Advertisement

Queueing Systems

, Volume 88, Issue 3–4, pp 389–407 | Cite as

Probabilistic selfish routing in parallel batch and single-server queues

  • A. Wang
  • I. Ziedins
Article

Abstract

We consider a network of parallel queues, operating under probabilistic routing, where users can choose to join either a batch service queue, or one of several FIFO single-server queues. Afimeimounga et al. (Queueing Syst 49:321–334, 2005) considered the 2-queue case, which is known to exhibit the Downs–Thomson paradox, where delays may increase as capacity is increased. We show that in larger parallel systems, with multiple single-server queues, the user equilibrium is always unique when the batch size is sufficiently large relative to the number of queues; no more than three equilibria exist; Braess paradox may appear when adding extra queues.

Keywords

Queueing network User equilibria Parallel queues Downs–Thomson paradox Wardrop’s equilibrium Braess paradox 

Mathematics Subject Classification

90B15 60K25 90B20 91A25 91A10 91A13 

Notes

Acknowledgements

The authors wish to thank Moshe Haviv and Mark Holmes for comments and suggestions in the preparation of this work. They are also very grateful to the anonymous referees whose suggestions and recommendations considerably improved both the content and the presentation of the paper. Alex Wang thanks the University of Auckland for providing support for this work with a University of Auckland Doctoral Scholarship. Both authors are grateful to the Marsden Fund for financial support.

References

  1. 1.
    Abraham, J.E., Hunt, J.D.: Transit system management, equilibrium mode split and the Downs–Thomson paradox. Technical report. Department of Civil Engineering, University of Calgary (2001)Google Scholar
  2. 2.
    Afimeimounga, H.: User optimal policies for a stochastic transportation network. Ph.D. Thesis. The University of Auckland (2011)Google Scholar
  3. 3.
    Afimeimounga, H., Solomon, W., Ziedins, I.: The Downs–Thomson paradox: existence, uniqueness and stability of user equilibria. Queueing Syst. 49, 321–334 (2005)CrossRefGoogle Scholar
  4. 4.
    Afimeimounga, H., Solomon, W., Ziedins, I.: User equilibria for a parallel queueing system with state dependent routing. Queueing Syst. 62, 169–193 (2010)CrossRefGoogle Scholar
  5. 5.
    Arnott, R., Small, K.: The economics of traffic congestion. Am. Sci. 82, 446–455 (1994)Google Scholar
  6. 6.
    Arvidsson, N.: The milk run revisited: a load factor paradox with economic and environmental implications for urban freight transport. Transp. Res. Part A Policy Pract. 51, 56–62 (2013)CrossRefGoogle Scholar
  7. 7.
    Bean, N.G., Kelly, F.P., Taylor, P.G.: Braess’s paradox in a loss network. J. Appl. Probab. 34, 155–159 (1997)CrossRefGoogle Scholar
  8. 8.
    Bell, C.E., Stidham, S.: Individual versus social optimization in the allocation of customers to alternative servers. Manag. Sci. 29, 831–839 (1983)CrossRefGoogle Scholar
  9. 9.
    Braess, D.: Über ein Paradox der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)Google Scholar
  10. 10.
    Braess, D., Koch, G.: On the existence of equilibria in asymmetrical multiclass-user transportation networks. Transp. Sci. 13, 56–63 (1979)CrossRefGoogle Scholar
  11. 11.
    Calvert, B.: The Downs–Thomson effect in a Markov process. Probab. Eng. Inf. Sci. 11, 327–340 (1997)CrossRefGoogle Scholar
  12. 12.
    Calvert, B., Solomon, W., Ziedins, I.: Braess’s paradox in a queueing network with state-dependent routing. J. Appl. Probab. 34, 134–154 (1997)CrossRefGoogle Scholar
  13. 13.
    Chen, Y., Holmes, M., Ziedins, I.: Monotonicity properties of user equilibrium policies for parallel batch systems. Queueing Syst. 70, 81–103 (2012)CrossRefGoogle Scholar
  14. 14.
    Cohen, J.E., Kelly, F.P.: A paradox of congestion in a queueing network. J. Appl. Probab. 27, 730–734 (1990)CrossRefGoogle Scholar
  15. 15.
    Downs, A.: The law of peak-hour expressway congestion. Traffic Q. 16, 393–409 (1962)Google Scholar
  16. 16.
    El-Zoghdy, S., Kameda, H., Li, J.: Numerical studies on a paradox for non-cooperative static load balancing in distributed computer systems. Comput. Oper. Res. 33, 345–355 (2006)CrossRefGoogle Scholar
  17. 17.
    Guo, P., Zhang, Z.: Strategic queueing behavior and its impact on system performance in service systems with the congestion-based staffing policy. Manuf. Serv. Oper. Manag. 15, 118–131 (2013)CrossRefGoogle Scholar
  18. 18.
    Guo, P., Lindsey, R., Zhang, Z.: On the Downs–Thomson paradox in a self-financing two-tier queuing system. Manuf. Serv. Oper. Manag. 16, 315–322 (2014)CrossRefGoogle Scholar
  19. 19.
    Hassin, R., Haviv, M.: To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems. Kluwer Academic, Dordrecht (2002)Google Scholar
  20. 20.
    Hassin, R.: Rational Queueing. Chapman & Hall, London (2016)CrossRefGoogle Scholar
  21. 21.
    Haviv, M., Roughgarden, T.: The price of anarchy in an exponential multi-server. Oper. Res. Lett. 35, 421–426 (2007)CrossRefGoogle Scholar
  22. 22.
    Knight, F.H.: Some fallacies in the interpretation of social cost. Q. J. Econ. 38, 382–606 (1924)CrossRefGoogle Scholar
  23. 23.
    Korilis, Y.A., Lazar, A.A., Orda, A.: Avoiding the Braess paradox in non-cooperative networks. J. Appl. Probab. 36, 211–222 (1999)CrossRefGoogle Scholar
  24. 24.
    Mogridge, M.J.H.: Planning for optimum urban efficiency: the relationship between congestion on the roads and public transport. Transp. Plan. Syst. 1, 11–19 (1990)Google Scholar
  25. 25.
    Nobel, R., Stolwijk, M.: The Downs-Thomson paradox revisited. In: Nadarajan, R., Lekshami, R.S., Sai Sundara Krishnan, G. (eds.) Computational and Mathematical Modelling, pp. 20–45. Narosa Publishing House, New Delhi (2011)Google Scholar
  26. 26.
    Patriksson, M.: The Traffic Assignment Problem: Models and Methods. VSP, Dordrecht (1994)Google Scholar
  27. 27.
    Roughgarden, T., Tardos, E.: How bad is selfish routing? J. ACM 49, 236–259 (2002)CrossRefGoogle Scholar
  28. 28.
    Thomson, J.M.: Great Cities and Their Traffic. Gollancz, London (1977)Google Scholar
  29. 29.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civil Eng. II 1, 325–378 (1952)Google Scholar
  30. 30.
    Watling, D.: Asymmetric problems and stochastic process models of traffic assignment. Transp. Res. 30, 339–357 (1996)CrossRefGoogle Scholar
  31. 31.
    Whitt, W.: Deciding which queue to join: some counterexamples. Oper. Res. 34, 55–62 (1986)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand

Personalised recommendations