Queueing Systems

, 63:3 | Cite as

The first Erlang century—and the next

  • J. F. C. Kingman


The history of queueing theory, particularly over the first sixty years after Erlang’s 1909 paper, is summarised and assessed, with particular reference to the influence of Pollaczek and Kendall. The interactions between the world of telephone traffic and that of applied probability and operational research are a significant factor. The history is followed by speculation about the directions in which the theory might now develop, in response to new problems and new possibilities. It is suggested that classical unsolved problems like the queue M/G/k might be revisited, and that non-renewal inputs might be handled by martingale techniques.

History of queueing theory Multi-server queues Teletraffic Poisson arrivals Martingales 

Mathematics Subject Classification (2000)



  1. 1.
    Asmussen, S.: Applied Probability and Queues. Springer, New York (2003) Google Scholar
  2. 2.
    Brockmeyer, E., Halstrøm, H.L., Jensen, A.: The Life and Works of A.K. Erlang. Akademiet for de Tekniske Videnskaber, København (1948) Google Scholar
  3. 3.
    Cohen, J.W.: The Single Server Queue. North-Holland, Amsterdam (1969) Google Scholar
  4. 4.
    Cox, D.R.: Some statistical models related to series of events. J. R. Stat. Soc. B 17, 129–164 (1955) Google Scholar
  5. 5.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1950) Google Scholar
  6. 6.
    Fry, T.C.: Probability and Its Engineering Uses. Van Nostrand, New York (1928) Google Scholar
  7. 7.
    Kelly, F.P.: Reversibility and Stochastic Networks. Wiley, London (1979) Google Scholar
  8. 8.
    Kelly, F.P., Zachary, S., Ziedins, I. (eds.): Stochastic Networks: Theory and Applications. Oxford University Press, Oxford (1996) Google Scholar
  9. 9.
    Kendall, D.G.: Some problems in the theory of queues. J. R. Stat. Soc. B 13, 151–173 (1951); discussion 173–185 Google Scholar
  10. 10.
    Kendall, D.G.: Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24, 338–354 (1953) CrossRefGoogle Scholar
  11. 11.
    Khinchin, A.Y.: Mathematical Methods in the Theory of Queueing. Griffin, London (1960) Google Scholar
  12. 12.
    Kingman, J.F.C.: On queues in which customers are served in random order. Proc. Camb. Philos. Soc. 58, 79–91 (1962) CrossRefGoogle Scholar
  13. 13.
    Kingman, J.F.C.: The use of Spitzer’s identity in the investigation of the busy period and other quantities in the queue GI/G/1. J. Aust. Math. Soc. 2, 345–356 (1962) CrossRefGoogle Scholar
  14. 14.
    Kingman, J.F.C.: On doubly stochastic Poisson processes. Proc. Camb. Philos. Soc. 60, 923–930 (1964) CrossRefGoogle Scholar
  15. 15.
    Kingman, J.F.C.: On the algebra of queues. J. Appl. Probab. 3, 285–326 (1966) CrossRefGoogle Scholar
  16. 16.
    Kingman, J.F.C.: Regenerative Phenomena. Wiley, London (1972) Google Scholar
  17. 17.
    Kingman, J.F.C.: Poisson Processes. Oxford University Press, Oxford (1993) Google Scholar
  18. 18.
    Lindley, D.V.: The theory of queues with a single server. Proc. Camb. Philos. Soc. 48, 277–289 (1952) CrossRefGoogle Scholar
  19. 19.
    Loynes, R.M.: The stability of a queue with non-independent interarrival and service times. Proc. Camb. Philos. Soc. 58, 497–520 (1962) CrossRefGoogle Scholar
  20. 20.
    Pollaczek, F.: Ueber eine Aufgabe der Wahrscheinlichkeitstheorie. Math. Z. 32, 64–100 (1930) 729–750 CrossRefGoogle Scholar
  21. 21.
    Pollaczek, F.: Problèmes stochastiques posés par le phénomène de formation d’une queue d’attente à un guichet et par des phénomènes apparentés. Mém. Sci. Math. 136, 1–122 (1957) Google Scholar
  22. 22.
    Pollaczek, F.: Théorie analytique des problèmes stochastiques relatifs à un groupe de lignes téléphoniques avec dispositif d’attente. Mém. Sci. Math. 150, 1–114 (1961) Google Scholar
  23. 23.
    Pollaczek, F.: Concerning an analytic method for the treatment of queueing problems. In: Smith, W.L., Wilkinson, W.E. (eds.) Congestion Theory, pp. 1–42. University of North Carolina, Chapel Hill (1964) Google Scholar
  24. 24.
    Pollaczek, F.: Order statistics of partial sums of mutually independent random variables. J. Appl. Probab. 12, 390–395 (1975) CrossRefGoogle Scholar
  25. 25.
    Saaty, T.L.: Stochastic network flows: advances in networks of queues. In: Smith, W.L., Wilkinson, W.E. (eds.) Congestion Theory, pp. 86–107. University of North Carolina, Chapel Hill (1964) Google Scholar
  26. 26.
    Sebag-Montefiore, H.: Enigma: The Battle for the Code. Weidenfeld, Nicolson (2000) Google Scholar
  27. 27.
    Smith, M., Briggs, K., Kelly, F.P. (eds.): Networks: modelling and control. Philos. Trans. R. Soc. A 366, 1875–2092 (2008) Google Scholar
  28. 28.
    Smith, W.L.: On the distribution of queueing times. Proc. Camb. Philos. Soc. 49, 449–461 (1953) CrossRefGoogle Scholar
  29. 29.
    Smith, W.L., Wilkinson, W.E. (eds.): Congestion Theory. University of North Carolina, Chapel Hill (1964) Google Scholar
  30. 30.
    Whitt, W.: The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution. Queueing Syst. Theory Appl. 36, 71–87 (2000) CrossRefGoogle Scholar
  31. 31.
    Whittle, P.: Systems in Stochastic Equilibrium. Wiley, London (1986) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.BristolUK

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