On the Analysis of Queues with Heavy Tails: A Non-Extensive Maximum Entropy Formalism and a Generalisation of the Zipf-Mandelbrot Distribution

  • Demetres D. Kouvatsos
  • Salam A. Assi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6821)


A critique of a non-extensive maximum entropy (NME) formalism is undertaken in conjunction with its application into the analysis of queues with heavy tails that are often observed in performance evaluation studies of heterogeneous networks exhibiting traffic burstiness, self-similarity and/or long range dependence (LRD). The credibility of the NME formalism, as a method of inductive inference, for the study of non-extensive systems with long-range interactions is explored in terms of four consistency axioms of extensive systems with short-range interactions. Focusing on a a general physical system and, as a special case, a single server queue with finite capacity, it is shown that the NME state probability is characterised by a generalisation of the Zipf-Mandelbrot (Z-M) type distribution depicting heavy tails and asymptotic power law behaviour. Typical numerical experiments are employed to illustrate the adverse combined impact of traffic burstiness and self-similarity on the behaviour of the queue. A reference to open issues relating to the NME formalism and open queueing networks is included.


Performance evaluation extensive maximum entropy (EME) formalism non-extensive maximum entropy (NME) formalism generalised exponential (GE) traffic burstiness self-similarity short-range dependence (SRD) long-range dependence (LRD) queueing systems Zipf-Mandelbrot (Z-M) distribution 


  1. 1.
    Assi, S.A.: An Investigation into Generalised Entropy Optimisation with Queueing Systems Applications. MSc Dissertation, Dept. of Computing, School of Informatics, University of Bradford (2000)Google Scholar
  2. 2.
    Bateman, H.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)Google Scholar
  3. 3.
    Beran, J.: Statistics for Long-Memory Processes. Chapman & Hall (1994) ISBN 0-412-04901-5 Google Scholar
  4. 4.
    Chakrabarti, C.G., Kajal, D.: Boltzmann-Gibbs Entropy: Axiomatic Characterisation and Application. Internat. J. Math. & Math. Sci. 23(4), 243–251 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Choudhury, G.L., Whitt, W.: Long-tail Buffer-content Distributions in Broadband Networks. Performance Evaluation 30, 177–190 (1997)CrossRefGoogle Scholar
  6. 6.
    Crovella, M.E., Lipsky, L.: Long-lasting Transient Conditions in Simulations with Heavy-Tailed Workloads. In: Proc. of Winter Simulation Conference, pp. 1005–1012 (1997)Google Scholar
  7. 7.
    Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions. Notebooks (23:22 January 29, 2007),,html
  8. 8.
    Havrda, J.H., Charvat, F.: Quantification Methods of Classificatory Processes: Concept of Structural Entropy. Kybernatica 3, 30–35 (1967)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Jaynes, E.T.: Information Theory and Statistical Mechanics. Physical Review 106, 620–630 (1957)Google Scholar
  10. 10.
    Karmeshu, Sharma, S.: Long Tail Behaviour of Queue Lengths in Broadband Networks: Tsallis Entropy Framework. Technical Report, School of Computing and System Sciences, J. Nehru University, New Delhi, India (August 2005)Google Scholar
  11. 11.
    Karmeshu, Sharma, S.: q-ExponentiaL Product-Form Solution of Packet Distribution in Queueing Networks: maximisation of Tsallis Entropy. IEEE Communication Letters 10(8), 585–587 (2006)CrossRefGoogle Scholar
  12. 12.
    Kouvatsos, D.D.: Entropy Maximization and Queueing Network Models. Annals of Operation Research 48, 63–126 (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kouvatsos, D.D., Awan, I., Fretwell, R., Dimakopoulos, G.: A Cost-Effective Approximation for SRD Traffic in Arbitrary Multi-Buffered Networks. Computer Networks 34, 97–113 (2000)CrossRefGoogle Scholar
  14. 14.
    Kouvatsos, D.D., Assi, S.A.: An Investigation into Generalised Entropy Optimisation with Queueing System Applications. In: Merabti, M. (ed.) The Proceedings of the 3rd Annual Postgraduate Symposium on the Convergence of Telecommunications, Networking and Broadcasting (PGNet 2002), pp. 409–414. Liverpool John Moores University Publisher (2002)Google Scholar
  15. 15.
    Kouvatsos, D.D., Assi, S.A.: On the Analysis of Queues with Long Range Dependent Traffic: An Extended Maximum Entropy Approach. In: Proceedings of the 3rd Euro-NGI Conference on Next Generation Internet Networks - Design and Engineering for Heterogeneity, Trodheim, Norway, pp. 226–233 (May 2007)Google Scholar
  16. 16.
    Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman, New York (1982)zbMATHGoogle Scholar
  17. 17.
    Norros, I.: A Storage Model with Self-similar Input. Queueing Systems and their Applications 16, 387–396 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Renyi, A.: On Measures of Entropy and Information. In: Proceedings of the 4th Berkely Symposium Math Stat And Probability, vol. 1, pp. 547–561 (1961)Google Scholar
  19. 19.
    Rezaul, K.M., Grout, V.: A Comparison of Methods for Estimating the Tail Index of Heavy-tailed Internet Traffic. In: Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications, pp. 219–222. Springer, Dordrecht (2007)CrossRefGoogle Scholar
  20. 20.
    Sahinoglu, Z., Tekinay, S.: On Multimedia Networks: Self-similar Traffic and Network Performance. IEEE Communication Magazine 37, 48–52 (1999)CrossRefGoogle Scholar
  21. 21.
    Shannon, C.E.: A Mathematical Theory of Communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Shore, J.E., Johnson, R.W.: Axiomatic Derivation of the Principle of ME and the Principle of Minimum Cross-Entropy. IEEE Transaction on Information Theory IT-26, 26–37 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Tsallis, C.: Possible Generalisation of Boltzmann-Gibbs Statistics. Journal of Statistical Physics 52(1-2), 479–487 (1988)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2011

Authors and Affiliations

  • Demetres D. Kouvatsos
    • 1
  • Salam A. Assi
    • 1
  1. 1.Networks and Performance Engineering Research Group (NetPEn)Informatics Research Institute (IRI), University of BradfordBradfordUK

Personalised recommendations