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Equilibrium customer behavior in the M/M/1 retrial queue with working vacations and a constant retrial rate

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A Correction to this article was published on 03 October 2019

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Abstract

In this paper, we investigate the M/M/1 retrial queue with working vacations and a constant retrial rate. In the queue, customers decide about the entry based on the information upon their arrival instants. Scenarios regarding the availability of information (i.e., the server is occupied or not, and the server is on the vacation or not) for customers are compared. We derive the closed form solution for the stationary probabilities of the queue. Social optimizing and Nash equilibrium strategies for joining the system are investigated. Based on numerical results, the social benefit rate is best when customers know all information about the server.

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Change history

  • 03 October 2019

    In the originally published article, the order of the affiliations of the first and the corresponding author was incorrectly indicated.

References

  • 3GPP TR 36.889—Feasibility Study on Licensed-Assisted Access to Unlicensed Spectrum, (2015)

  • Artalejo JR, Gómez-Corral A (2008) Retrial Queueing Systems. Springer, Berlin

    Book  Google Scholar 

  • Artalejo JR, Economou A, Gómez-Corral A (2007) Applications of maximum queue lengths to call center management. Comput. OR 34(4):983–996

    Article  Google Scholar 

  • Artalejo JR, Economou A, Gómez-Corral A (2008) Algorithmic analysis of the Geo/Geo/c. Eur. J. Oper. Res. 189(3):1042–1056

    Article  Google Scholar 

  • Boudali O, Economou A (2012) Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes. Eur. J. Oper. Res. 218(3):708–715

    Article  Google Scholar 

  • Burnetas A, Economou A (2007) Equilibrium customer strategies in asingle server Markovian queue with setup times. Queueing Syst. 56(3–4):213–228

    Article  Google Scholar 

  • Chakka R, Harrison PG (2001) A Markov modulated multi-server queue with negative customers - the MM CPP/GE/c/L G-queue. Acta Inform. 37:881–919

    Article  Google Scholar 

  • Do TV (2010) M/M/1 retrial queue with working vacations. Acta Inform. 47(1):67–75

  • Do TV, Papp D, Chakka R, Sztrik J, Wang J (2014) M/M/1 retrial queue with working vacations and negative customer arrivals. IJAIP 6(1):52–65

  • Doshi BT (1986) Queueing systems with vacations—a survey. Queueing Syst. Theory Appl. 1(1):29–66

    Article  Google Scholar 

  • Economou A, Kanta S (2008) Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs. Oper. Res. Lett. 36(6):696–699

    Article  Google Scholar 

  • Economou A, Kanta S (2008) Optimal balking strategies and pricing for the single server Markovian queue with compartmented waiting space. Queueing Syst. 59(3–4):237–269

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Economou A, Gmez-Corral A, Kanta S (2011) Optimal balking strategies in single-server queues with general service and vacation times. Perform. Eval. 68(10):967–982

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Hassin R, Haviv M (1996) Optimal and equilibrium retrial rates in a busy system. Probab. Eng. Inform. Sci. 10:223–227

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Li T, Wang Z, Liu Z (2012) Geo/Geo/1 retrial queue with working vacations and vacation interruption. J. Appl. Math. Comput. 39:131–143

    Article  Google Scholar 

  • Li L, Wang J, Zhang F (2013) Equilibrium customer strategies in Markovian queues with partial breakdowns. Comput. Ind. Eng. 66(4):751–757

    Article  Google Scholar 

  • Liu Z, Song Y (2013) Geo/Geo/1 retrial queue with non-persistent customers and working vacations. J. Appl. Math. Comput. 42(1–2):103–115

    Article  Google Scholar 

  • Liu W, Xu X, Tian N (2007) Stochastic decompositions in the M/M/1 queue with working vacations. Oper. Res. Lett. 35:595–600

    Article  Google Scholar 

  • Melikov AZ, Ponomarenko LA (2014) Multidimensional Queueing Models in Telecommunication Networks. Springer, Berlin

    Book  Google Scholar 

  • Mitrani I, Chakka R (1995) Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method. Perform. Eval. 23:241–260

    Article  Google Scholar 

  • Naor P (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15–24

    Article  Google Scholar 

  • Ponomarenko LA, Soong Kim C, Melikov AZ (2010) Performance Analysis and Optimization of Multi-Traffic on Communication Networks. Springer, Berlin

    Book  Google Scholar 

  • Servi LD, Finn SG (2002) M/M/1 queues with working vacations (M/M/1/WV). Perform. Eval. 50(1–4):41–52

    Article  Google Scholar 

  • Stidham S Jr (2009) Optimal Design of Queueing Systems, 1st edn. Chapman & Hall CRC, Boca Raton

    Book  Google Scholar 

  • Tao L, Liu Z, Wang Z (2012) M/M/1 retrial queue with collisions and working vacation interruption under n-policy. RAIRO Oper. Res. 46:355–371

    Article  Google Scholar 

  • Tian N, Zhang ZG (2003) Stationary distributions of GI/M/c queue with PH type vacations. Queueing Syst. Theory Appl. 44(2):183–202

    Article  Google Scholar 

  • Wang J, Zhang F (2011) Equilibrium analysis of the observable queues with balking and delayed repairs. Appl. Math. Comput. 218(6):2716–2729

    Google Scholar 

  • Wang J, Zhang F (2013) Strategic joining in M/M/1 retrial queues. Eur. J. Oper. Res. 230(1):76–87

    Article  Google Scholar 

  • Wu J, Liu Z, Peng Y (2011) A discrete-time Geo/G/1 retrial queue with preemptive resume and collisions. Appl. Math. Model. 35(2):837–847

    Article  Google Scholar 

  • Wüchner P, Sztrik J, de Meer H (2009) Finite-source M/M/S retrial queue with search for balking and impatient customers from the orbit. Comput. Netw. 53(8):1264–1273

    Article  Google Scholar 

  • Zhang, Z., Wang, J., Zhang, F.: Equilibrium customer strategies in the single-server constant retrial queue with breakdowns and repairs. Mathematical Problems in Engineering, 2014:14, (2014). Article ID: 379572

  • Zhang F, Wang J, Liu B (2012) On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. J. Ind. Manag. Optim. 8(4):861–875

    Google Scholar 

  • Zhang F, Wang J, Liu B (2013) Equilibrium balking strategies in Markovian queues with working vacations. Appl. Math. Model. 37(16–17):8264–8282

    Article  Google Scholar 

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Acknowledgements

The authors thank reviewers for critical comments that helped the authors to improve the presentation of this paper. The  research of N. H. Do is supported by the OTKA K123914 project. The research of T. V. Do has been carried out within the project Thematic Research Cooperation Establishing Innovative Informatic and Info-communication Solutions, which is partially financed by the European Social Fund under Grant Number EFOP-3.6.2-16-2017-00013.

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Authors and Affiliations

  1. Analysis, Design and Development of ICT systems (AddICT) Laboratory, Department of Networked Systems and Services, Budapest University of Technology and Economics, H-117, Magyar tudósok körútja 2, Budapest, Hungary

    Nam H. Do & Tien Van Do

  2. Division of Knowledge and System Engineering for ICT, Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tien Van Do

  3. Institute of Cybernetics National Academy of Sciences of Azerbaijan, Baku, Azerbaijan

    Agassi Melikov

Authors
  1. Nam H. Do
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  2. Tien Van Do
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  3. Agassi Melikov
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Correspondence to Tien Van Do.

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Do, N.H., Van 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 20, 627–646 (2020). https://doi.org/10.1007/s12351-017-0369-7

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  • DOI: https://doi.org/10.1007/s12351-017-0369-7

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