Queueing Systems

, Volume 76, Issue 1, pp 51–72 | Cite as

Efficiency of simulation in monotone hyper-stable queueing networks

  • Jonatha Anselmi
  • Bruno Gaujal


We consider Jackson queueing networks with finite buffer constraints (JQN) and analyze the efficiency of sampling from their stationary distribution. In the context of exact sampling, the monotonicity structure of JQNs ensures that such efficiency is of the order of the coupling time (or meeting time) of two extremal sample paths. In the context of approximate sampling, it is given by the mixing time. Under a condition on the drift of the stochastic process underlying a JQN, which we call hyper-stability, in our main result we show that the coupling time is polynomial in both the number of queues and buffer sizes. Then, we use this result to show that the mixing time of JQNs behaves similarly up to a given precision threshold. Our proof relies on a recursive formula relating the coupling times of trajectories that start from network states having “distance one”, and it can be used to analyze the coupling and mixing times of other Markovian networks, provided that they are monotone. An illustrative example is shown in the context of JQNs with blocking mechanisms.


Jackson queueing networks Finite buffer Perfect simulation  Coupling time Mixing time 

Mathematics Subject Classification

60J10 60K25 60K20 



This research was partially supported by grant MTM2010-17405 of the MICINN (Spain) and grant PI2010-2 of the Basque Government (Department of Education and Research).


  1. 1.
    Abate, J., Whitt, W.: Transient Behavior of the M/M/1 queue: starting at the origin. Queueing Syst. 2(1), 41–65 (1987)CrossRefGoogle Scholar
  2. 2.
    Andradóttir, S., Hosseini-Nasab, M.: Efficiency of time segmentation parallel simulation of finite markovian queueing networks. Oper. Res. 51(2), 272–280 (2003)CrossRefGoogle Scholar
  3. 3.
    Anselmi, J., Gaujal, B.: On the efficiency of perfect simulation in monotone queueing networks. SIGMETRICS Perform. Eval. Rev. ACM 39(2), 56–58 (2011)CrossRefGoogle Scholar
  4. 4.
    Balsamo, S., de Nitto Personé, V., Onvural, R.: Analysis of Queueing Networks with Blocking. International Series in Operations Research and Management Science. Kluwer (2001)Google Scholar
  5. 5.
    Baskett, F., Chandy, K.M., Muntz, R., Palacios, F.G.: Open, closed, and mixed networks of queues with different classes of customers. J. ACM 22(2), 248–260 (1975)CrossRefGoogle Scholar
  6. 6.
    Bolch, G., Greiner, S., de Meer, H., Trivedi, K.: Queueing Networks and Markov Chains. Wiley-Interscience (2005)Google Scholar
  7. 7.
    Bremaud, P.: Markov Chains, Gibbs Fields. Monte Carlo Simulation and Queues. Texts in Applied Mathematics. Springer, Berlin (1999)Google Scholar
  8. 8.
    Chen, W.L., O’Cinneide, C.A.: Towards a polynomial-time randomized algorithm for closed product-form networks. ACM Trans. Model. Comput. Simul. 8(3), 227–253 (1998)CrossRefGoogle Scholar
  9. 9.
    Dopper, J., Gaujal, B., Vincent, J.M.: Bounds for the coupling time in queueing networks perfect simulation. In: Celebration of the 100th Anniversary of Markov, pp. 117–136 (2006)Google Scholar
  10. 10.
    Jackson, J.R.: Job shop-like queueing systems. Manag. Sci. 10, 131 (1963)CrossRefGoogle Scholar
  11. 11.
    Kelly, F.: Reversibility and Stochastic, Networks (1979)Google Scholar
  12. 12.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. University Series in Undergraduate Mathematics, VanNostrand (1969)Google Scholar
  13. 13.
    Kijima, S., Matsui, T.: Approximation algorithm and perfect sampler for closed jackson networks with single servers. SIAM J. Comput. 38(4), 1484–1503 (2008)CrossRefGoogle Scholar
  14. 14.
    Kijima, S., Matsui, T.: Randomized approximation scheme and perfect sampler for closed jackson networks with multiple servers. Ann. OR 162(1), 35–55 (2008)CrossRefGoogle Scholar
  15. 15.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society (2008)Google Scholar
  16. 16.
    Massey, W.A.: An operator analytic approach to the Jackson network. J. Appl. Probab. 21, 379–393 (1984)CrossRefGoogle Scholar
  17. 17.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley (2002)Google Scholar
  18. 18.
    Narayan Bhat, U.: An Introduction to Queueing Theory: Modeling and Analysis in Applications. Birkhauser Verlag (2008)Google Scholar
  19. 19.
    Propp, J.G., Wilson, D.B.: Exact sampling with coupled markov chains and applications to statistical mechanics. Rand. Struct. Alg. 9(1–2), 223–252 (1996)CrossRefGoogle Scholar
  20. 20.
    Vincent, J.-M.: Perfect Generation, Monotonicity and Finite Queueing Networks. In: IEEE QEST, p. 319 (2008)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Basque Center for Applied Mathematics (BCAM) BilbaoSpain
  2. 2.INRIA and LIGMontBonnot Saint-MartinFrance

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