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


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.


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

Mathematics Subject Classification

90B15 60K25 90B20 91A25 91A10 91A13 



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.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand

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