Queueing Systems

, Volume 66, Issue 4, pp 369–412 | Cite as

Statistical estimation of delays in a multicast tree using accelerated EM



Tomography is one of the most promising techniques today to provide spatially localized information about internal network performance in a robust and scalable way. The key idea is to measure performance at the edge of the network, and to correlate these measurements to infer the internal network performance.

This paper focuses on a specific delay tomographic problem on a multicast diffusion tree, where end-to-end delays are observed at every leaf of the tree, and mean sojourn times are estimated for every node in the tree. The estimation is performed using the Maximum Likelihood Estimator (MLE) and the Expectation-Maximization (EM) algorithm.

Using queuing theory results, we carefully justify the model we use in the case of rare probing. We then give an explicit EM implementation in the case of i.i.d. exponential delays for a general tree. As we work with non-discretized delays and a full MLE, EM is known to be slow. We hence present a very simple but, in our case, very effective speed-up technique using Principal Component Analysis (PCA). MLE estimations are provided for a few different trees to evaluate our technique.


Network tomography Multicast trees Expectation-maximization (EM) algorithm Acceleration Principal Component Analysis (PCA) 

Mathematics Subject Classification (2000)

60K25 62F30 94A99 90B22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baccelli, F., Kauffmann, B., Veitch, D.: Inverse problems in queueing theory and internet probing. Queueing Syst. 63(1–4), 59–107 (2009) CrossRefGoogle Scholar
  2. 2.
    Chen, A., Cao, J., Bu, T.: Network tomography: identifiability and Fourier domain estimation. In: IEEE INFOCOM, pp. 1875–1883 (2007) Google Scholar
  3. 3.
    Chrétien, S., Hero, A.O.: Kullback proximal algorithms for maximum likelihood estimation. Inria report 3756 (2009) Google Scholar
  4. 4.
    Coates, M., Nowak, R.: Network tomography for internal delay estimation. In: Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP), Salt Lake City, Utah (2001) Google Scholar
  5. 5.
    Cramér, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946) Google Scholar
  6. 6.
    Dalibard, S., Laumond, J.P.: Control of probabilistic diffusion in motion planning. In: 8th International Workshop on the Algorithmic Foundations of Robotics (WAFR 2008) (2008) Google Scholar
  7. 7.
    Duffield, N., Presti, F.L.: Multicast inference of packet delay variance at interior network links. In: IEEE INFOCOM, Tel Aviv, Israel, pp. 1351–1360 (2000) Google Scholar
  8. 8.
    Huang, H.S., Yang, B.H., Hsu, C.N.: Triple jump acceleration for the EM algorithm. In: IEEE International Conference on Data Mining (ICDM’05) (2005) Google Scholar
  9. 9.
    Jamshidian, M., Jennrich, R.I.: Conjugate gradient acceleration of the EM algorithm. J. Am. Stat. Assoc. (1993) Google Scholar
  10. 10.
    Kauffmann, B., Baccelli, F., Veitch, D.: Towards multihop available bandwidth estimation—inverse problems in queueing networks. In: Proc. ACM Sigmetrics/Performance’09, Seattle, WA, USA (2009) Google Scholar
  11. 11.
    Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979) Google Scholar
  12. 12.
    Lawrence, E., Michailidis, G., Nair, V.N.: Network delay tomography using flexicast experiments. J. R. Stat. Soc., Ser. B 68, 785–813 (2006) CrossRefGoogle Scholar
  13. 13.
    Lawrence, E., Michailidis, G., Nair, V.N.: Statistical inverse problems in active network tomography. In: Complex Datasets and Inverse Problems: Tomography, Networks and Beyond. IMS Lecture Notes-Monograph Series, vol. 54, pp. 24–44. IMS, Beachwood (2007). doi: 10.1214/074921707000000049 CrossRefGoogle Scholar
  14. 14.
    Liang, G., Yu, B.: Maximum pseudo likelihood estimation in network tomography. IEEE Trans. Signal Process. 51(8), 2043–2053 (2003) (Special Issue on Data Networks) CrossRefGoogle Scholar
  15. 15.
    McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions, 2nd edn. Wiley Series in Probability and Statistics. Wiley, New York (2008) CrossRefGoogle Scholar
  16. 16.
    Presti, F.L., Duffield, N.G., Horowitz, J., Towsley, D.: Multicast-based inference of network-internal delay distributions. IEEE/ACM Trans. Netw. 10(6), 761–775 (2002) CrossRefGoogle Scholar
  17. 17.
    Salakhutdinov, R., Roweis, S.: Adaptive overrelaxed bound optimization methods. In: International Conference on Machine Learning (ICML-2003) (2003) Google Scholar
  18. 18.
    Shih, M.F., Hero, A.O.: Unicast-based inference of network link delay distributions with finite mixture models. IEEE Trans. Signal Process. 51(8), 2219–2228 (2003) (Special Issue on Data Networks) CrossRefGoogle Scholar
  19. 19.
    Tsang, Y., Coates, M., Nowak, R.: Network delay tomography. IEEE Trans. Signal Process. 51(8), 2125–2136 (2003) (Special Issue on Data Networks) CrossRefGoogle Scholar
  20. 20.
    Wu, C.J.: On the convergence properties of the EM algorithm. Ann. Stat. 11(1), 95–103 (1983) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Département d’InformatiqueÉcole Normale SupérieureParis CedexFrance
  2. 2.Centre for Ultra-Broadband Information Networks (CUBIN), Department of Electrical and Electronic EngineeringThe University of MelbourneParkvilleAustralia
  3. 3.INRIAParisFrance

Personalised recommendations