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International Teletraffic Congress

ITC 2007: Managing Traffic Performance in Converged Networks pp 495–506Cite as

Network Calculus Delay Bounds in Queueing Networks with Exact Solutions

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Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 4516))

Abstract

The purpose of this paper is to shed light on the accuracy of probabilistic delay bounds obtained with network calculus. In particular, by comparing calculus bounds with exact results in a series of M/M/1 queues with cross traffic, we show that reasonably accurate bounds are achieved when the percentage of cross traffic is low. We use recent results in network calculus and, in addition, propose novel bounds based on Doob’s maximal inequality for supermartingales. In the case of single M/M/1 and M/D/1 queues, our results improve existing bounds by \(\Omega\left(\frac{\log(1-\rho)^{-1}}{1-\rho}\right)\) when the utilization factor ρ converges to one.

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References

  1. Cruz, R.: A calculus for network delay, parts I and II. IEEE Transactions on Information Theory 37(1), 114–141 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chang, C.S.: Performance Guarantees in Communication Networks. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  3. Le Boudec, J.Y., Thiran, P.: Network Calculus. In: Thiran, P., Le Boudec, J.-Y. (eds.) Network Calculus. LNCS, vol. 2050, Springer, Heidelberg (2001)

    Google Scholar 

  4. Kurose, J.: On computing per-session performance bounds in high-speed multi-hop computer networks. In: ACM Sigmetrics, pp. 128–139. ACM Press, New York (1992)

    Google Scholar 

  5. Knightly, E.W.: Second moment resource allocation in multi-service networks. In: ACM Sigmetrics, Seattle, Washington, United States, pp. 181–191. ACM Press, New York (1997), doi:10.1145/258612.258687

    Google Scholar 

  6. Boorstyn, R., Burchard, A., Liebeherr, J., Oottamakorn, C.: Statistical service assurances for traffic scheduling algorithms. IEEE Journal on Selected Areas in Comm. Special Issue on Internet QoS 18, 2651–2664 (2000)

    Article  Google Scholar 

  7. Li, C., Burchard, A., Liebeherr, J.: A network calculus with effective bandwidth. Tech. Rep. CS-2003-20, University of Virginia (Nov 2003)

    Google Scholar 

  8. Yaron, O., Sidi, M.: Performance and stability of communication networks via robust exponential bounds. IEEE/ACM Trans. on Net. 1, 372–385 (1993)

    Article  Google Scholar 

  9. Starobinski, D., Sidi, M.: Stochastically bounded burstiness for communication networks. IEEE Transactions of Information Theory 46, 206–212 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ciucu, F., Burchard, A., Liebeherr, J.: Scaling properties of statistical end-to-end bounds in the network calculus. IEEE Transactions on Information Theory 52(6), 2300–2312 (2006)

    Article  MathSciNet  Google Scholar 

  11. Fidler, M.: An end-to-end probabilistic network calculus with moment generating functions. In: IEEE 14th Int. Workshop on Quality of Service, pp. 261–270. IEEE Computer Society Press, Los Alamitos (2006)

    Chapter  Google Scholar 

  12. Jiang, Y.: A basic stochastic network calculus. In: ACM Sigcomm, pp. 123–134 (2006)

    Google Scholar 

  13. Knightly, E.: Enforceable quality of service guarantees for bursty traffic streams. In: IEEE Infocom, pp. 635–642 (1998)

    Google Scholar 

  14. Kleinrock, L.: Queueing Systems, vol. 1. John Wiley and Sons, Chichester (1975)

    MATH  Google Scholar 

  15. Pandit, K., Schmitt, J., Steinmetz, R.: Network calculus meets queueing theory - a simulation based approach to bounded queues. In: IEEE 12th International Workshop on Quality of Service, pp. 114–118. IEEE Computer Society Press, Los Alamitos (2004)

    Chapter  Google Scholar 

  16. Kelly, F.: Notes on effective bandwidths. In: Kelly, F.P., Zachary, S., Ziedins, I.B. (eds.) Stochastic Networks: Theory and Applications. Statistical Society Lecture Notes Series, vol. 4, pp. 141–168. Oxford University Press, Oxford (1996)

    Google Scholar 

  17. Kingman, J.F.C.: A martingale inequality in the theory of queues. Cambridge Philos. Soc. 59, 359–361 (1964)

    Article  MathSciNet  Google Scholar 

  18. Buffet, E., Duffield, N.G.: Exponential upper bounds via martingales for multiplexers with Markovian arrivals. J. of App. Prob. 31, 1049–1060 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu, Z., Nain, P., Towsley, D.: Exponential bounds with applications to call admission. J. of the ACM 44, 366–394 (1997), doi:10.1145/258128.258129

    Article  MATH  MathSciNet  Google Scholar 

  20. Chang, C.S.: On the exponentiality of stochastic linear systems under the max-plus algebra. IEEE Trans. on Automatic Control 41, 1182–1188 (1996)

    Article  MATH  Google Scholar 

  21. Boudec, J.-Y.L.: Some properties of variable length packet shapers. IEEE/ACM Transactions on Networking 10, 329–337 (2002)

    Article  Google Scholar 

  22. Iversen, V.B., Staalhagen, L.: Waiting time distribution in M/D/1 queueing systems. Electronics Letters 35(25), 2184–2185 (1999)

    Article  Google Scholar 

  23. Burchard, A., Liebeherr, J., Ciucu, F.: On \(\Theta\left(H\log H\right)\) scaling of network delays. In: Proceedings of IEEE Infocom, IEEE Computer Society Press, Los Alamitos (2007)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Virginia, U.S.A.

    Florin Ciucu

Authors
  1. Florin Ciucu
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Lorne Mason Tadeusz Drwiega James Yan

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© 2007 Springer-Verlag Berlin Heidelberg

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Ciucu, F. (2007). Network Calculus Delay Bounds in Queueing Networks with Exact Solutions. In: Mason, L., Drwiega, T., Yan, J. (eds) Managing Traffic Performance in Converged Networks. ITC 2007. Lecture Notes in Computer Science, vol 4516. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72990-7_45

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  • DOI: https://doi.org/10.1007/978-3-540-72990-7_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72989-1

  • Online ISBN: 978-3-540-72990-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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