Queueing Systems

, Volume 55, Issue 1, pp 71–82 | Cite as

The M/G/∞ system revisited: finiteness, summability, long range dependence, and reverse engineering

  • Iddo Eliazar


We explore M/G/∞ systems ‘fed’ by Poissonian inflows with infinite arrival rates. Three processes – corresponding to the system's state, workload, and queue-size – are studied and analyzed. Closed form formulae characterizing the system's stationary structure and correlation structure are derived. And, the issues of queue finiteness, workload summability, and Long Range Dependence are investigated.

We then turn to devise a ‘reverse engineering’ scheme for the design of the system's correlation structure. Namely: how to construct an M/G/∞ system with a pre-desired ‘target’ workload/queue auto-covariance function. The ‘reverse engineering’ scheme is applied to various examples, including ones with infinite queues and non-summable workloads.


M/G/∞ systems Poisson point processes Infinite arrival rates Lévy inflows Workload processes Long Range Dependence (LRD) Reverse engineering 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Takacs, Introduction to the Theory of Queues, Oxford University Press, 1962.Google Scholar
  2. 2.
    L.I. Schiff, Statistical analysis of counter data, Phys. Rev. (2) 50 (1936) 88–96.CrossRefGoogle Scholar
  3. 3.
    C. Levert and W.L. Scheen, Probability fluctuations of discharges in a Geiger-Müller counter produced by cosmic radiation, Physica 10 (1943) 225–238.CrossRefGoogle Scholar
  4. 4.
    W. Feller, On probability problems in the theory of counters, Courant anniversary volume 1948 (pp. 105–115).Google Scholar
  5. 5.
    J.M. Hammersley, On counters with random dead time I, Proc. Camb. Phil. Soc. 49 (1953) 623–637.CrossRefGoogle Scholar
  6. 6.
    F. Pollaczek, Sur la théory stochastique des compteurs, C.R. Acad. Sci. Paris 238 (1954) 766–768.Google Scholar
  7. 7.
    L. Takacs, On processes of happenings generated by means of Poisson process, Acta. Math. Hung. 6 (1955) 81–99CrossRefGoogle Scholar
  8. 8.
    D.R. Cox and H.D. Miller, The theory of stochastic processes, Methuen (London), 1965.Google Scholar
  9. 9.
    D.R. Cox and V. Isham, Point processes, Chapman and Hall, 1980.Google Scholar
  10. 10.
    S.M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, 1970.Google Scholar
  11. 11.
    D.R. Cox, Long-range dependence: a review, in: H.A. David and H.T. David (Eds.), Statistics: An appraisal, Iowa State University Press, 1984 (pp. 55–74).Google Scholar
  12. 12.
    M.S. Taqqu and G. Oppenheim (Eds.), Theory and applications of long-range dependence, Birkhauser, 2002.Google Scholar
  13. 13.
    G. Rangarajan and M. Ding (Eds.), Processes with long-range correlations: theory and applications (lecture notes in physics, 621), Springer-Verlag, 2003.Google Scholar
  14. 14.
    W. Leland, M.S. Taqqu, W. Willinger, and D. Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Trans. Net. 2 (1994) 1–15.CrossRefGoogle Scholar
  15. 15.
    V. Paxson and S. Floyd, Wide area traffic: The failure of Poisson modeling, IEEE/ACM Trans. Net. 3 (1994) 226–244.CrossRefGoogle Scholar
  16. 16.
    M. Crovella and A. Bestavros, Self-similarity in World Wide Web traffic: Evidence and possible causes, Performance Evaluation Rev. 24 (1996) 160–169.CrossRefGoogle Scholar
  17. 17.
    W. Willinger, M.S. Taqqu, R. Sherman, and D. Wilson, Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level, IEEE/ACM Trans. Net. 5 (1997) 71–86.CrossRefGoogle Scholar
  18. 18.
    B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises, and applications, SIAM Rev. 10 (1968) 422–437.CrossRefGoogle Scholar
  19. 19.
    P. Embrechts and M. Maejima, Selfsimilar Processes, Princeton University Press, 2002.Google Scholar
  20. 20.
    W. Whitt, An introduction to stochastic-process limits and their applications to queues, Springer, 2002.Google Scholar
  21. 21.
    N. Likhanov, B. Tsybakov, and N.D. Georganas, Analysis of an ATM buffer with self-similar (“fractal”) input traffic, Proceedings of INFOCOM 1995, New York, (1995) 985–992.Google Scholar
  22. 22.
    M. Parulekar and A. Makowski, M/G/∞ input processes: A versatile class of models for traffic network, Proceedings of INFOCOM 1997, Kobe (Japan), (1997) 419–426.Google Scholar
  23. 23.
    M. Parulekar and A. Makowski, Tail probabilities for M/G/∞ processes (I): Preliminary asymptotics, Queueing Systems 27 (1997) 271–296.Google Scholar
  24. 24.
    P.R. Jelenkovic and A. Lazar, Asymptotic results for multiplexing subexponential on-off processes, Adv. Appl. Prob. 31 (1999) 394–421.CrossRefGoogle Scholar
  25. 25.
    S. Resnick and G. Samorodnitsky, Activity periods of an infinite server queue and performance of certain heavy tailed queues, Queueing Systems 33 (1999) 43–71.CrossRefGoogle Scholar
  26. 26.
    D. Heath, S.I. Resnick, and G. Samorodnitsky, How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails, Ann. Appl. Prob. 9 (1999) 352–375.CrossRefGoogle Scholar
  27. 27.
    S. Resnick and H. Rootzen, Self-similar communication models and very heavy tails, Ann. Appl. Prob. 10 (2000) 753–778.CrossRefGoogle Scholar
  28. 28.
    T. Mikosch, S. Resnick, H. Rootzen, and A. Stegeman, Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Prob. 12 (2002) 23–68.CrossRefGoogle Scholar
  29. 29.
    C. Guerin, H. Nyberg, O. Perrin, S. Resnick, H. Rootzen, and C. Starica, Empirical testing of the infinite source Poisson data traffic model, Stochastic Models 19 (2003) 151–200.CrossRefGoogle Scholar
  30. 30.
    K. Maulik and S. Resnick, Small and large time scale analysis of a network traffic model, Queueing Systems 43 (2003) 221–250.CrossRefGoogle Scholar
  31. 31.
    A. Suarez-Gonzalez, J.C. Lopez-Ardao, C. Lopez-Garcia, M. Fernandez-Veiga, R. Rodriguez-Rubio, and M.E. Sousa-Vieira, A new heavy-tailed discrete distribution for LRD M/G/∞ sample generation, Performance Evaluation 47 (2002) 197–219.CrossRefGoogle Scholar
  32. 32.
    M.E. Sousa-Vieira, A. Suarez-Gonzalez, C. Lopez-Garcia, M. Fernandez-Veiga, and J.C. Lopez-Ardao, A highly efficient M/G/∞ model for generating self-similar traces, Proceedings of the 2002 Winter Simulation Conferences (E. Yücesan et al. Eds.), 2002 (pp. 2003–2010).Google Scholar
  33. 33.
    B. Fristedt and L. Gray, A modern approach to probability theory, Birkhauser 1997.Google Scholar
  34. 34.
    B.B. Mandelbrot and J.R. Wallis, Noah, Joseph and operational hydrology, Water Resources Research 4 (1968) 909.CrossRefGoogle Scholar
  35. 35.
    J. Bertoin, Subordinators: examples and applications, Lecture notes in mathematics 1717, Springer, 1999.Google Scholar
  36. 36.
    O. Kella and W. Whitt, Queues with server vacations and Lévy processes with secondary jump input, Ann. Appl. Prob. 1 (1991) 104–117.Google Scholar
  37. 37.
    P. Dube, F. Guillemin, and R.R. Mazumdar, Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues, J. Appl. Prob. 41 (2004) 1145–1156.CrossRefGoogle Scholar
  38. 38.
    O. Kella, Parallel and tandem fluid networks with dependent Lévy inputs, Ann. Appl. Prob. 3 (1993) 682–695.Google Scholar
  39. 39.
    O. Kella, An exhaustive Lévy storage process with intermittent output, Comm. Statist. Stochastic Models 14 (1998) 979–992.CrossRefGoogle Scholar
  40. 40.
    I. Eliazar, The snowblower problem, Queueing Systems 45 (2003) 357–380.CrossRefGoogle Scholar
  41. 41.
    I. Eliazar, Gated polling systems with Lévy inflow and inter-dependent switchover times: a dynamical-systems approach, Queueing Systems 49 (2005) 49–72.CrossRefGoogle Scholar
  42. 42.
    J.F.C. Kingman, Poisson processes, Oxford University Press, 1993.Google Scholar
  43. 43.
    N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular variation, Cambridge University Press, 1987.Google Scholar
  44. 44.
    D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes (2nd edition), Springer, 2003.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.School of Mathematics & School of ChemistrySackler Faculty of Exact Sciences, Tel Aviv UniversityTel AvivIsrael

Personalised recommendations