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Analysis and Application of a Batch Arrival Queueing Model with the Second Optional Service and Randomized Vacation Policy

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HCI in Business, Government and Organizations (HCII 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14039))

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Abstract

This paper aims to investigate M[X]/(G1, G2)/1/VAC(J) queuing system with a random(p) vacation policy and optional second service, where X is the batch arrival number of customers. When no customers are in the system, the server immediately goes on vacation. And when the server returns from a vacation and finds that at least one customer is waiting in the system, the server will immediately provide the First Essential Service (FES). After customers complete the first essential service, some will continue to receive the Second Optional service (SOS). After the customer completes the FES, some customers will continue to choose to accept the second additional equipment adjustment or maintenance service (the probability is θ). In addition, when the server returns from vacation and finds that no customers are waiting for service in the system, the server will be idle in the system with a probability of p waiting for customers to enter the system for service, but there will be a probability of (1-p) to continue vacation. This pattern will continue until the number of server vacations reaches J times. Suppose the server returns to the system after the Jth vacation and finds that no customers are waiting for service in the system; the server will always be idle in the system waiting for customers to enter the system for service. This paper consider the servers are unreliable and can be repaired immediately, and establish the supplementary variables of the system as well as use the supplementary variables to construct the Kolmogorov forward equation that governs the system, and then use the supplementary variable techniques to derive the expected number of customers, the expected waiting time and other important system characteristics in the proposed queueing system. The relevant results can be used as the service performance evaluation and decision-making tools that require secondary optional services and regular maintenance in practical applications of queueing models.

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References

  1. Doshi, B.T.: Queueing system with vacation-a survey, Que. Syst. 1, 29–66 (1986)

    Google Scholar 

  2. Levy, Y., Yechiali, U.: Utilization of idle time in an M/G/1 queueing system. Manag. Sci. 22, 202–211 (1975)

    Google Scholar 

  3. Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation. Vol. I, Vacation And Priority Systems, Part I, North-Holland, Amsterdam (1991)

    Google Scholar 

  4. Servi, L.D., Finn, S.G.: M/M/1 queues with working vacations (m/m/1/wv). Perform. Eval. 50(1), 41–52 (2002)

    Article  Google Scholar 

  5. Wu, D.A., Takagi, H.: M/G/1 queue with multiple working vacations. Perform. Eval. 63(7), 654–681 (2006)

    Article  Google Scholar 

  6. Baba, Y.: On the M[X]/G/1 queue with vacation time. Operat. Res. Lett. 5, 93–98 (1986)

    Google Scholar 

  7. Kella, O.: The threshold policy in the M/G/1 queue with server vacations. Naval Res. Logist. 36, 111–123 (1989)

    Google Scholar 

  8. Zhang, Z.G., Tian, N.: Discrete time Geo/G/1 queue with multiple adaptive vacations. Que. Syst. 38, 419–429 (2001)

    Google Scholar 

  9. Ke, J.-C., Huang, K.-B., Pearn, W.L.: The randomized vacation policy for a batch arrival queue. Appl. Math. Model. 34(6), 1524–1538 (2009)

    Google Scholar 

  10. Ke, J.-C., Huang, K.B., Pearn, W.L.: Randomized policy of a Poisson input queue with J vacations. J. Syst. Sci. Syst. Eng. 19(1), 50–71 (2010)

    Google Scholar 

  11. Ke, J-C., Huang, K.-B., Pearn, W.L.: A batch arrival queue under randomized multi-vacation policy with unreliable server and repair. Int. J. Syst. Sci. 43(3), 552–565 (2012)

    Google Scholar 

  12. Ke, J.-C., Huang, K.-B.: Analysis of an unreliable server M[X]/G/1 system with a randomized vacation policy and delayed repair. Stoch. Model. 26(2), 212–241 (2010)

    Google Scholar 

  13. Ke, J.-C. Huang, K.-B., Pearn, W.L.: The performance measures and randomized optimization for an unreliable server M[X]/G/1 vacation system. Appl, Math. Comput. 217(21), 8277–8290 (2011)

    Google Scholar 

  14. Ke, J.-C., Huang, K.-B.: Analysis of batch arrival queue with randomized vacation policy and an un-reliable server. J. Syst. Sci. Complexity 25, 759–777 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ke, J.-C., Huang, K.-B., Kuo, C.-C.: Reliability-based measures for a batch-arrival queue with an unreliable server and delayed repair under randomised vacations. Int. J. Indust. Syst. Eng. 27(4), 500–525 (2017)

    Google Scholar 

  16. Ke, J.C., Chang, F.-M., Liu, T.-H.: M/M/c balking retrial queue with vacation. Qual. Technol. Quan. Manage. 26(1), 54–66 (2019)

    Google Scholar 

  17. Madan, K.C.: An M/G/1 queue with second optional service. Que. Syst. 34, 37–46 (2000)

    Google Scholar 

  18. Medhi, J.: A single server Poisson input queue with a second optional channel. Que. Syst. 42, 239–242 (2002)

    Google Scholar 

  19. Krishna Kumar, B., Vijayakumar, A., Arivudainambi, A.: An M/G/1 retrial queueing system with two phase service and preemptive resume. Ann. Oper. Res. 113 61–79 (2002)

    Google Scholar 

  20. Artalejo, J.R., Choudhury, G.: Steady state analysis of an M/G/1 queue with repeated attempts and two phase service. Qual. Technol. Quant. Manag. 1, 189–199 (2004)

    Google Scholar 

  21. Moreno, A.P.: Geo/G/1 retrial queue with second optional service. Int. J. Oper. Res. 1, 340–362 (2006)

    Google Scholar 

  22. Wang, J., Zhao, Q.: A discrete time Geo/G/1 retrial queue with starting failures and second optional service. Comput. Math. Appl. 53, 115–127 (2007)

    Google Scholar 

  23. Wang, J.: An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47, 1713–1723 (2004)

    Google Scholar 

  24. Choudhury, G., Tadj, L.: An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair. Appl. Math. Model. 33, 2699–2709 (2009)

    Google Scholar 

  25. Choudhury, G., Tadj, L., Deka, K.: A batch arrival retrial queueing system with two phases of service and service interruption. Comput. Math. Appl. 59(1), 437–450 (2010)

    Google Scholar 

  26. Oduol, V.K., Ardil, C.: Transient analysis of a single server queue with fixed size batch arrivals. Int. J. Electr. Comput. Eng. 6(2), 253–258 (2012)

    Google Scholar 

  27. Xu, X., Tian, N., Zhang, Z.: Analysis for the M/M/1 working vacation queue. Int. J. Inf. Manage. Sci. 20, 379–394 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Laxmi, P.V., Rajesh, P.: Analysis of variant working vacations on batch arrival queues. Opsearch 53(2), 303–316 (2015). https://doi.org/10.1007/s12597-015-0236-3

    Article  MathSciNet  MATH  Google Scholar 

  29. Vijaya Laxmi P., Rajesh, P., Kassahun, T.W.: Performance measures of variant working vacations on batch queue with server breakdowns. Int. J. Manag. Sci. Eng. Manag. 14(1), 53–63 (2018)

    Google Scholar 

  30. Chandrika, U.K., Kalaiselvi, C.: Batch arrival feedback queue with additional multi optional service and multiple vacation. Int. J. Sci. Res. Publ. 3(3): 1–8 (2013)

    Google Scholar 

  31. Kirupa, K., Chandrika, K.U.: Batch arrival retrial queue with negative customers, multi optional service and feedback. Commun. Appl. Electr. 2(4), 14–18 (2015)

    Article  Google Scholar 

  32. Suganya, S.: A batch arrival feedback queue with M-optional service and multiple vacations subject to random breakdown. Int. J. Sci. Res. 3(11), 1877–1881 (2014)

    Google Scholar 

  33. Vinnarasi, S., Maria Remona, J., Julia Rose Mary K.: Unreliable batch arrival queueing system with SWV. Int. J, Innov. Res. Sci. Eng. Technol. 5(3), 2884–2889 (2016)

    Google Scholar 

  34. Sree Parimala, R., Palaniammal, S.: An analysis of bulk service queueing model with servers’ various vacations. Int. J. Adv. Res. Technol. 4(2), 22–33 (2015)

    Google Scholar 

  35. Kalyanaraman, R., Marugan, S.P.B.: A single server queue with additional optional service in batches and server vacation. Appl. Math. Sci. 12(56), 2765– 2776 (2008)

    Google Scholar 

  36. Vijaya Laxmi, P., George, A.A., Girija Bhavani, E.: Performance of a single server batch queueing model with second optional service under transient and steady state. RT&A 4(65) (2021)

    Google Scholar 

  37. Cox, D.R.: The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Camb. Philos. Soc. 51, 433-441 (1955)

    Google Scholar 

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Authors and Affiliations

  1. Department of Business Administration, Fu Jen Catholic University, New Taipei, Taiwan

    Kai-Bin Huang

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  1. Kai-Bin Huang
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Correspondence to Kai-Bin Huang .

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Editors and Affiliations

  1. City University of Hong Kong, Kowloon, Hong Kong

    Fiona Nah

  2. City University of Hong Kong, Kowloon, Hong Kong

    Keng Siau

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Huang, KB. (2023). Analysis and Application of a Batch Arrival Queueing Model with the Second Optional Service and Randomized Vacation Policy. In: Nah, F., Siau, K. (eds) HCI in Business, Government and Organizations. HCII 2023. Lecture Notes in Computer Science, vol 14039. Springer, Cham. https://doi.org/10.1007/978-3-031-36049-7_24

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  • DOI: https://doi.org/10.1007/978-3-031-36049-7_24

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