A busy period approach to some queueing games

Abstract

The traditional approach when looking for a symmetric equilibrium behavior in queueing models with strategic customers who arrive according to some stationary arrival process is to look for a strategy which, if used by all, is also a best response of an individual customer under the resulting stochastic steady-state conditions. This description lacks a key component: a proper definition of the set of players. Hence, many of these models cannot be defined as proper noncooperative games. This limitation raises two main concerns. First, it is hard to formulate these models in a canonical way, and hence, results are typically limited to a specific model or to a narrow class of models. Second, game theoretic results cannot be applied directly in the analysis of such models and call for ad hoc adaptations. We suggest a different approach, one that is based on the stationarity of the underlying stochastic processes. In particular, instead of considering a system that is functioning from time immemorial and has already reached stochastic steady-state conditions, we look at the process as a series of isolated, a priori identical, and independent strategic situations with a random, yet finite, set of players. Each of the isolated games can be analyzed using existing tools from the classical game theoretic literature. Moreover, this approach suggests a canonic definition of strategic queueing models as properly defined mathematical objects. A significant advantage of our approach is its compatibility with models in the existing literature. This is exemplified in detail for the famous model of the unobservable “to queue or not to queue” problem and other related models.

This is a preview of subscription content, access via your institution.

Fig. 1

Notes

  1. 1.

    This assumption has no impact on the performance of the queue due to the memoryless property of the exponential distribution.

  2. 2.

    Assume that the distribution of N permits this interchangeability of differentiation and expectation.

  3. 3.

    Interestingly, nothing will be changed had he been informed.

References

  1. 1.

    Edelson, N.M., Hildebrand, D.K.: Congestion tolls for Poisson queueing processes. Econometrica 43, 81–92 (1975)

    Article  Google Scholar 

  2. 2.

    Harchol-Balter, M.: Performance Modeling and Design of Computer Systems: Queueing Theory in Action. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  3. 3.

    Hart, S.: Chapter 2: Games in extensive and strategic forms. Handbook of Game Theory with Economic Applications. Elsevier, Amsterdam (1992)

    Google Scholar 

  4. 4.

    Hassin, R.: Rational Queueing. CRC, Boca Raton (2016)

    Google Scholar 

  5. 5.

    Hassin, R., Haviv, M.: Equilibrium strategies for queues with impatient customers. Oper. Res. Lett. 17, 41–45 (1995)

    Article  Google Scholar 

  6. 6.

    Hassin, R., Haviv, M.: To Queue or not to Queue: Equilibrium Behavior in Queueing Systems. Kluwer, Dordrecht (2003)

    Google Scholar 

  7. 7.

    Haviv, M., Milchtaich, I.: Auctions with a random number of identical bidders. Econ. Lett. 111, 143–146 (2012)

    Article  Google Scholar 

  8. 8.

    Haviv, M., Oz, B.: Self-regulation of an unobservable queue. Manag. Sci. 64, 2380–2389 (2018)

    Article  Google Scholar 

  9. 9.

    Haviv, M., Oz, B.: Social cost of deviation: new and old results on optimal customer behavior in queues. Queueing Models Serv. Manag. 1, 31–58 (2018)

    Google Scholar 

  10. 10.

    Kerner, Y.: Equilibrium joining probabilities for an M/G/1 queue. Games Econ. Behav. 71, 521–526 (2011)

    Article  Google Scholar 

  11. 11.

    Little, J.D.C.: A proof for the queuing formula: \(L={\lambda } W\). Oper. Res. 9, 383–387 (1961)

    Article  Google Scholar 

  12. 12.

    McAfee, R.P., McMillan, J.: Auctions with a stochastic number of bidders. J. Econ. Theory 43, 1–19 (1987)

    Article  Google Scholar 

  13. 13.

    Milchtaich, I.: Random-player games. Games Econ. Behav. 47, 353–388 (2004)

    Article  Google Scholar 

  14. 14.

    Myerson, R.B.: Population uncertainty and Poisson games. Int. J. Game Theory 27, 375–392 (1997)

    Article  Google Scholar 

  15. 15.

    Naor, P.: The regulation of queue size by levying tolls. Econometrica 37, 15–24 (1969)

    Article  Google Scholar 

  16. 16.

    Ross, S.M.: Applied Probability Models with Optimization Applications. Holden-Day, San Francisco (1970)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Binyamin Oz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was partly supported by Israel Science Foundation Grant No. 1828/19 and 1512/19 and by the Zagagi Foundation.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haviv, M., Oz, B. A busy period approach to some queueing games. Queueing Syst (2021). https://doi.org/10.1007/s11134-021-09692-0

Download citation

Keywords

  • Strategic behavior in queues
  • Games with random number of players
  • Nash equilibrium
  • Renewal process
  • Non-cooperative games

Mathematics Subject Classification

  • 60K05
  • 60K25
  • 91A10