Skip to main content
SpringerLink
Log in

Statistical analysis of queueing systems

  • Invited Paper
  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper provides an overview of the literature on statistical analysis of queueing systems. Topics discussed include: model identification, estimation, hypothesis testing and other related aspects. Not all of these statistical problems are covered in books on queueing theory or stochastic processes. The bibliography is not exhaustive, but comprehensive enough to provide sources from the literature.

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

Similar content being viewed by others

References

  1. D.J. Aigner, Parameter estimation from cross-sectional observations on an elementary queueing system, Oper. Res. 22, 2(1974)422.

    Google Scholar 

  2. I.V. Basawa, private communication (1985).

  3. I.V. Basawa and N.U. Prabhu, Estimation in single server queues, Naval Res. Log. Quart. 28(1981)475.

    Google Scholar 

  4. I.V. Basawa and B.L.S. Prakasa Rao,Statistical Inference for Stochastic Processes (Academic Press, New York, 1980).

    Google Scholar 

  5. V.E. Beneš, A sufficient set of statistics for a simple telephone exchange model, Bell Syst. Tech. J. 36(1957)939.

    Google Scholar 

  6. V.E. Beneš, The covariance function of a simple trunk group with applications to traffic measurement, Bell. Syst. Tech. J. 40(1961)117.

    Google Scholar 

  7. U.N. Bhat,Elements of Applied Stochastic Processes, 2nd ed. (Wiley, New York, 1984).

    Google Scholar 

  8. U.N. Bhat, A sequential technique for the control of traffic intensity in Markovian queues, Ann. Oper. Res. 8(1987)151.

    Google Scholar 

  9. U.N. Bhat and S.S. Rao, A statistical technique for the control of traffic intensity in queueing systems M/G/1 and GI/M/1, Oper. Res. 20(1972)955.

    Google Scholar 

  10. P. Billingsley,Statistical Inference for Markov Chains (University of Chicago Press, Chicago, 1961).

    Google Scholar 

  11. A. Birnbaum, Statistical methods for Poisson processes and exponential populations, J. Amer. Statis. Soc. 49(1954)254.

    Google Scholar 

  12. N. Blomqvist, The covariance function of the M/G/1 queueing system, Skand. Aktuar. 50 (1967)157.

    Google Scholar 

  13. N. Blomqvist, Estimation of waiting time parameters in the GI/G/1 queueing systems, Part I: General results, Skand. Aktuar. 51(1968)178.

    Google Scholar 

  14. N. Blomqvist, Estimation of waiting time parameters in the GI/G/1 queueing systems, Part II: Heavy traffic approximations, Skand. Aktuar. 52(1969)125.

    Google Scholar 

  15. J.V. Bradley,Distribution-Free Statistical Tests (Prentice Hall, Englewood Cliffs, N.J., 1968).

    Google Scholar 

  16. A.B. Clarke, Maximum likelihood estimates in a simple queue, Ann. Math. Stat. 28(1957) 1036.

    Google Scholar 

  17. W.J. Conover,Practical Nonparametric Statistics (Wiley, New York, 1971).

    Google Scholar 

  18. D.R. Cox,Renewal Theory (Methuen and Co., London, 1962).

    Google Scholar 

  19. D.R. Cox, Some problems of statistical analysis connected with congestion,Proc. Symp. on Congestion Theory, ed. W.L. Smith and W.B. Wilkinson, University of North Carolina, Chapel Hill, N.C. (1965).

    Google Scholar 

  20. D.R. Cox and P.A.W. Lewis,The Statistical Analysis of Series of Events (Meuthen and Co., London, 1966).

    Google Scholar 

  21. B.D. Craven, Asymptotic correlation in a queue, J. Appl. Prob., 6(1969)573.

    Google Scholar 

  22. D.J. Daley, Monte Carlo estimation of mean queue size in a stationary GI/M/1 queue, Oper. Res. 16(1968)1002.

    Google Scholar 

  23. D.J. Daley, The serial correlation coefficients of waiting times in a stationary single server queue, J. Austr. Math. Soc. 8(1968)683.

    Google Scholar 

  24. A. Descloux, On the accuracy of loss estimates, Bell Syst. Tech. J. 44, 6(1965)1139.

    Google Scholar 

  25. G.S. Fishman,Principles of Discrete Event Simulation (Wiley, New York, 1978).

    Google Scholar 

  26. A.V. Gafarian and C.J. Ancker, Mean value estimation from digital computer simulations, Oper. Res. 14(1966)25.

    Google Scholar 

  27. D.P. Gaver and J.P. Lehoczky, Random parameter Markov population process models and their likelihood, Bayes, and empirical Bayes analysis, NPS55-85-020, Monterey, California: Naval Postgraduate School (1985).

    Google Scholar 

  28. R.F. Gebhard, A limiting distribution of an estimator of mean queue length, Oper. Res. 11 (1963)1000.

    Google Scholar 

  29. B.V. Gnedenko, Y.K. Belyayev and A.D. Solovyev,Mathematical Methods of Reliability Theory (Academic Press, New York, 1969).

    Google Scholar 

  30. D. Gross and C.M. Harris,Fundamentals of Queueing Theory, 2nd ed. (Wiley, New York, 1985).

    Google Scholar 

  31. K. Harishchandra and S.S. Rao, Statistical inference about the traffic intensity parameter of M/E k /1 and E k /M/1 queues, Report, Indian Institute of Management, Bangalore, India (1984).

    Google Scholar 

  32. C.M. Harris, Some new results in the statistical analysis of queues,Proc. Conf. on Math. Methods in Queueing Theory, ed. A.B. Clarke, Vol. 98, Lecture Notes in Economics and Mathematical Systems (Springer-Verlag, New York, 1974) p. 157.

    Google Scholar 

  33. M. Jacobsen,Statistical Analysis of Counting Processes (Springer-Verlag, New York, 1982).

    Google Scholar 

  34. J.H. Jenkins, The relative efficiency of direct and maximum likelihood estimates of mean waiting time in the simple queue M/M/1, J. Appl. Prob. 9(1972)396.

    Google Scholar 

  35. D.G. Kendall, Stochastic processes occurring in the theory of queues and their analysis by the method of imbedded Markov chains, Ann. Math. Stat. 24(1953)338.

    Google Scholar 

  36. L. Kosten, J.R. Manning and F. Garwood, On the accuracy of measurements of probabilities of loss in telephone systems, J. Roy. Statis. Soc. B11(1949)54.

    Google Scholar 

  37. J.A. Koziol and A.F. Nemec, On a Cramér-von Mises type statistic for testing bivariate independence, Can. J. Stat. 7(1979)43.

    Google Scholar 

  38. P.A.W. Lewis, Recent results in the statistical analysis of univariate point processes, in:Stochastic Point Processes, ed. P.A.W. Lewis (Wiley, New York, 1972) p. 1.

    Google Scholar 

  39. H.W. Lilliefors, Some confidence intervals for queues, Oper. Res. 14(1966)723.

    Google Scholar 

  40. M.F. McGrath and N.D. Singpurwalla, A subjective Bayesian approach to the theory of queues: Part I — Modeling, Part II — Inference and information, GWU/lRRA/Serial TR-85/14,15, The George Washington University, Washington D.C. (1985).

    Google Scholar 

  41. P.W. Mielke, Jr., A note on some squared rank tests with existing ties, Technometrics 9 (1967)312.

    Google Scholar 

  42. S.C. Moore, Approximate techniques for non-stationary queues, Ph.D. Dissertation, Computer Science/Operations Research Center, Southern Methodist University, Dallas, Texas (1972).

    Google Scholar 

  43. S.R. Neal and A. Kuczura, A theory of traffic-mesurement errors for loss systems with renewal input,Proc. Conf. on Math. Methods in Queueing Theory, ed. A.B. Clarke, Vol. 98, Lecture Notes in Economics and Mathematical Systems (Springer-Verlag, New York, 1974) p. 199.

    Google Scholar 

  44. J. Putter, The treatment of ties in some nonparametric tests, Ann. Math. Stat. 26(1955)268.

    Google Scholar 

  45. R.H. Randies and D.A. Wolffe,Introduction to the Theory of Non-parametric Statistics (Wiley, New York, 1979).

    Google Scholar 

  46. S.S. Rao, U.N. Bhat and K. Harishchandra, Control of traffic intensity in a queue — A method based on SPRT, Opsearch 21(1984)63.

    Google Scholar 

  47. J.F. Reynolds, Asymptotic properties of mean length estimators for finite Markov queue, Oper. Res. 20, 1(1972)52.

    Google Scholar 

  48. J.F. Reynolds, The covariance structure of queues and related processes: A survey of recent work, Adv. Appl. Prob. 7(1975)383.

    Google Scholar 

  49. J. Riordan,Stochastic Service Systems (Wiley, New York, 1962).

    Google Scholar 

  50. D.A. Stanford, B. Pagurek and C.W. Woodside, Optimal prediction of times and queue lengths in the GI/M/1 queue, Oper. Res. 31(1983)322.

    Google Scholar 

  51. R. Syski,Introduction to Congenstion Theory in Telephone Systems (Oliver and Boyd, London, 1960).

    Google Scholar 

  52. T.R. Thiagarajan and C.M. Harris, Statistical tests for exponential service from M/G/1 waiting-time data, Naval Res. Log. Quart. 26, 3(1979)511.

    Google Scholar 

  53. R.W. Wolff, Problems of statistical inference for birth-and-death queueing models, Oper. Res. 13(1965)343.

    Google Scholar 

  54. C.M. Woodside, D.A. Stanford and B. Pagurek, Optimal prediction of queue lengths and delays in GI/M/m multiserver queues, Oper. Res. 32(1984)809.

    Google Scholar 

  55. P.J. Burke, The output of a queueing system, Oper. Res. 4(1956)699.

    Google Scholar 

  56. E. Reich, Waiting times when queues are in tandem, Ann. Math. Stat. 28(1957)768.

    Google Scholar 

Download references

Author information

Author notes

  1. S. Subba Rao

    Present address: Department of Information Systems and Operations Management, University of Toledo, 43606, Toledo, Ohio, USA

Authors and Affiliations

  1. Department of Statistics, Southern Methodist University, 75275, Dallas, Texas, USA

    U. Narayan Bhat

  2. Department of Management and Systems, Washington State University, 99164, Pullman, Washington, USA

    S. Subba Rao

Authors
  1. U. Narayan Bhat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Subba Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bhat, U.N., Rao, S.S. Statistical analysis of queueing systems. Queueing Syst 1, 217–247 (1987). https://doi.org/10.1007/BF01149536

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01149536

Keywords and phrases

Search

Navigation