Advertisement

Queueing Systems

, Volume 29, Issue 2–4, pp 313–336 | Cite as

Analysis of a single-server queue interacting with a fluid reservoir

  • I.J.B.F. Adan
  • E.A. van Doorn
  • J.A.C. Resing
  • W.R.W. Scheinhardt
Article

Abstract

We consider a single-server queueing system with Poisson arrivals in which the speed of the server depends on whether an associated fluid reservoir is empty or not. Conversely, the rate of change of the content of the reservoir is determined by the state of the queueing system, since the reservoir fills during idle periods and depletes during busy periods of the server. Our interest focuses on the stationary joint distribution of the number of customers in the system and the content of the fluid reservoir, from which various performance measures such as the steady-state sojourn time distribution of a customer may be obtained. We study two variants of the system. For the first, in which the fluid reservoir is infinitely large, we present an exact analysis. The variant in which the fluid reservoir is finite is analysed approximatively through a discretization technique. The system may serve as a mathematical model for a traffic regulation mechanism - a two-level traffic shaper - at the edge of an ATM network, regulating a very bursty source. We present some numerical results showing the effect of the mechanism.

single-server queue fluid queue traffic regulator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I.J.B.F. Adan and J.A.C. Resing, Simple analysis of a fluid queue driven by an M/M/1 queue, Queueing Systems 22 (1996) 171–174.CrossRefGoogle Scholar
  2. [2]
    I.J.B.F. Adan and J.A.C. Resing, A two-level traffic shaper for an on-off source, in preparation.Google Scholar
  3. [3]
    D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data-handling system with multiple sources, Bell System Tech. J. 61 (1982) 1871–1894.Google Scholar
  4. [4]
    A.W. Berger, Performance analysis of a rate-control throttle where tokens and jobs queue, IEEE J. Select. Areas Commun. 9 (1991) 165–170.CrossRefGoogle Scholar
  5. [5]
    A.W. Berger and W. Whitt, The impact of a job buffer in a token-bank rate-control throttle, Stochastic Models 8 (1992) 685–717.Google Scholar
  6. [6]
    A.W. Berger and W. Whitt, The pros and cons of a job buffer in a token-bank rate-control throttle, IEEE Trans. Commun. 42 (1994) 857–861.CrossRefGoogle Scholar
  7. [7]
    T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).Google Scholar
  8. [8]
    A.I. Elwalid and D. Mitra, Analysis and design of rate-based congestion control of high speed networks, Part I: Stochastic fluid models, access regulation, Queueing Systems 9 (1991) 29–64.CrossRefGoogle Scholar
  9. [9]
    D.P. Kroese and W.R.W. Scheinhardt, A Markov-modulated fluid system with two interacting reservoirs, Memorandum No. 1365, Faculty of Mathematical Sciences, University of Twente, Enschede, The Netherlands (1997).Google Scholar
  10. [10]
    D.P. Kroese and W.R.W. Scheinhardt, A stochastic fluid model with two interacting reservoirs, submitted.Google Scholar
  11. [11]
    K.K. Leung, B. Sengupta and R.W. Yeung, A credit manager for traffic regulation in high-speed networks: A queueing analysis, IEEE/ACM Trans. Networking 1 (1993) 236–245.CrossRefGoogle Scholar
  12. [12]
    Z. Liu and D. Towsley, Burst reduction properties of rate-control throttles: Downstream queue behaviour, IEEE/ACM Trans. Networking 3 (1995) 82–90.CrossRefGoogle Scholar
  13. [13]
    D. Mitra, Stochastic theory of a fluid model of producers and consumers coupled by a buffer, Adv. in Appl. Probab. 20 (1988) 646–676.CrossRefGoogle Scholar
  14. [14]
    M. Miyazawa, Rate conservation laws: A survey, Queueing Systems 15 (1994) 1–58.CrossRefGoogle Scholar
  15. [15]
    B.V. Patel and C.C. Bisdikian, On the performance behavior of ATM end-stations, in: Proc. of the 14th Annual Joint Conf. of the IEEE Computer and Communication Societies — IEEE INFOCOM '95, Boston, MA, USA (2–6 April, 1995) (IEEE Computer Soc. Press, Los Alamitos, 1995) pp. 188–196.Google Scholar
  16. [16]
    J. Roberts, U. Mocci and J. Virtamo, eds., Broadband Network Teletraffic — Final Report of Action COST 242 (Springer, Berlin, 1996).Google Scholar
  17. [17]
    G. Sansigre and G. Valent, A large family of semi-classical polynomials: The perturbed Tchebichev, J. Comput. Appl. Math. 57 (1995) 271–281.CrossRefGoogle Scholar
  18. [18]
    M. Sidi, W.-Z. Liu, I. Cidon and I. Gopal, Congestion control through input rate regulation, IEEE Trans. Commun. 41 (1993) 471–477.CrossRefGoogle Scholar
  19. [19]
    T.E. Stern and A.I. Elwalid, Analysis of separable Markov-modulated rate models for information-handling systems, Adv. in Appl. Probab. 23 (1991) 105–139.CrossRefGoogle Scholar
  20. [20]
    E.A. van Doorn, A.A. Jagers and J.S.J. de Wit, A fluid reservoir regulated by a birth-death process, Stochastic Models 4 (1988) 457–472.Google Scholar
  21. [21]
    E.A. van Doorn and W.R.W. Scheinhardt, A fluid queue driven by an infinite-state birth-death process, in: Proc. of the 15th Internat. Teletraffic Congress on Teletraffic Contributions for the Information Age, Washington, DC, USA (22–27 June, 1997), eds. V. Ramaswami and P.E. Wirth (Elsevier, Amsterdam, 1997) pp. 465–475.Google Scholar
  22. [22]
    J. Virtamo and I. Norros, Fluid queue driven by an M/M/1 queue, Queueing Systems 16 (1994) 373–386.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • I.J.B.F. Adan
    • 1
  • E.A. van Doorn
    • 2
  • J.A.C. Resing
    • 1
  • W.R.W. Scheinhardt
    • 2
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Faculty of Mathematical SciencesUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations