Skip to main content
SpringerLink
Log in

Optimal design of measurements on queueing systems

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We examine the optimal design of measurements on queues with particular reference to the M/M/1 queue. Using the statistical theory of design of experiments, we calculate numerically the Fisher information matrix for an estimator of the arrival rate and the service rate to find optimal times to measure the queue when the number of measurements is limited for both interfering and non-interfering measurements. We prove that in the non-interfering case, the optimal design is equally spaced. For the interfering case, optimal designs are not necessarily equally spaced. We compute optimal designs for a variety of queuing situations and give results obtained under the \(D\)- and \(D_s\)-optimality criteria.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Abate, J., Whitt, W.: Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Syst. 25(1), 173–233 (1997)

    Article  Google Scholar 

  2. Acharya, S.: On normal approximation for maximum likelihood estimation from single server queues. Queueing Syst. 40(3), 313 (1999)

    Google Scholar 

  3. Aigner, D.: Parameter estimation from cross-sectional observations on an elementary queuing system. Oper. Res. 22(2), 422–428 (1974)

    Article  Google Scholar 

  4. Atkinson, A., Donev, A., Tobias, R.: Optimum Experimental Designs, with SAS. Oxford University Press, Oxford (2007)

    Google Scholar 

  5. Baccelli, F., Machiraju, S., Veitch, D., Bolot, J.: The role of PASTA in network measurement. ACM SIGCOMM Comput. Commun. Rev. 36(4), 231–242 (2006)

    Article  Google Scholar 

  6. Baccelli, F., Machiraju, S., Veitch, D., Bolot, J.: On optimal probing for delay and loss measurement. In: Proceedings of the 7th ACM SIGCOMM conference on Internet measurement, pp. 291–302. ACM (2007)

  7. Baccelli, F., Machiraju, S., Veitch, D., Bolot, J.: Probing for loss: the case against probe trains. Commun. Lett. IEEE 15(5), 590–592 (2011)

    Article  Google Scholar 

  8. Basawa, I., Bhat, U., Lund, R.: Maximum likelihood estimation for single server queues from waiting time data. Queueing Syst. 24, 155–167 (1997)

    Article  Google Scholar 

  9. Basawa, I., Bhat, U., Zhou, J.: Parameter Estimation in Queueing Systems using Partial Information. Technical Report. Ohio State University, Columbus (2006)

    Google Scholar 

  10. Basawa, I., Prabhu, N.: Estimation in single server queues. Nav. Res. Logist. Q. 28(3), 475–487 (1981)

    Article  Google Scholar 

  11. Basawa, I., Prabhu, N.: Large sample inference from single server queues. Queueing Syst. Theory Appl. 3(4), 289–304 (1988)

    Article  Google Scholar 

  12. Billingsley, P.: Statistical Inference for Markov Processes. The University of Chicago Press, Chicago (1962)

    Google Scholar 

  13. Bodas, S., Shah, D., Wischik, D.: Congestion control meets medium access: throughput, delay, and complexity. In: Proceedings of the 12th ACM SIGMETRICS/PERFORMANCE joint international conference on Measurement and Modeling of Computer Systems, pp. 399–400 (2012)

  14. Chaloner, K., Verdinelli, I.: Bayesian experimental design: a review. Stat. Sci. 10(3), 273–304 (1995)

    Article  Google Scholar 

  15. Chen, T.: Parameter estimation for partially observed queues. IEEE Trans. Commun. 42(9), 2730–2739 (1994)

    Article  Google Scholar 

  16. Chernoff, H.: Locally optimal designs for estimating parameters. Ann. Math. Stat. 24(4), 586–602 (1953)

    Article  Google Scholar 

  17. Clarke, A.: Maximum likelihood estimates in a simple queue. Ann. Math. Stat. 28(4), 1036–1040 (Dec 1957)

  18. Coffman, E., Robert, P., Simatos, F., Tarumi, S., Zussman, G.: A performance analysis of channel fragmentation in dynamic spectrum access systems. Queueing Syst. 71(3), 293–320 (2012)

  19. Grossglauser, M., Tse, D.N.: A time-scale decomposition approach to measurement-based admission control. IEEE/ACM Trans. Netw. 11(4), 550–563 (2003)

    Article  Google Scholar 

  20. Jenkins, J.: The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue M/M/l. J. Appl. Prob. 9(2), 396–403 (1972)

    Article  Google Scholar 

  21. Khisti, A., Huitema, C., Dube, A.: Controlling admission of data streams onto a network based on end-to-end measurements. US Patent 7,239,611, 3 July 2007

  22. Kleinrock, L.: Queueing Systems: Theory, vol. 1. John Wiley & Sons Inc, New York (1975)

    Google Scholar 

  23. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J. Optim. 9, 112–147 (1998)

    Article  Google Scholar 

  24. Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Trans. Netw. 2(1), 1–15 (1994)

    Article  Google Scholar 

  25. Love, S., Pollock, G.S., Goldsack, P., Kirshenbaum, E.R.: System and method for monitoring communication networks using data stream characterization. US Patent 6,904,020, 7 June 2005

  26. Morse, P.M.: Stochastic properties of waiting lines. J. Oper. Res. Soc. Am. 3(3), 255–261 (1955). (Copyright 1955 INFORMS)

    Google Scholar 

  27. Nilsson, M.: Measuring available path capacity using short probe trains. In IEEE Network Operations and Management Symposium (NOMS), pp. 910–913 (2010)

  28. Norros, I.: A storage model with self-similar input. Queueing Syst. 16(3), 387–396 (1994)

    Article  Google Scholar 

  29. Pagendam, D., Pollett, P.: Optimal sampling and problematic likelihood functions in a simple population model. Environ. Model. Assess. 14(6), 759–767 (2009)

    Article  Google Scholar 

  30. Pagendam, D., Pollett, P.: Robust optimal observation of a metapopulation. Ecol. Model. 221(21), 2521–2525 (2010)

    Article  Google Scholar 

  31. Parker, B., Gilmour, S., Schormans, J.: Measurement of packet loss probability by optimal design of packet probing experiments. IET Commun. 3(6), 979 (2009)

    Article  Google Scholar 

  32. Parker, B.M., Gilmour, S.G., Schormans, J.A.: Design of experiments for categorical repeated measurements in packet communication networks. Technometrics 53(4), 339–352 (2011)

    Article  Google Scholar 

  33. Raiciu, C., Barre, S., Pluntke, C., Greenhalgh, A., Wischik, D., Handley, M.: Improving datacenter performance and robustness with multipath TCP. ACM SIGCOMM Comput. Commun. Rev. 41, 266–277 (2011)

    Article  Google Scholar 

  34. Reynolds, J.: The covariance structure of queues and related processes: a survey of recent work. Adv. Appl. Prob. 7(2), 383–415 (1975)

    Article  Google Scholar 

  35. Ross, J., Pagendam, D., Pollett, P.: On parameter estimation in population models II: multi-dimensional processes and transient dynamics. Theor. Popul. Biol. 75(2–3), 123–132 (2009)

    Article  Google Scholar 

  36. Roughan, M.: Fundamental bounds on the accuracy of network performance measurements. In: Proceedings of the 2005 ACM SIGMETRICS international conference on Measurement and modeling of computer systems, pp. 253–264 (2005)

  37. Roughan, M.: A comparison of poisson and uniform sampling for active measurements. IEEE J. Sel. Areas Commun. 24(12), 2299–2312 (2006)

    Article  Google Scholar 

  38. Shah, D., Wischik, D.: Fluid models of congestion collapse in overloaded switched networks. Queueing Syst. 69(2), 121–143 (2011)

    Article  Google Scholar 

  39. Shah, D., Wischik, D.: Log-weight scheduling in switched networks. Queueing Syst. 71(1–2), 97–136 (2012)

  40. Spall, J.: Introduction to Stochastic Search and Optimization: Estimation, Simulation and Control. John Wiley & Sons Inc, New Jersey (2003)

    Book  Google Scholar 

  41. The ISP Column. The QoS emperor’s wardrobe. http://www.internetsociety.org/publications/isp-column-june-2012-qos-emperors-wardrobe-1 (2012). Accessed 26 Sept 2014

  42. Wolff, R.: Poisson arrivals see time averages. Oper. Res. 30(2), 223–231 (1982)

    Article  Google Scholar 

Download references

Acknowledgments

The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton, in the completion of this work.

Author information

Authors and Affiliations

  1. Southampton Statistical Sciences Research Institute, University of Southampton, Southampton, UK

    Ben M. Parker & Steven Gilmour

  2. School of Electronic Engineering and Computer Science, Queen Mary University of London, London, UK

    John Schormans

  3. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Hugo Maruri-Aguilar

Authors
  1. Ben M. Parker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven Gilmour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. John Schormans
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hugo Maruri-Aguilar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben M. Parker.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parker, B.M., Gilmour, S., Schormans, J. et al. Optimal design of measurements on queueing systems. Queueing Syst 79, 365–390 (2015). https://doi.org/10.1007/s11134-014-9421-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-014-9421-y

Keywords

Mathematics Subject Classification

Search

Navigation