Stochastic Bounds for Switched Bernoulli Batch Arrivals Observed Through Measurements

  • Farah Aït-Salaht
  • Hind Castel-Taleb
  • Jean-Michel Fourneau
  • Nihal PekerginEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10378)


We generalise to non stationary traffics an approach that we have previously proposed to derive performance bounds of a queue under histogram-based input traffics. We use strong stochastic ordering to derive stochastic bounds on the queue length and the output traffic. These bounds are valid for transient distributions of these measures and also for the steady-state distributions when they exist. We provide some numerical techniques under arrivals modelled by a Switched Batch Bernoulli Process (SBBP). Unlike approximate methods, these bounds can be used to check if the Quality of Service constraints are satisfied or not. Our approach provides a tradeoff between the accuracy of results and the computational complexity and it is much faster than the histogram-based simulation proposed in the literature.


  1. 1.
    Aït-Salaht, F., Castel-Taleb, H., Fourneau, J.-M., Pekergin, N.: Stochastic bounds and histograms for network performance analysis. In: Balsamo, M.S., Knottenbelt, W.J., Marin, A. (eds.) EPEW 2013. LNCS, vol. 8168, pp. 13–27. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40725-3_3CrossRefGoogle Scholar
  2. 2.
    Aït-Salaht, F., Castel Taleb, H., Fourneau, J.-M., Pekergin, N.: Performance analysis of a queue by combining stochastic bounds, real traffic traces and histograms. Comput. J. 59(12), 1817–1830 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aït-Salaht, F., Cohen, J., Castel Taleb, H., Fourneau, J.M., Pekergin, N.: Accuracy vs. complexity: the stochastic bound approach. In: 11th International Workshop on Discrete Event Systems, pp. 343–348 (2012)CrossRefGoogle Scholar
  4. 4.
    Fischer, W., Hellstern, K.M.: The Markov-modulated Poisson process (MMPP) cookbook. Perform. Eval. 18, 149–171 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gupta, V., Harchol-Balter, M., Dai, J.G., Zwart, B.: On the inapproximability of M/G/K: why two moments of job size distribution are not enough. Queueing Syst. 64(1), 5–48 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gusak, O., Dayar, T., Fourneau, J.-M.: Iterative disaggregation for a class of lumpable discrete-time stochastic automata networks. Perform. Eval. 53(1), 43–69 (2003)CrossRefGoogle Scholar
  7. 7.
    Hashida, O., Takahashi, Y., Shimogawa, S.: Switched batch Bernoulli process (SBBP) and the discrete-time SBBP/G/1 queue with application to statistical multiplexer performance. IEEE J. Select. Areas Commun. 9(3), 394–401 (1991)CrossRefGoogle Scholar
  8. 8.
    Hernández-Orallo, E., Vila-Carbó, J.: Network performance analysis based on histogram workload models. In: MASCOTS, pp. 209–216 (2007)Google Scholar
  9. 9.
    Hernández-Orallo, E., Vila-Carbó, J.: Web server performance analysis using histogram workload models. Comput. Netw. 53(15), 2727–2739 (2009)CrossRefGoogle Scholar
  10. 10.
    Hernández-Orallo, E., Vila-Carbó, J.: Network queue and loss analysis using histogram-based traffic models. Comput. Commun. 33(2), 190–201 (2010)CrossRefGoogle Scholar
  11. 11.
    Horváth, G., Telek, M., Buchholz, P.: A map fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation. In: QEST, pp. 124–133. IEEE Computer Society (2005)Google Scholar
  12. 12.
    Klemm, A., Lindemann, C., Lohmann, M.: Traffic modelling of IP networks using the batch Markovian arrival process. Perform. Eval. 54(25), 149–173 (2003)CrossRefGoogle Scholar
  13. 13.
    Muller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)zbMATHGoogle Scholar
  14. 14.
    Muscarielloa, L., Melliaa, M., Meoa, M., Ajmone, M.M., Lo Cignob, R.: Markov models of internet traffic and a new hierarchical MMPP model. Comput. Commun. 28, 1835–1851 (2005)CrossRefGoogle Scholar
  15. 15.
    Skelly, P., Schwartz, M., Dixit, S.S.: A histogram-based model for video traffic behaviour in an ATM multiplexer. IEEE/ACM Trans. Netw. 1(4), 446–459 (1993)CrossRefGoogle Scholar
  16. 16.
    Sony, K.C., Cho, K.: Traffic data repository at the wide project. In: Proceedings of USENIX 2000 Annual Technical Conference: FREENIX Track, pp. 263–270 (2000)Google Scholar
  17. 17.
    Stewart, W.: Introduction to the numerical Solution of Markov Chains. Princeton University Press, New Jersey (1995)Google Scholar
  18. 18.
    Wittevrongel, S., Bruneel, H.: Discrete-time queues with correlated arrivals and constant service times. Comput. Oper. Res. 26, 93–108 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhou, W., Wang, A.: Discrete-time queue with Bernoulli bursty source arrival and generally distributed service times. Appl. Math. Model. 3, 2223–2240 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Farah Aït-Salaht
    • 1
  • Hind Castel-Taleb
    • 2
  • Jean-Michel Fourneau
    • 3
  • Nihal Pekergin
    • 4
    Email author
  1. 1.LIP6, EnsaiRennesFrance
  2. 2.SAMOVAR, UMR 5157, Télécom Sud ParisEvryFrance
  3. 3.DAVID, UVSQ, Univ. Paris SaclayVersaillesFrance
  4. 4.LACL, Univ. Paris EstCréteilFrance

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