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Metaheuristics and strategic behavior of markovian retrial queue under breakdown, vacation and bernoulli feedback

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Abstract

This research article addresses the performance analysis of Markovian retrial queueing system with two types of customers, unreliable server, and Bernoulli feedback. Both regular customers (RC) and prime customers (PC) may either join, or balk from the system based on the trade-off between service profit and delay cost. When the system is busy, the regular customers have to choose whether to join a retrial orbit and make re-attempts or leave the system. Furthermore, due to congestion among regular customers, the server may discontinue the service during breakdown. Due to the unavailability of the service process, customers may experience dissatisfaction. Therefore, our objective is to introduce a Bernoulli feedback service process to enhance service quality, ensuring that customers are successfully served with a certain probability. To analyze the proposed model mathematically, Chapman-Kolmogorov (C-K) inflow-outflow balanced equations have been framed. Then, the probability generating function (PGF) method employed to explicitly derive the queue size distribution, throughput, and other performance metrics. These performance measures provide critical insights into system behavior, which are then incorporated to determine the equilibrium strategies for two types of joining strategies: (i) non-cooperative strategies and (ii) cooperative strategies. Finally, optimization approaches are employed to determine the optimum cost and make tactical decisions regarding the quality of service (QoS) in an integrated manner. The cost optimization is done using metaheuristic optimization techniques such as PSO and GWO. The analytic results established are validated by numerical simulation. The effect of various parameters on the performance indices are examined by cost optimization and sensitivity analysis. The comparison of both algorithms, including average fitness, standard deviation, and convergence analysis, were used and combined with Wilcoxon rank-sum test.

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References

  1. Dimitriou I (2013) A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations. Appl Math Model 37:1295–1309. https://doi.org/10.1016/j.apm.2012.04.011

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammar S, Rajadurai P (2019) Performance analysis of preemptive priority retrial queueing system with disaster under working breakdown services. Symmetry (Basel) 11:419. https://doi.org/10.3390/sym11030419

    Article  MATH  Google Scholar 

  3. Bhagat A, Jain M (2020) Retrial queue with multiple repairs, multiple services and non preemptive priority. Opsearch 57:787–814. https://doi.org/10.1007/s12597-020-00443-y

    Article  MathSciNet  MATH  Google Scholar 

  4. Ayyappan G, Udayageetha J, Somasundaram B (2020) Analysis of non-pre-emptive priority retrial queueing system with two-way communication, Bernoulli vacation, collisions, working breakdown, immediate feedback and reneging. Int J Math Oper Res 16:480–498. https://doi.org/10.1504/IJMOR.2020.108420

    Article  MathSciNet  MATH  Google Scholar 

  5. Shajin D, Dudin AN, Dudina O, Krishnamoorthy A (2020) Two-priority single server retrial queue with additional items. J Ind Manag Optim 16:2891–2912. https://doi.org/10.3934/jimo.2019085

    Article  MathSciNet  MATH  Google Scholar 

  6. Dhibar S, Jain M (2023) Strategic behaviour for M/M/1 double orbit retrial queue with imperfect service and vacation. Int J Math Oper Res 23:369–385. https://doi.org/10.1504/IJMOR.2022.10048415

    Article  MathSciNet  MATH  Google Scholar 

  7. Jain M, Jain A (2014) Batch arrival priority queueing model with second optional service and server breakdown. Int J Oper Res 11:112–130

    MathSciNet  MATH  Google Scholar 

  8. Liou CD (2015) Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers. Int J Syst Sci 46:2165–2182. https://doi.org/10.1080/00207721.2013.859326

    Article  MathSciNet  MATH  Google Scholar 

  9. Lan S, Tang Y (2017) Performance analysis of a discrete-time queue with working breakdowns and searching for the optimum service rate in working breakdown period. J Syst Sci Inf 5:176–192. https://doi.org/10.21078/jssi-2017-176-17

    Article  MATH  Google Scholar 

  10. Yang DY, Chen YH (2018) Computation and optimization of a working breakdown queue with second optional service. J Ind Prod Eng 35:181–188. https://doi.org/10.1080/21681015.2018.1439113

    Article  MATH  Google Scholar 

  11. Jiang T, Xin B (2019) Computational analysis of the queue with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy. Commun Stat - Theory Methods 48:926–941. https://doi.org/10.1080/03610926.2017.1422756

    Article  MathSciNet  MATH  Google Scholar 

  12. Yang DY, Chen YH, Wu CH (2020) Modelling and optimisation of a two-server queue with multiple vacations and working breakdowns. Int J Prod Res 58:3036–3048. https://doi.org/10.1080/00207543.2019.1624856

    Article  MATH  Google Scholar 

  13. Yang DY, Chung CH, Wu CH (2021) Sojourn times in a Markovian queue with working breakdowns and delayed working vacations. Comput Ind Eng 156:107239. https://doi.org/10.1016/j.cie.2021.107239

    Article  MATH  Google Scholar 

  14. Liu T-H, Chang F-M, Ke J-C, Sheu S-H (2022) Optimization of retrial queue with unreliable servers subject to imperfect coverage and reboot delay. Qual Technol Quant Manag 19:428–453. https://doi.org/10.1080/16843703.2021.2020952

    Article  MATH  Google Scholar 

  15. Wang J, Li WW (2016) Noncooperative and cooperative joining strategies in cognitive radio networks with random access. IEEE Trans Veh Technol 65:5624–5636. https://doi.org/10.1109/TVT.2015.2470115

    Article  MATH  Google Scholar 

  16. Wang J, Wang F, Li WW (2017) Strategic behavior and admission control of cognitive radio systems with imperfect sensing. Comput Commun 113:53–61. https://doi.org/10.1016/j.comcom.2017.09.015

    Article  MATH  Google Scholar 

  17. Devarajan K, Senthilkumar M (2021) On the retrial-queuing model for strategic access and equilibrium-joining strategies of cognitive users in cognitive-radio networks with energy harvesting. Energies 14:2088. https://doi.org/10.3390/en14082088

    Article  MATH  Google Scholar 

  18. Tian R, Wang Y (2019) Analysis of equilibrium strategies in markovian queues with negative customers and working breakdowns. IEEE Access 7:159868–159878. https://doi.org/10.1109/ACCESS.2019.2950268

    Article  MATH  Google Scholar 

  19. Aghsami A, Jolai F (2020) Equilibrium threshold strategies and social benefits in the fully observable Markovian queues with partial breakdowns and interruptible setup/closedown policy. Qual Technol Quant Manag 17:685–722. https://doi.org/10.1080/16843703.2020.1736365

    Article  MATH  Google Scholar 

  20. Van Heesch M, Wissink PLJ, Ranji R et al (2020) Combining cooperative with non-cooperative game theory to model wi-fi congestion in apartment blocks. IEEE Access 8:64603–64616. https://doi.org/10.1109/ACCESS.2020.2984535

    Article  Google Scholar 

  21. Gao S, Dong H, Wang X (2021) Equilibrium and pricing analysis for an unreliable retrial queue with limited idle period and single vacation. Oper Res 21:621–643. https://doi.org/10.1007/s12351-018-0437-7

    Article  MATH  Google Scholar 

  22. Wang Z, Liu L, Shao Y, Zhao YQ (2021) Joining strategies under two kinds of games for a multiple vacations retrial queue with N-policy and breakdowns. AIMS Math 6:9075–9099. https://doi.org/10.3934/math.2021527

    Article  MathSciNet  MATH  Google Scholar 

  23. Meziani K, Rahmoune F, Radjef MS (2022) Service pricing and customer behaviour strategies of Stackelberg’s equilibrium in an unobservable Markovian queue with unreliable server and delayed repairs. Int J Math Oper Res 21:281–304. https://doi.org/10.1504/IJMOR.2022.122216

    Article  MathSciNet  MATH  Google Scholar 

  24. Jain M, Dhibar S (2023) ANFIS and metaheuristic optimization for strategic joining policy with re-attempt and vacation. Math Comput Simul 211:57–84. https://doi.org/10.1016/j.matcom.2023.03.024

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu J, Wang J (2017) Strategic joining rules in a single server Markovian queue with Bernoulli vacation. Oper Res 17:413–434. https://doi.org/10.1007/s12351-016-0231-3

    Article  MATH  Google Scholar 

  26. Sun K, Wang J (2021) Equilibrium joining strategies in the single-server constant retrial queues with Bernoulli vacations. RAIRO - Oper Res 55:S481–S502. https://doi.org/10.1051/ro/2019087

    Article  MathSciNet  MATH  Google Scholar 

  27. Fackrell M, Taylor P, Wang J (2021) Strategic customer behavior in an M/M/1 feedback queue. Queueing Syst 97:223–259. https://doi.org/10.1007/s11134-021-09693-z

    Article  MathSciNet  MATH  Google Scholar 

  28. Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In Proceedings of ICNN’95 - International Conference on Neural Networks 4:1942–1948

  29. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007

    Article  MATH  Google Scholar 

  30. Alabert A, Berti A, Caballero R, Ferrante M (2015) No-Free-Lunch theorems in the continuum. Theoretical Computer Science 600:98–106. https://doi.org/10.1016/j.tcs.2015.07.029

    Article  MathSciNet  MATH  Google Scholar 

  31. Gupta S, Deep K (2019) A novel Random Walk Grey Wolf Optimizer. Swarm Evol Comput 44:101–112. https://doi.org/10.1016/j.swevo.2018.01.001

    Article  MATH  Google Scholar 

  32. Preeti, Deep K (2022) A random walk Grey wolf optimizer based on dispersion factor for feature selection on chronic disease prediction. Exp Syst Appl 206:117864. https://doi.org/10.1016/j.eswa.2022.117864

  33. Awotunde AA, Naranjo C (2014) Well placement optimization constrained to minimum well spacing. In SPE Latin Am Caribb Pet Eng Conf 1:325–350. https://doi.org/10.2118/169272-MS

    Article  MATH  Google Scholar 

  34. Zhou M, Liu L, Chai X, Wang Z (2019) Equilibrium strategies in a constant retrial queue with setup time and the N-policy. Commun Stat - Theory Methods 49:1695–1711. https://doi.org/10.1080/03610926.2019.1565779

    Article  MathSciNet  MATH  Google Scholar 

  35. Rani S, Jain M, Meena RK (2023) Queueing modeling and optimization of a fault-tolerant system with reboot, recovery, and vacationing server operating under admission control policy. Math Comput Simul 209:408–425. https://doi.org/10.1016/j.matcom.2023.02.015

    Article  MathSciNet  MATH  Google Scholar 

  36. Dhibar S, Jain M (2024) Particle swarm optimization and FM/FM/1/WV retrial queues with catastrophes: application to cloud storage. J Supercomput 80:15429–15463. https://doi.org/10.1007/s11227-024-06068-y

    Article  MATH  Google Scholar 

  37. Wang J, Zhang X, Huang P (2017) Strategic behavior and social optimization in a constant retrial queue with the N-policy. Eur J Oper Res 256:841–849. https://doi.org/10.1016/j.ejor.2016.06.034

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang J, Zhang Y, Li WW (2017) Strategic joining and optimal pricing in the cognitive radio system with delay-sensitive secondary users. IEEE Trans Cogn Commun Netw 3:298–312. https://doi.org/10.1109/TCCN.2017.2723900

    Article  MATH  Google Scholar 

  39. Alhammadi A, Roslee M, Alias MY (2017) Analysis of sspectrum handoff schemes in cognitive radio network using particle swarm optimization. 2016 IEEE 3rd Int Symp Telecommun Technol ISTT 2016 103–107.https://doi.org/10.1109/ISTT.2016.7918093

  40. Zhu S, Wang J, Li WW (2018) Optimal service rate in cognitive radio networks with different queue length information. IEEE Access 6:51577–51586. https://doi.org/10.1109/ACCESS.2018.2867049

    Article  MATH  Google Scholar 

  41. Sumathi D, Manivannan SS (2021) Stochastic approach for channel selection in cognitive radio networks using optimization techniques. Telecommun Syst 76:167–186. https://doi.org/10.1007/s11235-020-00705-6

    Article  MATH  Google Scholar 

  42. Phung-Duc T, Akutsu K, Kawanishi K et al (2022) Queueing models for cognitive wireless networks with sensing time of secondary users. Ann Oper Res 310:641–660. https://doi.org/10.1007/s10479-021-04118-9

    Article  MathSciNet  MATH  Google Scholar 

  43. Gross D, Shortle JF, Thompson JM, Harris CM (2008) Fundamentals of Queueing Theory. Fourth Edition. John Wiley & Sons, Inc., Doi, In Fundamentals of Queueing Theory. https://doi.org/10.1002/9781119453765

    Book  MATH  Google Scholar 

  44. Caraffini F, Iacca G (2020) The SOS platform: designing, tuning and statistically benchmarking optimisation algorithms. Mathematics 8(5):785. https://doi.org/10.3390/math8050785

    Article  MATH  Google Scholar 

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Acknowledgements

We would like to thank the Editor-in-Chief and the anonymous referees for their valuable comments and feedback, which have greatly contributed to the improvement of this research work. The first author, Sibasish Dhibar, is grateful to the Ministry of Education, India, for supporting the present research work through a Senior Research Fellowship (SRF), Grant MHC01-23-200-428.

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Sibasish Dhibar: Model design, computational results, analysis verification, and manuscript writes up. Dr. Madhu Jain: Model design, analysis verification and manuscript write-up.

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Correspondence to Madhu Jain.

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Dhibar, S., Jain, M. Metaheuristics and strategic behavior of markovian retrial queue under breakdown, vacation and bernoulli feedback. Appl Intell 55, 273 (2025). https://doi.org/10.1007/s10489-024-05978-x

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