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On Equilibrium in a Constant Retrial Queuing System with Reserved Time and Vacations

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Abstract

This paper investigates an M/M/1 constant retrial queue with reserved time and vacations. A new arriving customer will take up the server and accept service immediately if the server is idle. Otherwise, if the server is busy or on vacation, customers have to join a retrial orbit and wait for retry. Once a service is completed, the server will reserve a random time to seek a customer from the orbit at a constant retrial rate. If there is no arrivals (from the orbit or outside) during the idle period, to save energy, the server will take a vacation. This paper studies the fully unobservable case. First, the steady-state condition of the system is analyzed by using the Foster’s criterion, and the customers’ expected waiting time is obtained based on the generating function technique. And then, by introducing an appropriate revenue structure, the equilibrium strategies of customers and the socially optimal strategy are all derived. Furthermore, a comparison between them is made and the effect of some main system parameters is studied.

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References

  1. Falin, G.I., Templeton, J.G.C.: Retrial Queues. Springer, US (1997)

    Book  MATH  Google Scholar 

  2. Artalejo, J.R.: A queueing system with returning customers and waiting line. Oper. Res. Lett. 17, 191–199 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Artalejo, J.R., Gomez-Corral, A.: Steady state solution of a single-server queue with linear repeated requests. J. Appl. Probab. 34, 223–233 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Artalejo, J.R., Gomez-Corral, A.: Analysis of a stochastic clearing system with repeated attempts. Stoch. Models 14, 623–645 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fayolle, G.: A simple telephone exchange with delayed feedbacks. In: Proceedings of the International Seminar on Teletraffic Analysis and Computer Performance Evaluation, pp. 245-253. (1986)

  6. Falin, G.I.: The M/M/1 retrial queue with retrials due to server failures. Queu. Syst. 58, 155–160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems: A Computational Approach, vol. 318. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  8. Do, N., Do, T., Melikov, A.: Equilibrium customer behavior in the M/M/1 retrial queue with working vacations and a constant retrial rate. Oper. Res. Int. J. (2018). https://doi.org/10.1007/s12351-017-0369-7

    Article  Google Scholar 

  9. Phungduc, T.: M/M/1/1 Retrial Queues with Setup Time. Queueing Theory and Network Applications. Springer International Publishing, Berlin (2016)

    Google Scholar 

  10. Phung-Duc, T.: Single server retrial queues with setup time. J. Ind. Manag. Optim. 3, 1329–1345 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhou, M., Liu, L., Chai, X.: Zhen Wang Equilibrium strategies in a constant retrial queue with setup time and the N-policy. Commun. Stat. Theory Methods (2019). https://doi.org/10.1080/03610926.2019.1565779

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Edelson Noel, M., Hilderbrand, D.K.: Congestion tolls for Poisson queuing processes. Econometrica 43(1), 81–92 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sitopu, J.W., Mawengkang, H., Lubis, R.S.: An improved search approach for solving non-convex mixed-integer non linear programming problems. IOP Conf. Ser. Mater. Sci. Eng. 300, 012022 (2018)

    Article  Google Scholar 

  15. Lotfi, R., Mostafaeipour, A., Mardani, N., Mardani, S.: Investigation of wind farm location planning by considering budget constraints. Int. J. Sustain. En. 37(8), 799–817 (2018)

    Article  Google Scholar 

  16. Maccio, V.J., Down, D.G.: On optimal policies for energy-aware servers. Perform. Eval. 90, 36–52 (2015)

    Article  Google Scholar 

  17. Yutaka, S., Yoshitaka, T., Yutaka, T., et al.: A composite queue with vacation and set-up and close-down times for SVCC in IP over ATM networks. J. Oper. Res. Soc. Jpn. 41(1), 68–80 (1998)

    MathSciNet  MATH  Google Scholar 

  18. Frey, A., Takahashi, Y.: An MX/GI/1/N queue with close-down and vacation times. Int. J. Stoch. Anal 12(1), 63–83 (1999)

    MATH  Google Scholar 

  19. Wang, J., Zhang, F.: Monopoly pricing in a retrial queue with delayed vacations for local area network applications[J]. IMA J. Manag. Math. 27(2), 315–334 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, Y., Wang, J.: Equilibrium pricing in an M/G/1 retrial queue with reserved idle time and setup time[J]. Appl. Math. Model. 49, 514–530 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Barroso, L.A., Holzle, U.: The case for energy-proportional computing. Computer 40, 33–37 (2007)

    Article  Google Scholar 

  22. Sennott, L.I., Humblet, P.A., Tweedie, R.L.: Technical note-mean drifts and the non-ergodicity of Markov chains. Oper. Res. 31(4), 783–789 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Economou, A., Kanta, S.: Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue. Nav. Res. Logist. 58(2), 107–122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hassin, R., Haviv, M.: To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

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Wang, LL., Liu, LW., Chai, XD. et al. On Equilibrium in a Constant Retrial Queuing System with Reserved Time and Vacations. J. Oper. Res. Soc. China 10, 785–800 (2022). https://doi.org/10.1007/s40305-019-00290-9

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  • DOI: https://doi.org/10.1007/s40305-019-00290-9

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