Advertisement

Queueing Systems

, Volume 56, Issue 2, pp 79–92 | Cite as

A fluid system with coupled input and output, and its application to bottlenecks in ad hoc networks

  • Michel Mandjes
  • Frank Roijers
Open Access
Article

Abstract

This paper studies a fluid queue with coupled input and output. Flows arrive according to a Poisson process, and when n flows are present, each of them transmits traffic into the queue at a rate c/(n+1), where the remaining c/(n+1) is used to serve the queue. We assume exponentially distributed flow sizes, so that the queue under consideration can be regarded as a system with Markov fluid input. The rationale behind studying this queue lies in ad hoc networks: bottleneck links have roughly this type of sharing policy. We consider four performance metrics: (i) the stationary workload of the queue, (ii) the queueing delay, i.e., the delay of a ‘packet’ (a fluid particle) that arrives at the queue at an arbitrary point in time, (iii) the flow transfer delay, i.e., the time elapsed between arrival of a flow and the epoch that all its traffic has been put into the queue, and (iv) the sojourn time, i.e., the flow transfer time increased by the time it takes before the last fluid particle of the flow is served. For each of these random variables we compute the Laplace transform. The corresponding tail probabilities decay exponentially, as is shown by a large-deviations analysis.

Keywords

Fluid queues Flow-level analysis Sojourn times Ad hoc networks 

Mathematics Subject Classification (2000)

60K25 

References

  1. 1.
    Anick, D., Mitra, D., Sondhi, M.: Stochastic theory of a data-handling system with multiple sources. Bell Syst. Tech. J. 61, 1871–1894 (1982) Google Scholar
  2. 2.
    Asmussen, S.: Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability, vol. 2. World Scientific, London (2000) Google Scholar
  3. 3.
    van den Berg, H., Mandjes, M., Roijers, F.: Performance modeling of a bottleneck node in an IEEE 802.11 ad-hoc network. In: Kunz, T., Ravi, S.S. (eds.) Ad Hoc Now 2006. 5th International Conference on Ad hoc Networks & Wireless “Ad Hoc Now”, Ottawa, Canada. Lecture Notes in Computer Science (LNCS) Series, vol. 4104, pp. 321–336 (2006) Google Scholar
  4. 4.
    Bheema Reddy, T., Karthigeyan, I., Manoj, B.S., Siva Ram Murthy, C.: Quality-of-Service provisioning in ad hoc wireless networks: a survey of issues and solutions. J. Ad Hoc Netw. 4, 83–124 (2006) CrossRefGoogle Scholar
  5. 5.
    Borst, S., Boxma, O., Hegde, N.: Sojourn times in finite-capacity processor-sharing queues, In: Proceedings 1st EURO-NGI Conference, Rome, Italy, 2005 Google Scholar
  6. 6.
    Brandt, A., Brandt, M.: On the distribution of the number of packets in the fluid flow approximation of packet arrival streams. Queueing Syst. 17, 275–315 (1994) CrossRefGoogle Scholar
  7. 7.
    Cohen, J.: Superimposed renewal processes and storage with gradual input. Stoch. Proc. Appl. 2, 31–58 (1974) CrossRefGoogle Scholar
  8. 8.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. 2nd edn. Springer, New York (1998) Google Scholar
  9. 9.
    van Doorn, E., Jagers, A., de Wit, J.: A fluid reservoir regulated by a birth-death process. Stoch. Models 4, 457–472 (1988) Google Scholar
  10. 10.
    Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985) Google Scholar
  11. 11.
    Elwalid, A., Mitra, D.: Effective bandwidth of general Markovian traffic sources and admission control of high-speed networks. IEEE/ACM Trans. Netw. 1, 329–343 (1993) CrossRefGoogle Scholar
  12. 12.
    Fayolle, G., Iasnogorodski, R.: Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verwandte Geb. 47, 325–351 (1979) CrossRefGoogle Scholar
  13. 13.
    Guillemin, F., Simonian, A.: Transient characteristics of an M/M/∞ system. Adv. Appl. Probab. 27, 862–888 (1995) CrossRefGoogle Scholar
  14. 14.
    Kesidis, G., Walrand, J., Chang, C.S.: Effective bandwidths for multiclass Markov fluids and other ATM sources. IEEE/ACM Trans. Netw. 1, 424–428 (1993) CrossRefGoogle Scholar
  15. 15.
    Kosten, L.: Stochastic theory of data-handling systems with groups of multiple sources. In: Rudin H., Bux W. (eds.) Performance of Computer Communication Systems, pp. 321–331. Elsevier, Amsterdam (1984) Google Scholar
  16. 16.
    Kim, J.G., Krunz, M.: Fluid analysis of delay and packet discard performance for QoS support in wireless networks. IEEE J. Sel. Areas Commun. 19, 384–395 (2001) CrossRefGoogle Scholar
  17. 17.
    Mandjes, M., Mitra, D., Scheinhardt, W.: Models of network access using feedback fluid queues. Queueing Syst. 44, 365–398 (2003) CrossRefGoogle Scholar
  18. 18.
    Mandjes, M., Ridder, A.: Finding the conjugate of Markov fluid processes. Probab. Eng. Inform. Sci. 9, 297–315 (1995) CrossRefGoogle Scholar
  19. 19.
    Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Rockleigh (1964) Google Scholar
  20. 20.
    Mitra, D.: Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Probab. 20, 646–676 (1988) CrossRefGoogle Scholar
  21. 21.
    Preater, J.: M/M/∞ transience revisited. J. Appl. Probab. 34, 1061–1067 (1997) CrossRefGoogle Scholar
  22. 22.
    Reich, E.: On the integro-differential equation of Takács. I. Ann. Math. Stat. 29, 563–570 (1958) Google Scholar
  23. 23.
    Roijers, F., Mandjes, M., van den Berg, H.: Analysis of congestion periods in an M/M/∞-queue. Perform. Eval. 64(7–8), 737–754 (2007) CrossRefGoogle Scholar
  24. 24.
    Scheinhardt, W., van Foreest, N., Mandjes, M.: Continuous feedback fluid queues. Oper. Res. Lett. 33, 551–559 (2005) CrossRefGoogle Scholar
  25. 25.
    Sengupta, B., Jagerman, D.: A conditional response time of the M/M/1 processor-sharing queue. AT&T Techn. J. 2, 409–421 (1985) Google Scholar
  26. 26.
    Virtamo, J., Norros, I.: Fluid queue driven by an M/M/1 queue. Queueing Syst. 16, 373–386 (1994) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Korteweg-de Vries InstituteUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.EURANDOMEindhovenThe Netherlands
  4. 4.TNO ICTDelftThe Netherlands

Personalised recommendations