.Aix Marseille Université, Université de ToulonMarseilleFrance
  3. 3.Inst. of the Information Transmission ProblemsRASMoscowRussia
  4. 4.Department of MathematicsUT AustinAustinUSA

Personalised recommendations